# an introduction to business statistics

```SUBJECT: BUSINESS STATISTICS
COURSE CODE: MC-106
LESSON: 01
AUTHOR: SURINDER KUNDU
VETTER: DR. B. S. BODLA
OBJECTIVE:
The aim of the present lesson is to enable the students to understand
the meaning, definition, nature, importance and limitations of statistics.
“A knowledge of statistics is like a knowledge of foreign
language of algebra; it may prove of use at any time under
any circumstance”……………………………………...Bowley.
STRUCTURE:
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
Introduction
Meaning and Definitions of Statistics
Types of Data and Data Sources
Types of Statistics
Scope of Statistics
Limitations of statistics
Summary
Self-Test Questions
1.1
INTRODUCTION
For a layman, ‘Statistics’ means numerical information expressed in quantitative
terms. This information may relate to objects, subjects, activities, phenomena, or
regions of space. As a matter of fact, data have no limits as to their reference,
coverage, and scope. At the macro level, these are data on gross national product and
shares of agriculture, manufacturing, and services in GDP (Gross Domestic Product).
1
At the micro level, individual firms, howsoever small or large, produce extensive
statistics on their operations. The annual reports of companies contain variety of data
on sales, production, expenditure, inventories, capital employed, and other activities.
These data are often field data, collected by employing scientific survey techniques.
Unless regularly updated, such data are the product of a one-time effort and have
limited use beyond the situation that may have called for their collection. A student
knows statistics more intimately as a subject of study like economics, mathematics,
chemistry, physics, and others. It is a discipline, which scientifically deals with data,
and is often described as the science of data. In dealing with statistics as data,
statistics has developed appropriate methods of collecting, presenting, summarizing,
and analysing data, and thus consists of a body of these methods.
1.2
MEANING AND DEFINITIONS OF STATISTICS
In the beginning, it may be noted that the word ‘statistics’ is used rather curiously in
two senses plural and singular. In the plural sense, it refers to a set of figures or data.
In the singular sense, statistics refers to the whole body of tools that are used to
collect data, organise and interpret them and, finally, to draw conclusions from them.
It should be noted that both the aspects of statistics are important if the quantitative
data are to serve their purpose. If statistics, as a subject, is inadequate and consists of
poor methodology, we could not know the right procedure to extract from the data the
information they contain. Similarly, if our data are defective or that they are
inadequate or inaccurate, we could not reach the right conclusions even though our
subject is well developed.
A.L. Bowley has defined statistics as: (i) statistics is the science of counting, (ii)
Statistics may rightly be called the science of averages, and (iii) statistics is the
science of measurement of social organism regarded as a whole in all its mani-
2
festations. Boddington defined as: Statistics is the science of estimates and
probabilities. Further, W.I. King has defined Statistics in a wider context, the science
of Statistics is the method of judging collective, natural or social phenomena from the
results obtained by the analysis or enumeration or collection of estimates.
Seligman explored that statistics is a science that deals with the methods of collecting,
classifying, presenting, comparing and interpreting numerical data collected to throw
some light on any sphere of enquiry. Spiegal defines statistics highlighting its role in
decision-making particularly under uncertainty, as follows: statistics is concerned
with scientific method for collecting, organising, summa rising, presenting and
analyzing data as well as drawing valid conclusions and making reasonable decisions
on the basis of such analysis. According to Prof. Horace Secrist, Statistics is the
aggregate of facts, affected to a marked extent by multiplicity of causes, numerically
expressed, enumerated or estimated according to reasonable standards of accuracy,
collected in a systematic manner for a pre-determined purpose, and placed in relation
to each other.
From the above definitions, we can highlight the major characteristics of statistics as
follows:
(i)
Statistics are the aggregates of facts. It means a single figure is not statistics.
For example, national income of a country for a single year is not statistics but
the same for two or more years is statistics.
(ii)
Statistics are affected by a number of factors. For example, sale of a product
depends on a number of factors such as its price, quality, competition, the
income of the consumers, and so on.
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(iii)
Statistics must be reasonably accurate. Wrong figures, if analysed, will lead to
erroneous conclusions. Hence, it is necessary that conclusions must be based
on accurate figures.
(iv)
Statistics must be collected in a systematic manner. If data are collected in a
haphazard manner, they will not be reliable and will lead to misleading
conclusions.
(v)
Collected in a systematic manner for a pre-determined purpose
(vi)
Lastly, Statistics should be placed in relation to each other. If one collects data
unrelated to each other, then such data will be confusing and will not lead to
any logical conclusions. Data should be comparable over time and over space.
1.3
TYPES OF DATA AND DATA SOURCES
Statistical data are the basic raw material of statistics. Data may relate to an activity of
our interest, a phenomenon, or a problem situation under study. They derive as a
result of the process of measuring, counting and/or observing. Statistical data,
therefore, refer to those aspects of a problem situation that can be measured,
quantified, counted, or classified. Any object subject phenomenon, or activity that
generates data through this process is termed as a variable. In other words, a variable
is one that shows a degree of variability when successive measurements are recorded.
In statistics, data are classified into two broad categories: quantitative data and
qualitative data. This classification is based on the kind of characteristics that are
measured.
Quantitative data are those that can be quantified in definite units of measurement.
These refer to characteristics whose successive measurements yield quantifiable
observations. Depending on the nature of the variable observed for measurement,
quantitative data can be further categorized as continuous and discrete data.
4
Obviously, a variable may be a continuous variable or a discrete variable.
(i)
Continuous data represent the numerical values of a continuous variable. A
continuous variable is the one that can assume any value between any two
points on a line segment, thus representing an interval of values. The values
are quite precise and close to each other, yet distinguishably different. All
characteristics such as weight, length, height, thickness, velocity, temperature,
tensile strength, etc., represent continuous variables. Thus, the data recorded
on these and similar other characteristics are called continuous data. It may be
noted that a continuous variable assumes the finest unit of measurement.
Finest in the sense that it enables measurements to the maximum degree of
precision.
(ii)
Discrete data are the values assumed by a discrete variable. A discrete
variable is the one whose outcomes are measured in fixed numbers. Such data
are essentially count data. These are derived from a process of counting, such
as the number of items possessing or not possessing a certain characteristic.
The number of customers visiting a departmental store everyday, the incoming
flights at an airport, and the defective items in a consignment received for sale,
are all examples of discrete data.
Qualitative data refer to qualitative characteristics of a subject or an object. A
characteristic is qualitative in nature when its observations are defined and noted in
terms of the presence or absence of a certain attribute in discrete numbers. These data
are further classified as nominal and rank data.
(i)
Nominal data are the outcome of classification into two or more categories of
items or units comprising a sample or a population according to some quality
characteristic. Classification of students according to sex (as males and
5
females), of workers according to skill (as skilled, semi-skilled, and unskilled),
and of employees according to the level of education (as matriculates,
such basis of classification, it is always possible to assign each item to a
particular class and make a summation of items belonging to each class. The
count data so obtained are called nominal data.
(ii)
Rank data, on the other hand, are the result of assigning ranks to specify order
in terms of the integers 1,2,3, ..., n. Ranks may be assigned according to the
level of performance in a test. a contest, a competition, an interview, or a
show. The candidates appearing in an interview, for example, may be assigned
ranks in integers ranging from I to n, depending on their performance in the
interview. Ranks so assigned can be viewed as the continuous values of a
variable involving performance as the quality characteristic.
Data sources could be seen as of two types, viz., secondary and primary. The two can
be defined as under:
(i)
Secondary data: They already exist in some form: published or unpublished in an identifiable secondary source. They are, generally, available from
published source(s), though not necessarily in the form actually required.
(ii)
Primary data: Those data which do not already exist in any form, and thus
have to be collected for the first time from the primary source(s). By their very
nature, these data require fresh and first-time collection covering the whole
population or a sample drawn from it.
1.4
TYPES OF STATISTICS
There are two major divisions of statistics such as descriptive statistics and inferential
statistics. The term descriptive statistics deals with collecting, summarizing, and
6
simplifying data, which are otherwise quite unwieldy and voluminous. It seeks to
achieve this in a manner that meaningful conclusions can be readily drawn from the
data. Descriptive statistics may thus be seen as comprising methods of bringing out
and highlighting the latent characteristics present in a set of numerical data. It not
only facilitates an understanding of the data and systematic reporting thereof in a
manner; and also makes them amenable to further discussion, analysis, and
interpretations.
The first step in any scientific inquiry is to collect data relevant to the problem in
hand. When the inquiry relates to physical and/or biological sciences, data collection
is normally an integral part of the experiment itself. In fact, the very manner in which
an experiment is designed, determines the kind of data it would require and/or
generate. The problem of identifying the nature and the kind of the relevant data is
thus automatically resolved as soon as the design of experiment is finalized. It is
possible in the case of physical sciences. In the case of social sciences, where the
required data are often collected through a questionnaire from a number of carefully
selected respondents, the problem is not that simply resolved. For one thing,
designing the questionnaire itself is a critical initial problem. For another, the number
of respondents to be accessed for data collection and the criteria for selecting them
has their own implications and importance for the quality of results obtained. Further,
the data have been collected, these are assembled, organized, and presented in the
form of appropriate tables to make them readable. Wherever needed, figures,
diagrams, charts, and graphs are also used for better presentation of the data. A useful
tabular and graphic presentation of data will require that the raw data be properly
classified in accordance with the objectives of investigation and the relational analysis
to be carried out.
.
7
A well thought-out and sharp data classification facilitates easy description of the
hidden data characteristics by means of a variety of summary measures. These include
measures of central tendency, dispersion, skewness, and kurtosis, which constitute the
essential scope of descriptive statistics. These form a large part of the subject matter
of any basic textbook on the subject, and thus they are being discussed in that order
here as well.
Inferential statistics, also known as inductive statistics, goes beyond describing a
given problem situation by means of collecting, summarizing, and meaningfully
presenting the related data. Instead, it consists of methods that are used for drawing
inferences, or making broad generalizations, about a totality of observations on the
basis of knowledge about a part of that totality. The totality of observations about
which an inference may be drawn, or a generalization made, is called a population or
a universe. The part of totality, which is observed for data collection and analysis to
gain knowledge about the population, is called a sample.
The desired information about a given population of our interest; may also be
collected even by observing all the units comprising the population. This total
coverage is called census. Getting the desired value for the population through census
is not always feasible and practical for various reasons. Apart from time and money
considerations making the census operations prohibitive, observing each individual
unit of the population with reference to any data characteristic may at times involve
even destructive testing. In such cases, obviously, the only recourse available is to
employ the partial or incomplete information gathered through a sample for the
purpose. This is precisely what inferential statistics does. Thus, obtaining a particular
value from the sample information and using it for drawing an inference about the
entire population underlies the subject matter of inferential statistics. Consider a
8
situation in which one is required to know the average body weight of all the college
students in a given cosmopolitan city during a certain year. A quick and easy way to
do this is to record the weight of only 500 students, from out of a total strength of,
say, 10000, or an unknown total strength, take the average, and use this average based
on incomplete weight data to represent the average body weight of all the college
students. In a different situation, one may have to repeat this exercise for some future
year and use the quick estimate of average body weight for a comparison. This may
be needed, for example, to decide whether the weight of the college students has
undergone a significant change over the years compared.
Inferential statistics helps to evaluate the risks involved in reaching inferences or
generalizations about an unknown population on the basis of sample information. for
example, an inspection of a sample of five battery cells drawn from a given lot may
reveal that all the five cells are in perfectly good condition. This information may be
used to conclude that the entire lot is good enough to buy or not.
Since this inference is based on the examination of a sample of limited number of
cells, it is equally likely that all the cells in the lot are not in order. It is also possible
that all the items that may be included in the sample are unsatisfactory. This may be
used to conclude that the entire lot is of unsatisfactory quality, whereas the fact may
indeed be otherwise. It may, thus, be noticed that there is always a risk of an inference
about a population being incorrect when based on the knowledge of a limited sample.
The rescue in such situations lies in evaluating such risks. For this, statistics provides
the necessary methods. These centres on quantifying in probabilistic term the chances
of decisions taken on the basis of sample information being incorrect. This requires an
understanding of the what, why, and how of probability and probability distributions
to equip ourselves with methods of drawing statistical inferences and estimating the
9
degree of reliability of these inferences.
1.5
SCOPE OF STATISTICS
Apart from the methods comprising the scope of descriptive and inferential branches
of statistics, statistics also consists of methods of dealing with a few other issues of
specific nature. Since these methods are essentially descriptive in nature, they have
been discussed here as part of the descriptive statistics. These are mainly concerned
with the following:
(i)
It often becomes necessary to examine how two paired data sets are related.
For example, we may have data on the sales of a product and the expenditure
incurred on its advertisement for a specified number of years. Given that sales
and advertisement expenditure are related to each other, it is useful to examine
the nature of relationship between the two and quantify the degree of that
relationship. As this requires use of appropriate statistical methods, these falls
under the purview of what we call regression and correlation analysis.
(ii)
Situations occur quite often when we require averaging (or totalling) of data
on prices and/or quantities expressed in different units of measurement. For
example, price of cloth may be quoted per meter of length and that of wheat
per kilogram of weight. Since ordinary methods of totalling and averaging do
not apply to such price/quantity data, special techniques needed for the
purpose are developed under index numbers.
(iii)
Many a time, it becomes necessary to examine the past performance of an
activity with a view to determining its future behaviour. For example, when
engaged in the production of a commodity, monthly product sales are an
important measure of evaluating performance. This requires compilation and
analysis of relevant sales data over time. The more complex the activity, the
10
more varied the data requirements. For profit maximising and future sales
planning, forecast of likely sales growth rate is crucial. This needs careful
collection and analysis of past sales data. All such concerns are taken care of
under time series analysis.
(iv)
Obtaining the most likely future estimates on any aspect(s) relating to a
business or economic activity has indeed been engaging the minds of all
concerned. This is particularly important when it relates to product sales and
demand, which serve the necessary basis of production scheduling and
planning. The regression, correlation, and time series analyses together help
develop the basic methodology to do the needful. Thus, the study of methods
and techniques of obtaining the likely estimates on business/economic
variables comprises the scope of what we do under business forecasting.
Keeping in view the importance of inferential statistics, the scope of statistics may
finally be restated as consisting of statistical methods which facilitate decision-making under conditions of uncertainty. While the term statistical methods is often
used to cover the subject of statistics as a whole, in particular it refers to methods by
which statistical data are analysed, interpreted, and the inferences drawn for decisionmaking.
Though generic in nature and versatile in their applications, statistical methods have
come to be widely used, especially in all matters concerning business and economics.
These are also being increasingly used in biology, medicine, agriculture, psychology,
and education. The scope of application of these methods has started opening and
expanding in a number of social science disciplines as well. Even a political scientist
finds them of increasing relevance for examining the political behaviour and it is, of
course, no surprise to find even historians statistical data, for history is essentially past
11
data presented in certain actual format.
1.6
There are three major functions in any business enterprise in which the statistical
methods are useful. These are as follows:
(i)
The planning of operations: This may relate to either special projects or to
the recurring activities of a firm over a specified period.
(ii)
The setting up of standards: This may relate to the size of employment,
volume of sales, fixation of quality norms for the manufactured product,
norms for the daily output, and so forth.
(iii)
The function of control: This involves comparison of actual production
achieved against the norm or target set earlier. In case the production has
fallen short of the target, it gives remedial measures so that such a deficiency
does not occur again.
A worth noting point is that although these three functions-planning of operations,
setting standards, and control-are separate, but in practice they are very much
interrelated.
Different authors have highlighted the importance of Statistics in business. For
instance, Croxton and Cowden give numerous uses of Statistics in business such as
project planning, budgetary planning and control, inventory planning and control,
quality control, marketing, production and personnel administration. Within these also
they have specified certain areas where Statistics is very relevant. Another author,
Irwing W. Burr, dealing with the place of statistics in an industrial organisation,
specifies a number of areas where statistics is extremely useful. These are: customer
wants and market research, development design and specification, purchasing,
12
production, inspection, packaging and shipping, sales and complaints, inventory and
maintenance, costs, management control, industrial engineering and research.
Statistical problems arising in the course of business operations are multitudinous. As
such, one may do no more than highlight some of the more important ones to
emphasis the relevance of statistics to the business world. In the sphere of production,
for example, statistics can be useful in various ways.
Statistical quality control methods are used to ensure the production of quality goods.
Identifying and rejecting defective or substandard goods achieve this. The sale targets
can be fixed on the basis of sale forecasts, which are done by using varying methods
of forecasting. Analysis of sales affected against the targets set earlier would indicate
the deficiency in achievement, which may be on account of several causes: (i) targets
were too high and unrealistic (ii) salesmen's performance has been poor (iii)
emergence of increase in competition (iv) poor quality of company's product, and so
on. These factors can be further investigated.
Another sphere in business where statistical methods can be used is personnel
management. Here, one is concerned with the fixation of wage rates, incentive norms
and performance appraisal of individual employee. The concept of productivity is
very relevant here. On the basis of measurement of productivity, the productivity
bonus is awarded to the workers. Comparisons of wages and productivity are
undertaken in order to ensure increases in industrial productivity.
Statistical methods could also be used to ascertain the efficacy of a certain product,
say, medicine. For example, a pharmaceutical company has developed a new
medicine in the treatment of bronchial asthma. Before launching it on commercial
basis, it wants to ascertain the effectiveness of this medicine. It undertakes an
experimentation involving the formation of two comparable groups of asthma
13
patients. One group is given this new medicine for a specified period and the other
one is treated with the usual medicines. Records are maintained for the two groups for
the specified period. This record is then analysed to ascertain if there is any
significant difference in the recovery of the two groups. If the difference is really
significant statistically, the new medicine is commercially launched.
1.7
LIMITATIONS OF STATISTICS
Statistics has a number of limitations, pertinent among them are as follows:
(i)
There are certain phenomena or concepts where statistics cannot be used. This
is because these phenomena or concepts are not amenable to measurement.
For example, beauty, intelligence, courage cannot be quantified. Statistics has
no place in all such cases where quantification is not possible.
(ii)
Statistics reveal the average behaviour, the normal or the general trend. An
application of the 'average' concept if applied to an individual or a particular
situation may lead to a wrong conclusion and sometimes may be disastrous.
For example, one may be misguided when told that the average depth of a
river from one bank to the other is four feet, when there may be some points in
between where its depth is far more than four feet. On this understanding, one
may enter those points having greater depth, which may be hazardous.
(iii)
Since statistics are collected for a particular purpose, such data may not be
relevant or useful in other situations or cases. For example, secondary data
(i.e., data originally collected by someone else) may not be useful for the other
person.
(iv)
Statistics are not 100 per cent precise as is Mathematics or Accountancy.
Those who use statistics should be aware of this limitation.
14
(v)
In statistical surveys, sampling is generally used as it is not physically possible
to cover all the units or elements comprising the universe. The results may not
be appropriate as far as the universe is concerned. Moreover, different surveys
based on the same size of sample but different sample units may yield
different results.
(vi)
At times, association or relationship between two or more variables is studied
in statistics, but such a relationship does not indicate cause and effect'
relationship. It simply shows the similarity or dissimilarity in the movement of
the two variables. In such cases, it is the user who has to interpret the results
carefully, pointing out the type of relationship obtained.
(vii)
A major limitation of statistics is that it does not reveal all pertaining to a
certain phenomenon. There is some background information that statistics
does not cover. Similarly, there are some other aspects related to the problem
on hand, which are also not covered. The user of Statistics has to be well
informed and should interpret Statistics keeping in mind all other aspects
having relevance on the given problem.
Apart from the limitations of statistics mentioned above, there are misuses of it. Many
people, knowingly or unknowingly, use statistical data in wrong manner. Let us see
what the main misuses of statistics are so that the same could be avoided when one
has to use statistical data. The misuse of Statistics may take several forms some of
which are explained below.
(i)
Sources of data not given: At times, the source of data is not given. In the
absence of the source, the reader does not know how far the data are reliable.
Further, if he wants to refer to the original source, he is unable to do so.
15
(ii)
Defective data: Another misuse is that sometimes one gives defective data.
This may be done knowingly in order to defend one's position or to prove a
particular point. This apart, the definition used to denote a certain
phenomenon may be defective. For example, in case of data relating to unemployed persons, the definition may include even those who are employed,
though partially. The question here is how far it is justified to include partially
employed persons amongst unemployed ones.
(iii)
Unrepresentative sample: In statistics, several times one has to conduct a
survey, which necessitates to choose a sample from the given population or
universe. The sample may turn out to be unrepresentative of the universe. One
may choose a sample just on the basis of convenience. He may collect the
desired information from either his friends or nearby respondents in his
neighbourhood even though such respondents do not constitute a
representative sample.
(iv)
Inadequate sample: Earlier, we have seen that a sample that is
unrepresentative of the universe is a major misuse of statistics. This apart, at
times one may conduct a survey based on an extremely inadequate sample.
For example, in a city we may find that there are 1, 00,000 households. When
we have to conduct a household survey, we may take a sample of merely 100
households comprising only 0.1 per cent of the universe. A survey based on
such a small sample may not yield right information.
(v)
Unfair Comparisons: An important misuse of statistics is making unfair
comparisons from the data collected. For instance, one may construct an index
of production choosing the base year where the production was much less.
Then he may compare the subsequent year's production from this low base.
16
Such a comparison will undoubtedly give a rosy picture of the production
though in reality it is not so. Another source of unfair comparisons could be
when one makes absolute comparisons instead of relative ones. An absolute
comparison of two figures, say, of production or export, may show a good
increase, but in relative terms it may turnout to be very negligible. Another
example of unfair comparison is when the population in two cities is different,
but a comparison of overall death rates and deaths by a particular disease is
attempted. Such a comparison is wrong. Likewise, when data are not properly
classified or when changes in the composition of population in the two years
are not taken into consideration, comparisons of such data would be unfair as
(vi)
Unwanted conclusions: Another misuse of statistics may be on account of
unwarranted conclusions. This may be as a result of making false assumptions.
For example, while making projections of population in the next five years,
one may assume a lower rate of growth though the past two years indicate
otherwise. Sometimes one may not be sure about the changes in business
environment in the near future. In such a case, one may use an assumption that
may turn out to be wrong. Another source of unwarranted conclusion may be
the use of wrong average. Suppose in a series there are extreme values, one is
too high while the other is too low, such as 800 and 50. The use of an
arithmetic average in such a case may give a wrong idea. Instead, harmonic
mean would be proper in such a case.
(vii)
Confusion of correlation and causation: In statistics, several times one has
to examine the relationship between two variables. A close relationship between the
two variables may not establish a cause-and-effect-relationship in the sense that one
17
variable is the cause and the other is the effect. It should be taken as something that
measures degree of association rather than try to find out causal relationship..
1.8
SUMMARY
In a summarized manner, ‘Statistics’ means numerical information expressed in
quantitative terms. As a matter of fact, data have no limits as to their reference,
coverage, and scope. At the macro level, these are data on gross national product and
shares of agriculture, manufacturing, and services in GDP (Gross Domestic Product).
At the micro level, individual firms, howsoever small or large, produce extensive
statistics on their operations. The annual reports of companies contain variety of data
on sales, production, expenditure, inventories, capital employed, and other activities.
These data are often field data, collected by employing scientific survey techniques.
Unless regularly updated, such data are the product of a one-time effort and have
limited use beyond the situation that may have called for their collection. A student
knows statistics more intimately as a subject of study like economics, mathematics,
chemistry, physics, and others. It is a discipline, which scientifically deals with data,
and is often described as the science of data. In dealing with statistics as data,
statistics has developed appropriate methods of collecting, presenting, summarizing,
and analysing data, and thus consists of a body of these methods.
1.9
SELF-TEST QUESTIONS
1.
Define Statistics. Explain its types, and importance to trade, commerce and
2.
“Statistics is all-pervading”. Elucidate this statement.
3.
Write a note on the scope and limitations of Statistics.
4.
What are the major limitations of Statistics? Explain with suitable examples.
5.
Distinguish between descriptive Statistics and inferential Statistics.
18
1.
Gupta, S. P. : Statistical Methods, Sultan chand and Sons, New Delhi.
2.
Hooda, R. P.: Statistics for Business and Economics, Macmillan, New Delhi.
3.
Hein, L. W. Quantitative Approach to Managerial Decisions, Prentice Hall,
NJ.
4.
Levin, Richard I. and David S. Rubin: Statistics for Management, Prentice
Hall, New Delhi.
5.
Lawrance B. Moore: Statistics for Business & Economics, Harper Collins,
NY.
6.
Watsman Terry J. and Keith Parramor: Quantitative Methods in Finance
19
COURSE CODE: MC-106
LESSON: 02
AUTHOR:
VETTER:
SURINDER KUNDU
PROF. M. S. TURAN
AN OVERVIEW OF CENTRAL TENDENCY
OBJECTIVE:
The present lesson imparts understanding of the calculations and main
properties of measures of central tendency, including mean, mode,
median, quartiles, percentiles, etc.
STRUCTURE:
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
Introduction
Arithmetic Mean
Median
Mode
Relationships of the Mean, Median and Mode
The Best Measure of Central Tendency
Geometric Mean
Harmonic Mean
Summary
Self-Test Questions
2.1
INTRODUCTION
The description of statistical data may be quite elaborate or quite brief depending on
two factors: the nature of data and the purpose for which the same data have been
collected. While describing data statistically or verbally, one must ensure that the
description is neither too brief nor too lengthy. The measures of central tendency
enable us to compare two or more distributions pertaining to the same time period or
within the same distribution over time. For example, the average consumption of tea
in two different territories for the same period or in a territory for two years, say, 2003
and 2004, can be attempted by means of an average.
20
2.2
ARITHMETIC MEAN
Adding all the observations and dividing the sum by the number of observations
results the arithmetic mean. Suppose we have the following observations:
10, 15,30, 7, 42, 79 and 83
These are seven observations. Symbolically, the arithmetic mean, also called simply
mean is
x
= ∑x/n, where x is simple mean.
=
10 + 15 + 30 + 7 + 42 + 79 + 83
7
=
266
= 38
7
It may be noted that the Greek letter μ is used to denote the mean of the population
and n to denote the total number of observations in a population. Thus the population
mean μ = ∑x/n. The formula given above is the basic formula that forms the
definition of arithmetic mean and is used in case of ungrouped data where weights are
not involved.
2.2.1 UNGROUPED DATA-WEIGHTED AVERAGE
In case of ungrouped data where weights are involved, our approach for calculating
arithmetic mean will be different from the one used earlier.
Example 2.1: Suppose a student has secured the following marks in three tests:
Mid-term test 30
Laboratory
25
Final
20
The simple arithmetic mean will be
30 + 25 + 20
= 25
3
21
However, this will be wrong if the three tests carry different weights on the basis of
their relative importance. Assuming that the weights assigned to the three tests are:
Mid-term test
2 points
Laboratory
3 points
Final
5 points
Solution: On the basis of this information, we can now calculate a weighted mean as
shown below:
Table 2.1: Calculation of a Weighted Mean
Type of Test
Relative Weight (w)
Marks (x)
(wx)
Mid-term
2
30
60
Laboratory
3
25
75
Final
5
20
100
Total
∑ w = 10
x=
235
∑ wx w1 x1 + w2 x 2 + w3 x 3
=
∑w
w1 + w2 + w3
=
60 + 75 + 100
= 23.5 marks
2+3+5
It will be seen that weighted mean gives a more realistic picture than the simple or
unweighted mean.
Example 2.2: An investor is fond of investing in equity shares. During a period of
falling prices in the stock exchange, a stock is sold at Rs 120 per share on one day, Rs
105 on the next and Rs 90 on the third day. The investor has purchased 50 shares on
the first day, 80 shares on the second day and 100 shares on the third' day. What
average price per share did the investor pay?
22
Solution:
Table 2.2: Calculation of Weighted Average Price
Day
Price per Share (Rs) (x)
No of Shares Purchased (w)
Amount Paid (wx)
1
120
50
6000
2
105
80
8400
3
90
100
9000
Total
-
230
23,400
Weighted average
=
=
w1 x1 + w2 x 2 + w3 x 3 ∑ wx
=
w1 + w2 + w3
∑w
6000 + 8400 + 9000
= 101.7 marks
50 + 80 + 100
Therefore, the investor paid an average price of Rs 101.7 per share.
It will be seen that if merely prices of the shares for the three days (regardless of the
number of shares purchased) were taken into consideration, then the average price
would be
Rs.
120 + 105 + 90
= 105
3
This is an unweighted or simple average and as it ignores the-quantum of shares
purchased, it fails to give a correct picture. A simple average, it may be noted, is also
a weighted average where weight in each case is the same, that is, only 1. When we
use the term average alone, we always mean that it is an unweighted or simple
average.
2.2.2 GROUPED DATA-ARITHMETIC MEAN
For grouped data, arithmetic mean may be calculated by applying any of the
following methods:
(i) Direct method,
(ii) Short-cut method , (iii) Step-deviation method
23
In the case of direct method, the formula x = ∑fm/n is used. Here m is mid-point of
various classes, f is the frequency of each class and n is the total number of
frequencies. The calculation of arithmetic mean by the direct method is shown below.
Example 2.3: The following table gives the marks of 58 students in Statistics.
Calculate the average marks of this group.
Marks
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Total
No. of Students
4
8
11
15
12
6
2
58
Solution:
Table 2.3: Calculation of Arithmetic Mean by Direct Method
Marks
Mid-point m
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
15
25
35
45
55
65
No. of Students
f
4
8
11
15
12
6
2
fm
20
120
275
525
540
330
130
∑fm = 1940
Where,
x=
∑ fm = 1940 =
n
58
33.45 marks or 33 marks approximately.
It may be noted that the mid-point of each class is taken as a good approximation of
the true mean of the class. This is based on the assumption that the values are
distributed fairly evenly throughout the interval. When large numbers of frequency
occur, this assumption is usually accepted.
24
In the case of short-cut method, the concept of arbitrary mean is followed. The
formula for calculation of the arithmetic mean by the short-cut method is given
below:
x= A+
∑ fd
n
Where A = arbitrary or assumed mean
f = frequency
d = deviation from the arbitrary or assumed mean
When the values are extremely large and/or in fractions, the use of the direct method
would be very cumbersome. In such cases, the short-cut method is preferable. This is
because the calculation work in the short-cut method is considerably reduced
particularly for calculation of the product of values and their respective frequencies.
However, when calculations are not made manually but by a machine calculator, it
may not be necessary to resort to the short-cut method, as the use of the direct method
may not pose any problem.
As can be seen from the formula used in the short-cut method, an arbitrary or assumed
mean is used. The second term in the formula (∑fd ÷ n) is the correction factor for the
difference between the actual mean and the assumed mean. If the assumed mean turns
out to be equal to the actual mean, (∑fd ÷ n) will be zero. The use of the short-cut
method is based on the principle that the total of deviations taken from an actual mean
is equal to zero. As such, the deviations taken from any other figure will depend on
how the assumed mean is related to the actual mean. While one may choose any value
as assumed mean, it would be proper to avoid extreme values, that is, too small or too
high to simplify calculations. A value apparently close to the arithmetic mean should
be chosen.
25
For the figures given earlier pertaining to marks obtained by 58 students, we calculate
the average marks by using the short-cut method.
Example 2.4:
Table 2.4: Calculation of Arithmetic Mean by Short-cut Method
Marks
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Mid-point
m
5
15
25
35
45
55
65
f
d
fd
4
8
11
15
12
6
2
-30
-20
-10
0
10
20
30
-120
-160
-110
0
120
120
60
∑fd = -90
It may be noted that we have taken arbitrary mean as 35 and deviations from
midpoints. In other words, the arbitrary mean has been subtracted from each value of
mid-point and the resultant figure is shown in column d.
x= A+
∑ fd
n
⎛ − 90 ⎞
= 35 + ⎜
⎟
⎝ 58 ⎠
= 35 - 1.55 = 33.45 or 33 marks approximately.
Now we take up the calculation of arithmetic mean for the same set of data using the
step-deviation method. This is shown in Table 2.5.
Table 2.5: Calculation of Arithmetic Mean by Step-deviation Method
Marks
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Mid-point
5
15
25
35
45
55
65
f
4
8
11
15
12
6
2
d
-30
-20
-10
0
10
20
30
26
d’= d/10
-3
-2
-1
0
1
2
3
Fd’
-12
-16
-11
0
12
12
6
∑fd’ =-9
x = A+
∑ fd ' × C
n
⎛ − 9 × 10 ⎞
= 35 + ⎜
⎟
⎝ 58 ⎠
= 33.45 or 33 marks approximately.
It will be seen that the answer in each of the three cases is the same. The stepdeviation method is the most convenient on account of simplified calculations. It may
also be noted that if we select a different arbitrary mean and recalculate deviations
from that figure, we would get the same answer.
Now that we have learnt how the arithmetic mean can be calculated by using different
methods, we are in a position to handle any problem where calculation of the
arithmetic mean is involved.
Example 2.6: The mean of the following frequency distribution was found to be 1.46.
No. of Accidents
0
1
2
3
4
5
No. of Days (frequency)
46
?
?
25
10
5
Total
200 days
Calculate the missing frequencies.
Solution:
Here we are given the total number of frequencies and the arithmetic mean. We have
to determine the two frequencies that are missing. Let us assume that the frequency
against 1 accident is x and against 2 accidents is y. If we can establish two
simultaneous equations, then we can easily find the values of X and Y.
Mean =
(0.46) + (1 . x) + (2 . y) + (3 . 25) + (4 . l0) + (5 . 5)
200
27
1.46 =
x + 2y + 140
200
x + 2y + 140 = (200) (1.46)
x + 2y = 152
x + y=200- {46+25 + 1O+5}
x + y = 200 - 86
x + y = 114
Now subtracting equation (ii) from equation (i), we get
x + 2y =
x+y =
- y
=
152
114
38
Substituting the value of y = 38 in equation (ii) above, x + 38 = 114
Therefore, x = 114 - 38 = 76
Hence, the missing frequencies are:
Against accident 1 : 76
Against accident 2 : 38
2.2.3 CHARACTERISTICS OF THE ARITHMETIC MEAN
Some of the important characteristics of the arithmetic mean are:
1.
The sum of the deviations of the individual items from the arithmetic mean is
always zero. This means I: (x - x ) = 0, where x is the value of an item and x is
the arithmetic mean. Since the sum of the deviations in the positive direction
is equal to the sum of the deviations in the negative direction, the arithmetic
mean is regarded as a measure of central tendency.
2.
The sum of the squared deviations of the individual items from the arithmetic
mean is always minimum. In other words, the sum of the squared deviations
taken from any value other than the arithmetic mean will be higher.
28
3.
As the arithmetic mean is based on all the items in a series, a change in the
value of any item will lead to a change in the value of the arithmetic mean.
4.
In the case of highly skewed distribution, the arithmetic mean may get
distorted on account of a few items with extreme values. In such a case, it
may cease to be the representative characteristic of the distribution.
2.3
MEDIAN
Median is defined as the value of the middle item (or the mean of the values of the
two middle items) when the data are arranged in an ascending or descending order of
magnitude. Thus, in an ungrouped frequency distribution if the n values are arranged
in ascending or descending order of magnitude, the median is the middle value if n is
odd. When n is even, the median is the mean of the two middle values.
Suppose we have the following series:
15, 19,21,7, 10,33,25,18 and 5
We have to first arrange it in either ascending or descending order. These figures are
arranged in an ascending order as follows:
5,7,10,15,18,19,21,25,33
Now as the series consists of odd number of items, to find out the value of the middle
item, we use the formula
Where
n +1
2
Where n is the number of items. In this case, n is 9, as such
n +1
= 5, that is, the size
2
of the 5th item is the median. This happens to be 18.
Suppose the series consists of one more items 23. We may, therefore, have to include
23 in the above series at an appropriate place, that is, between 21 and 25. Thus, the
series is now 5, 7, 10, 15, 18, 19, and 21,23,25,33. Applying the above formula, the
29
median is the size of 5.5th item. Here, we have to take the average of the values of 5th
and 6th item. This means an average of 18 and 19, which gives the median as 18.5.
It may be noted that the formula
n +1
itself is not the formula for the median; it
2
merely indicates the position of the median, namely, the number of items we have to
count until we arrive at the item whose value is the median. In the case of the even
number of items in the series, we identify the two items whose values have to be
averaged to obtain the median. In the case of a grouped series, the median is
calculated by linear interpolation with the help of the following formula:
M = l1
l 2 + l1
(m − c)
f
Where M = the median
l1 = the lower limit of the class in which the median lies
12 = the upper limit of the class in which the median lies
f = the frequency of the class in which the median lies
m = the middle item or (n + 1)/2th, where n stands for total number of
items
c = the cumulative frequency of the class preceding the one in which the median lies
Example 2.7:
Monthly Wages (Rs)
800-1,000
1,000-1,200
1,200-1,400
1,400-1,600
1,600-1,800
1,800-2,000
No. of Workers
18
25
30
34
26
10
Total
143
In order to calculate median in this case, we have to first provide cumulative
frequency to the table. Thus, the table with the cumulative frequency is written as:
30
Monthly Wages Frequency
800 -1,000
1,000 -1,200
1,200 -1,400
1,400 -1,600
1,600 -1,800
1.800 -2,000
Cumulative Frequency
18
25
30
34
26
10
M = l1
18
43
73
107
133
143
l 2 + l1
(m − c)
f
M = n + 1 = 143 + 1 = 72
2
2
It means median lies in the class-interval Rs 1,200 - 1,400.
Now, M = 1200 + 1400 − 1200 (72 − 43)
30
= 1200 +
200
(29)
30
= Rs 1393.3
At this stage, let us introduce two other concepts viz. quartile and decile. To
understand these, we should first know that the median belongs to a general class of
statistical descriptions called fractiles. A fractile is a value below that lays a given
fraction of a set of data. In the case of the median, this fraction is one-half (1/2).
Likewise, a quartile has a fraction one-fourth (1/4). The three quartiles Q1, Q2 and Q3
are such that 25 percent of the data fall below Q1, 25 percent fall between Q1 and Q2,
25 percent fall between Q2 and Q3 and 25 percent fall above Q3 It will be seen that Q2
is the median. We can use the above formula for the calculation of quartiles as well.
The only difference will be in the value of m. Let us calculate both Q1 and Q3 in
respect of the table given in Example 2.7.
Q1
=
l1
l 2 − l1
(m − c)
f
31
Here, m will be
n + 1 143 + 1
=
= 36
4
4
=
Q1 = 1000 +
= 1000 +
1200 − 1000
(36 − 18)
25
200
(18)
25
= Rs. 1,144
In the case of Q3, m will be 3 =
Q1 = 1600 +
= 1600 +
n + 1 3×144
=
= 108
4
4
1800 − 1600
(108 − 107)
26
200
(1)
26
Rs. 1,607.7 approx
In the same manner, we can calculate deciles (where the series is divided into 10
parts) and percentiles (where the series is divided into 100 parts). It may be noted that
unlike arithmetic mean, median is not affected at all by extreme values, as it is a
positional average. As such, median is particularly very useful when a distribution
happens to be skewed. Another point that goes in favour of median is that it can be
computed when a distribution has open-end classes. Yet, another merit of median is
that when a distribution contains qualitative data, it is the only average that can be
used. No other average is suitable in case of such a distribution. Let us take a couple
of examples to illustrate what has been said in favour of median.
32
Example 2.8:Calculate the most suitable average for the following data:
Size of the Item Below 50
Frequency
50-100
15
100-150
20
150-200
36
40
200 and above
10
Solution: Since the data have two open-end classes-one in the beginning (below 50) and the
other at the end (200 and above), median should be the right choice as a measure of central
tendency.
Table 2.6: Computation of Median
Size of Item
Frequency
Below 50
50-100
100-150
150-200
200 and above
15
15
20
36
35
71
40
10
Median is the size of
=
Cumulative Frequency
111
121
n +1
th item
2
121 + 1
= 61st item
2
Now, 61st item lies in the 100-150 class
l 2 − l1
(m − c)
f
Median =
11 = l1
= 100 +
150 − 100
(61 − 35)
36
= 100 + 36.11 = 136.11 approx.
Example 2.9: The following data give the savings bank accounts balances of nine sample
households selected in a survey. The figures are in rupees.
745
2,000 1,500 68,000 461 549
3750 1800 4795
(a) Find the mean and the median for these data; (b) Do these data contain an outlier? If so,
exclude this value and recalculate the mean and median. Which of these summary measures
33
has a greater change when an outlier is dropped?; (c) Which of these two summary measures
is more appropriate for this series?
Solution:
Mean = Rs.
745 + 2,000 + 1,500 + 68,000 + 461 + 549 + 3,750 + 1,800 + 4,795
9
=
Median =
=
Rs 83,600
= Rs 9,289
9
Size of
n + 1
th item
2
9 + 1
= 5th item
2
Arranging the data in an ascending order, we find that the median is Rs 1,800.
(b) An item of Rs 68,000 is excessively high. Such a figure is called an 'outlier'. We
exclude this figure and recalculate both the mean and the median.
Mean =
=
Rs.
Rs
Median = Size of
=
83,600 − 68,000
8
15,600
= Rs. 1,950
8
n + 1
th item
2
8 + 1
= 4.5th item.
2
= Rs.
1,500 − 1,800
= Rs. 1,650
2
It will be seen that the mean shows a far greater change than the median when the
outlier is dropped from the calculations.
(c) As far as these data are concerned, the median will be a more appropriate measure
than the mean.
Further, we can determine the median graphically as follows:
34
Example 2.10: Suppose we are given the following series:
Class interval 0-10
Frequency
6
10-20 20-30 30-40 40-50 50-60 60-70
12
22
37
17
8
5
We are asked to draw both types of ogive from these data and to determine the
median.
Solution:
First of all, we transform the given data into two cumulative frequency distributions,
one based on ‘less than’ and another on ‘more than’ methods.
Table A
Frequency
Less than 10
Less than 20
Less than 30
Less than 40
Less than 50
Less than 60
Less than 70
6
18
40
77
94
102
107
Table B
Frequency
107
101
89
67
30
13
5
More than 0
More than 10
More than 20
More than 30
More than 40
More than 50
More than 60
It may be noted that the point of
intersection of the two ogives gives the
value of the median. From this point of
intersection A, we draw a straight line to
35
meet the X-axis at M. Thus, from the point of origin to the point at M gives the value
of the median, which comes to 34, approximately. If we calculate the median by
applying the formula, then the answer comes to 33.8, or 34, approximately. It may be
pointed out that even a single ogive can be used to determine the median. As we have
determined the median graphically, so also we can find the values of quartiles, deciles
or percentiles graphically. For example, to determine we have to take size of {3(n +
1)} /4 = 81st
item. From this point on the Y-axis, we can draw a perpendicular to
meet the 'less than' ogive from which another straight line is to be drawn to meet the
X-axis. This point will give us the value of the upper quartile. In the same manner,
other values of Q1 and deciles and percentiles can be determined.
2.3.1 CHARACTERISTICS OF THE MEDIAN
1.
Unlike the arithmetic mean, the median can be computed from open-ended
distributions. This is because it is located in the median class-interval, which
would not be an open-ended class.
2.
The median can also be determined graphically whereas the arithmetic mean
cannot be ascertained in this manner.
3.
As it is not influenced by the extreme values, it is preferred in case of a
distribution having extreme values.
4.
In case of the qualitative data where the items are not counted or measured but
are scored or ranked, it is the most appropriate measure of central tendency.
2.4
MODE
The mode is another measure of central tendency. It is the value at the point around
which the items are most heavily concentrated. As an example, consider the following
series: 8,9, 11, 15, 16, 12, 15,3, 7, 15
36
There are ten observations in the series wherein the figure 15 occurs maximum
number of times three. The mode is therefore 15. The series given above is a discrete
series; as such, the variable cannot be in fraction. If the series were continuous, we
could say that the mode is approximately 15, without further computation.
In the case of grouped data, mode is determined by the following formula:
Mode= l1 +
Where,
f1 − f 0
×i
( f1 − f 0 ) + ( f1 − f 2 )
l1 = the lower value of the class in which the mode lies
fl = the frequency of the class in which the mode lies
fo = the frequency of the class preceding the modal class
f2 = the frequency of the class succeeding the modal class
i = the class-interval of the modal class
While applying the above formula, we should ensure that the class-intervals are
uniform throughout. If the class-intervals are not uniform, then they should be made
uniform on the assumption that the frequencies are evenly distributed throughout the
class. In the case of inequal class-intervals, the application of the above formula will
Example 2.11:
Let us take the following frequency distribution:
Class intervals (1)
Frequency (2)
30-40
4
40-50
6
50-60
8
60-70
12
70-80
9
80-90
7
90-100
4
We have to calculate the mode in respect of this series.
Solution: We can see from Column (2) of the table that the maximum frequency of
12 lies in the class-interval of 60-70. This suggests that the mode lies in this classinterval. Applying the formula given earlier, we get:
37
Mode = 60 +
= 60 +
12 - 8
× 10
12 - 8 (12 - 8) + (12 - 9)
4
× 10
4+3
= 65.7 approx.
In several cases, just by inspection one can identify the class-interval in which the
mode lies. One should see which the highest frequency is and then identify to which
class-interval this frequency belongs. Having done this, the formula given for
calculating the mode in a grouped frequency distribution can be applied.
At times, it is not possible to identify by inspection the class where the mode lies. In
such cases, it becomes necessary to use the method of grouping. This method consists
of two parts:
(i)
Preparation of a grouping table: A grouping table has six columns, the first
column showing the frequencies as given in the problem. Column 2 shows
frequencies grouped in two's, starting from the top. Leaving the first
frequency, column 3 shows frequencies grouped in two's. Column 4 shows the
frequencies of the first three items, then second to fourth item and so on.
Column 5 leaves the first frequency and groups the remaining items in three's.
Column 6 leaves the first two frequencies and then groups the remaining in
three's. Now, the maximum total in each column is marked and shown either
in a circle or in a bold type.
(ii)
Preparation of an analysis table: After having prepared a grouping table, an
analysis table is prepared. On the left-hand side, provide the first column for
column numbers and on the right-hand side the different possible values of
mode. The highest values marked in the grouping table are shown here by a
bar or by simply entering 1 in the relevant cell corresponding to the values
38
they represent. The last row of this table will show the number of times a
particular value has occurred in the grouping table. The highest value in the
analysis table will indicate the class-interval in which the mode lies. The
procedure of preparing both the grouping and analysis tables to locate the
modal class will be clear by taking an example.
Example 2.12: The following table gives some frequency data:
Size of Item
Frequency
10-20
20-30
30-40
40-50
50-60
60-70
10
18
25
26
17
4
Solution:
Size of item
10-20
1
2
Grouping Table
3
4
5
6
10
28
20-30
18
53
43
30-40
25
69
51
40-50
26
68
43
50-60
17
47
21
60-70
4
Analysis table
Col. No.
1
2
3
4
5
10-20
1
Size of item
20-30
30-40
1
1
1
1
1
1
1
39
40-50
50-60
1
1
1
1
1
6
Total
1
3
1
1
1
5
5
2
This is a bi-modal series as is evident from the analysis table, which shows that the
two classes 30-40 and 40-50 have occurred five times each in the grouping. In such a
situation, we may have to determine mode indirectly by applying the following
formula:
Mode = 3 median - 2 mean
Median = Size of (n + l)/2th item, that is, 101/2 = 50.5th item. This lies in the class
30-40. Applying the formula for the median, as given earlier, we get
40 - 30
(50.5 − 28)
25
=
30 +
=
30 + 9 = 39
Now, arithmetic mean is to be calculated. This is shown in the following table.
Class- interval
10-20
20-30
30-40
40-50
50-60
60-70
Total
Frequency
10
18
25
26
17
4
100
Mid- points
15
25
35
45
55
65
Deviation is taken from arbitrary mean = 35
∑ fd ' × i
Mean =
A+
=
35 +
=
38.4
Mode =
n
34
× 10
100
3 median - 2 mean
=
(3 x 39) - (2 x 38.4)
=
117 -76.8
40
d
-20
-10
0
10
20
30
d' = d/10
-2
-I
0
1
2
3
fd'
-20
-18
0
26
34
12
34
=
40.2
This formula, Mode = 3 Median-2 Mean, is an empirical formula only. And it can
give only approximate results. As such, its frequent use should be avoided. However,
when mode is ill defined or the series is bimodal (as is the case in the present
example) it may be used.
2.5
RELATIONSHIPS OF THE MEAN, MEDIAN AND MODE
Having discussed mean, median and mode, we now turn to the relationship amongst
these three measures of central tendency. We shall discuss the relationship assuming
that there is a unimodal frequency distribution.
(i)
When a distribution is symmetrical, the mean, median and mode are the same,
as is shown below in the following figure.
In case, a distribution is
skewed to the right, then
mean>
median>
Generally,
income
mode.
distri-
bution is skewed to the right where a large number of families have relatively
low income and a small number of families have extremely high income. In
such a case, the mean is pulled up by the extreme high incomes and the
relation among these three measures is as shown in Fig. 6.3. Here, we find that
mean> median> mode.
(ii)
When a distribution is skewed to
the left, then mode> median>
mean. This is because here mean is
pulled down
below the median
by extremely low values. This is
41
shown as in the figure.
(iii)
Given the mean and median of a unimodal distribution, we can determine
whether it is skewed to the
right or left. When mean>
median, it is skewed to the
right; when median> mean, it
is skewed to the left. It may be noted that the median is always in the middle
between mean and mode.
2.6
THE BEST MEASURE OF CENTRAL TENDENCY
At this stage, one may ask as to which of these three measures of central tendency the
best is. There is no simple answer to this question. It is because these three measures
are based upon different concepts. The arithmetic mean is the sum of the values
divided by the total number of observations in the series. The median is the value of
the middle observation that divides the series into two equal parts. Mode is the value
around which the observations tend to concentrate. As such, the use of a particular
measure will largely depend on the purpose of the study and the nature of the data;
For example, when we are interested in knowing the consumers preferences for
different brands of television sets or different kinds of advertising, the choice should
go in favour of mode. The use of mean and median would not be proper. However,
the median can sometimes be used in the case of qualitative data when such data can
be arranged in an ascending or descending order. Let us take another example.
Suppose we invite applications for a certain vacancy in our company. A large number
of candidates apply for that post. We are now interested to know as to which age or
age group has the largest concentration of applicants. Here, obviously the mode will
be the most appropriate choice. The arithmetic mean may not be appropriate as it may
42
be influenced by some extreme values. However, the mean happens to be the most
commonly used measure of central tendency as will be evident from the discussion in
the subsequent chapters.
2.7
GEOMETRIC MEAN
Apart from the three measures of central tendency as discussed above, there are two
other means that are used sometimes in business and economics. These are the
geometric mean and the harmonic mean. The geometric mean is more important than
the harmonic mean. We discuss below both these means. First, we take up the
geometric mean. Geometric mean is defined at the nth root of the product of n
observations of a distribution.
Symbolically, GM = n x1 ....x 2 .....x n ... If we have only two observations, say, 4 and
4 × 16 = 64 = 8. Similarly, if there are three observations, then we
16 then GM =
have to calculate the cube root of the product of these three observations; and so on.
When the number of items is large, it becomes extremely difficult to multiply the
numbers and to calculate the root. To simplify calculations, logarithms are used.
Example 2.13: If we have to find out the geometric mean of 2, 4 and 8, then we find
Log GM
GM
=
∑ log x
i
n
=
Log 2 + Log 4 + Log 8
3
=
0.3010 + 0.6021 + 0.9031
3
=
1.8062
= 0.60206
3
=
Antilog 0.60206
= 4
43
When the data are given in the form of a frequency distribution, then the geometric
mean can be obtained by the formula:
Log GM
=
=
f 1 . log xl + f 2 . log x 2 + ... + f n . log x
f 1 + f 2 + .......... fn
n
∑ f . log x
f 1 + f 2 + .......... fn
Then, GM = Antilog n
The geometric mean is most suitable in the following three cases:
1.
Averaging rates of change.
2.
The compound interest formula.
3.
Discounting, capitalization.
Example 2.14: A person has invested Rs 5,000 in the stock market. At the end of the
first year the amount has grown to Rs 6,250; he has had a 25 percent profit. If at the
end of the second year his principal has grown to Rs 8,750, the rate of increase is 40
percent for the year. What is the average rate of increase of his investment during the
two years?
Solution:
GM = 1.25 × 1.40 = 1.75. = 1.323
The average rate of increase in the value of investment is therefore 1.323 - 1 = 0.323,
which if multiplied by 100, gives the rate of increase as 32.3 percent.
Example 2.15: We can also derive a compound interest formula from the above set of
data. This is shown below:
Solution: Now, 1.25 x 1.40 = 1.75. This can be written as 1.75 = (1 + 0.323)2.
Let P2 = 1.75, P0 = 1, and r = 0.323, then the above equation can be written as P2 = (1
+ r)2 or P2 = P0 (1 + r)2.
44
Where P2 is the value of investment at the end of the second year, P0 is the initial
investment and r is the rate of increase in the two years. This, in fact, is the familiar
compound interest formula. This can be written in a generalised form as Pn = P0(1 +
r)n. In our case Po is Rs 5,000 and the rate of increase in investment is 32.3 percent.
Let us apply this formula to ascertain the value of Pn, that is, investment at the end of
the second year.
Pn = 5,000 (1 + 0.323)2
= 5,000 x 1.75
= Rs 8,750
It may be noted that in the above example, if the arithmetic mean is used, the resultant
figure will be wrong. In this case, the average rate for the two years is
per year, which comes to 32.5. Applying this rate, we get Pn =
25 + 40
percent
2
165
x 5,000
100
= Rs 8,250
This is obviously wrong, as the figure should have been Rs 8,750.
Example 2.16: An economy has grown at 5 percent in the first year, 6 percent in the
second year, 4.5 percent in the third year, 3 percent in the fourth year and 7.5 percent
in the fifth year. What is the average rate of growth of the economy during the five
years?
Solution:
Year
1
2
3
4
5
Rate of Growth
( percent)
5
6
4.5
3
7.5
Value at the end of the
Log x
Year x (in Rs)
105
2.02119
106
2.02531
104.5
2.01912
103
2.01284
107.5
2.03141
∑ log X = 10.10987
45
⎛ ∑ log x ⎞
⎟
GM = Antilog ⎜
⎜
⎟
n
⎝
⎠
⎛ 10.10987 ⎞
= Antilog ⎜
⎟
5
⎝
⎠
= Antilog 2.021974
= 105.19
Hence, the average rate of growth during the five-year period is 105.19 - 100 = 5.19
percent per annum. In case of a simple arithmetic average, the corresponding rate of
growth would have been 5.2 percent per annum.
2.7.1
DISCOUNTING
The compound interest formula given above was
Pn=P0(1+r)n This can be written as P0 =
Pn
(1 + r ) n
This may be expressed as follows:
If the future income is Pn rupees and the present rate of interest is 100 r percent, then
the present value of P n rupees will be P0 rupees. For example, if we have a machine
that has a life of 20 years and is expected to yield a net income of Rs 50,000 per year,
and at the end of 20 years it will be obsolete and cannot be used, then the machine's
present value is
50,000
50,000 50,000 50,000
+
+.................
+
n
2
3
(1 + r )
(1 + r )
(1 + r )
(1 + r ) 20
This process of ascertaining the present value of future income by using the interest
rate is known as discounting.
In conclusion, it may be said that when there are extreme values in a series, geometric
mean should be used as it is much less affected by such values. The arithmetic mean
in such cases will give misleading results.
46
Before we close our discussion on the geometric mean, we should be aware of its
1.
Geometric mean is based on each and every observation in the data set.
2.
It is rigidly defined.
3.
It is more suitable while averaging ratios and percentages as also in calculating
growth rates.
4.
As compared to the arithmetic mean, it gives more weight to small values and
less weight to large values. As a result of this characteristic of the geometric
mean, it is generally less than the arithmetic mean. At times it may be equal to
the arithmetic mean.
5.
It is capable of algebraic manipulation. If the geometric mean has two or more
series is known along with their respective frequencies. Then a combined
geometric mean can be calculated by using the logarithms.
2.7.3 LIMITATIONS OF G.M.
1.
As compared to the arithmetic mean, geometric mean is difficult to
understand.
2.
Both computation of the geometric mean and its interpretation are rather
difficult.
3.
When there is a negative item in a series or one or more observations have
zero value, then the geometric mean cannot be calculated.
In view of the limitations mentioned above, the geometric mean is not frequently
used.
2.8
HARMONIC MEAN
47
The harmonic mean is defined as the reciprocal of the arithmetic mean of the
reciprocals of individual observations. Symbolically,
HM=
n
∑1 / x
= Re ciprocal
n
1/ x1 + 1/ x 2 + 1/ x 3 + . .. + 1/ x n
The calculation of harmonic mean becomes very tedious when a distribution has a
large number of observations. In the case of grouped data, the harmonic mean is
calculated by using the following formula:
HM = Reciprocal of
⎛
n
∑ ⎜⎜ f
i −1
⎝
i
1
xi
⎞
⎟⎟
⎠
×
1
xi
×
or
n
n
⎛
∑ ⎜⎜ f
i −1
⎝
i
⎞
⎟⎟
⎠
Where n is the total number of observations.
Here, each reciprocal of the original figure is weighted by the corresponding
frequency (f).
The main advantage of the harmonic mean is that it is based on all observations in a
distribution and is amenable to further algebraic treatment. When we desire to give
greater weight to smaller observations and less weight to the larger observations, then
the use of harmonic mean will be more suitable. As against these advantages, there
are certain limitations of the harmonic mean. First, it is difficult to understand as well
as difficult to compute. Second, it cannot be calculated if any of the observations is
zero or negative. Third, it is only a summary figure, which may not be an actual
observation in the distribution.
It is worth noting that the harmonic mean is always lower than the geometric mean,
which is lower than the arithmetic mean. This is because the harmonic mean assigns
48
lesser importance to higher values. Since the harmonic mean is based on reciprocals,
it becomes clear that as reciprocals of higher values are lower than those of lower
values, it is a lower average than the arithmetic mean as well as the geometric mean.
Example 2.17: Suppose we have three observations 4, 8 and 16. We are required to
calculate the harmonic mean. Reciprocals of 4,8 and 16 are:
Since HM =
1 1 1
, ,
respectively
4 8 16
n
1/ x 1 + 1/ x 2 + 1/ x 3
=
3
1/ 4 + 1/ 8 + 1/ 16
=
3
0.25 + 0.125 + 0.0625
=
6.857 approx.
Example 2.18: Consider the following series:
Class-interval
2-4
4-6
6-8
8-10
Frequency
20
40
30
10
Solution:
Let us set up the table as follows:
Class-interval
2-4
4-6
6-8
8-10
n
⎛
∑ ⎜⎜ f
=
=
i −1
⎝
i
×
1
xi
Mid-value
3
5
7
9
Frequency
20
40
30
10
⎞
⎟⎟
⎠
n
100
= 4.984 approx.
20.0641
49
Reciprocal of MV
0.3333
0.2000
0.1429
0.1111
Total
f x 1/x
6.6660
8.0000
4.2870
1.1111
20.0641
Example 2.19: In a small company, two typists are employed. Typist A types one
page in ten minutes while typist B takes twenty minutes for the same. (i) Both are
asked to type 10 pages. What is the average time taken for typing one page? (ii) Both
are asked to type for one hour. What is the average time taken by them for typing one
page?
Solution: Here Q-(i) is on arithmetic mean while Q-(ii) is on harmonic mean.
(i)
M=
HM
(10 × 10) + (20 × 20)(min utes)
10 × 2( pages)
=
15 minutes
=
60 × (min utes)
60 / 10 + 60 / 20( pages)
=
120
40
=
= 13 min utes and 20 seconds.
120 + 60
3
20
Example 2.20: It takes ship A 10 days to cross the Pacific Ocean; ship B takes 15
days and ship C takes 20 days. (i) What is the average number of days taken by a ship
to cross the Pacific Ocean? (ii) What is the average number of days taken by a cargo
to cross the Pacific Ocean when the ships are hired for 60 days?
Solution:
Here again Q-(i) pertains to simple arithmetic mean while Q-(ii) is
concerned with the harmonic mean.
(i)
M
=
10 + 15 + 20
= 15 days
3
(ii)
HM
=
60 × 3(days) _
60 / 10 + 60 / 15 + 60 / 20
=
180
360 + 240 + 180
60
50
=
2.9
13.8 days approx.
We have seen earlier that the geometric mean is the antilogarithm of the arithmetic
mean of the logarithms, and the harmonic mean is the reciprocal of the arithmetic
mean of the reciprocals. Likewise, the quadratic mean (Q) is the square root of the
arithmetic mean of the squares. Symbolically,
Q=
x12 + x 22 + ...... + 2n
n
Instead of using original values, the quadratic mean can be used while averaging
deviations when the standard deviation is to be calculated. This will be used in the
next chapter on dispersion.
2.9.1
Relative Position of Different Means
The relative position of different means will always be:
Q> x >G>H provided that all the individual observations in a series are positive and
all of them are not the same.
2.9.2
Composite Average or Average of Means
Sometimes, we may have to calculate an average of several averages. In such cases,
we should use the same method of averaging that was employed in calculating the
original averages. Thus, we should calculate the arithmetic mean of several values of
x, the geometric mean of several values of GM, and the harmonic mean of several
values of HM. It will be wrong if we use some other average in averaging of means.
2.10 SUMMARY
It is the most important objective of statistical analysis is to get one single value that
describes the characteristics of the entire mass of cumbersome data. Such a value is
finding out, which is known as central value to serve our purpose.
51
2.11 SELF-TEST QUESTIONS
1.
What are the desiderata (requirements) of a good average? Compare the mean,
the median and the mode in the light of these desiderata? Why averages are
called measures of central tendency?
2.
"Every average has its own peculiar characteristics. It is difficult to say which
average is the best." Explain with examples.
3.
What do you understand .by 'Central Tendency'? Under what conditions is the
median more suitable than other measures of central tendency?
4.
The average monthly salary paid to all employees in a company was Rs 8,000.
The average monthly salaries paid to male and female employees of the
company were Rs 10,600 and Rs 7,500 respectively. Find out the percentages
of males and females employed by the company.
5.
Calculate the arithmetic mean from the following data:
Class
10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89
Frequency
6.
2
4
9
12
6
4
2
Calculate the mean, median and mode from the following data:
Height in Inches
Number of Persons
62-63
63-64
64-65
65-66
66-67
67-68
68-69
Total
7.
11
2
6
14
16
8
3
1
50
A number of particular articles have been classified according to their weights.
After drying for two weeks, the same articles have again been weighed and
similarly classified. It is known that the median weight in the first weighing
52
was 20.83 gm while in the second weighing it was 17.35 gm. Some
frequencies a and b in the first weighing and x and y in the second are missing.
It is known that a = 1/3x and b = 1/2 y. Find out the values of the missing
frequencies.
Class
8
Frequencies
First Weighing
Second Weighing
0- 5
a
z
5-10
b
y
10-15
11
40
15-20
52
50
20-25
75
30
25-30
22
28
Cities A, Band C are equidistant from each other. A motorist travels from A to
B at 30 km/h; from B to C at 40 km/h and from C to A at 50 km/h. Determine
his average speed for the entire trip.
9
10
Calculate the harmonic mean from the following data:
Class-Interval
2-4
4-6
6-8
8-10
Frequency
20
40
30
10
A vehicle when climbing up a gradient, consumes petrol @ 8 km per litre.
While coming down it runs 12 km per litre. Find its average consumption for
to and fro travel between two places situated at the two ends of 25 Ian long
53
1.
Levin, Richard I. and David S. Rubin: Statistics for Management, Prentice
Hall, New Delhi.
2.
Watsman Terry J. and Keith Parramor: Quantitative Methods in Finance
3.
Hooda, R. P.: Statistics for Business and Economics, Macmillan, New Delhi.
4.
Hein, L. W. Quantitative Approach to Managerial Decisions, Prentice Hall,
NJ.
54
COURSE CODE: MC-106
LESSON: 03
AUTHOR:
VETTER:
SURINDER KUNDU
PROF. M. S. TURAN
DISPERSION AND SKEWNESS
OBJECTIVE:
The objective of the present lesson is to impart the knowledge of
measures of dispersion and skewness and to enable the students to
distinguish between average, dispersion, skewness, moments and
kurtosis.
STRUCTURE:
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
Introduction
Meaning and Definition of Dispersion
Significance and Properties of Measuring Variation
Measures of Dispersion
Range
Interquartile Range or Quartile Deviation
Mean Deviation
Standard Deviation
Lorenz Curve
Skewness: Meaning and Definitions
Tests of Skewness
Measures of Skewness
Moments
Kurtosis
Summary
Self-Test Questions
3.1
INTRODUCTION
In the previous chapter, we have explained the measures of central tendency. It may
be noted that these measures do not indicate the extent of dispersion or variability in a
distribution. The dispersion or variability provides us one more step in increasing our
understanding of the pattern of the data. Further, a high degree of uniformity (i.e. low
degree of dispersion) is a desirable quality. If in a business there is a high degree of
variability in the raw material, then it could not find mass production economical.
55
Suppose an investor is looking for a suitable equity share for investment. While
examining the movement of share prices, he should avoid those shares that are highly
fluctuating-having sometimes very high prices and at other times going very low.
Such extreme fluctuations mean that there is a high risk in the investment in shares.
The investor should, therefore, prefer those shares where risk is not so high.
3.2
MEANING AND DEFINITIONS OF DISPERSION
The various measures of central value give us one single figure that represents the
entire data. But the average alone cannot adequately describe a set of observations,
unless all the observations are the same. It is necessary to describe the variability or
dispersion of the observations. In two or more distributions the central value may be
the same but still there can be wide disparities in the formation of distribution.
Measures of dispersion help us in studying this important characteristic of a
distribution.
Some important definitions of dispersion are given below:
1.
"Dispersion is the measure of the variation of the items."
2.
"The degree to which numerical data tend to spread about an average value is
called the variation of dispersion of the data."
3.
-Spiegel
Dispersion or spread is the degree of the scatter or variation of the variable
4.
-A.L. Bowley
-Brooks & Dick
"The measurement of the scatterness of the mass of figures in a series about an
average is called measure of variation or dispersion."
-Simpson & Kajka
It is clear from above that dispersion (also known as scatter, spread or variation)
measures the extent to which the items vary from some central value. Since measures
of dispersion give an average of the differences of various items from an average,
they are also called averages of the second order. An average is more meaningful
when it is examined in the light of dispersion. For example, if the average wage of the
56
workers of factory A is Rs. 3885 and that of factory B Rs. 3900, we cannot
necessarily conclude that the workers of factory B are better off because in factory B
there may be much greater dispersion in the distribution of wages. The study of
dispersion is of great significance in practice as could well be appreciated from the
following example:
Series A
Series B
Series C
100
100
1
100
105
489
100
102
2
100
103
3
100
90
5
Total
500
500
500
x
100
100
100
Since arithmetic mean is the same in all three series, one is likely to conclude that
these series are alike in
nature.
But
a
close
examination shall reveal
that
distributions
differ
widely from one another.
In series A, (In Box-3.1)
each and every item is
perfectly represented by the
arithmetic mean or in other
words none of the items of
series A deviates from the
57
arithmetic mean and hence there is no dispersion. In series B, only one item is
perfectly represented by the arithmetic mean and the other items vary but the variation
is very small as compared to series C. In series C. not a single item is represented by
the arithmetic mean and the items vary widely from one another. In series C,
dispersion is much greater compared to series B. Similarly, we may have two groups
of labourers with the same mean salary and yet their distributions may differ widely.
The mean salary may not be so important a characteristic as the variation of the items
from the mean. To the student of social affairs the mean income is not so vitally
important as to know how this income is distributed. Are a large number receiving the
mean income or are there a few with enormous incomes and millions with incomes far
below the mean? The three figures given in Box 3.1 represent frequency distributions
with some of the characteristics. The two curves in diagram (a) represent two
distractions with the same mean X , but with different dispersions. The two curves in
(b) represent two distributions with the same dispersion but with unequal means X
l
and X 2, (c) represents two distributions with unequal dispersion. The measures of
central tendency are, therefore insufficient. They must be supported and supplemented
with other measures.
In the present chapter, we shall be especially concerned with the measures of
variability or spread or dispersion. A measure of variation or dispersion is one that
measures the extent to which there are differences between individual observation and
some central or average value. In measuring variation we shall be interested in the
amount of the variation or its degree but not in the direction. For example, a measure
of 6 inches below the mean has just as much dispersion as a measure of six inches
above the mean.
58
Literally meaning of dispersion is ‘scatteredness’. Average or the measures of central
tendency gives us an idea of the concentration of the observations about the central
part of the distribution. If we know the average alone, we cannot form a complete idea
about the distribution. But with the help of dispersion, we have an idea about
homogeneity or heterogeneity of the distribution.
3.3
SIGNIFICANCE
AND
PROPERTIES
OF
MEASURING
VARIATION
Measures of variation are needed for four basic purposes:
1.
Measures of variation point out as to how far an average is representative of
the mass. When dispersion is small, the average is a typical value in the sense
that it closely represents the individual value and it is reliable in the sense that
it is a good estimate of the average in the corresponding universe. On the other
hand, when dispersion is large, the average is not so typical, and unless the
sample is very large, the average may be quite unreliable.
2.
Another purpose of measuring dispersion is to determine nature and cause of
variation in order to control the variation itself. In matters of health variations
in body temperature, pulse beat and blood pressure are the basic guides to
diagnosis. Prescribed treatment is designed to control their variation. In
industrial production efficient operation requires control of quality variation
the causes of which are sought through inspection is basic to the control of
causes of variation. In social sciences a special problem requiring the
measurement of variability is the measurement of "inequality" of the
distribution of income or wealth etc.
3.
Measures of dispersion enable a comparison to be made of two or more series
with regard to their variability. The study of variation may also be looked
59
upon as a means of determining uniformity of consistency. A high degree of
variation would mean little uniformity or consistency whereas a low degree of
variation would mean great uniformity or consistency.
4.
Many powerful analytical tools in statistics such as correlation analysis. the
testing of hypothesis, analysis of variance, the statistical quality control,
regression analysis is based on measures of variation of one kind or another.
A good measure of dispersion should possess the following properties
3.4
1.
It should be simple to understand.
2.
It should be easy to compute.
3.
It should be rigidly defined.
4.
It should be based on each and every item of the distribution.
5.
It should be amenable to further algebraic treatment.
6.
It should have sampling stability.
7.
Extreme items should not unduly affect it.
MEAURES OF DISPERSION
There are five measures of dispersion: Range, Inter-quartile range or Quartile
Deviation, Mean deviation, Standard Deviation, and Lorenz curve. Among them, the
first four are mathematical methods and the last one is the graphical method. These
are discussed in the ensuing paragraphs with suitable examples.
3.5
RANGE
The simplest measure of dispersion is the range, which is the difference between the
maximum value and the minimum value of data.
Example 3.1: Find the range for the following three sets of data:
Set 1:
05
15
15
05
15
05
15
15
15
15
Set 2:
8
7
15
11
12
5
13
11
15
9
60
Set 3:
5
5
5
5
5
5
5
5
5
5
Solution: In each of these three sets, the highest number is 15 and the lowest number
is 5. Since the range is the difference between the maximum value and the minimum
value of the data, it is 10 in each case. But the range fails to give any idea about the
dispersal or spread of the series between the highest and the lowest value. This
becomes evident from the above data.
In a frequency distribution, range is calculated by taking the difference between the
upper limit of the highest class and the lower limit of the lowest class.
Example 3.2: Find the range for the following frequency distribution:
Size of Item
Frequency
20- 40
7
40- 60
11
60- 80
30
80-100
17
100-120
5
Total
70
Solution: Here, the upper limit of the highest class is 120 and the lower limit of the
lowest class is 20. Hence, the range is 120 - 20 = 100. Note that the range is not
influenced by the frequencies. Symbolically, the range is calculated b the formula L S, where L is the largest value and S is the smallest value in a distribution. The
coefficient of range is calculated by the formula: (L-S)/ (L+S). This is the relative
measure. The coefficient of the range in respect of the earlier example having three
sets of data is: 0.5.The coefficient of range is more appropriate for purposes of
comparison as will be evident from the following example:
Example 3.3: Calculate the coefficient of range separately for the two sets of data
given below:
Set 1
8
10
20
9
15
10
13
28
Set 2
30
35
42
50
32
49
39
33
61
Solution: It can be seen that the range in both the sets of data is the same:
Set 1
28 - 8 = 20
Set 2
50 - 30 = 20
Coefficient of range in Set 1 is:
28 – 8 = 0.55
28+8
Coefficient of range in set 2 is:
50 – 30
= 0.25
50 +30
3.5.1 LIMITATIONS OF RANGE
There are some limitations of range, which are as follows:
1.
It is based only on two items and does not cover all the items in a distribution.
2.
It is subject to wide fluctuations from sample to sample based on the same
population.
3.
It fails to give any idea about the pattern of distribution. This was evident from
the data given in Examples 1 and 3.
4.
Finally, in the case of open-ended distributions, it is not possible to compute
the range.
Despite these limitations of the range, it is mainly used in situations where one wants
to quickly have some idea of the variability or' a set of data. When the sample size is
very small, the range is considered quite adequate measure of the variability. Thus, it
is widely used in quality control where a continuous check on the variability of raw
materials or finished products is needed. The range is also a suitable measure in
weather forecast. The meteorological department uses the range by giving the
maximum and the minimum temperatures. This information is quite useful to the
common man, as he can know the extent of possible variation in the temperature on a
particular day.
62
3.6
INTERQUARTILE RANGE OR QUARTILE DEVIATION
The interquartile range or the quartile deviation is a better measure of variation in a
distribution than the range. Here, avoiding the 25 percent of the distribution at both
the ends uses the middle 50 percent of the distribution. In other words, the
interquartile range denotes the difference between the third quartile and the first
quartile.
Symbolically, interquartile range = Q3- Ql
Many times the interquartile range is reduced in the form of semi-interquartile range
or quartile deviation as shown below:
Semi interquartile range or Quartile deviation = (Q3 – Ql)/2
When quartile deviation is small, it means that there is a small deviation in the central
50 percent items. In contrast, if the quartile deviation is high, it shows that the central
50 percent items have a large variation. It may be noted that in a symmetrical
distribution, the two quartiles, that is, Q3 and QI are equidistant from the median.
Symbolically,
M-QI = Q3-M
However, this is seldom the case as most of the business and economic data are
asymmetrical. But, one can assume that approximately 50 percent of the observations
are contained in the interquartile range. It may be noted that interquartile range or the
quartile deviation is an absolute measure of dispersion. It can be changed into a
relative measure of dispersion as follows:
Coefficient of QD =
Q3 –Q1
Q3 +Q1
The computation of a quartile deviation is very simple, involving the computation of
upper and lower quartiles. As the computation of the two quartiles has already been
explained in the preceding chapter, it is not attempted here.
63
3.6.1 MERITS OF QUARTILE DEVIATION
The following merits are entertained by quartile deviation:
1.
As compared to range, it is considered a superior measure of dispersion.
2.
In the case of open-ended distribution, it is quite suitable.
3.
Since it is not influenced by the extreme values in a distribution, it is
particularly suitable in highly skewed or erratic distributions.
3.6.2 LIMITATIONS OF QUARTILE DEVIATION
1.
Like the range, it fails to cover all the items in a distribution.
2.
It is not amenable to mathematical manipulation.
3.
It varies widely from sample to sample based on the same population.
4.
Since it is a positional average, it is not considered as a measure of dispersion.
It merely shows a distance on scale and not a scatter around an average.
In view of the above-mentioned limitations, the interquartile range or the quartile
deviation has a limited practical utility.
3.7
MEAN DEVIATION
The mean deviation is also known as the average deviation. As the name implies, it is
the average of absolute amounts by which the individual items deviate from the mean.
Since the positive deviations from the mean are equal to the negative deviations,
while computing the mean deviation, we ignore positive and negative signs.
Symbolically,
MD =
∑| x |
n
Where MD = mean deviation, |x| = deviation of an item
from the mean ignoring positive and negative signs, n = the total number of
observations.
64
Example 3.4:
Size of Item
2-4
4-6
6-8
8-10
Frequency
20
40
30
10
Solution:
Size of Item Mid-points (m) Frequency (f) fm d from x f |d|
2-4
4-6
6-8
8-10
x
=
∑ fm
MD ( x ) =
n
∑f
3
5
7
9
Total
=
60
200
210
90
560
-2.6
-0.6
1.4
3.4
52
24
42
34
152
560
= 5 .6
100
|d |
n
20
40
30
10
100
=
152
= 1.52
100
3.7.1 MERITS OF MEAN DEVIATION
1.
A major advantage of mean deviation is that it is simple to understand and
easy to calculate.
2.
It takes into consideration each and every item in the distribution. As a result,
a change in the value of any item will have its effect on the magnitude of mean
deviation.
3.
The values of extreme items have less effect on the value of the mean
deviation.
4.
As deviations are taken from a central value, it is possible to have meaningful
comparisons of the formation of different distributions.
3.7.2 LIMITATIONS OF MEAN DEVIATION
1.
It is not capable of further algebraic treatment.
65
2.
At times it may fail to give accurate results. The mean deviation gives best
results when deviations are taken from the median instead of from the mean.
But in a series, which has wide variations in the items, median is not a
satisfactory measure.
3.
Strictly on mathematical considerations, the method is wrong as it ignores the
algebraic signs when the deviations are taken from the mean.
In view of these limitations, it is seldom used in business studies. A better measure
known as the standard deviation is more frequently used.
3.8
STANDARD DEVIATION
The standard deviation is similar to the mean deviation in that here too the deviations
are measured from the mean. At the same time, the standard deviation is preferred to
the mean deviation or the quartile deviation or the range because it has desirable
mathematical properties.
Before defining the concept of the standard deviation, we introduce another concept
viz. variance.
Example 3.5:
X
20
15
19
24
16
14
108
X-μ
20-18=12
15-18= -3
19-18 = 1
24-18 = 6
16-18 = -2
14-18 = -4
Total
Solution:
Mean =
108
= 18
6
66
(X-μ)2
4
9
1
36
4
16
70
The second column shows the deviations from the mean. The third or the last column
shows the squared deviations, the sum of which is 70. The arithmetic mean of the
squared deviations is:
∑ (x − μ )
N
2
= 70/6=11.67 approx.
This mean of the squared deviations is known as the variance. It may be noted that
this variance is described by different terms that are used interchangeably: the
variance of the distribution X; the variance of X; the variance of the distribution; and
just simply, the variance.
Symbolically, Var (X) =
It is also written as σ
2
∑ (x − μ )
2
N
∑ (x
=
i
− μ)
2
N
Where σ2 (called sigma squared) is used to denote the variance.
Although the variance is a measure of dispersion, the unit of its measurement is
(points). If a distribution relates to income of families then the variance is (Rs)2 and
not rupees. Similarly, if another distribution pertains to marks of students, then the
unit of variance is (marks)2. To overcome this inadequacy, the square root of variance
is taken, which yields a better measure of dispersion known as the standard deviation.
Taking our earlier example of individual observations, we take the square root of the
variance
SD or σ = Variance = 11.67 = 3.42 points
Symbolically, σ = ∑ (x i − μ )
2
N
In applied Statistics, the standard deviation is more frequently used than the variance.
This can also be written as:
67
(∑ x )
−
2
∑x
σ =
2
i
i
N
N
We use this formula to calculate the standard deviation from the individual
observations given earlier.
Example 7.6:
X
X2
20
15
400
225
19
24
16
14
361
576
256
196
108
2014
Solution:
∑x
σ =
σ =
σ =
∑x
= 2014
2
i
2014 −
i
= 108
(108)2
6
6
Or,
12084 − 11664
6
Or,
6
70
6
Or,
σ =
N=6
11664
6
6
2014 −
σ =
420
6
6
σ =
11.67
σ = 3.42
Example 3.7:
The following distribution relating to marks obtained by students in an examination:
Marks
0- 10
10- 20
20- 30
30- 40
40- 50
50- 60
Number of Students
1
3
6
10
12
11
68
60- 70
70- 80
80- 90
90-100
6
3
2
1
Solution:
Marks
0- 10
10- 20
20- 30
30- 40
40- 50
50- 60
60- 70
70- 80
80- 90
90-100
Total
Frequency (f)
1
3
6
10
12
11
6
3
2
1
55
Mid-points
5
15
25
35
45
55
65
75
85
95
Deviations (d)/10=d’
-5
-4
-3
-2
-1
0
1
2
3
4
Total
Fd’ fd'2
-5 25
-12 48
-18 54
-20 40
-12 12
0
0
6
6
6
12
6
18
4
16
-45 231
In the case of frequency distribution where the individual values are not known, we
use the midpoints of the class intervals. Thus, the formula used for calculating
the standard deviation is as given below:
K
σ =
∑ fi(m − μ )
2
i
i =1
N
Where mi is the mid-point of the class intervals μ is the mean of the distribution, fi is
the frequency of each class; N is the total number of frequency and K is the number of
classes. This formula requires that the mean μ be calculated and that deviations (mi μ) be obtained for each class. To avoid this inconvenience, the above formula can be
modified as:
K
σ =
∑
i =1
K
⎞
2⎛
fid i ⎜ ∑ fd i ⎟
⎝ i =1
⎠
N
Where C is the class interval: fi is the frequency of the ith class and di is the deviation
of the of item from an assumed origin; and N is the total number of observations.
Applying this formula for the table given earlier,
σ = 10
231 ⎛ − 45 ⎞
−⎜
⎟
55 ⎝ 55 ⎠
69
2
= 10 4.2 − 0.669421
=18.8 marks
When it becomes clear that the actual mean would turn out to be in fraction,
calculating deviations from the mean would be too cumbersome. In such cases,
an assumed mean is used and the deviations from it are calculated. While midpoint of any class can be taken as an assumed mean, it is advisable to choose
the mid-point of that class that would make calculations least cumbersome.
Guided by this consideration, in Example 3.7 we have decided to choose 55 as
the mid-point and, accordingly, deviations have been taken from it. It will be
seen from the calculations that they are considerably simplified.
3.8.1
USES OF THE STANDARD DEVIATION
The standard deviation is a frequently used measure of dispersion. It enables us to
determine as to how far individual items in a distribution deviate from its mean. In a
symmetrical, bell-shaped curve:
(i)
About 68 percent of the values in the population fall within: + 1 standard
deviation from the mean.
(ii)
About 95 percent of the values will fall within +2 standard deviations from the
mean.
(iii)
About 99 percent of the values will fall within + 3 standard deviations from
the mean.
The standard deviation is an absolute measure of dispersion as it measures variation in
the same units as the original data. As such, it cannot be a suitable measure while
comparing two or more distributions. For this purpose, we should use a relative
measure of dispersion. One such measure of relative dispersion is the coefficient of
variation, which relates the standard deviation and the mean such that the standard
deviation is expressed as a percentage of mean. Thus, the specific unit in which the
standard deviation is measured is done away with and the new unit becomes percent.
70
Symbolically, CV (coefficient of variation) =
σ
x 100
μ
Example 3.8: In a small business firm, two typists are employed-typist A and typist
B. Typist A types out, on an average, 30 pages per day with a standard deviation of 6.
Typist B, on an average, types out 45 pages with a standard deviation of 10. Which
typist shows greater consistency in his output?
Solution:
Coefficient of variation for A =
σ
x 100
μ
Or A =
Or
Coefficient of variation for B =
6
x 100
30
20%
and
σ
x 100
μ
B=
10
x 100
45
or 22.2 %
These calculations clearly indicate that although typist B types out more pages, there
is a greater variation in his output as compared to that of typist A. We can say this in a
different way: Though typist A's daily output is much less, he is more consistent than
typist B. The usefulness of the coefficient of variation becomes clear in comparing
two groups of data having different means, as has been the case in the above example.
3.8.2 STANDARDISED VARIABLE, STANDARD SCORES
The variable Z = (x - x )/s or (x - μ)/μ, which measures the deviation from the mean
in units of the standard deviation, is called a standardised variable. Since both the
numerator and the denominator are in the same units, a standardised variable is
independent of units used. If deviations from the mean are given in units of the
standard deviation, they are said to be expressed in standard units or standard scores.
71
Through this concept of standardised variable, proper comparisons can be made
between individual observations belonging to two different distributions whose
compositions differ.
Example 3.9: A student has scored 68 marks in Statistics for which the average
marks were 60 and the standard deviation was 10. In the paper on Marketing, he
scored 74 marks for which the average marks were 68 and the standard deviation was
15. In which paper, Statistics or Marketing, was his relative standing higher?
Solution: The standardised variable Z = (x - x ) ÷ s measures the deviation of x from
the mean x in terms of standard deviation s. For Statistics, Z = (68 - 60) ÷ 10 = 0.8
For Marketing, Z = (74 - 68) ÷ 15 = 0.4
Since the standard score is 0.8 in Statistics as compared to 0.4 in Marketing, his
relative standing was higher in Statistics.
Example 3.10: Convert the set of numbers 6, 7, 5, 10 and 12 into standard scores:
Solution:
X
X2
6
7
5
10
12
36
49
25
100
144
∑ X = 40
∑X
2
= 354
x = ∑ x ÷ N = 40 ÷ 5 = 8
(∑ X )
2
σ =
∑x
2
−
N
N
or,
σ =
354 −
= 354 − 320 = 2.61 approx.
5
72
(40)2
5
5
Z=
x−x
σ
=
6−8
= -0.77 (Standard score)
2.61
Applying this formula to other values:
(i)
7−8
2.61
= -0.38
(ii)
5−8
2.61
(iii)
10 − 8
= 0.77
2.61
(iv)
12 − 8
= 1.53
2.61
= -1.15
Thus the standard scores for 6,7,5,10 and 12 are -0.77, -0.38, -1.15, 0.77 and 1.53,
respectively.
3.9
LORENZ CURVE
This measure of dispersion is graphical. It is known as the Lorenz curve named after
Dr. Max Lorenz. It is generally used to show the extent of concentration of income
and wealth. The steps involved in plotting the Lorenz curve are:
1.
Convert a frequency distribution into a cumulative frequency table.
2.
Calculate percentage for each item taking the total equal to 100.
3.
Choose a suitable scale and plot the cumulative percentages of the persons and
income. Use the horizontal axis of X to depict percentages of persons and the
vertical axis of Y to depict percent ages of income.
4.
Show the line of equal distribution, which will join 0 of X-axis with 100 of Yaxis.
5.
The curve obtained in (3) above can now be compared with the straight line of
equal distribution obtained in (4) above. If the Lorenz curve is close to the line
of equal distribution, then it implies that the dispersion is much less. If, on the
73
contrary, the Lorenz curve is farther away from the line of equal distribution,
it implies that the dispersion is considerable.
The Lorenz curve is a simple graphical device to show the disparities of distribution
in any phenomenon. It is, used in business and economics to represent inequalities in
income, wealth, production, savings, and so on.
Figure 3.1 shows two Lorenz curves by way of illustration. The straight line AB is a
line of equal distribution, whereas AEB shows complete inequality. Curve ACB and
curve ADB are the Lorenz curves.
F
A
Figure 3.1: Lorenz Curve
As curve ACB is nearer to the line of equal distribution, it has more equitable
distribution of income than curve ADB. Assuming that these two curves are for the
same company, this may be interpreted in a different manner. Prior to taxation, the
curve ADB showed greater inequality in the income of its employees. After the
taxation, the company’s data resulted into ACB curve, which is closer to the line of
equal distribution. In other words, as a result of taxation, the inequality has reduced.
3.10 SKEWNESS: MEANING AND DEFINITIONS
In the above paragraphs, we have discussed frequency distributions in detail. It may
be repeated here that frequency distributions differ in three ways: Average value,
Variability or dispersion, and Shape. Since the first two, that is, average value and
74
variability or dispersion have already been discussed in previous chapters, here our
main spotlight will be on the shape of frequency distribution. Generally, there are two
comparable characteristics called skewness and kurtosis that help us to understand a
distribution. Two distributions may have the same mean and standard deviation but
may differ widely in their overall appearance as can be seen from the following:
In both these distributions the value of
mean and standard deviation is the same
( X = 15, σ = 5). But it does not imply
that the distributions are alike in nature.
The distribution on the left-hand side is
a symmetrical one whereas the distribution on the right-hand side is symmetrical or
skewed. Measures of skewness help us to distinguish between different types of
distributions.
Some important definitions of skewness are as follows:
1.
"When a series is not symmetrical it is said to be asymmetrical or skewed."
-Croxton & Cowden.
2.
"Skewness refers to the asymmetry or lack of symmetry in the shape of a
frequency distribution."
3.
-Morris Hamburg.
"Measures of skewness tell us the direction and the extent of skewness. In
symmetrical distribution the mean, median and mode are identical. The more
the mean moves away from the mode, the larger the asymmetry or skewness."
-Simpson & Kalka
4.
"A distribution is said to be 'skewed' when the mean and the median fall at
different points in the distribution, and the balance (or centre of gravity) is
shifted to one side or the other-to left or right."
75
-Garrett
The above definitions show that the term 'skewness' refers to lack of symmetry" i.e.,
when a distribution is not symmetrical (or is asymmetrical) it is called a skewed
distribution.
The concept of skewness will be clear from the following three diagrams showing a
symmetrical distribution. a positively skewed distribution and a negatively skewed
distribution.
1.
Symmetrical Distribution. It is clear from the diagram (a) that in a sym-
metrical distribution the values of mean, median and mode coincide. The
spread of the frequencies is the same on
both sides of the centre point of the curve.
2.
Asymmetrical
Distribution.
A
distribution, which is not symmetrical, is
called a skewed distribution and such a
distribution could either be positively
skewed or negatively skewed as would be
clear from the diagrams (b) and (c).
3.
Positively Skewed Distribution. In the
positively skewed distribution the value of
the mean is maximum and that of mode least-the median lies in between the
two as is clear from the diagram (b).
4.
Negatively Skewed Distribution. The following is the shape of negatively
skewed distribution. In a negatively skewed distribution the value of mode is
maximum and that of mean least-the median lies in between the two. In the
positively skewed distribution the frequencies are spread out over a greater
76
range of values on the high-value end of the curve (the right-hand side) than
they are on the low-value end. In the negatively skewed distribution the
position is reversed, i.e. the excess tail is on the left-hand side. It should be
noted that in moderately symmetrical distributions the interval between the
mean and the median is approximately one-third of the interval between the
mean and the mode. It is this relationship, which provides a means of
measuring the degree of skewness.
3.11 TESTS OF SKEWNESS
In order to ascertain whether a distribution is skewed or not the following tests may
be applied. Skewness is present if:
1.
The values of mean, median and mode do not coincide.
2.
When the data are plotted on a graph they do not give the normal bellshaped form i.e. when cut along a vertical line through the centre the two
halves are not equal.
3.
The sum of the positive deviations from the median is not equal to the sum
of the negative deviations.
4.
Quartiles are not equidistant from the median.
5.
Frequencies are not equally distributed at points of equal deviation from
the mode.
On the contrary, when skewness is absent, i.e. in case of a symmetrical distribution,
the following conditions are satisfied:
1.
The values of mean, median and mode coincide.
2.
Data when plotted on a graph give the normal bell-shaped form.
3.
Sum of the positive deviations from the median is equal to the sum of the
negative deviations.
77
4.
Quartiles are equidistant from the median.
5.
Frequencies are equally distributed at points of equal deviations from the
mode.
3.12 MEASURES OF SKEWNESS
There are four measures of skewness, each divided into absolute and relative
measures. The relative measure is known as the coefficient of skewness and is more
frequently used than the absolute measure of skewness. Further, when a comparison
between two or more distributions is involved, it is the relative measure of skewness,
which is used. The measures of skewness are: (i) Karl Pearson's measure, (ii)
Bowley’s measure, (iii) Kelly’s measure, and (iv) Moment’s measure. These
measures are discussed briefly below:
3.12.1 KARL PEARON’S MEASURE
The formula for measuring skewness as given by Karl Pearson is as follows:
Skewness = Mean - Mode
Coefficient of skewness =
Mean – Mode
Standard Deviation
In case the mode is indeterminate, the coefficient of skewness is:
Mean - (3 Median - 2 Mean)
Skp = Standard deviation
Skp =
3(Mean - Median)
Standard deviation
Now this formula is equal to the earlier one.
Mean - Mode
Standard deviation
3(Mean - Median)
Standard deviation
Or 3 Mean - 3 Median = Mean - Mode
Or Mode = Mean - 3 Mean + 3 Median
Or Mode = 3 Median - 2 Mean
The direction of skewness is determined by ascertaining whether the mean is greater
than the mode or less than the mode. If it is greater than the mode, then skewness is
78
positive. But when the mean is less than the mode, it is negative. The difference
between the mean and mode indicates the extent of departure from symmetry. It is
measured in standard deviation units, which provide a measure independent of the
unit of measurement. It may be recalled that this observation was made in the
preceding chapter while discussing standard deviation. The value of coefficient of
skewness is zero, when the distribution is symmetrical. Normally, this coefficient of
skewness lies between +1. If the mean is greater than the mode, then the coefficient of
skewness will be positive, otherwise negative.
Example 3.11: Given the following data, calculate the Karl Pearson's coefficient of
skewness: ∑x = 452 ∑x2= 24270
Mode = 43.7 and N = 10
Solution:
Pearson's coefficient of skewness is:
Mean - Mode
Standard deviation
SkP =
Mean ( x )= ∑ X
N
SD (σ ) = ∑
x2
N
(σ ) =
=
452
= 45.2
10
⎛∑x⎞
⎟ (σ ) =
− ⎜⎜
⎟
⎝ N ⎠
24270 ⎛ 452 ⎞
−⎜
⎟
10
⎝ 10 ⎠
2
2
∑
N
x2
⎛∑x⎞
⎟
− ⎜⎜
⎟
⎝ N ⎠
2
2
= 2427 − (45.2) = 19.59
Applying the values of mean, mode and standard deviation in the above formula,
Skp = 45.2 – 43.7
19.59
=0.08
This shows that there is a positive skewness though the extent of skewness is
marginal.
Example 3.12: From the following data, calculate the measure of skewness using the
mean, median and standard deviation:
X
f
10 - 20
18
20 - 30
30
30 - 40
40
79
40 - 50
55
50-60
38
60 - 70 70 - 80
20
16
Solution:
x
10 - 20
20 - 30
30 - 40
40-50
50 - 60
60 - 70
70 - 80
MVx
15
25
35
45=a
55
65
75
dx
-3
-2
-1
0
1
2
3
Total
f
18
30
40
55
38
20
16
217
fdx
-54
-60
-40
0
38
40
48
-28
fdX2
162
120
40
0
38
80
144
584
cf
18
48
88
143
181
201
217
a = Assumed mean = 45, cf = Cumulative frequency, dx = Deviation from assumed
mean, and i = 10
x=a+
= 45 −
∑ fdx × i
N
28
× 10 = 43.71
217
Median= l1 +
l 2 − l1
(m − c)
f1
Where m = (N + 1)/2th item
= (217 + 1)/2 = 109th item
Median = 40 −
= 40 +
50 − 40
(109 − 88)
55
10
× 21
55
= 43.82
SD
Skewness
=
∑ fd
∑f
=
2.69 - 0.016 × 10 = 16.4
2
x
⎛ ∑ fd x
−⎜
⎜ ∑f
⎝
2
⎞
⎟ × 10 =
⎟
⎠
=
3 (Mean - Median)
=
3 (43.71 - 43.82)
=
3 x -0.011
80
2
584 ⎛ − 28 ⎞
−⎜
⎟ × 10
217 ⎝ 217 ⎠
=
-0.33
Coefficient of skewness
Skewness
SD
=
=
or
-0.33
16.4
-0.02
The result shows that the distribution is negatively skewed, but the extent of skewness
is extremely negligible.
3.12.2 Bowley's Measure
Bowley developed a measure of skewness, which is based on quartile values. The
formula for measuring skewness is:
Skewness =
Q3 + Q1 − 2 M
Q3 − Q1
Where Q3 and Q1 are upper and lower quartiles and M is the median. The value of this
skewness varies between +1. In the case of open-ended distribution as well as where
extreme values are found in the series, this measure is particularly useful. In a
symmetrical distribution, skewness is zero. This means that Q3 and Q1 are positioned
equidistantly from Q2 that is, the median. In symbols, Q3 - Q2 = Q2 – Q1' In contrast,
when the distribution is skewed, then Q3 - Q2 will be different from Q2 – Q1' When Q3
- Q2 exceeds Q2 – Q1' then skewness is positive. As against this; when Q3 - Q2 is less
than Q2 – Q1' then skewness is negative. Bowley’s measure of skewness can- be
written as:
Skewness = (Q3 - Q2) - (Q2 – Q1
Or
or
Q3 - Q2 - Q2 + Q1
Q3 + Q1 - 2Q2 (2Q2 is 2M)
However, this is an absolute measure of skewness. As such, it cannot be used while
comparing two distributions where the units of measurement are different. In view of
this limitation, Bowley suggested a relative measure of skewness as given below:
81
Relative Skewness
=
=
=
=
(Q3 − Q2 ) − (Q2 − Q1 )
(Q3 − Q2 ) + (Q2 − Q1 )
Q3 − Q2 − Q2 − Q1
Q3 − Q2 + Q2 − Q1
Q3 − Q1 − 2Q 2
Q3 − Q1
Q3 − Q1 − 2M
Q3 − Q1
Example 3.13: For a distribution, Bowley’s coefficient of skewness is - 0.56,
Q1=16.4 and Median=24.2. What is the coefficient of quartile deviation?
Solution:
Bowley's coefficient of skewness is:
SkB =
Q3 − Q1 − 2 M
Q3 − Q1
Substituting the values in the above formula,
SkB =
Q 3 + 16.4 - (2 x 24.2)
Q3 − 16.4
− 0.56 =
Q 3 + 16.4 - 48.4
Q3 − 16.4
or
- 0.56 (Q3-16.4)
=
Q3-32
or
- 0.56 Q3 + 9.184
=
Q3-32
or
- 0.56 Q3 - Q3
=
-32 - 9.184
- 1.56 Q3
=
- 41.184
=
− 41.184
= 26.4
1.56
Q3
Now, we have the values of both the upper and the lower quartiles.
Coefficient of quartile deviation =
=
Q3 − Q1
Q3 + Q1
26.4 − 16.4
10
=
= 0.234 Approx.
26.4 + 16.4 42.8
Example 3.14: Calculate an appropriate measure of skewness from the following
data:
82
Value in Rs
Frequency
Less than 50
40
50 - 100
80
100 - 150
130
150 – 200
60
200 and above
30
Solution: It should be noted that the series given in the question is an open-ended
series. As such, Bowley's coefficient of skewness, which is based on quartiles, would
be the most appropriate measure of skewness in this case. In order to calculate the
quartiles and the median, we have to use the cumulative frequency. The table is
reproduced below with the cumulative frequency.
Value in Rs
Q1 = l1 +
Now
Less than 50
40
40
50 - 100
80
120
100 - 150
130
250
150 - 200
60
310
200 and above
30
340
l 2 − l1
(m − c)
f1
m=(
Q1 = 50 +
M=(
Frequency Cumulative Frequency
341
n +1
= 85.25, which lies in 50 - 100 class
) item =
4
4
100 − 50
(85.25 − 40) = 78.28
80
341
n +1
= 170.25, which lies in 100 - 150 class
) item =
4
4
83
M= 100 +
Q3 = l1 +
150 − 100
(170.5 − 120) = 119.4
130
l 2 − l1
(m − c)
f1
m = 3(341) ÷ 4 = 255.75
Q3 = 150 +
200 − 150
(255.75 − 250) = 154.79
60
Bowley's coefficient of skewness is:
Q3 + QI - 2M
Q3 - QI =
154.79+ 78.28 - (2 x 119.4) -5.73
=
154.79 -78.28
76.51
= - 0.075 approx.
This shows that there is a negative skewness, which has a very negligible magnitude.
3.12.3 Kelly's Measure
Kelly developed another measure of skewness, which is based on percentiles. The
formula for measuring skewness is as follows:
Coefficient of skewness =
P90 − 2 P50 + P10
P90 − P10
Or,
D1 + D9 − 2 M
D9 − D1
Where P and D stand for percentile and decile respectively. In order to calculate the
coefficient of skewness by this formula, we have to ascertain the values of 10th, 50th
and 90th percentiles. Somehow, this measure of skewness is seldom used. All the
same, we give an example to show how it can be calculated.
Example 3.15: Use Kelly's measure to calculate skewness.
Class Intervals
f
cf
10 - 20
20 - 30
18
30
18
48
84
30- 40
40- 50
50 - 60
60 – 70
70 - 80.
40
55
38
20
16
88
143
181
201
217
Solution: Now we have to calculate P10 P30 and P90.
PIO = l1 +
l 2 − l1
(m − c) , where m = (n + 1)/10th item
f1
217 + 1
= 21.8th item
10
This lies in the 20 - 30 class.
20 +
30 − 20
10 × 3.8
(21.8 − 18) = 20 +
= 21.27approx.
30
30
P50 (median): where m = (n + 1)/2th item =
217 + 1
= 109th item
2
This lies in the class 40 - 50. Applying the above formula:
40 +
50 − 40
10 × 21
(109 − 88) = 40 +
× 21 = 43.82approx.
55
55
P90: here m = 90 (217 + 1)/100th item = 196.2th item
This lies in the class 60 - 70. Applying the above formula:
60 +
70 − 60
10 × 15.2
(196.2 − 181) = 60 +
= 67.6approx.
20
20
Kelley's skewness
SkK
P90 − 2 P50 + P10
P90 − P10
=
67.6 - (2 x 43.82) + 21.27
67.6 - 21.27
=
88.87 - 87.64
46.63
=
0.027
85
This shows that the series is positively skewed though the extent of skewness is
extremely negligible. It may be recalled that if there is a perfectly symmetrical
distribution, then the skewness will be zero. One can see that the above answer
is very close to zero.
3.13 MOMENTS
In mechanics, the term moment is used to denote the rotating effect of a force. In
Statistics, it is used to indicate peculiarities of a frequency distribution. The utility of
moments lies in the sense that they indicate different aspects of a given distribution.
Thus, by using moments, we can measure the central tendency of a series, dispersion
or variability, skewness and the peakedness of the curve. The moments about the
actual arithmetic mean are denoted by μ. The first four moments about mean or
central moments are as follows:
First moment
μ1
=
1
N
∑ (x
−x
)
Second moment
μ2
=
1
N
∑ (x
−x
)
Third moment
μ3
=
1
N
∑ (x
−x
)
Fourth moment
μ3
=
1
N
∑ (x
−x
)
1
1
1
1
2
3
4
These moments are in relation to individual items. In the case of a frequency
distribution, the first four moments will be:
First moment
μ1
=
1
N
∑ fi(x
−x
)
Second moment
μ2
=
1
N
∑ fi(x
−x
)
Third moment
μ3
=
1
N
∑ fi(x
−x
)
86
1
1
1
2
3
μ3
Fourth moment
=
1
N
∑ fi(x
1
−x
)
4
It may be noted that the first central moment is zero, that is, μ= 0.
The second central moment is μ2=σ, indicating the variance.
The third central moment μ3 is used to measure skewness. The fourth central moment
gives an idea about the Kurtosis.
Karl Pearson suggested another measure of skewness, which is based on the third and
second central moments as given below:
β1 =
μ 32
μ 23
Example 3.16: Find the (a) first, (b) second, (c) third and (d) fourth moments for the
set of numbers 2,3,4,5 and 6.
Solution:
∑ x = 2 + 3 + 4 + 5 + 6 = 20 = 4
(a)
x=
(b)
∑x
x=
2
2 2 + 32 + 4 2 + 5 2 + 6 2
=
5
3
=
N
2 3 + 33 + 4 3 + 5 3 + 6 3
5
8 + 27 + 64 + 125 + 216
= 88
5
x=
=
5
4 + 9 + 16 + 25 + 36
= 18
5
∑x
x=
=
(d)
5
N
=
(c)
N
∑x
N
4
=
2 4 + 34 + 4 4 + 5 4 + 6 4
5
16 + 81 + 256 + 625 + 1296
= 454.8
5
87
Example 3.17: Using the same set of five figures as given in Example 3.7, find the
(a) first, (b) second, (c) third and (d) fourth moments about the mean.
Solution:
= ( x − x) =
m1
N
5
- 2 -1 + 0 + 1 + 2
=0
5
=
m2 = ( x − x) 2 =
∑ ( x − x) = (2 − 4) + (3 − 4) + (4 − 4) + (5 − 4) + (6 − 4)
∑ ( x − x)
2
N
=
(2 − 4) 2 + (3 − 4) 2 + (4 − 4) 2 + (5 − 4) 2 + (6 − 4) 2
5
=
(-2) 2 + (_1) 2 + 0 2 + 12 + 2 2
5
=
4 +1+ 0 +1+ 4
= 2. It may be noted that m2 is the variance
5
m3= = ( x − x) 3 =
=
∑ ( x − x)
3
=
N
(2 − 4) 3 + (3 − 4) 3 + (4 − 4) 3 + (5 − 4) 3 + (6 − 4) 3
5
(-2) 3 + (_1) 3 + 0 3 + 13 + 2 3 - 8 - 1 + 0 + 1 + 8
=
=0
5
5
m4= = ( x − x)
4
∑ ( x − x)
=
4
N
4
(2 − 4) + (3 − 4) 4 + (4 − 4) 4 + (5 − 4) 4 + (6 − 4) 4
=
5
=
(-2) 4 + (_1) 4 + 0 4 + 14 + 2 4
5
=
16 + 1 + 0 + 1 + 016
= 6.8
5
Example 3.18: Calculate the first four central moments from the following data:
Class interval
50-60
60-70
70-80
80-90
90-100
Frequency
5
12
20
7
6
Solution:
Class Interval
50- 60
f
5
MV
55
d from 75
-20
d/10
-2
88
fd
-10
fd2
20
fd3
-40
fd4
80
60- 70
70- 80
80- 90
90-100
Total
μ1 ' =
12
20
7
6
50
65
75
85
95
-10
0
10
20
-1
0
1
2
-12
0
7
12
-3
12
0
7
24
-12
0
7
48
-4
12
0
7
96
195
∑ fd × i = − 3 × 10 = −0.6
N
μ2
∑ fd
'=
μ2
∑ fd
'=
μ2
∑ fd
'=
50
2
×i
N
3
×i
N
4
×i
N
=
63 × 10
= 12.6
50
=
− 4 × 10
= 0 .8
50
=
195 × 10
= 19
50
μ1=μ1’ - μ1’= -0.6-(-0.6) = 0
μ2=μ2’ - μ1’2=10-( -0.6)2= 10-3.6=6.4
μ3=μ3’ - 3μ2’μ’1+2μ1’3=-0.8-3(12.6)(-0.6)+2(-0.6)3
= -0.8 + 22.68 + 0.432 = 22.312
μ4=μ4’ - 4μ3’μ’1+6μ2μ1’2-3μ1’4
= 19 + 4(-0.8)(-0.6) + 6(10)(-0.6)2- 3(-0.6)4
= 19 + 1.92 + 21.60 - 0.3888
= 42.1312
3.14 KURTOSIS
Kurtosis is another measure of the shape of a frequency curve. It is a Greek word,
which means bulginess. While skewness signifies the extent of asymmetry, kurtosis
measures the degree of peakedness of a frequency distribution. Karl Pearson classified
curves into three types on the basis of the shape of their peaks. These are mesokurtic,
leptokurtic and platykurtic. These three types of curves are shown in figure below:
89
It will be seen from Fig.
3.2 that mesokurtic curve is
neither too much flattened
nor too much peaked. In
fact, this is the frequency
curve
of
distribution.
a
normal
Leptokurtic
curve is a more peaked than the normal curve. In contrast, platykurtic is a relatively
flat curve. The coefficient of kurtosis as given by Karl Pearson is β2=μ4/μ22. In case of
a normal distribution, that is, mesokurtic curve, the value of β2=3. If β2 turn out to be
> 3, the curve is called a leptokurtic curve and is more peaked than the normal curve.
Again, when β2 < 3, the curve is called a platykurtic curve and is less peaked than the
normal curve. The measure of kurtosis is very helpful in the selection of an
appropriate average. For example, for normal distribution, mean is most appropriate;
for a leptokurtic distribution, median is most appropriate; and for platykurtic
distribution, the quartile range is most appropriate.
Example 3.19: From the data given in Example 3.18, calculate the kurtosis.
Solution: For this, we have to calculate β2 This can be done by using the formula
β2=μ4/μ22. In the preceding example, values of μ4 and μ2 are given. Hence, β2 =
42.1312 ÷ (6.4)2 = 1.03.
As β2. < 3, the distribution is platykurtic.
Another measure of kurtosis is based on both quartiles and percentiles and is given by
the following formula:
K=
Q
P90 − P10
90
Where K = kurtosis, Q = ½ (Q3 – Q1) is the semi-interquartile range; P90 is 90th
percentile and P10 is the 10th percentile. This is also known as the percentile
coefficient of kurtosis. In case of the normal distribution, the value of K is 0.263.
Example 3.20: From the data given below, calculate the percentile coefficient of
kurtosis.
Daily Wages in Rs.
Number of Workers
cf
50- 60
10
10
60-70
14
24
70-80
18
42
80 - 90
24
66
90-100
16
82
100 -110
12
94
110 - 120
6
100
Total
100
Solution: It may be noted that the question involved first two columns and in order to
calculate quartiles and percentiles, cumulative frequencies have been shown in
column three of the above table.
Q1
=
l1
l 2 − l1
(m − c) , where m = (n + 1)/4th item, which is = 25.25th item
f1
This falls in 70 - 80 class interval.
Q3
=
70 +
80 − 70
(25.25 − 24) = 70.69
18
=
l1 +
l 2 − l1
(m − c) , where m = 75.75
f1
This falls in 90 - 100 class interval.
=
90 +
100 − 90
(75.75 - 66) = 96.09
16
91
PI0
=
l1 +
l 2 − l1
(m − c) , where m = 10.1
f1
This falls in 60 - 70 class interval.
P90
=
60 +
=
l1 +
70 − 60
(10.01 -10) = 60.07
14
l 2 − l1
(m − c) , where m = 90.9
f1
This falls in 100 - 110 class interval.
= 100 +
K=
110 − 100
(90.9 - 82) = 107.41
12
Q
P90 − P10
=
1 / 2(Q3 − Q1 )
P90 − P10
=
½ (96.09 - 70.69)
107.41 - 60.07
= 0.268
It will be seen that the above distribution is very close to normal distribution as the
value of K is 0.268, which is extremely close to 0.263.
3.15
SUMMARY
The average value cannot adequately describe a set of observations, unless all the
observations are the same. It is necessary to describe the variability or
dispersion of the observations. In two or more distributions the central value
may be the same but still there can be wide disparities in the formation of
distribution. Therefore, we have to use the measures of dispersion.
Further, two distributions may have the same mean and standard deviation but may
differ widely in their overall appearance in terms of symmetry and skewness. To
92
distinguish between different types of distributions, we may use the measures of
skewness.
3.16
SELF TEST QUESTIONS
1.
What do you mean by dispersion? What are the different measures of dispersion?
2.
“Variability is not an important factor because even though the outcome is more
certain, you still have an equal chance of falling either above or below the median.
Therefore, on an average, the outcome will be the same.” Do you agree with this
3.
Why is the standard deviation the most widely used measure of dispersion? Explain.
4.
Define skewness and Dispersion.
5.
Define Kurtosis and Moments.
6.
What are the different measures of skewness? Which one is repeatedly used?
7.
Measures of dispersion and skewness are complimentary to one another in
understanding a frequency distribution." Elucidate the statement.
8.
Calculate Karl Pearson's coefficient of skewness from the following data:
Weekly Sales (Rs lakh)
10-12
12 – 14
14 – 16
16 - 18
18 – 20
22-24
24-26
Number of Companies
12
18
35
42
50
30
8
9. For a distribution, the first four moments about zero are 1,7,38 and 155 respectively.
(i) Compute the moment coefficients of skewness and kurtosis. (ii) Is the distribution
mesokurtic? Give reason.
10. The first four moments of a distribution about the value 4 are 1,4, 10 and 45. Obtain
various characteristics of the distribution on the basis of the information given.
Comment upon the nature of the distribution.
93
11. Define kurtosis. If β1=1 and β2 =4 and variance = 9, find the values of β3 and β4 and
comment upon the nature of the distribution.
12. Calculate the first four moments about the mean from the following data. Also
calculate the values of β1 and β2
Marks
0-10
10 – 20 20-30
No. of students
5
12
18
30 – 40 40 – 50 50 - 60 60 - 70
40
15
7
3
1.
Levin, Richard I. and David S. Rubin: Statistics for Management, Prentice
Hall, New Delhi.
2.
Watsman terry J. and Keith Parramor: Quantitative Methods in Finance
3.
Hooda, R. P.: Statistics for Business and Economics, Macmillan, New
Delhi.
4.
Hein, L. W. Quantitative Approach to Managerial Decisions, Prentice
Hall, NJ.
94
Course:
Course Code: MC-106
Lesson:
Author:
Anil Kumar
Vetter :
Prof. Harbhajan Bansal
04
CORRELATION ANALYSIS
Objectives :
The overall objective of this lesson is to give you an understanding of
bivariate linear correlation, there by enabling you to understand the
importance as well as the limitations of correlation analysis.
Structure
4.1
4.2
4.3
4.4
4.5
4.6
Introduction
What is Correlation?
Correlation Analysis
4.3.1 Scatter Diagram
4.3.2 Correlation Graph
4.3.3 Pearson’s Coefficient of Correlation
4.3.4 Spearman’s Rank Correlation
4.3.5 Concurrent Deviation Method
Limitations of Correlation Analysis
Self-Assessment Questions
95
...if we have information on more than one variables, we might be interested in seeing if
there is any connection - any association - between them.
4.1 INTRODUCTION
Statistical methods of measures of central tendency, dispersion, skewness and kurtosis are
helpful for the purpose of comparison and analysis of distributions involving only one
variable i.e. univariate distributions. However, describing the relationship between two or
more variables, is another important part of statistics.
In many business research situations, the key to decision making lies in understanding the
relationships between two or more variables. For example, in an effort to predict the behavior
of the bond market, a broker might find it useful to know whether the interest rate of bonds is
related to the prime interest rate. While studying the effect of advertising on sales, an account
executive may find it useful to know whether there is a strong relationship between
advertising dollars and sales dollars for a company.
The statistical methods of Correlation (discussed in the present lesson) and Regression (to be
discussed in the next lesson) are helpful in knowing the relationship between two or more
variables which may be related in same way, like interest rate of bonds and prime interest
rate; advertising expenditure and sales; income and consumption; crop-yield and fertilizer
used; height and weights and so on.
In all these cases involving two or more variables, we may be interested in seeing:
¾ if there is any association between the variables;
¾ if there is an association, is it strong enough to be useful;
¾ if so, what form the relationship between the two variables takes;
¾ how we can make use of that relationship for predictive purposes, that is, forecasting;
and
¾ how good such predictions will be.
96
Since these issues are inter related, correlation and regression analysis, as two sides of a
single process, consists of methods of examining the relationship between two or more
variables. If two (or more) variables are correlated, we can use information about one (or
more) variable(s) to predict the value of the other variable(s), and can measure the error
of estimations - a job of regression analysis.
4.2 WHAT IS CORRELATION?
Correlation is a measure of association between two or more variables. When two or more
variables very in sympathy so that movement in one tends to be accompanied by
corresponding movements in the other variable(s), they are said to be correlated.
“The correlation between variables is a measure of the nature and degree of
association between the variables”.
As a measure of the degree of relatedness of two variables, correlation is widely used in
exploratory research when the objective is to locate variables that might be related in some
way to the variable of interest.
4.2.1
TYPES OF CORRELATION
Correlation can be classified in several ways. The important ways of classifying correlation
are:
(i)
Positive and negative,
(ii)
Linear and non-linear (curvilinear) and
(iii)
Simple, partial and multiple.
Positive and Negative Correlation
If both the variables move in the same direction, we say that there is a positive correlation,
i.e., if one variable increases, the other variable also increases on an average or if one variable
decreases, the other variable also decreases on an average.
97
On the other hand, if the variables are varying in opposite direction, we say that it is a case of
negative correlation; e.g., movements of demand and supply.
Linear and Non-linear (Curvilinear) Correlation
If the change in one variable is accompanied by change in another variable in a constant ratio,
it is a case of linear correlation. Observe the following data:
X : 10
20
30
Y :
50
75
25
40
50
100 125
The ratio of change in the above example is the same. It is, thus, a case of linear correlation.
If we plot these variables on graph paper, all the points will fall on the same straight line.
On the other hand, if the amount of change in one variable does not follow a constant ratio
with the change in another variable, it is a case of non-linear or curvilinear correlation. If a
couple of figures in either series X or series Y are changed, it would give a non-linear
correlation.
Simple, Partial and Multiple Correlation
The distinction amongst these three types of correlation depends upon the number of
variables involved in a study. If only two variables are involved in a study, then the
correlation is said to be simple correlation. When three or more variables are involved in a
study, then it is a problem of either partial or multiple correlation. In multiple correlation,
three or more variables are studied simultaneously. But in partial correlation we consider only
two variables influencing each other while the effect of other variable(s) is held constant.
Suppose we have a problem comprising three variables X, Y and Z. X is the number of hours
studied, Y is I.Q. and Z is the number of marks obtained in the examination. In a multiple
correlation, we will study the relationship between the marks obtained (Z) and the two
variables, number of hours studied (X) and I.Q. (Y). In contrast, when we study the
98
relationship between X and Z, keeping an average I.Q. (Y) as constant, it is said to be a study
involving partial correlation.
In this lesson, we will study linear correlation between two variables.
4.2.2 CORRELATION DOES NOT NECESSARILY MEAN CAUSATION
The correlation analysis, in discovering the nature and degree of relationship between
variables, does not necessarily imply any cause and effect relationship between the variables.
Two variables may be related to each other but this does not mean that one variable causes
the other. For example, we may find that logical reasoning and creativity are correlated, but
that does not mean if we could increase peoples’ logical reasoning ability, we would produce
greater creativity. We need to conduct an actual experiment to unequivocally demonstrate a
causal relationship. But if it is true that influencing someones’ logical reasoning ability does
influence their creativity, then the two variables must be correlated with each other. In other
words, causation always implies correlation, however converse is not true.
Let us see some situations1. The correlation may be due to chance particularly when the data pertain to a small
sample. A small sample bivariate series may show the relationship but such a
relationship may not exist in the universe.
2. It is possible that both the variables are influenced by one or more other variables.
For example, expenditure on food and entertainment for a given number of
households show a positive relationship because both have increased over time.
But, this is due to rise in family incomes over the same period. In other words, the
two variables have been influenced by another variable - increase in family
incomes.
99
3. There may be another situation where both the variables may be influencing each
other so that we cannot say which is the cause and which is the effect. For
example, take the case of price and demand. The rise in price of a commodity may
lead to a decline in the demand for it. Here, price is the cause and the demand is
the effect. In yet another situation, an increase in demand may lead to a rise in
price. Here, the demand is the cause while price is the effect, which is just the
reverse of the earlier situation. In such situations, it is difficult to identify which
variable is causing the effect on which variable, as both are influencing each
other.
The foregoing discussion clearly shows that correlation does not indicate any causation or
functional relationship. Correlation coefficient is merely a mathematical relationship and
this has nothing to do with cause and effect relation. It only reveals co-variation between
two variables. Even when there is no cause-and-effect relationship in bivariate series and one
interprets the relationship as causal, such a correlation is called spurious or non-sense
correlation. Obviously, this will be misleading. As such, one has to be very careful in
correlation exercises and look into other relevant factors before concluding a cause-and-effect
relationship.
4.3
CORRELATION ANALYSIS
Correlation Analysis is a statistical technique used to indicate the nature and degree of
relationship existing between one variable and the other(s). It is also used along with
regression analysis to measure how well the regression line explains the variations of the
dependent variable with the independent variable.
The commonly used methods for studying linear relationship between two variables involve
both graphic and algebraic methods. Some of the widely used methods include:
1.
Scatter Diagram
2.
Correlation Graph
100
3.
Pearson’s Coefficient of Correlation
4.
Spearman’s Rank Correlation
5.
Concurrent Deviation Method
4.3.1 SCATTER DIAGRAM
This method is also known as Dotogram or Dot diagram. Scatter diagram is one of the
simplest methods of diagrammatic representation of a bivariate distribution. Under this
method, both the variables are plotted on the graph paper by putting dots. The diagram so
obtained is called "Scatter Diagram". By studying diagram, we can have rough idea about the
nature and degree of relationship between two variables. The term scatter refers to the
spreading of dots on the graph. We should keep the following points in mind while
interpreting correlation:
¾ if the plotted points are very close to each other, it indicates high degree of
correlation. If the plotted points are away from each other, it indicates low degree of
correlation.
101
Figure 4-1
Scatter Diagrams
¾ if the points on the diagram reveal any trend (either upward or downward), the
variables are said to be correlated and if no trend is revealed, the variables are
uncorrelated.
¾ if there is an upward trend rising from lower left hand corner and going upward to the
upper right hand corner, the correlation is positive since this reveals that the values of
the two variables move in the same direction. If, on the other hand, the points depict a
downward trend from the upper left hand corner to the lower right hand corner, the
correlation is negative since in this case the values of the two variables move in the
opposite directions.
¾ in particular, if all the points lie on a straight line starting from the left bottom and
going up towards the right top, the correlation is perfect and positive, and if all the
points like on a straight line starting from left top and coming down to right bottom,
the correlation is perfect and negative.
The various diagrams of the scattered data in Figure 4-1 depict different forms of correlation.
Example 4-1
102
Given the following data on sales (in thousand units) and expenses (in thousand rupees) of a
firm for 10 month:
Month :
J
F
M
A
M
J
J
A
S
O
Sales:
50
50
55
60
62
65
68
60
60
50
Expenses:
11
13
14
16
16
15
15
14
13
13
a) Make a Scatter Diagram
b) Do you think that there is a correlation between sales and expenses of the
firm? Is it positive or negative? Is it high or low?
Solution:(a) The Scatter Diagram of the given data is shown in Figure 4-2
Expenses
20
15
10
5
0
0
20
40
60
80
Sales
Figure 4.2
Scatter Diagram
(b) Figure 4-2 shows that the plotted points are close to each other and reveal an upward
trend. So there is a high degree of positive correlation between sales and expenses of the firm.
4.3.2 CORRELATION GRAPH
This method, also known as Correlogram is very simple. The data pertaining to two series are
plotted on a graph sheet. We can find out the correlation by examining the direction and
closeness of two curves. If both the curves drawn on the graph are moving in the same
direction, it is a case of positive correlation. On the other hand, if both the curves are moving
in opposite direction, correlation is said to be negative. If the graph does not show any
definite pattern on account of erratic fluctuations in the curves, then it shows an absence of
correlation.
103
Example 4-2
Find out graphically, if there is any correlation between price yield per plot (qtls); denoted by
Y and quantity of fertilizer used (kg); denote by X.
Plot No.:
1
2
3
4
5
6
7
8
9
10
Y:
3.5
4.3
5.2
5.8
6.4
7.3
7.2
7.5
7.8
8.3
X:
6
8
9
12
10
15
17
20
18
24
Solution: The Correlogram of the given data is shown in Figure 4-3
X and Y
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
Plot Number
Figure 4-3
Correlation Graph
Figure 4-3 shows that the two curves move in the same direction and, moreover, they are very
close to each other, suggesting a close relationship between price yield per plot (qtls) and
quantity of fertilizer used (kg)
Remark:
Both the Graphic methods - scatter diagram and correlation graph provide a
‘feel for’ of the data – by providing visual representation of the association between the
variables. These are readily comprehensible and enable us to form a fairly good, though
rough idea of the nature and degree of the relationship between the two variables. However,
these methods are unable to quantify the relationship between them. To quantify the extent of
correlation, we make use of algebraic methods - which calculate correlation coefficient.
4.3.3 PEARSON’S COEFFICIENT OF CORRELATION
A mathematical method for measuring the intensity or the magnitude of linear relationship
104
between two variables was suggested by Karl Pearson (1867-1936), a great British
Biometrician and Statistician and, it is by far the most widely used method in practice.
Karl Pearson’s measure, known as Pearsonian correlation coefficient between two variables X
and Y, usually denoted by r(X,Y) or rxy or simply r is a numerical measure of linear
relationship between them and is defined as the ratio of the covariance between X and Y, to
the product of the standard deviations of X and Y.
Symbolically
rxy =
Cov ( X , Y )
S x .S y
…………(4.1)
when, ( X 1 , Y1 ); ( X 2 , Y2 );..................( X n , Yn ) are N pairs of observations of the variables X and
Y in a bivariate distribution,
Cov( X , Y ) =
Sx =
and
Sy =
∑ ( X − X )(Y − Y )
…………(4.2a)
N
∑(X − X )
2
…………(4.2b)
N
∑ (Y − Y )
2
…………(4.2c)
N
Thus by substituting Eqs. (4.2) in Eq. (4.1), we can write the Pearsonian correlation
coefficient as
rxy =
rxy =
1
N
1
N
∑ ( X − X )(Y − Y )
∑(X − X )
2
1
N
∑ (Y − Y )
∑ ( X − X )(Y − Y )
∑ ( X − X ) ∑ (Y − Y )
2
2
…………(4.3)
2
105
If we denote, d x = X − X and d y = Y − Y
Then
rxy =
∑d
∑d
x
dy
2
x
dy
…………(4.3a)
2
We can further simply the calculations of Eqs. (4.2)
We have
Cov( X , Y ) =
1
N
∑ ( X − X )(Y − Y )
=
1
N
∑ XY − X Y
=
1
N
∑ XY −
=
1
N ∑ XY − ∑ X ∑ Y
N2
and
∑ X ∑Y
N
[
S x2 =
1
N
∑(X − X )
=
1
N
∑X
1
=
N
=
N
]
…………(4.4)
2
−( X ) 2
2
⎛∑X
∑ X −⎜⎜ N
⎝
2
⎞
⎟
⎟
⎠
2
[
1
2
N ∑ X 2 − (∑ X )
2
N
]
…………(4.5a)
Similarly, we have
S y2 =
[
1
2
N ∑ Y 2 − (∑ Y )
2
N
]
…………(4.5b)
So Pearsonian correlation coefficient may be found as
rxy =
or
rxy =
[
]
1
N ∑ XY − ∑ X ∑ Y
N2
1
1
2
2
N ∑ X 2 − (∑ X )
N ∑ Y 2 − (∑ Y )
2
2
N
N
[
]
[
N ∑ XY − ∑ X ∑ Y
N ∑ X − (∑ X )
2
2
N ∑ Y − (∑ Y )
2
2
106
]
…………(4.6)
Remark:
Eq. (4.3) or Eq. (4.3a) is quite convenient to apply if the means X and
Y come out to be integers. If X or/and Y is (are) fractional then the Eq. (4.3) or Eq. (4.3a) is
quite cumbersome to apply, since the computations of
∑ ( X − X )(Y − Y )
∑(X − X )
2
,
∑ (Y − Y )
2
and
are quite time consuming and tedious. In such a case Eq. (4.6) may be
used provided the values of X or/ and Y are small. But if X and Y assume large values, the
calculation of
∑ X , ∑Y
2
2
and
∑ XY
is again quite time consuming.
Thus if (i) X and Y are fractional and (ii) X and Y assume large values, the Eq. (4.3) and Eq.
(4.6) are not generally used for numerical problems. In such cases, the step deviation method
where we take the deviations of the variables X and Y from any arbitrary points is used. We
will discuss this method in the properties of correlation coefficient.
4.3.3.1 Properties of Pearsonian Correlation Coefficient
The following are important properties of Pearsonian correlation coefficient:
1. Pearsonian correlation coefficient cannot exceed 1 numerically. In other words it lies
between –1 and +1. Symbolically,
-1 ≤ r ≤1
Remarks:
(i) This property provides us a check on our calculations. If in any problem,
the obtained value of r lies outside the limits + 1, this implies that there is some mistake in
our calculations.
(ii)
The sign of r indicate the nature of the correlation. Positive value of r indicates
positive correlation, whereas negative value indicates negative correlation. r = 0 indicate
absence of correlation.
(iii)
The following table sums up the degrees of correlation corresponding to various
values of r:
107
Value of r
Degree of correlation
±1
perfect correlation
±0.90 or more
very high degree of correlation
±0.75 to ±0.90
sufficiently high degree of
correlation
±0.60 to ±0.75
moderate degree of correlation
less than ±0.30
only the possibility of a
correlation
possibly no correlation
0
absence of correlation
±0.30 to ±0.60
2. Pearsonian Correlation coefficient is independent of the change of origin and scale.
Mathematically, if given variables X and Y are transformed to new variables U and V
by change of origin and scale, i. e.
U=
X−A
h
and
V =
Y −B
k
Where A, B, h and k are constants and h > 0, k > 0; then the correlation coefficient
between X and Y is same as the correlation coefficient between U and V i.e.,
r(X,Y) = r(U, V) => rxy = ruv
Remark:
This is one of the very important properties of the correlation coefficient and
Eq.(4.6) become quite tedious to use in numerical problems if X and/or Y are in fractions or if
X and Y are large. In such cases we can conveniently change the origin and scale (if possible)
in X or/and Y to get new variables U and V and compute the correlation between U and V by
the Eq. (4.7)
rxy = ruv =
N ∑ UV − ∑ U ∑ V
N ∑ U − (∑ U )
2
2
N ∑ V − (V )
2
2
…………(4.7)
3. Two independent variables are uncorrelated but the converse is not true
108
If X and Y are independent variables then
rxy = 0
However, the converse of the theorem is not true i.e., uncorrelated variables need not
necessarily be independent. As an illustration consider the following bivariate
distribution.
X
:
1
2
3
-3
-2
-1
Y
:
1
4
9
9
4
1
For this distribution, value of r will be 0.
Hence in the above example the variable X and Y are uncorrelated. But if we examine
the data carefully we find that X and Y are not independent but are connected by the
relation Y = X2. The above example illustrates that uncorrelated variables need not be
independent.
Remarks:
One should not be confused with the words uncorrelation and independence.
rxy = 0 i.e., uncorrelation between the variables X and Y simply implies the absence of any
linear (straight line) relationship between them. They may, however, be related in some other
form other than straight line e.g., quadratic (as we have seen in the above example),
logarithmic or trigonometric form.
4. Pearsonian coefficient of correlation is the geometric mean of the two regression
coefficients, i.e.
rxy = ± bxy .b yx
The signs of both the regression coefficients are the same, and so the value of r will
also have the same sign.
This property will be dealt with in detail in the next lesson on Regression Analysis.
109
5. The square of Pearsonian correlation coefficient is known as the coefficient of
determination.
Coefficient of determination, which measures the percentage variation in the
dependent variable that is accounted for by the independent variable, is a much better
and useful measure for interpreting the value of r. This property will also be dealt
with in detail in the next lesson.
4.3.3.2 Probable Error of Correlation Coefficient
The correlation coefficient establishes the relationship of the two variables. After ascertaining
this level of relationship, we may be interested to find the extent upto which this coefficient is
dependable. Probable error of the correlation coefficient is such a measure of testing the
reliability of the observed value of the correlation coefficient, when we consider it as
satisfying the conditions of the random sampling.
If r is the observed value of the correlation coefficient in a sample of N pairs of observations
for the two variables under consideration, then the Probable Error, denoted by PE (r) is
expressed as
PE (r ) = 0.6745 SE (r )
or
PE (r ) = 0.6745
1− r2
N
There are two main functions of probable error:
1. Determination of limits: The limits of population correlation coefficient are r ±
PE(r), implying that if we take another random sample of the size N from the same
population, then the observed value of the correlation coefficient in the second
sample can be expected to lie within the limits given above, with 0.5 probability.
When sample size N is small, the concept or value of PE may lead to wrong
110
conclusions. Hence to use the concept of PE effectively, sample size N it should be
fairly large.
2. Interpretation of 'r': The interpretation of 'r' based on PE is as under:
¾ If r < PE(r), there is no evidence of correlation, i.e. a case of insignificant
correlation.
¾ If r > 6 PE(r), correlation is significant. If r < 6 PE(r), it is insignificant.
¾ If the probable error is small, correlation exist where r > 0.5
Example 4-3
Find the Pearsonian correlation coefficient between sales (in thousand units) and expenses (in
thousand rupees) of the following 10 firms:
Firm:
1
2
3
4
5
6
7
8
9
10
Sales:
50
50
55
60
65
65
65
60
60
50
Expenses:
11
13
14
16
16
15
15
14
13
13
Solution: Let sales of a firm be denoted by X and expenses be denoted by Y
Calculations for Coefficient of Correlation
{Using Eq. (4.3) or (4.3a)}
Firm
X
Y
dx =X−X
dy =Y −Y
d x2
d y2
dx.dy
1
50
11
-8
-3
64
9
24
2
50
13
-8
-1
64
1
8
3
55
14
-3
0
9
0
0
4
60
16
2
2
4
4
4
5
65
16
7
2
49
4
14
6
65
15
7
1
49
1
7
7
65
15
7
1
49
1
7
8
60
14
2
0
4
0
0
9
60
13
2
-1
4
1
-2
10
50
13
-8
-1
64
1
8
∑ X ∑Y
∑d
111
2
x
∑d
2
y
∑d
x
dy
X=
=
=
580
140
∑X =
=360
580
= 58
10
N
Y=
and
=22
=70
∑ Y = 140 = 14
10
N
Applying the Eq. (4.3a), we have, Pearsonian coefficient of correlation
rxy =
rxy =
rxy =
∑d
∑d
x
dy
2
x
dy
2
70
360 x 22
70
7920
rxy = 0.78
The value of rxy = 0.78 , indicate a high degree of positive correlation between sales and expenses.
Example 4-4
The data on price and quantity purchased relating to a commodity for 5 months is given
below:
Month :
January
February
March
April
May
Prices(Rs):
10
10
11
12
12
Quantity(Kg):
5
6
4
3
3
Find the Pearsonian correlation coefficient between prices and quantity and comment on its
sign and magnitude.
Solution: Let price of the commodity be denoted by X and quantity be denoted by Y
Calculations for Coefficient of Correlation
{Using Eq. (4.6)}
Month
X
Y
X2
Y2
XY
1
10
5
100
25
50
2
10
6
100
36
60
3
11
4
121
16
44
4
12
3
144
9
36
5
12
3
144
9
36
112
∑X
=55
∑ Y =21
∑X
2
= 609
∑Y
2
= 95
∑ XY = 226
Applying the Eq. (4.6), we have, Pearsonian coefficient of correlation
N ∑ XY − ∑ X ∑ Y
rxy =
N ∑ X 2 − (∑ X )
2
2
5 x 226 − 55 x 21
rxy =
rxy =
N ∑ Y 2 − (∑ Y )
(5 x609 − 55 x55)(5 x95 − 21x 21)
1130 − 1155
rxy =
20 x34
− 25
680
rxy = −0.98
The negative sign of r indicate negative correlation and its large magnitude indicate a very
high degree of correlation. So there is a high degree of negative correlation between prices
and quantity demanded.
Example 4-5
Find the Pearsonian correlation coefficient from the following series of marks obtained by 10
students in a class test in mathematics (X) and in Statistics (Y):
X:
45
70
65
30
90
40
50
75
85
60
Y:
35
90
70
40
95
40
60
80
80
50
Also calculate the Probable Error.
Solution:
Calculations for Coefficient of Correlation
{Using Eq. (4.7)}
X
Y
U
V
U2
V2
45
35
-3
-6
9
36
18
70
90
2
5
4
25
10
65
70
1
1
1
1
1
113
UV
30
40
-6
-5
36
25
30
90
95
6
6
36
36
36
40
40
-4
-5
16
25
20
50
60
-2
-1
4
1
2
75
80
3
3
9
9
9
85
80
5
3
25
9
15
60
50
0
-3
0
∑U = 2 ∑V = −2 ∑U
2
9
= 140
∑V
2
= 176
0
∑UV
= 141
We have, defined variables U and V as
U=
X − 60
5
V=
and
Y − 65
5
Applying the Eq. (4.7)
rxy = ruv =
N ∑ UV − (∑ U ∑ V )
N ∑ U 2 − (∑ U )
2
=
2
10 x141 − 2 x(−2)
=
=
N ∑ V 2 − (∑ V )
10 x140 − 2 x 2 10 x176 − (−2) x(−2)
1410 + 4
1400 − 4 1760 − 4
1414
2451376
= 0.9
So there is a high degree of positive correlation between marks obtained in Mathematics and
in Statistics.
Probable Error, denoted by PE (r) is given as
PE (r ) = 0.6745
1− r2
N
114
PE (r ) = 0.6745
1 − (0.9 )
2
10
PE (r ) = 0.0405
So the value of r is highly significant.
4.3.4 SPEARMAN’S RANK CORRELATION
Sometimes we come across statistical series in which the variables under consideration are
not capable of quantitative measurement but can be arranged in serial order. This happens
when we are dealing with qualitative characteristics (attributes) such as honesty, beauty,
character, morality, etc., which cannot be measured quantitatively but can be arranged
serially. In such situations Karl Pearson’s coefficient of correlation cannot be used as such.
Charles Edward Spearman, a British Psychologist, developed a formula in 1904, which
consists in obtaining the correlation coefficient between the ranks of N individuals in the two
attributes under study.
Suppose we want to find if two characteristics A, say, intelligence and B, say, beauty are
related or not. Both the characteristics are incapable of quantitative measurements but we can
arrange a group of N individuals in order of merit (ranks) w.r.t. proficiency in the two
characteristics. Let the random variables X and Y denote the ranks of the individuals in the
characteristics A and B respectively. If we assume that there is no tie, i.e., if no two
individuals get the same rank in a characteristic then, obviously, X and Y assume numerical
values ranging from 1 to N.
The Pearsonian correlation coefficient between the ranks X and Y is called the rank
correlation coefficient between the characteristics A and B for the group of individuals.
Spearman’s rank correlation coefficient, usually denoted by ρ(Rho) is given by the equation
ρ =1 −
6∑ d 2
…………(4.8)
N ( N 2 − 1)
115
Where d is the difference between the pair of ranks of the same individual in the two
characteristics and N is the number of pairs.
Example 4-6
Ten entries are submitted for a competition. Three judges study each entry and list the ten in
rank order. Their rankings are as follows:
Entry:
A
B
C
D
E
F
G
H
I
J
Judge J1:
9
3
7
5
1
6
2
4
10
8
Judge J2:
9
1
10
4
3
8
5
2
7
6
Judge J3:
6
3
8
7
2
4
1
5
9
10
(i)
Which pair of judges agrees the most?
(ii)
Which pair of judges disagrees the most?
Solution:
Calculations for Coefficient of Rank Correlation
{Using Eq.(4.8)}
Entry
Rank by
Judges
Difference in Ranks
J1
J2
J3
d(J1&J2)
d2
d(J1&J3)
d2
d(J2&J3)
d2
A
9
9
6
0
0
+3
9
+3
9
B
3
1
3
+2
4
0
0
-2
4
C
7
10
8
-3
9
-1
1
+2
4
D
5
4
7
+1
1
-2
4
-3
9
E
1
3
2
-2
4
-1
1
+1
1
F
6
8
4
-2
4
+2
4
+4
16
G
2
5
1
-3
9
+1
1
+4
16
H
4
2
5
+2
4
-1
1
-3
9
I
10
7
9
+3
9
+1
1
-2
4
J
8
6
10
+2
4
-2
4
-4
16
∑d2 =48
ρ (J1 & J2) = 1 −
6∑ d 2
N ( N 2 − 1)
116
∑d2 =26
∑d2 =88
=1 −
6 x 48
10(10 2 − 1)
=1 −
288
990
=1 – 0.29
= +0.71
ρ (J1 & J3)
=1 −
6∑ d 2
N ( N 2 − 1)
=1 −
6 x 26
10(10 2 − 1)
=1 −
156
990
=1 – 0.1575
= +0.8425
ρ (J2 & J3)
=1 −
6∑ d 2
N ( N 2 − 1)
=1 −
6 x 88
10(10 2 − 1)
=1 −
528
990
=1 – 0.53
= +0.47
So
(i)
Judges J1 and J3 agree the most
(ii)
Judges J2 and J3 disagree the most
Spearman’s rank correlation Eq.(4.8) can also be used even if we are dealing with variables,
which are measured quantitatively, i.e. when the actual data but not the ranks relating to two
variables are given. In such a case we shall have to convert the data into ranks. The highest
(or the smallest) observation is given the rank 1. The next highest (or the next lowest)
observation is given rank 2 and so on. It is immaterial in which way (descending or
ascending) the ranks are assigned. However, the same approach should be followed for all the
variables under consideration.
117
Example 4-7
Calculate the rank coefficient of correlation from the following data:
X:
75
88
95
70
60
80
81
50
Y:
120
134
150
115
110
140
142
100
Solution:
Calculations for Coefficient of Rank Correlation
{Using Eq.(4.8)}
Ranks RX
Y
Ranks RY
d = RX -RY
d2
75
5
120
5
0
0
88
2
134
4
-2
4
95
1
150
1
0
0
70
6
115
6
0
0
60
7
110
7
0
0
80
4
140
3
+1
1
81
3
142
2
+1
1
50
8
100
8
0
0
X
∑d2 = 6
ρ
= 1−
6∑ d 2
N ( N 2 − 1)
= 1−
6x6
8(8 2 − 1)
= 1−
36
504
= 1 – 0.07
= + 0.93
Hence, there is a high degree of positive correlation between X and Y
Repeated Ranks
In case of attributes if there is a tie i.e., if any two or more individuals are placed together in
any classification w.r.t. an attribute or if in case of variable data there is more than one item
with the same value in either or both the series then Spearman’s Eq.(4.8) for calculating the
rank correlation coefficient breaks down, since in this case the variables X [the ranks of
118
individuals in characteristic A (1st series)] and Y [the ranks of individuals in characteristic B
(2nd series)] do not take the values from 1 to N.
In this case common ranks are assigned to the repeated items. These common ranks are the
arithmetic mean of the ranks, which these items would have got if they were different from
each other and the next item will get the rank next to the rank used in computing the common
rank. For example, suppose an item is repeated at rank 4. Then the common rank to be
assigned to each item is (4+5)/2, i.e., 4.5 which is the average of 4 and 5, the ranks which
these observations would have assumed if they were different. The next item will be assigned
the rank 6. If an item is repeated thrice at rank 7, then the common rank to be assigned to
each value will be (7+8+9)/3, i.e., 8 which is the arithmetic mean of 7,8 and 9 viz., the ranks
these observations would have got if they were different from each other. The next rank to be
assigned will be 10.
If only a small proportion of the ranks are tied, this technique may be applied together with
Eq.(4.8). If a large proportion of ranks are tied, it is advisable to apply an adjustment or a
correction factor to Eq.(4.8)as explained below:
“In the Eq.(4.8) add the factor
m(m 2 − 1)
12
to
∑d
2
…………(4.8a)
; where m is the number of times an item is repeated. This correction factor is to be
added for each repeated value in both the series”.
Example 4-8
For a certain joint stock company, the prices of preference shares (X) and debentures (Y) are
given below:
X:
73.2
85.8
78.9
75.8
77.2
81.2
83.8
Y:
97.8
99.2
98.8
98.3
98.3
96.7
97.1
119
Use the method of rank correlation to determine the relationship between preference prices
and debentures prices.
Solution:
Calculations for Coefficient of Rank Correlation
{Using Eq. (4.8) and (4.8a)}
X
Y
Rank of X (XR)
Rank of Y (YR)
d = XR – YR
d2
73.2
97.8
7
5
2
4
85.8
99.2
1
1
0
0
78.9
98.8
4
2
2
4
75.8
98.3
6
3.5
2.5
6.25
77.2
98.3
5
3.5
1.5
2.25
81.2
96.7
3
7
-4
16
83.8
97.1
2
6
-4
16
∑d = 0
∑d
2
= 48.50
In this case, due to repeated values of Y, we have to apply ranking as average of 2 ranks,
which could have been allotted, if they were different values. Thus ranks 3 and 4 have been
allotted as 3.5 to both the values of Y = 98.3. Now we also have to apply correction factor
m(m 2 − 1)
to
12
∑d
2
, where m in the number of times the value is repeated, here m = 2.
(
ρ
=
=
)
⎡
m m2 −1 ⎤
6 ⎢∑ d 2 +
⎥
2
⎣
⎦
2
N ( N − 1)
2(4 − 1) ⎤
⎡
6⎢48.5 +
12 ⎥⎦
⎣
7(7 2 − 1)
6 x 49
7 x 48
=
1-
=
0.125
Hence, there is a very low degree of positive correlation, probably no correlation,
between preference share prices and debenture prices.
120
Remarks on Spearman’s Rank Correlation Coefficient
∑ d = 0 , which provides a check for numerical calculations.
1.
We always have
2.
Since Spearman’s rank correlation coefficient, ρ, is nothing but Karl Pearson’s
correlation coefficient, r, between the ranks, it can be interpreted in the same way
as the Karl Pearson’s correlation coefficient.
3.
Karl Pearson’s correlation coefficient assumes that the parent population from
which sample observations are drawn is normal. If this assumption is violated then
we need a measure, which is distribution free (or non-parametric). Spearman’s ρ
is such a distribution free measure, since no strict assumption are made about the
from of the population from which sample observations are drawn.
4.
Spearman’s formula is easy to understand and apply as compared to Karl
Pearson’s formula. The values obtained by the two formulae, viz Pearsonian r and
Spearman’s ρ are generally different. The difference arises due to the fact that
when ranking is used instead of full set of observations, there is always some loss
of information. Unless many ties exist, the coefficient of rank correlation should
be only slightly lower than the Pearsonian coefficient.
5.
Spearman’s formula is the only formula to be used for finding correlation
coefficient if we are dealing with qualitative characteristics, which cannot be
measured quantitatively but can be arranged serially. It can also be used where
actual data are given. In case of extreme observations, Spearman’s formula is
preferred to Pearson’s formula.
6.
Spearman’s formula has its limitations also. It is not practicable in the case of
bivariate frequency distribution. For N >30, this formula should not be used unless
the ranks are given.
121
4.3.5 CONCURRENT DEVIATION METHOD
This is a casual method of determining the correlation between two series when we are not
very serious about its precision. This is based on the signs of the deviations
(i.e. the
direction of the change) of the values of the variable from its preceding value and does not
take into account the exact magnitude of the values of the variables. Thus we put a plus (+)
sign, minus (-) sign or equality (=) sign for the deviation if the value of the variable is greater
than, less than or equal to the preceding value respectively. The deviations in the values of
two variables are said to be concurrent if they have the same sign (either both deviations are
positive or both are negative or both are equal). The formula used for computing correlation
coefficient rc by this method is given by
⎛ 2c − N ⎞
rc = + +⎜
⎟
⎝ N ⎠
…………(4.9)
Where c is the number of pairs of concurrent deviations and N is the number of pairs of
deviations. If (2c-N) is positive, we take positive sign in and outside the square root in Eq.
(4.9) and if (2c-N) is negative, we take negative sign in and outside the square root in Eq.
(4.9).
Remarks:
(i)
It should be clearly noted that here N is not the number of pairs of
observations but it is the number of pairs of deviations and as such it is one less than the
number of pairs of observations.
(ii) Coefficient of concurrent deviations is primarily based on the following
principle:
“If the short time fluctuations of the time series are positively correlated or in other
words, if their deviations are concurrent, their curves would move in the same
direction and would indicate positive correlation between them”
Example 4-9
122
Calculate coefficient of correlation by the concurrent deviation method
Supply:
112
125
126
118
118
121
125
125
131
135
Price:
106
102
102
104
98
96
97
97
95
90
Solution:
Calculations for Coefficient of Concurrent Deviations
{Using Eq. (4.9)}
Supply
Sign of deviation from
Price
Sign of deviation
Concurrent
(X)
preceding value (X)
(Y)
preceding value (Y)
deviations
112
106
125
+
102
-
126
+
102
=
118
-
104
+
118
=
98
-
121
+
96
-
125
+
97
+
+(c)
125
=
97
=
= ( c)
131
+
95
-
135
+
90
-
We have
Number of pairs of deviations, N =10 – 1 = 9
c = Number of concurrent deviations
= Number of deviations having like signs
=2
Coefficient of correlation by the method of concurrent deviations is given by:
⎛ 2c − N ⎞
rc = + +⎜
⎟
⎝ N ⎠
⎛ 2x 2 − 9 ⎞
rc = + +⎜
⎟
⎝ 9 ⎠
rc = + +(− 0.5556 )
Since 2c – N = -5 (negative), we take negative sign inside and outside the square root
123
rc = − − (− 0.5556 )
rc = − 0.5556
rc = −0.7
Hence there is a fairly good degree of negative correlation between supply and price.
4.4
LIMITATIONS OF CORRELATION ANALYSIS
As mentioned earlier, correlation analysis is a statistical tool, which should be properly used
so that correct results can be obtained. Sometimes, it is indiscriminately used by
management, resulting in misleading conclusions. We give below some errors frequently
made in the use of correlation analysis:
1. Correlation analysis cannot determine cause-and-effect relationship. One should not
assume that a change in Y variable is caused by a change in X variable unless one is
reasonably sure that one variable is the cause while the other is the effect. Let us take
an example.
.
Suppose that we study the performance of students in their graduate examination and
their earnings after, say, three years of their graduation. We may find that these two
variables are highly and positively related. At the same time, we must not forget that
both the variables might have been influenced by some other factors such as quality of
teachers, economic and social status of parents, effectiveness of the interviewing
process and so forth. If the data on these factors are available, then it is worthwhile to
use multiple correlation analysis instead of bivariate one.
2. Another mistake that occurs frequently is on account of misinterpretation of the
coefficient of correlation. Suppose in one case r = 0.7, it will be wrong to interpret
that correlation explains 70 percent of the total variation in Y. The error can be seen
easily when we calculate the coefficient of determination. Here, the coefficient of
124
determination r2 will be 0.49. This means that only 49 percent of the total variation in
Y is explained.
Similarly, the coefficient of determination is misinterpreted if it is also used to
indicate causal relationship, that is, the percentage of the change in one variable is due
to the change in another variable.
3. Another mistake in the interpretation of the coefficient of correlation occurs when one
concludes a positive or negative relationship even though the two variables are
actually unrelated. For example, the age of students and their score in the examination
have no relation with each other. The two variables may show similar movements but
there does not seem to be a common link between them.
To sum up, one has to be extremely careful while interpreting coefficient of correlation. Before one concludes a causal relationship, one has to consider other relevant factors that might
have any influence on the dependent variable or on both the variables. Such an approach will
avoid many of the pitfalls in the interpretation of the coefficient of correlation. It has been
rightly said that the coefficient of correlation is not only one of the most widely used, but
also one of the widely abused statistical measures.
4.5
SELF-ASSESSMENT QUESTIONS
1. “Correlation and Regression are two sides of the same coin”. Explain.
2. Explain the meaning and significance of the concept of correlation. Does correlation
always signify casual relationships between two variables? Explain with illustration
on what basis can the following correlation be criticized?
(a)
Over a period of time there has been an increased financial aid to under
developed countries and also an increase in comedy act television shows. The
correlation is almost perfect.
125
(b)
The correlation between salaries of school teachers and amount of liquor sold
during the period 1940 – 1980 was found to be 0.96
3. Write short not on the following
(a)
Spurious correlation
(b)
Positive and negative correlation
(c)
Linear and non-linear correlation
(d)
Simple, multiple and partial correlation
4. What is a scatter diagram? How does it help in studying correlation between two
variables, in respect of both its nature and extent?
5. Write short note on the following
(a)
Karl Pearson’s coefficient of correlation
(b)
Probable Error
(c)
Spearman’s Rank Correlation Coefficient
(d)
Coefficient of Concurrent Deviation
6. Draw a scatter diagram from the data given below and interpret it.
X:
10
20
30
40
50
60
70
80
Y:
32
20
24
36
40
28
38
44
7. Calculate Karl Pearson’s coefficient of correlation between expenditure on
advertising (X) and sales (Y) from the data given below:
X:
39
65
62
90
82
75
25
98
36
78
Y:
47
53
58
86
62
68
60
91
51
84
8. To study the effectiveness of an advertisement a survey is conducted by calling
and the number of items purchased (Y) in that week.
X:
5
10
4
0
2
7
3
6
Y:
10
12
5
2
1
3
4
8
126
Calculate the correlation coefficient and comment on the result.
9. Calculate coefficient of correlation between X and Y series from the following data
and calculate its probable error also.
X:
78
89
96
69
59
79
68
61
Y:
125
137
156
112
107
136
123
108
10. In two set of variables X and Y, with 50 observations each, the following data are
observed:
X
=
10,
SD of X = 3
Y
=
6,
SD of Y = 2
rxy = 0.3
However, on subsequent verification, it was found that one value of X (=10) and one
value of Y (= 6) were inaccurate and hence weeded out with the remaining 49 pairs of
values. How the original value of is rxy = 0.3 affected?
11. Calculate coefficient of correlation r between the marks in statistics (X) and
Accountancy (Y) of 10 students from the following:
X:
52
74
93
55
41
23
92
64
40
71
Y:
45
80
63
60
35
40
70
58
43
64
Also determine the probable error or r.
12. The coefficient of correlation between two variables X and Y is 0.48. The covariance
is 36. The variance of X is 16. Find the standard deviation of Y.
13. Twelve entries in painting competition were ranked by two judges as shown below:
Entry:
A
B
C
D
E
F
G
H
I
J
Judge I: 5
2
3
4
1
6
8
7
10
9
Judge II: 4
5
2
1
6
7
10
9
3
8
Find the coefficient of rank correlation.
sales (Y) from the following data:
127
X:
39
65
62
90
82
75
25
98
36
78
Y:
47
53
58
86
62
68
60
91
51
84
15. An examination of eight applicants for a clerical post was taken by a firm. From the
marks obtained by the applicants in the Accountancy (X) and Statistics (Y) paper,
compute rank coefficient of correlation.
Applicant:
A
B
C
D
E
F
G
H
X:
15
20
28
12
40
60
20
80
Y:
40
30
50
30
20
10
30
60
16. Calculate the coefficient of concurrent deviation from the following data:
Year:
1993 1994 1995 1996 1997 1998 1999 2000
Supply:
160
164
172
182
166
170
178
192
Price:
222
280
260
224
266
254
230
190
17. Obtain a suitable measure of correlation from the following data regarding changes in
price index of the shares A and B during nine months of a year:
Month:
A
M
J
J
A
S
O
N
D
A:
+4
+3
+2
-1
-3
+4
-5
+1
+2
B:
-2
+5
+3
-2
-1
-3
+4
-1
-3
18. The cross-classification table shows the marks obtained by 105 students in the
subjects of Statistics and Finance:
Marks in Finance
Marks in Statistics
50-54
55-59
50-59
4
6
60-69
-
70-79
60-64
65-74
Total
8
7
25
10
12
13
35
16
9
20
-
45
80-89
-
-
-
-
-
Total
20
25
40
20
105
Find the coefficient of correlation between marks obtained in two subjects.
128
4.6
1.
Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2.
Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3.
Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4.
Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5.
Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6.
Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7.
Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8.
Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
129
Course:
Course Code: MC-106
Lesson:
Author:
Anil Kumar
Vetter:
Prof. Harbhajan Bansal
05
REGRESSION ANALYSIS
Objectives: The overall objective of this lesson is to give you an understanding of
linear regression, there by enabling you to understand the importance
and also the limitations of regression analysis.
Structure
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
Introduction
What is Regression?
Linear Regression
5.3.1 Regression Line of Y on X
5.3.1.1 Scatter Diagram
5.3.1.2 Fitting a Straight Line
5.3.1.3 Predicting an Estimate and its Preciseness
5.3.1.4 Error of Estimate
5.3.2 Regression Line of X on Y
Properties of Regression Coefficients
Regression Lines and Coefficient of Correlation
Coefficient of Determination
Correlation Analysis Versus Regression Analysis
Solved Problems
Self-Assessment Questions
130
...if we find any association between two or more variables, we might be interested in
estimating the value of one variable for known value(s) of another variable(s)
5.1
INTRODUCTION
In business, several times it becomes necessary to have some forecast so that the management
can take a decision regarding a product or a particular course of action. In order to make a
forecast, one has to ascertain some relationship between two or more variables relevant to a
particular situation. For example, a company is interested to know how far the demand for
television sets will increase in the next five years, keeping in mind the growth of population
in a certain town. Here, it clearly assumes that the increase in population will lead to an
increased demand for television sets. Thus, to determine the nature and extent of relationship
between these two variables becomes important for the company.
In the preceding lesson, we studied in some depth linear correlation between two variables.
Here we have a similar concern, the association between variables, except that we develop it
further in two respects. First, we learn how to build statistical models of relationships
between the variables to have a better understanding of their features. Second, we extend the
models to consider their use in forecasting.
For this purpose, we have to use the technique - regression analysis - which forms the
subject-matter of this lesson.
5.2
WHAT IS REGRESSION?
In 1889, Sir Francis Galton, a cousin of Charles Darwin published a paper on heredity,
“Natural Inheritance”. He reported his discovery that sizes of seeds of sweet pea plants
appeared to “revert” or “regress”, to the mean size in successive generations. He also reported
results of a study of the relationship between heights of fathers and heights of their sons. A
straight line was fit to the data pairs: height of father versus height of son. Here, too, he found
a “regression to mediocrity” The heights of the sons represented a movement away from their
131
fathers, towards the average height. We credit Sir Galton with the idea of statistical
regression.
While most applications of regression analysis may have little to do with the
“regression to the mean” discovered by Galton, the term “regression” remains. It
now refers to the statistical technique of modeling the relationship between two or
more variables. In general sense, regression analysis means the estimation or
prediction of the unknown value of one variable from the known value(s) of the other
variable(s). It is one of the most important and widely used statistical techniques in
almost all sciences - natural, social or physical.
In this lesson we will focus only on simple regression –linear regression involving only two
variables: a dependent variable and an independent variable. Regression analysis for studying
more than two variables at a time is known as multiple regressions.
5.2.1 INDEPENDENT AND DEPENDENT VARIABLES
Simple regression involves only two variables; one variable is predicted by another variable.
The variable to be predicted is called the dependent variable. The predictor is called the
independent variable, or explanatory variable. For example, when we are trying to predict
the demand for television sets on the basis of population growth, we are using the demand for
television sets as the dependent variable and the population growth as the independent or
predictor variable.
The decision, as to which variable is which sometimes, causes problems. Often the choice is
obvious, as in case of demand for television sets and population growth because it would
make no sense to suggest that population growth could be dependent on TV demand! The
population growth has to be the independent variable and the TV demand the dependent
variable.
132
If we are unsure, here are some points that might be of use:
¾ if we have control over one of the variables then that is the independent. For example,
a manufacturer can decide how much to spend on advertising and expect his sales to
be dependent upon how much he spends
¾ it there is any lapse of time between the two variables being measured, then the latter
must depend upon the former, it cannot be the other way round
¾ if we want to predict the values of one variable from your knowledge of the other
variable, the variable to be predicted must be dependent on the known one
5.3
LINEAR REGRESSION
The task of bringing out linear relationship consists of developing methods of fitting a
straight line, or a regression line as is often called, to the data on two variables.
The line of Regression is the graphical or relationship representation of the best estimate of
one variable for any given value of the other variable. The nomenclature of the line depends
on the independent and dependent variables. If X and Y are two variables of which
relationship is to be indicated, a line that gives best estimate of Y for any value of X, it is
called Regression line of Y on X. If the dependent variable changes to X, then best estimate
of X by any value of Y is called Regression line of X on Y.
5.3.1 REGRESSION LINE OF Y ON X
For purposes of illustration as to how a straight line relationship is obtained, consider the
sample paired data on sales of each of the N = 5 months of a year and the marketing
expenditure incurred in each month, as shown in Table 5-1
Table 5-1
Month
Sales
(Rs lac)
Marketing Expenditure
(Rs thousands)
133
April
Y
14
X
10
May
17
12
June
23
15
July
21
20
August
25
23
Let Y, the sales, be the dependent variable and X, the marketing expenditure, the independent
variable. We note that for each value of independent variable X, there is a specific value of
the dependent variable Y, so that each value of X and Y can be seen as paired observations.
5.3.1.1 Scatter Diagram
Before obtaining a straight-line relationship, it is necessary to discover whether the
relationship between the two variables is linear, that is, the one which is best explained by a
straight line. A good way of doing this is to plot the data on X and Y on a graph so as to yield
a scatter diagram, as may be seen in Figure 5-1. A careful reading of the scatter diagram
reveals that:
¾ the overall tendency of the points is to move upward, so the relationship is positive
¾ the general course of movement of the various points on the diagram can be best
explained by a straight line
¾ there is a high degree of correlation between the variables, as the points are very close
to each other
134
Figure 5-1
Scatter Diagram with Line of Best Fit
5.3.1.2 Fitting a Straight Line on the Scatter Diagram
If the movement of various points on the scatter diagram is best described by a straight line,
the next step is to fit a straight line on the scatter diagram. It has to be so fitted that on the
whole it lies as close as possible to every point on the scatter diagram. The necessary
requirement for meeting this condition being that the sum of the squares of the vertical
deviations of the observed Y values from the straight line is minimum.
As shown in Figure 5-1, if dl, d2,..., dN are the vertical deviations' of observed Y values from
the straight line, fitting a straight line requires that
N
d12 + d 22 + ..................... + d N2 = ∑ d 2j
j =1
is the minimum. The deviations dj have to be squared to avoid negative deviations canceling
out the positive deviations. Since a straight line so fitted best approximates all the points on
the scatter diagram, it is better known as the best approximating line or the line of best fit. A
line of best fit can be fitted by means of:
1. Free hand drawing method, and
2. Least square method
Free Hand Drawing:
Free hand drawing is the simplest method of fitting a straight line. After a careful
inspection of the movement and spread of various points on the scatter diagram, a
straight line is drawn through these points by using a transparent ruler such that on the
135
whole it is closest to every point. A straight line so drawn is particularly useful when
future approximations of the dependent variable are promptly required.
Whereas the use of free hand drawing may yield a line nearest to the line of best fit, the major
drawback is that the slope of the line so drawn varies from person to person because of the
influence of subjectivity. Consequently, the values of the dependent variable estimated on the
basis of such a line may not be as accurate and precise as those based on the line of best fit.
Least Square Method:
The least square method of fitting a line of best fit requires minimizing the sum of the
squares of vertical deviations of each observed Y value from the fitted line. These deviations,
such as d1 and d3, are shown in Figure 5-1 and are given by Y - Yc, where Y is the observed
value and Yc the corresponding computed value given by the fitted line
Yc = a + bX i
…………(5.1)
for the ith value of X.
The straight line relationship in Eq.(5.1), is stated in terms of two constants a and b
¾ The constant a is the Y-intercept; it indicates the height on the vertical axis from
where the straight line originates, representing the value of Y when X is zero.
¾ Constant b is a measure of the slope of the straight line; it shows the absolute change
in Y for a unit change in X. As the slope may be positive or negative, it indicates the
nature of relationship between Y and X. Accordingly, b is also known as the
regression coefficient of Y on X.
Since a straight line is completely defined by its intercept a and slope b, the task of fitting the
same reduces only to the computation of the values of these two constants. Once these two
values are known, the computed Yc values against each value of X can be easily obtained by
substituting X values in the linear equation.
136
In the method of least squares the values of a and b are obtained by solving simultaneously
the following pair of normal equations
∑ Y = aN + b∑ X
…………(5.2)
∑ XY = a∑ X + b∑ X
2
…………(5.2)
∑ X , ∑ Y , ∑ XY and ∑ X
The value of the expressions -
2
can be obtained from the given
observations and then can be substituted in the above equations to obtain the value of a and b.
Since simultaneous solving the two normal equations for a and b may quite often be
cumbersome and time consuming, the two values can be directly obtained as
a = Y − bX
…………(5.3)
and
b=
N ∑ XY − ∑ X ∑ Y
…………(5.4)
N ∑ X 2 − (∑ X )
2
Note: Eq. (5.3) is obtained simply by dividing both sides of the first of Eqs. (5.2) by N and
Eq.(5.4) is obtained by substituting ( Y − b X ) in place of a in the second of Eqs. (5.2)
Instead of directly computing b, we may first compute value of a as
∑ Y ∑ X − ∑ X ∑ XY
a=
N ∑ X − (∑ X )
2
…………(5.5)
2
2
and
b=
Y −a
X
…………(5.6)
Note: Eq. (5.5) is obtained by substituting
N ∑ XY − ∑ X ∑ Y
N ∑ X 2 − (∑ X )
2
for b in Eq. (5.3) and Eq.
(5.6) is obtained simply by rearranging Eq. (5.3)
Table 5-2
Computation of a and b
Y
X
XY
137
X2
Y2
14
10
140
100
196
17
12
204
144
289
23
15
345
225
529
21
20
420
400
441
25
23
575
529
625
∑ Y = 100 ∑ X = 80 ∑ XY = 1684 ∑ X
2
= 1398
∑Y
So using Eqs. (5.5) and (5.4)
a=
=
=
100 x1398 − 80 x1684
5 x1398 − (80 )
2
139800 − 134720
6990 − 6400
5080
590
= 8.6101695
and
b=
5 x1684 − 80 x100
5 x1398 − (80)
=
8420 − 8000
6990 − 6400
=
420
590
2
= 0.7118644
Now given
a = 8.61
and
b = 0.71
The regression Eq.(5.1) takes the form
Yc = 8.61 + 0.71X
…………(5.1a)
138
2
= 2080
Figure 5-2
Regression Line of Y on X
Then, to fit the line of best fit on the scatter diagram, only two computed Yc values are
needed. These can be easily obtained by substituting any two values of X in Eq. (5.1a). When
these are plotted on the diagram against their corresponding values of X, we get two points,
by joining which (by means of a straight line) gives us the required line of best fit, as shown
in Figure 5-2
Some Important Relationships
We can have some important relationships for data analysis, involving other measures such as
X , Y , Sx, Sy and the correlation coefficient rxy.
Substituting Y − b X [from Eq.(5.3)] for a in Eq.(5.1)
Yc = ( Y − b X ) +bX
or
Yc - Y = b(X- X )
…………(5.7)
Dividing the numerator and denominator of Eq.(5.4) by N2, we get
∑ XY − ⎛⎜ ∑ X ⎞⎟⎛⎜ ∑Y ⎞⎟
b=
or
b=
or
b=
⎜ N ⎟⎜
⎝
⎠⎝
2
∑ X − ⎛⎜ ∑ X
⎜ N
N
⎝
N
N ⎟⎠
2
⎞
⎟
⎟
⎠
∑ XY − X Y
N
S x2
Cov( X , Y )
S x2
…………(5.8)
We know, coefficient of correlation, rxy is given by
rxy =
Cov( X , Y )
Sx Sy
139
or Cov( X , Y ) = rxy S x S y
So Eq. (5.8) becomes
b = rxy
b = rxy
SxSy
S x2
Sy
…………(5.9)
Sx
Substituting rxy
Sy
Sx
Yc - Y = rxy
for b in Eq.(5.7), we get
Sy
Sx
(X- X )
…………(5.10)
These are important relationships for data analysis.
5.3.1.3 Predicting an Estimate and its Preciseness
The main objective of regression analysis is to know the nature of relationship between two
variables and to use it for predicting the most likely value of the dependent variable
corresponding to a given, known value of the independent variable. This can be done by
substituting in Eq.(5.1a) any known value of X corresponding to which the most likely
estimate of Y is to be found.
For example, the estimate of Y (i.e. Yc), corresponding to X = 15 is
Yc = 8.61 + 0.71(15)
= 8.61 + 10.65
= 19.26
It may be appreciated that an estimate of Y derived from a regression equation will not be
exactly the same as the Y value which may actually be observed. The difference between
estimated Yc values and the corresponding observed Y values will depend on the extent of
scatter of various points around the line of best fit.
140
The closer the various paired sample points (Y, X) clustered around the line of best fit, the
smaller the difference between the estimated Yc and observed Y values, and vice-versa. On the
whole, the lesser the scatter of the various points around, and the lesser the vertical distance
by which these deviate from the line of best fit, the more likely it is that an estimated Yc value
is close to the corresponding observed Y value.
The estimated Yc values will coincide the observed Y values only when all the points on the
scatter diagram fall in a straight line. If this were to be so, the sales for a given marketing
expenditure could have been estimated with l00 percent accuracy. But such a situation is too
rare to obtain. Since some of the points must lie above and some below the straight line,
perfect prediction is practically non-existent in the case of most business and economic
situations.
This means that the estimated values of one variable based on the known values of the other
variable are always bound to differ. The smaller the difference, the greater the precision of
the estimate, and vice-versa. Accordingly, the preciseness of an estimate can be obtained only
through a measure of the magnitude of error in the estimates, called the error of estimate.
5.3.1.4 Error of Estimate
A measure of the error of estimate is given by the standard error of estimate of Y on X,
denoted as Syx and defined as
Syx =
∑ (Y − Y )
2
c
…………(5.11)
N
Syx measures the average absolute amount by which observed Y values depart from the
corresponding computed Yc values.
Computation of Syx becomes little cumbersome where the number of observations N is large.
In such cases Syx may be computed directly by using the equation:
141
Syx =
∑Y
2
− a (∑ Y ) − b∑ XY
…………(5.12)
N
By substituting the values of
∑ Y , ∑ Y , and ∑ XY
2
from the Table 5-2, and the calculated
values of a and b
We have
Syx
=
2080 − 8.61x100 − 0.71x1684
5
=
2080 − 861 − 1195.64
5
=
23.36
5
=
4.67
= 2.16
Interpretations of Syx
A careful observation of how the standard error of estimate is computed reveals the
following:
1. Syx is a concept statistically parallel to the standard deviation Sy . The only difference
between the two being that the standard deviation measures the dispersion around the
mean; the standard error of estimate measures the dispersion around the regression
line. Similar to the property of arithmetic mean, the sum of the deviations of different
Y values from their corresponding estimated Yc values is equal to zero. That is
∑( Yi -Y ) = ∑ ( Yi - Yc) = 0 where i = 1, 2, ..., N.
2. Syx tells us the amount by which the estimated Yc values will, on an average, deviate
from the observed Y values. Hence it is an estimate of the average amount of error in
the estimated Yc values. The actual error (the residual of Y and Yc) may, however, be
smaller or larger than the average error. Theoretically, these errors follow a normal
distribution. Thus, assuming that n ≥ 30, Yc ± 1.Syx means that 68.27% of the estimates
142
based on the regression equation will be within 1.Syx Similarly, Yc ± 2.Syx means that
95.45% of the estimates will fall within 2.Syx
Further, for the estimated value of sales against marketing expenditure of Rs 15
thousand being Rs 19.26 lac, one may like to know how good this estimate is. Since
Syx is estimated to be Rs 2.16 lac, it means there are about 68 chances (68.27) out of
100 that this estimate is in error by not more than Rs 2.16 lac above or below Rs
19.26 lac. That is, there are 68% chances that actual sales would fall between (19.26 2.16) = Rs 17.10 lac and (19.26 + 2.16) = Rs 21.42 lac.
3. Since Syx measures the closeness of the observed Y values and the estimated Yc values,
it also serves as a measure of the reliability of the estimate. Greater the closeness
between the observed and estimated values of Y, the lesser the error and,
consequently, the more reliable the estimate. And vice-versa.
4. Standard error of estimate Syx can also be seen as a measure of correlation insofar as it
expresses the degree of closeness of scatter of observed Y values about the regression
line. The closer the observed Y values scattered around the regression line, the higher
the correlation between the two variables.
A major difficulty in using Syx as a measure of correlation is that it is expressed in the
same units of measurement as the data on the dependent variable. This creates
problems in situations requiring comparison of two or more sets of data in terms of
correlation. It is mainly due to this limitation that the standard error of estimate is not
generally used as a measure of correlation. However, it does serve as the basis of
evolving the coefficient of determination, denoted as r2, which provides an alternate
method of obtaining a measure of correlation.
5.3.2 REGRESSION LINE OF X ON Y
143
So far we have considered the regression of Y on X, in the sense that Y was in the role of
dependent and X in the role of an independent variable. In their reverse position, such that X
is now the dependent and Y the independent variable, we fit a line of regression of X on Y.
The regression equation in this case will be
Xc = a’ + b’Y
…………(5.13)
Where Xc denotes the computed values of X against the corresponding values of Y. a’ is the
X-intercept and b’ is the slope of the straight line.
Two normal equations to solve a’and b’ are
∑ X = a ' N + b' ∑ Y
…………(5.14)
∑ XY = a' ∑ Y + b' ∑ Y
2
…………(5.14)
The value of a’ and b’ can also be obtained directly
a’ = X - b’Y
…………(5.15)
and
b' =
N ∑ XY − ∑ X ∑ Y
…………(5.16)
N ∑ Y 2 − (∑ Y )
2
or
∑ X ∑ Y − ∑ Y ∑ XY
a' =
N ∑ Y − (∑ Y )
2
2
…………(5.17)
2
and
b' =
X − a'
Y
…………(5.18)
b' =
Cov(Y , X )
S y2
…………(5.19)
b' = ryx
Sx
Sy
…………(5.20)
So, Regression equation of X on Y may also be written as
144
Xc - X = b’ (Y- Y )
Xc - X = ryx
…………(5.21)
Sx
(Y - Y )
Sy
…………(5.22)
As before, once the values of a’ and b’ have been found, their substitution in Eq.(5.13) will
enable us to get an estimate of X corresponding to a known value of Y
Standard Error of estimate of X on Y i.e. Sxy will be
Sxy =
( X − X c )2
…………(5.23)
N
or
Sxy =
∑X
2
− a ' ∑ X − b' ∑ XY
…………(5.24)
N
For example, if we want to estimate the marketing expenditure to achieve a sale target of Rs
40 lac, we have to obtain regression line of X on Y i. e.
Xc = a’ + b’Y
So using Eqs. (5.17) and (5.16), and substituting the values of
from Table 5-2, we have
a' =
80 x 2080 − 100 x1684
2
5 x 2080 − (100)
=
166400 − 168400
10400 − 10000
=
− 2000
400
= -5.00
and
b' =
=
5 x1684 − 80 x100
2
5 x 2080 − (100)
8420 − 8000
10400 − 10000
145
∑ X , ∑ Y , ∑ Y and ∑ XY
2
=
420
400
= 1.05
Now given that a’= -5.00 and b’=1.05, Regression equation (5.13) takes the form
Xc = -5.00 +1.05Y
So when Y = 40(Rs lac), the corresponding X value is
Xc = -5.00+1.05x40
= -5 + 42
= 37
That is to achieve a sale target of Rs 40 lac, there is a need to spend Rs 37 thousand on
marketing.
5.4
PROPERTIES OF REGRESSION COEFFICIENTS
As explained earlier, the slope of regression line is called the regression coefficient. It tells
the effect on dependent variable if there is a unit change in the independent variable. Since
for a paired data on X and Y variables, there are two regression lines: regression line of Y on X
and regression line of X on Y, so we have two regression coefficients:
a.
Regression coefficient of Y on X, denoted by byx [b in Eq.(5.1)]
b.
Regression coefficient of X on Y, denoted by bxy [b’ in Eq.(5.13)]
The following are the important properties of regression coefficients that are helpful in data
analysis
1. The value of both the regression coefficients cannot be greater than 1. However, value
of both the coefficients can be below 1 or at least one of them must be below 1, so
that the square root of the product of two regression coefficients must lie in the limit
±1.
2. Coefficient of correlation is the geometric mean of the regression coefficients, i.e.
146
r = ± b. b'
…………(5.25)
The signs of both the regression coefficients are the same, and so the value of r will
also have the same sign.
3. The mean of both the regression coefficients is either equal to or greater than the
coefficient of correlation, i.e.
b + b'
≥r
2
3. Regression coefficients are independent of change of origin but not of change of
scale. Mathematically, if given variables X and Y are transformed to new variables U
and V by change of origin and scale, i. e.
U=
X−A
h
and
V =
Y −B
k
Where A, B, h and k are constants, h > 0, k > 0 then
Regression coefficient of Y on X = k/h (Regression coefficient of V on U)
b yx =
k
bvu
h
and
Regression coefficient of X on Y = h/k (Regression coefficient of U on V)
bxy =
h
buv
k
5. Coefficient of determination is the product of both the regression coefficients i.e.
r2 = b.b’
5.5
REGRESSION LINES AND COEFFICIENT OF CORRELATION
The two regression lines indicate the nature and extent of correlation between the variables.
The two regression lines can be represented as
Y- Y = r
Sy
Sx
(X - X )
and
X- X = r
147
Sx
(Y - Y )
Sy
We can write the slope of these lines, as
b= r
Sy
and
Sx
b’ = r
Sx
Sy
If θ is the angle between these lines, then
tan θ
=
b − b'
1 + bb'
Sx S y ⎛ r 2 −1⎞
⎟
⎜
= 2
S x + S y2 ⎜⎝ r ⎟⎠
⎡ SxSy
or θ = tan –1 ⎢ 2
2
⎢⎣ S x + S y
⎛ r 2 − 1 ⎞⎤
⎜⎜
⎟⎟⎥
⎝ r ⎠⎥⎦
…………(5.26)
148
Figure 5-3
Regression Lines and Coefficient of Correlation
Eq. (5.26) reveals the following:
¾ In case of perfect positive correlation (r = +1) and in case of perfect negative
correlation (r = -1), θ = 0, so the two regression lines will coincide, i.e. we have only
one line, see (a) and (b) in Figure 5-3.
The farther the two regression lines from each other, lesser will be the degree of
correlation and nearer the two regression lines, more will be the degree of correlation,
see (c) and (d) in Figure 5-3.
¾ If the variables are independent i.e. r = 0, the lines of regression will cut each other at
right angle. See (g) in Figure 5-3.
Note : Both the regression lines cut each other at mean value of X and mean value of Y i.e. at
X and Y .
5.6
COEFFICIENT OF DETERMINATION
Coefficient of determination gives the percentage variation in the dependent variable that is
accounted for by the independent variable. In other words, the coefficient of determination
gives the ratio of the explained variance to the total variance. The coefficient of
determination is given by the square of the correlation coefficient, i.e. r2. Thus,
Coefficient of determination
r2 =
Explained Variance
Total Variance
2
∑ (Y − Y )
∑ (Y − Y )
2
r =
c
…………(5.27)
2
149
We can calculate another coefficient K2, known as coefficient of Non-Determination, which
is the ratio of unexplained variance to the total variance.
K2 =
2
K =
Un exp lained Variance
Total Variance
∑ (Y − Y )
∑ (Y − Y )
2
c
…………(5.28)
2
Explained Variance
Total Variance
K2 = 1-
= 1 - r2
…………(5.29)
The square root of the coefficient of non-determination, i.e. K gives the coefficient of
alienation
K = ± 1− r2
…………(5.30)
Relation Between Syx and r:
A simple algebraic operation on Eq. (5.30) brings out some interesting points about the
relation between Syx and r. Thus, since
∑ (Y − Y )
c
2
= N S yx2
∑ (Y − Y )
2
and
= N S y2
So we have coefficient of Non-determination
K
2
∑ (Y − Y )
=
∑ (Y − Y )
2
N S yx2
K2 =
N S y2
=
So
or
1–r
2
S yx
Sy
2
c
=
S yx2
S y2
S yx2
S y2
= 1− r 2
…………(5.31)
150
If coefficient of correlation, r, is defined as the under root of the coefficient of determination
r=
r2
2
r = 1−
r = 1−
S yx2
S y2
S yx
…………(5.32)
S y2
On carefully observing Eq. (5.32), it will be noticed that the ratio Syx/Sy will be large if the
coefficient of determination is small, and it will be small when the coefficient of
determination is large. Thus
9 if r2 = r = 0, Syx/Sy =1, which means that Syx = Sy.
9 if r2 = r = 1, Syx/Sy =0, which means that Syx = 0.
9 when r = 0.865, Syx = 0.427 Sy means that Syx is 42.7% of Sy.
Eq. (5.32) also implies that Syx is generally less than Sy. The two can at the most be equal, but
only in the extreme situation when r = 0.
Interpretations of r2:
1. Even though the coefficient of determination, whose under root measures the degree
of correlation, is based on Syx,; it is expressed as 1 - ( Syx/Sy ). As it is a dimensionless
pure number, the unit in which Syx is measured becomes irrelevant. This facilitates
comparison between the two sets of data in terms of their coefficient of determination
r2 (or the coefficient of correlation r). This was not possible in terms of Sy x as the
units of measurement could be different.
2. The value of r2 can range between 0 and 1. When r2 = 1, all the points on the scatter
diagram fall on the regression line and the entire variations are explained by the
straight line. On the other hand, when r2 = 0, none of the points on the scatter diagram
falls on the regression line, meaning thereby that there is no relationship between the
two variables. However, being always non-negative coefficient of determination does
151
not tell us about the direction of the relationship (whether it is positive or negative)
between the two variables.
3. When r2 = 0.7455 (or any other value), 74.55% of the total variations in sales are
explained by the marketing expenditure used. What remains is the coefficient of nondetermination K2 (= 1 - r2) = 0.2545. It means 25.45% of the total variations remain
unexplained, which are due to factors other than the changes in the marketing
expenditure.
4. r2 provides the necessary link between regression and correlation which are the two
related aspects of a single problem of the analysis of relationship between two
variables. Unlike regression, correlation quantifies the degrees of relationship
between the variables under study, without making a distinction between the
dependent and independent ones. Nor does it, therefore, help in predicting the value of
one variable for a given value of the other.
5. The coefficient of correlation overstates the degree of relationship and it’s meaning is
not as explicit as that of the coefficient of determination. The coefficient of
correlation r = 0.865, as compared to r2 = 0.7455, indicates a higher degree of
correlation between sales and marketing expenditure. Therefore, the coefficient of'
determination is a more objective measure of the degree of relationship.
6. The sum of r and K never adds to one, unless one of the two is zero. That is, r + K can
be unity either when there is no correlation or when there is perfect correlation.
Except in these two extreme situations, (r + K) > 1.
5.7
CORRELATION ANALYSIS VERSUS REGRESSION ANALYSIS
Correlation and Regression are the two related aspects of a single problem of the analysis of
the relationship between the variables. If we have information on more than one variable, we
might be interested in seeing if there is any connection - any association - between them. If
152
we found such a association, we might again be interested in predicting the value of one
variable for the given and known values of other variable(s).
1. Correlation literally means the relationship between two or more variables that vary in
sympathy so that the movements in one tend to be accompanied by the corresponding
movements in the other(s). On the other hand, regression means stepping back or
returning to the average value and is a mathematical measure expressing the average
relationship between the two variables.
2. Correlation coefficient rxy between two variables X and Y is a measure of the direction
and degree of the linear relationship between two variables that is mutual. It is
symmetric, i.e., ryx = rxy and it is immaterial which of X and Y is dependent variable
and which is independent variable.
Regression analysis aims at establishing the functional relationship between the two(
or more) variables under study and then using this relationship to predict or estimate
the value of the dependent variable for any given value of the independent variable(s).
It also reflects upon the nature of the variable, i.e., which is dependent variable and
which is independent variable. Regression coefficient are not symmetric in X and Y,
i.e., byx ≠ bxy.
3. Correlation need not imply cause and effect relationship between the variable under
study. However, regression analysis clearly indicates the cause and effect relationship
between the variables. The variable corresponding to cause is taken as independent
variable and the variable corresponding to effect is taken as dependent variable.
4. Correlation coefficient rxy is a relative measure of the linear relationship between X
and Y and is independent of the units of measurement. It is a pure number lying
between ±1.
153
On the other hand, the regression coefficients, byx and bxy are absolute measures
representing the change in the value of the variable Y (or X), for a unit change in the
value of the variable X (or Y). Once the functional form of regression curve is known;
by substituting the value of the independent variable we can obtain the value of the
dependent variable and this value will be in the units of measurement of the
dependent variable.
5. There may be non-sense correlation between two variables that is due to pure chance
and has no practical relevance, e.g., the correlation, between the size of shoe and the
intelligence of a group of individuals. There is no such thing like non-sense
regression.
5.8
SOLVED PROBLEMS
Example 5-1
The following table shows the number of motor registrations in a certain territory for
a term of 5 years and the sale of motor tyres by a firm in that territory for the same
period.
Year
Motor Registrations
No. of Tyres Sold
1
600
1,250
2
630
1,100
3
720
1,300
4
750
1,350
5
800
1,500
Find the regression equation to estimate the sale of tyres when the motor registration
is known. Estimate sale of tyres when registration is 850.
Solution: Here the dependent variable is number of tyres; dependent on motor registrations.
Hence we put motor registrations as X and sales of tyres as Y and we have to establish the
regression line of Y on X.
Calculations of values for the regression equation are given below:
154
∑X
X
Y
dx = X- X
dy = Y-Y
dx2
dx dy
600
1,250
-100
-50
10,000
5,000
630
1,100
-70
-200
4,900
14,000
720
1,300
20
0
400
0
750
1,350
50
50
2,500
2,500
800
1,500
100
200
10,000
20,000
∑ Y = 6,500 ∑ d
= 3,500
X=
∑X =
N
x
=0
3,500
=700
5
∑d
y
=0
and
∑d
2
x
= 27,800
Y=
∑Y =
N
∑d
x
d y = 41,500
6,500
= 1,300
5
byx = Regression coefficient of Y on X
byx =
∑ (X − X )(Y − Y ) = ∑ d d
∑d
∑ (X − X )
x
2
2
x
y
=
4,1500
= 1.4928
2,7800
Now we can use these values for the regression line
Y-Y
or
=
byx (X- X )
Y – 1300
=
1.4928 (X - 700)
Y
=
1.4928 X + 255.04
When X = 850, the value of Y can be calculated from the above equation, by putting X = 850
in the equation.
Y
=
1.4928 x 850 + 255. 04
=
1523.92
=
1,524 Tyres
Example 5-2
A panel of Judges A and B graded seven debators and independently awarded the
following marks:
Debator
Marks by A
Marks by B
1
40
32
2
34
39
155
3
28
26
4
30
30
5
44
38
6
38
34
7
31
28
An eighth debator was awarded 36 marks by judge A, while Judge B was not present. If
Judge B were also present, how many marks would you expect him to award to the eighth
debator, assuming that the same degree of relationship exists in their judgement?
Solution: Let us use marks from Judge A as X and those from Judge B as Y. Now we have to
work out the regression line of Y on X from the calculation below:
Debtor
X
Y
U = X-35
V = Y-30
U2
V2
1
40
32
5
2
25
4
10
2
34
39
-1
9
1
81
-9
3
28
26
-7
-4
49
16
28
4
30
30
-5
0
25
0
0
5
44
38
9
8
81
64
72
6
38
34
3
4
9
16
12
7
31
28
-4
-2
16
4
8
∑ U = 0 ∑ V = 17
N=7
X = A+
∑ U = 35 +
N
byx = bvu =
=
0
= 35
7
∑U
N ∑ UV − (∑ U ∑ V )
)
2
7 x121 - 0 x17
= 0.587
7 x 206 - 0
Hence regression equation can be written as
Y- Y
Y – 32.43
= 206
Y = A+
and
N ∑ U 2 − (∑ U
2
= byx (X- X )
= 0.587 (X-35)
156
∑V
UV
= 185
∑ UV
∑ V = 30 + 17
= 32.43
N
2
7
= 121
or
Y
= 0.587X + 11.87
When X = 36 (awarded by Judge A)
Y
= 0.587 x 36 + 11.87
= 33
Thus if Judge B were present, he would have awarded 33 marks to the eighth debator.
Example 5-3
For some bivariate data, the following results were obtained.
Mean value of variable X
=
53.2
Mean value of variable Y
=
27.9
Regression coefficient of Y on X
=
- 1.5
Regression coefficient of X on Y
=
- 0.2
What is the most likely value of Y, when X = 60?
What is the coefficient of correlation between X and Y?
Solution: Given data indicate
X
=
53.2
Y
=
27.9
byx
=
-1.5
bxy
=
-0.2
To obtain value of Y for X = 60, we establish the regression line of Y on X,
or
Y- Y
=
byx (X- X )
Y – 27.9
=
-1.5 (X-53.2)
Y
=
-1.5X + 107.7
Putting value of X = 60, we obtain
Y
=
-1.5 x 60 + 107.7
=
17.7
Coefficient of correlation between X and Y is given by G.M. of byx and bxy
r2
=
byx bxy
157
= (-1.5) x (–0.2)
= 0.3
r
So
= ± 0.3 = ± 0.5477
Since both the regression coefficients are negative, we assign negative value to the
correlation coefficient
r
= - 0.5477
Example 5-4
Write regression equations of X on Y and of Y on X for the following data
X:
45
48
50
55
65
70
75
72
80
85
Y:
25
30
35
30
40
50
45
55
60
65
Solution: We prepare the table for working out the values for the regression lines.
X
Y
U = X-65
V = Y-45
U2
UV
45
25
-20
-20
400
400
400
48
30
-17
-15
289
255
225
50
35
-15
-10
225
150
100
55
30
-10
-15
100
150
225
65
40
0
-5
0
0
25
70
50
5
5
25
25
25
75
45
10
0
100
0
0
72
55
7
5
49
35
25
80
60
15
15
225
225
225
85
65
20
20
400
∑X
= 645∑ Y = 435
∑ U = 5 ∑ V = −20 ∑ U
2
= 1813
400
∑V
We have,
X=
∑X =
N
645
= 64.5
10
byx =
N ∑ UV − (∑ U ∑ V )
and
Y=
∑Y =
N
N ∑ U 2 − (∑ U )
2
158
V2
435
= 43.5
10
2
= 1415
400
∑ UV
= 1675
=
(10) x 1415 - (5) x (-20)
(10) x 1813 - (5) 2
=
14150 + 100 14250
=
= 0.787
18130 - 25 18105
Regression equation of Y on X is
or
Y-Y
=
byx (X- X )
Y – 43.5
=
0.787 (X-64.5)
Y
=
0.787X + 7.26
Similarly bxy can be calculated as
bxy =
N ∑ UV − (∑ U ∑ V )
N ∑ V 2 − (∑ V )
2
=
(10) x 1415 - (5) x (-20)
(10) x 1675 - (-20) 2
=
14150 + 100 14250
=
= 0.87
16750 - 400 16350
Regression equation of X on Y will be
or
X-X
=
bxy (Y-Y )
X – 64.5
=
0.87 (Y-43.5)
X
=
0.87Y + 26.65
Example 5-5
The lines of regression of a bivariate population are
8X – 10Y + 66 = 0
40X – 18Y = 214
The variance of X is 9. Find
(i)
The mean value of X and Y
(ii)
Correlation coefficient between X and Y
(iii)
Standard deviation of Y
159
Solution: The regression lines given are
8X – 10Y + 66 = 0
40X – 18Y = 214
Since both the lines of regression pass through the mean values, the point ( X , Y ) will satisfy
both the equations.
Hence these equations can be written as
8 X - 10 Y + 66 = 0
40 X - 18Y - 214 = 0
Solving these two equations for X and Y , we obtain
X = 13
(ii)
and
Y = 17
For correlation coefficient between X and Y, we have to calculate the values of byx and
bxy
Rewriting the equations
10Y = 8X + 66
byx = + 8/10 = + 4/5
Similarly,
40X = 18Y + 214
bxy = 18/40 = 9/20
By these values, we can now work out the correlation coefficient.
r2 = byx . bxy
= 4/5 x 9/20 = 9/25
So
r = + 9 / 25
= + 0.6
Both the values of the regression coefficients being positive, we have to consider only the
positive value of the correlation coefficient. Hence r = 0.6
(iii)
We have been given variance of X i.e
160
Sx2 = 9
Sx = ± 3
We consider Sx = 3 as SD is always positive
Since
byx = r Sy /Sx
Substituting the values of byx, r and Sx we obtain,
Sy = 4/5 x 3/0.6
= 4
Example 5-6
The height of a child increases at a rate given in the table below. Fit the straight line
using the method of least-square and calculate the average increase and the standard
error of estimate.
Month:
1
2
3
4
5
6
7
8
9
Height:
52.5
58.7
65
70.2
75.4
81.1
87.2
95.5
10
102.2 108.4
Solution: For Regression calculations, we draw the following table
Month (X)
Height (Y)
X2
XY
1
52.5
1
52.5
2
58.7
4
117.4
3
65.0
9
195.0
4
70.2
16
280.8
5
75.4
25
377.0
6
81.1
36
486.6
7
87.2
49
610.4
8
95.5
64
764.0
9
102.2
81
919.8
10
108.4
100
1084.0
∑X
=55
∑ Y =796.2
∑X
2
= 385
∑ XY = 4887.5
Considering the regression line as Y = a + bX, we can obtain the values of a and b from the
above values.
161
∑ Y ∑ X − ∑ X ∑ XY
N ∑ X − (∑ X )
2
a=
=
2
2
796.2 x 385 - 55 x 4887.5
10 x 385 - 55 x 55
= 45.73
b=
=
N ∑ XY − ∑ X ∑ Y
N ∑ X 2 − (∑ X )
2
10 x 4887.5 - 55 x 796.2
10 x 385 - 55 x 55
= 6.16
Hence the regression line can be written as
Y = 45.73 + 6.16X
For standard error of estimation, we note the calculated values of the variable against the
observed values,
When X = 1, Y1 = 45.73 + 6.16 = 51.89
for X = 2, Y2 = 45.73 + 616 x 2 = 58.05
Other values for X = 3 to X = 10 are calculated and are tabulated as follows:
Month (X)
Height (Y)
Yi
Y-Yi
(Y-Yi) 2
1
52.5
51.89
0.61
0.372
2
58.7
58.05
0.65
0.423
3
65.0
64.21
0.79
0.624
4
70.2
70.37
-0.17
0.029
5
75.4
76.53
-1.13
1.277
6
81.1
82.69
-1.59
2.528
7
87.2
88.85
-1.65
2.723
8
95.5
95.01
0.49
0.240
9
102.2
101.17
1.03
1.061
10
108.4
107.33
1.07
1.145
162
∑ (Y − Y )
i
2
= 10.421
Standard error of estimation
S yx
=
1
N
=
10.421
10
∑ (Y − Y )
2
i
= 1.02
Example 5-7
Given X = 4Y+5 and Y = kX + 4 are the lines of regression of X on Y and of Y on X
respectively. If k is positive, prove that it cannot exceed ¼.
If k = 1/16, find the means of the two variables and coefficient of correlation between them.
Solution: Line X = 4Y + 5 is regression line of X on Y
So
bxy = 4
Similarly from regression line of Y on X , Y = kX + 4,
We get
byx = k
Now
r2 = bxy. byx
= 4k
Since 0 ≤ r 2 ≤ 1, we obtain 0 ≤ 4k ≤ 1,
Or
Now for k =
0≤k ≤
1
,
4
1
,
16
r 2 = 4x
1 1
=
16 4
r=+½
= ½ since byx and byx are positive
163
When k =
1
, the regression line of Y on X becomes
16
Y=
Or
1
X+4
16
X – 16Y + 64 = 0
Since line of regression pass through the mean values of the variables, we obtain revised
equations as
X - 4Y - 5 = 0
X - 16 Y + 64 = 0
Solving these two equations, we get
X = 28
and
Y = 5.75
Example 5-8
A firm knows from its past experience that its monthly average expenses (X) on
advertisement are Rs 25,000 with standard deviation of Rs 25.25. Similarly, its average
monthly product sales (Y) have been Rs 45,000 with standard deviation of Rs 50.50. Given
this information and also the coefficient of correlation between sales and advertisement
expenditure as 0.75, estimate
(i)
the most appropriate value of sales against an advertisement expenditure of Rs
50,000
(ii)
Rs 80,000
Solution: Given the following
X = Rs 25,000
Sx = Rs 25.25
Y = Rs 45,000
Sy = Rs 50.50
r = 0.75
164
(i)
Using equation Yc -Y = r
Sy
Sx
(X- X ), the most appropriate value of sales Yc for an
Yc – 45,000 = 0.75
50.50
(50,000 – 25,000)
25.25
Yc = 45,000 + 37,500
= Rs 82,500
(ii)
Using equation Xc - X = r
Sx
Sy
expenditure Xc for achieving a sales target Y= Rs 80,000 is
Xc – 25,000 = 0.75
25.25
(80,000 – 45,000)
50.50
Xc = 13,125 + 25,000
= Rs 38,125
1.8
1.
SELF-ASSESSMENT QUESTIONS
Explain clearly the concept of Regression. Explain with suitable examples its role in
2.
What do you understand by linear regression?
3.
What is meant by ‘regression’? Why should there be in general, two lines of
regression for each bivariate distribution? How the two regression lines are useful in
studying correlation between two variables?
4.
Why is the regression line known as line of best fit?
5.
Write short note on
(i)
Regression Coefficients
(ii)
Regression Equations
(iii)
Standard Error of Estimate
(iv)
Coefficient of Determination
165
(v)
6.
Coefficient of Non-determination
Distinguish clearly between correlation and regression as concept used in statistical
analysis.
7.
Fit a least-square line to the following data:
(i)
Using X as independent variable
(ii)
Using X as dependent variable
X
:
1
3
4
8
9
11
14
Y
:
1
2
4
5
7
8
9
Hence obtain
c) The regression coefficients of Y on X and of X on Y
d) X and Y
e) Coefficient of correlation between and X and Y
f) What is the estimated value of Y when X = 10 and of X when Y = 5?
8.
What are regression coefficients? Show that r2 = byx. bxy where the symbols have their
usual meanings. What can you say about the angle between the regression lines when
(i) r = 0, (ii) r = 1 (iii) r increases from 0 to 1?
9.
Obtain the equations of the lines of regression of Y on X from the following data.
X
:
12
18
24
30
36
42
48
Y
:
5.27
5.68
6.25
7.21
8.02
8.71
8.42
Estimate the most probable value of Y, when X = 40.
10. The following table gives the ages and blood pressure of 9 women.
Age (X) :
56 42
Blood Pressure(Y) 147 125
36
47
49
42
60
72
63
118
128
145
140
155
160
149
Find the correlation coefficient between X and Y.
(i)
Determine the least square regression equation of Y on X.
166
(ii)
11.
Estimate the blood pressure of a woman whose age is 45 years.
Given the following results for the height (X) and weight (Y) in appropriate units of
1,000 students:
X = 68,
Y = 150,
S x = 2.5,
S y = 20 and r = 0.6.
Obtain the equations of the two lines of regression. Estimate the height of a student A
who weighs 200 units and also estimate the weight of the student B whose height is
60 units.
12.
From the following data, find out the probable yield when the rainfall is 29”.
Rainfall
Yield
Mean
25”
40 units per hectare
Standard Deviation
3”
6 units per hectare
Correlation coefficient between rainfall and production = 0.8.
13.
A study of wheat prices at two cities yielded the following data:
City A
City B
Average Price
Rs 2,463
Rs 2,797
Standard Deviation
Rs 0.326
Rs 0.207
Coefficient of correlation r is 0.774. Estimate from the above data the most likely
price of wheat
14.
(i)
at City A corresponding to the price of Rs 2,334 at City B
(ii)
at city B corresponding to the price of Rs 3.052 at City A
Find out the regression equation showing the regression of capacity utilisation on
production from the following data:
Average
Standard Deviation
Production (in lakh units)
35.6
10.5
Capacity Utilisation (in percentage)
84.8
8.5
r = 0.62
167
Estimate the production, when capacity utilisation is 70%.
15.
The following table shows the mean and standard deviation of the prices of two shares
in a stock exchange.
Share
Mean (in Rs)
Standard Deviation (in Rs)
A Ltd.
39.5
10.8
B Ltd.
47.5
16.0
If the coefficient of correlation between the prices of two shares is 0.42, find the most
likely price of share A corresponding to a price of Rs 55, observed in the case of share
B.
16.
Find out the regression coefficients of Y on X and of X on Y on the basis of following
data:
∑X
= 50,
∑Y
X = 5,
= 60,
∑ XY
Y = 6,
= 350
Variance of X = 4, Variance of Y = 9
17.
Find the regression equation of X and Y and the coefficient of correlation from the
following data:
∑X
= 60,
∑Y
= 40,
∑ XY
= 1150,
∑X
2
= 4160,
∑Y
2
= 1720 and N = 10.
18.
By using the following data, find out the two lines of regression and from them
compute the Karl Pearson’s coefficient of correlation.
∑ X = 250, ∑Y = 300, ∑ XY = 7900, ∑ X 2 = 6500, ∑ Y 2 = 10000, N = 10
19.
The equations of two regression lines between two variables are expressed as
2X – 3Y = 0 and 4Y – 5X-8 = 0.
20.
(i)
Identify which of the two can be called regression line of Y on X and of X on Y.
(ii)
Find X and Y and correlation coefficient r from the equations
If the two lines of regression are
4X - 5Y + 30 = 0 and
20X – 9Y – 107 = 0
Which of these is the lines of regression of X and Y. Find rxy and Sy when Sx = 3
168
21.
The regression equation of profits (X) on sales (Y) of a certain firm is 3Y – 5X +108 =
0. The average sales of the firm were Rs 44,000 and the variance of profits is 9/16th of
the variance of sales. Find the average profits and the coefficient of correlation
between the sales and profits.
5.10
9.
Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
10.
Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
11.
Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
12.
Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
13.
Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
14.
Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
15.
Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
16.
Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
169
Course:
Course Code: MC-106
Lesson:
Author:
Anil Kumar
Vetter:
Prof Harbhajan Bansal
06
INDEX NUMBERS
Objectives : The overall objective of this lesson is to give an understanding of Index
Numbers. After successful completion of the lesson, the students will be
able to understand the concepts, techniques and the problems involved in
constructing and using index numbers.
Structure
6.1
Introduction
6.2
What are Index Numbers?
6.3
Uses of Index Numbers
6.4
Types of Index Numbers
6.5
Simple Index Numbers
6.6
Composite Index Numbers
6.6.1 Simple Aggregative Price/Quantity Index
6.6.2 Index of Average of Price/Quantity Relatives
6.6.3 Weighted Aggregative Price/Quantity Index
6.6.4 Index of Weighted Average of Price/Quantity Relatives
6.1
6.6
Test of Adequacy of Index Numbers
6.7
Special Issues in the Construction of Index Numbers
6.9
Problems of Constructing Index Numbers
6.10
Self-Assessment Question
6.11
INTRODUCTION
In business, managers and other decision makers may be concerned with the way in which
the values of variables change over time like prices paid for raw materials, numbers of
170
employees and customers, annual income and profits, and so on. Index numbers are one way
of describing such changes.
If we turn to any journal devoted to economic and financial matters, we are very likely to
come across an index number of one or the other type. It may be an index number of share
prices or a wholesale price index or a consumer price index or an index of industrial
production. The objective of these index numbers is to measure the changes that have
occurred in prices, cost of living, production, and so forth. For example, if a wholesale price
index number for the year 2000 with base year 1990 was 170; it shows that wholesale prices,
in general, increased by 70 percent in 2000 as compared to those in 1990. Now, if the same
index number moves to 180 in 2001, it shows that there has been 80 percent increase in
wholesale prices in 2001 as compared to those in 1990.
With the help of various index numbers, economists and businessmen are able to describe
and appreciate business and economic situations quantitatively. Index numbers were
originally developed by economists for monitoring and comparing different groups of goods.
It is necessary in business to understand and manipulate the different published index
serieses, and to construct index series of your own. Having constructed your own index, it
can then be compared to a national one such as the RPI, a similar index for your industry as a
whole and also to indexes for your competitors. These comparisons are a very useful tool for
171
Figure 6-1 The Indexes of the Volume of Sales
For example, an accountant of a supermarket chain could construct an index of the company's
own sales and compare it to the index of the volume of sales for the general supermarket
industry. A graph of the two indexes will illustrate the company's performance within the
sector. It is immediately clear from Figure 6-1 that, after initially lagging behind the general
market, the supermarket company caught up and then overtook it. In the later stages, the
company was having better results than the general market but that, as with the whole
Our focus in this lesson will be on the discussion related to the methodology of index number
construction. The scope of the lesson is rather limited and as such, it does not discuss a large
number of index numbers that are presently compiled and published by different departments
of the Government of India.
6.2
WHAT ARE INDEX NUMBERS?
“Index numbers are statistical devices designed to measure the relative changes in the level
of a certain phenomenon in two or more situations”. The phenomenon under consideration
may be any field of quantitative measurements. It may refer to a single variable or a group of
distinct but related variables. In Business and Economics, the phenomenon under
consideration may be:
9 the prices of a particular commodity like steel, gold, leather, etc. or a group of
commodities like consumer goods, cereals, milk and milk products, cosmetics, etc.
9 volume of trade, factory production, industrial or agricultural production, imports or
exports, stocks and shares, sales and profits of a business house and so on.
172
9 the national income of a country, wage structure of workers in various sectors, bank
deposits, foreign exchange reserves, cost of living of persons of a particular
community, class or profession and so on.
The various situations requiring comparison may refer to either
9 the changes occurring over a time, or
9 the difference(s) between two or more places, or
9 the variations between similar categories of objects/subjects, such as persons, groups
of persons, organisations etc. or other characteristics such as income, profession, etc.
The utility of index numbers in facilitating comparison may be seen when, for example we
are interested in studying the general change in the price level of consumer goods, i.e. good
or commodities consumed by the people belonging to a particular section of society, say, low
income group or middle income group or labour class and so on. Obviously these changes are
not directly measurable as the price quotations of the various commodities are available in
different units, e.g., cereals (wheat, rice, pulses, etc) are quoted in Rs per quintal or kg; water
in Rs per gallon; milk, petrol, kerosene, etc. in Rs per liter; cloth in Rs per miter and so on.
Further, the prices of some of the commodities may be increasing while those of others may
be decreasing during the two periods and the rates of increase or decrease may be different
for different commodities. Index number is a statistical device, which enables us to arrive at a
single representative figure that gives the general level of the price of the phenomenon
(commodities) in an extensive group. According to Wheldon:
“Index number is a statistical device for indicating the relative movements of
the data where measurement of actual movements is difficult or incapable of
FY Edgeworth gave the classical definition of index numbers as follows:
173
“Index number shows by its variations the changes in a magnitude which is
not susceptible either of accurate measurement in itself or of direct valuation
in practice.”
On the basis of above discussion, the following characteristics of index numbers are apparent:
1. Index Numbers are specialized averages: An average is a summary figure measuring
the central tendency of the data, representing a group of figures. Index number has all
these functions to perform. L R Connor states, "in its simplest form, it (index number)
represents a special case of an average, generally a weighted average compiled from
a sample of items judged to be representative of the whole". It is a special type of
average – it averages variables having different units of measurement.
2. Index Numbers are expressed in percentages: Index numbers are expressed in terms
of percentages so as to show the extent of change. However, percentage sign (%) is
never used.
3. Index Numbers measure changes not capable of direct measurement: The technique
of index numbers is utilized in measuring changes in magnitude, which are not
capable of direct measurement. Such magnitudes do not exist in themselves.
Examples of such magnitudes are 'price level', 'cost of living', 'business or economic
activity' etc. The statistical methods used in the construction of index numbers are
largely methods for combining a number of phenomena representing a particular
magnitude in such a manner that the changes in that magnitude may be measured in a
meaningful way without introduction of serious bias.
4. Index Numbers are for comparison: The index numbers by their nature are
comparative. They compare changes taking place over time or between places or
between like categories.
174
In brief, index number is a statistical technique used in measuring the composite change in
several similar economic variables over time. It measures only the composite change, because
some of the variables included may be showing an increase, while some others may be
showing a decrease. It synthesizes the changes taking place in different directions and by
varying extents into the one composite change. Thus, an index number is a device to simplify
comparison to show relative movements of the data concerned and to replace what may be
complicated figures by simple ones calculated on a percentage basis.
6.3
USES OF INDEX NUMBER
The first index number was constructed by an Italian, Mr G R Carli, in 1764 to compare the
changes in price for the year 1750 (current year) with the price level in 1500 (base year) in
order to study the effect of discovery of America on the price level in Italy. Though originally
designed to study the general level of prices or accordingly purchasing power of money,
today index numbers are extensively used for a variety of purposes in economics, business,
management, etc., and for quantitative data relating to production, consumption, profits,
personnel and financial matters etc., for comparing changes in the level of phenomenon for
two periods, places, etc. In fact there is hardly any field or quantitative measurements where
index numbers are not constructed. They are used in almost all sciences – natural, social and
physical. The main uses of index numbers can be summarized as follows:
1. Index Numbers as Economic Barometers
Index numbers are indispensable tools for the management personnel of any
formulation of executive decisions. The indices of prices (wholesale & retail), output
(volume of trade, import and export, industrial and agricultural production) and bank
deposits, foreign exchange and reserves etc., throw light on the nature of, and
175
variation in the general economic and business activity of the country. They are the
indicators of business environment. A careful study of these indices gives us a fairly
good appraisal of the general trade, economic development and business activity of
the country. In the world of G Simpson and F Kafka:
“Index numbers are today one of the most widely used statistical devices. They are used to
take the pulse of the economy and they have come to be used as indicators of inflationary or
deflationary tendencies.”
Like barometers, which are used in Physics and Chemistry to measure atmospheric
pressure, index numbers are rightly termed as “economic barometers”, which measure
the pressure of economic and business behaviour.
2. Index Numbers Help in Studying Trends and Tendencies
Since the index numbers study the relative change in the level of a phenomenon at
different periods of time, they are especially useful for the study of the general trend
for a group phenomenon in time series data. The indices of output (industrial and
agricultural production), volume of trade, import and export, etc., are extremely useful
for studying the changes in the level of phenomenon due to the various components of
a time series, viz. secular trend, seasonal and cyclical variations and irregular
components and reflect upon the general trend of production and business activity. As
a measure of average change in extensive group, the index numbers can be used to
forecast future events. For instance, if a businessman is interested in establishing a
new undertaking, the study of the trend of changes in the prices, wages and incomes
in different industries is extremely helpful to him to frame a general idea of the
comparative courses, which the future holds for different undertakings.
3. Index Numbers Help in Formulating Decisions and Policies
176
Index numbers of the data relating to various business and economic variables serve
an important guide to the formulation of appropriate policy. For example, the cost of
living index numbers are used by the government and, the industrial and business
concerns for the regulation of dearness allowance (D.A.) or grant of bonus to the
workers so as to enable them to meet the increased cost of living from time to time.
The excise duty on the production or sales of a commodity is regulated according to
the index numbers of the consumption of the commodity from time to time. Similarly,
the indices of consumption of various commodities help in the planning of their future
production. Although index numbers are now widely used to study the general
economic and business conditions of the society, they are also applied with advantage
by sociologists (population indices), psychologists (IQs’), health and educational
authorities etc., for formulating and revising their policies from time to time.
4. Price Indices Measure the Purchasing Power of Money
A traditional use of index numbers is in measuring the purchasing power of money.
Since the changes in prices and purchasing power of money are inversely related, an
increase in the general price index indicates that the purchasing power of money has
gone down.
In general, the purchasing power of money may be computed as
1
x100
General Price Index
Accordingly, if the consumer price index for a given year is 150, the purchasing
power of a rupee is (1/150) 100 = 0.67. That is, the purchasing power of a rupee in the
given year is 67 paise as compared to the base year.
177
With the increase in prices, the amount of goods and services which money wages can
buy (or the real wages) goes on decreasing. Index numbers tell us the change in real
wages, which are obtained as
Real Wage =
Money Wage
x100
Consumer Price Index
A real wage index equal to, say, 120 corresponding to money wage index of 160 will
indicate an increase in real wages by only 20 per cent as against 60 per cent increase
in money wages.
Index numbers also serve as the basis of determining the terms of exchange. The
terms of exchange are the parity rate at which one set of commodities is exchanged
for another set of commodities. It is determined by taking the ratio of the price index
for the two groups of commodities and expressing it in percentage.
For example, if A and B are the two groups of commodities with 120 and 150 as their
price index in a particular year, respectively, the ratio 120/150 multiplied by 100 is 80
per cent. It means that prices of A group of commodities in terms of those in group B
are lower by 20 per cent.
5. Index Numbers are Used for Deflation
Consumer price indices or cost of living index numbers are used for deflation of net
national product, income value series in national accounts. The technique of obtaining
real wages from the given nominal wages (as explained in use 4 above) can be used to
find real income from inflated money income, real sales from nominal sales and so on
by taking into account appropriate index numbers.
178
5.4
TYPES OF INDEX NUMBERS
Index numbers may be broadly classified into various categories depending upon the type of
the phenomenon they study. Although index numbers can be constructed for measuring
relative changes in any field of quantitative measurement, we shall primarily confine the
discussion to the data relating to economics and business i.e., data relating to prices,
production (output) and consumption. In this context index numbers may be broadly
classified into the following three categories:
1. Price Index Numbers: The price index numbers measure the general changes in the
prices. They are further sub-divided into the following classes:
(i)
Wholesale Price Index Numbers: The wholesale price index numbers reflect
the changes in the general price level of a country.
(ii)
Retail Price Index Numbers: These indices reflect the general changes in the
retail prices of various commodities such as consumption goods, stocks and shares,
bank deposits, government bonds, etc.
(iii) Consumer Price Index: Commonly known as the Cost of living Index, CPI is
a specialized kind of retail price index and enables us to study the effect of changes
in the price of a basket of goods or commodities on the purchasing power or cost of
living of a particular class or section of the people like labour class, industrial or
agricultural worker, low income or middle income class etc.
2. Quantity Index Numbers: Quantity index numbers study the changes in the volume
of goods produced (manufactured), consumed or distributed, like: the indices of
agricultural production, industrial production, imports and exports, etc. They are
extremely helpful in studying the level of physical output in an economy.
179
3. Value Index Numbers: These are intended to study the change in the total value
(price multiplied by quantity) of output such as indices of retail sales or profits or
inventories. However, these indices are not as common as price and quantity indices.
Notations Used
Since index numbers are computed for prices, quantities, and values, these are denoted by the lower case
letters:
p, q, and v represent respectively the price, the quantity, and the value of an individual
commodity.
Subscripts 0, 1, 2,… i, ... are attached to these lower case letters to distinguish price, quantity, or value in
any one period from those in the other. Thus,
p0 denotes the price of a commodity in the base period,
p1 denotes the price of a commodity in period 1, or the current period, and
pi denotes the price of a commodity in the ith period, where i = 1,2,3, ...
Similar meanings are assigned to q0, q1, ... qi, ... and v0, v1, … vi, …
Capital letters P, Q and V are used to represent the price, quantity, and value index numbers, respectively.
Subscripts attached to P, Q, and V indicates the years compared. Thus,
POI means the price index for period 1 relative to period 0,
P02 means the price index for period 2 relative to period 0,
PI2 means the price index for period 2 relative to period 1, and so on.
Similar meanings are assigned to quantity Q and value V indices. It may be noted that all indices are
expressed in percent with 100 as the index for the base period, the period with which comparison is to be
Various indices can also be distinguished on the basis of the number of commodities that go
into the construction of an index. Indices constructed for individual commodities or variable
are termed as simple index numbers. Those constructed for a group of commodities or
variables are known as aggregative (or composite) index numbers.
180
Here, in this lesson, we will develop methods of constructing simple as well as composite
indices.
6.5
SIMPLE INDEX NUMBERS
A simple price index number is based on the price or quantity of a single commodity. To
construct a simple index, we first have to decide on the base period and then find ratio of the
value at any subsequent period to the value in that base period - the price/quantity relative.
This ratio is then finally converted to a percentage
Value in Period i
x100
Value in Base Year
i.e. Simple Price Index for period i = 1,2,3 ... will be
Index for any Period i =
pi
x100
p0
P0i =
…………(6-1)
Similarly, Simple Quantity Index for period i = 1,2,3 ... will be
Q0 i =
qi
x100
q0
…………(6-2)
Example 6-1
Given are the following price-quantity data of fish, with price quoted in Rs per kg and production in
qtls.
Year
:
1980 1981 1982 1983 1984 1985
Price
:
15
17
16
18
22
20
Production
:
500
550
480
610
650
600
Construct:
(a)
the price index for each year taking price of 1980 as base,
(b)
the quantity index for each year taking quantity of 1980 as base.
Solution:
Simple Price and Quantity Indices of Fish
(Base Year = 1980)
Year
Price
(pi)
Quantity Price Index
p
(qi)
P0i = i x100
p0
181
Quantity Index
q
Q0i = i x100
q0
1980
15
500
100.00
100.00
1981
17
550
113.33
110.00
1982
16
480
106.66
96.00
1983
18
610
120.00
122.00
1984
22
650
146.66
130.00
1985
20
600
133.33
120.00
These simple indices facilitate comparison by transforming absolute quantities/prices into
percentages. Given such an index, it is easy to find the percent by which the price/quantity
may have changed in a given period as compared to the base period. For example, observing
the index computed in Example 6-1, one can firmly say that the output of fish was 30 per cent
more in 1984 as compared to 1980.
It may also be noted that given the simple price/quantity for the base year and the index for
the period i = 1, 2, 3, …; the actual price/quantity for the period i = 1, 2, 3, … may easily be
obtained as:
and
⎛P ⎞
pi = p 0 ⎜ 0i ⎟
⎝ 100 ⎠
…………(6-3)
⎛Q ⎞
qi = q 0 ⎜ 0 i ⎟
⎝ 100 ⎠
…………(6-4)
For example, with i = 1983, Q0i = 122.00, and q0 = 500,
⎛ 122.00 ⎞
qi = 500⎜
⎟
⎝ 100 ⎠
= 610
6.6
COMPOSITE INDEX NUMBERS
The preceding discussion was confined to only one commodity. What about
price/quantity changes in several commodities? In such cases, composite index
182
numbers are used. Depending upon the method used for constructing an index,
composite indices may be:
1.
Simple Aggregative Price/ Quantity Index
2.
Index of Average of Price/Quantity Relatives
3.
Weighted Aggregative Price/ Quantity Index
4.
Index of Weighted Average of Price/Quantity Relatives
6.6.1 SIMPLE AGGREGATIVE PRICE/ QUANTITY INDEX
Irrespective of the units in which prices/quantities are quoted, this index for given
prices/quantities, of a group of commodities is constructed in the following three steps:
(i)
Find the aggregate of prices/quantities of all commodities for each period (or
place).
(ii)
Selecting one period as the base, divide the aggregate prices/quantities
corresponding to each period (or place) by the aggregate of prices/ quantities
in the base period.
(iii)
Express the result in percent by multiplying by 100.
The computation procedure contained in the above steps can be expressed as:
P0i =
∑p
∑p
i
x100
…………(6-5)
x100
…………(6-6)
0
Q0 i =
and
∑q
∑q
i
0
Example 6-2
Given are the following price-quantity data, with price quoted in Rs per kg and
production in qtls.
1980
1985
Item
Price
Production
Price
Production
Fish
15
500
20
600
Mutton
18
590
23
640
183
Chicken
Find
22
450
24
(a)
Simple Aggregative Price Index with 1980 as the base.
(b)
Simple Aggregative Quantity Index with 1980 as the base.
500
Solution:
Calculations for
Simple Aggregative Price and Quantity Indices
(Base Year = 1980)
Prices
Item
1980(p0)
1985(pi)
1980(q0)
1985(qi)
15
20
500
600
Mutton
18
23
590
640
Chicken
22
24
450
500
→
55
67
1540
1740
Fish
Sum
(a)
Quantities
Simple Aggregative Price Index with 1980 as the base
P0i =
∑p
∑p
i
x100
0
P0i =
67
x100
55
P0i = 121.82
(b)
Simple Aggregative Quantity Index with 1980 as the base
Q0 i =
∑q
∑q
i
x100
0
Q0 i =
1740
x100
1540
Q0i = 112.98
Although Simple Aggregative Index is simple to calculate, it has two important limitations:
First, equal weights get assigned to every item entering into the construction of this index
irrespective of relative importance of each individual item being different. For example,
184
items like pencil and milk are assigned equal importance in the construction of this index.
This limitation renders the index of no practical utility.
Second, different units in which the prices are quoted also sometimes unduly affect this
index. Prices quoted in higher weights, such as price of wheat per bushel as compared to a
price per kg, will have unduly large influence on this index. Consequently, the prices of only
a few commodities may dominate the index. This problem no longer exists when the units in
which the prices of various commodities are quoted have a common base.
Even the condition of common base will provide no real solution because commodities with
relatively high prices such as gold, which is not as important as milk, will continue to
dominate this index excessively. For example, in the Example 6-2 given above chicken prices
are relatively higher than those of fish, and hence chicken prices tend to influence this index
relatively more than the prices of fish.
6.6.2 INDEX OF AVERAGE OF PRICE/QUANTITY RELATIVES
This index makes an improvement over the index of simple aggregative prices/quantities as it
is not affected by the difference in the units of measurement in which prices/quantities are
expressed. However, this also suffers from the problem of equal importance getting assigned
to all the commodities.
Given the prices/quantities of a number of commodities that enter into the construction of this
index, it is computed in the following two steps:
(i)
After selecting the base year, find the price relative/quantity relative of each
commodity for each year with respect to the base year price/quantity. As
defined earlier, the price relative/quantity relative of a commodity for a given
period is the ratio of the price/quantity of that commodity in the given period
to its price/quantity in the base period.
185
(ii)
Multiply the result for each commodity by 100, to get simple price/quantity
indices for each commodity.
(iii)
Take the average of the simple price/quantity indices by using arithmetic
mean, geometric mean or median.
Thus it is computed as:
⎞
⎛p
P0i = Average of ⎜⎜ i x100 ⎟⎟
⎠
⎝ p0
⎞
⎛q
Q0i = Average of ⎜⎜ i x100 ⎟⎟
⎠
⎝ q0
and
Using arithmetic mean
⎛ pi
P0i =
∑ ⎜⎜ p
⎝
0
⎞
x100 ⎟⎟
⎠
…………(6-7)
⎞
x100 ⎟⎟
⎠
…………(6-8)
N
⎛ qi
Q0 i =
and
∑ ⎜⎜ q
⎝
0
N
Using geometric mean
and
⎡1
P0i = Anti log ⎢
⎣N
∑ log⎜⎜ p
⎡1
Q0i = Anti log ⎢
⎣N
∑ log⎜⎜ q
⎛ pi
⎝
0
⎛ qi
⎝
0
⎞⎤
x100 ⎟⎟⎥
⎠⎦
…………(6-9)
⎞⎤
x100 ⎟⎟⎥
⎠⎦
…………(6-10)
Example 6-3
From the data in Example 6.2 find:
(a) Index of Average of Price Relatives (base year 1980); using mean, median and
geometric mean.
(b) Index of Average of Quantity Relatives (base year 1980); using mean, median and
geometric mean.
Solution:
186
Calculations for
Index of Average of Price Relatives and Quantity Relatives
(Base Year = 1980)
Price Relative
⎞
⎛p
= ⎜⎜ i x100 ⎟⎟
⎠
⎝ p0
⎞
⎛p
log⎜⎜ i x100 ⎟⎟
⎠
⎝ p0
Fish
133.33
2.1248
120.00
2.0792
Mutton
127.77
2.1063
108.47
2.0354
Chicken
109.09
2.0378
111.11
2.0457
370.19
6.2689
339.58
6.1603
Item
Sum
(a)
→
Quantity Relative
⎞
⎛q
= ⎜⎜ i x100 ⎟⎟
⎠
⎝ q0
Index of Average of Price Relatives (base year 1980)
⎛ pi
Using arithmetic mean
P0i =
=
∑ ⎜⎜ p
⎝
0
⎞
x100 ⎟⎟
⎠
N
370.19
3
= 123.39
Using Median
⎛ N +1⎞
P0i = Size of ⎜
⎟th item
⎝ 2 ⎠
⎛ 3 +1⎞
= Size of ⎜
⎟th item
⎝ 2 ⎠
= Size of 2nd item
= 127.77
Using geometric mean
⎡1
⎛p
⎞⎤
P0i = Anti log ⎢ ∑ log⎜⎜ i x100 ⎟⎟⎥
⎝ p0
⎠⎦
⎣N
⎡1
⎤
= Anti log ⎢ (6.2689 )⎥
⎣3
⎦
= Anti log[2.08963]
= 122.92
(b)
Index of Average of Quantity Relatives (base year 1980)
187
⎞
⎛q
log⎜⎜ i x100 ⎟⎟
⎠
⎝ q0
⎛ qi
Q0 i =
Using arithmetic mean
=
∑ ⎜⎜ q
⎝
0
⎞
x100 ⎟⎟
⎠
N
339.58
3
= 113.19
⎛ N + 1⎞
Q0i = Size of ⎜
⎟th item
⎝ 2 ⎠
Using Median
⎛ 3 +1⎞
= Size of ⎜
⎟th item
⎝ 2 ⎠
= Size of 2nd item
= 111.11
Using geometric mean
⎡1
⎛q
⎞⎤
Q0i = Anti log ⎢ ∑ log⎜⎜ i x100 ⎟⎟⎥
⎝ q0
⎠⎦
⎣N
⎡1
⎤
= Anti log ⎢ (6.1603)⎥
⎣3
⎦
= Anti log[2.05343]
= 113.09
Apart from the inherent drawback that this index accords equal importance to all
items entering into its construction, a simple arithmetic mean and median are not
appropriate average to be applied to ratios. Because it is generally believed that a
simple average injects an upward bias in the index. So geometric mean is considered a
more appropriate average for ratios and percentages.
6.6.3 WEIGHTED AGGREGATIVE PRICE/QUANTITY INDICES
We have noted that the simple aggregative price/quantity indices do not take care of the
differences in the weights to be assigned to different commodities that enter into their
construction. It is primarily because of this limitation that the simple aggregative indices are
of very limited use. Weighted aggregative Indices make up this deficiency by assigning
proper weights to individual items.
188
Among several ways of assigning weights, two widely used ways are:
(i)
to use base period quantities/prices as weights, popularly known as Laspeyre's
Index, and
(ii)
to use the given (current) period quantities/prices as weights, popularly known
as Paasche's Index.
6.6.3.1 Laspeyre’s Index
Laspeyre’s Price Index, using base period quantities as weights is obtained as
P0La
i =
∑pq
∑p q
i
0
0
0
…………(6-11)
x100
Laspeyre’s Quantity Index, using base period prices as weights is obtained as
Q0Lai =
∑q p
∑q p
i
0
0
0
…………(6-12)
x100
6.6.3.2 Paasche’s Index
Paasche’s Price Index, using base period quantities as weights is obtained as
P0Pa
i =
∑pq
∑p q
i
i
0
i
…………(6-13)
x100
Paasche’s Quantity Index, using base period prices as weights is obtained as
Q0Pai =
∑q p
∑q p
i
i
0
i
…………(6-14)
x100
Example 6-4
From the data in Example 6.2 find:
(a) Laspeyre’s Price Index for 1985, using 1980 as the base
(b) Laspeyre’s Quantity Index for 1985, using 1980 as the base
(c) Paasche’s Price Index for 1985, using 1980 as the base
189
(d) Paasche’s Quantity Index for 1985, using 1980 as the base
Solution:
Calculations for
Laspeyre’s and Paasche’s Indices
(Base Year = 1980)
Item
p0 q0
p1 q0
p0 q1
p1 q1
Fish
7500
10000
9000
12000
Mutton
10620
13570
11520
14720
Chicken
9900
10800
11000
12000
28020
34370
31520
38720
Sum
(a)
→
Laspeyre’s Price Index for 1985, using 1980 as the base
P0La
i =
=
∑pq
∑p q
i
0
0
0
x100
34370
x100
28020
= 122.66
(b)
Laspeyre’s Quantity Index for 1985, using 1980 as the base
Q0Lai =
=
∑q p
∑q p
i
0
0
0
x100
31520
x100
28020
= 112.49
(c)
Paasche’s Price Index for 1985, using 1980 as the base
P0Pa
i =
=
∑pq
∑p q
i
i
0
i
x100
38720
x100
31520
= 122.84
(d)
Paasche’s Quantity Index for 1985, using 1980 as the base
190
Q0Pai =
=
∑q p
∑q p
i
i
0
i
x100
38720
x100
34370
= 112.66
Interpretations of Laspeyre's Index
On close examination it will be clear that the Laspeyre's Price Index offers the
following precise interpretations:
1. It compares the cost of collection of a fixed basket of goods selected in the base
period with the cost of collecting the same basket of goods in the given (current)
period.
Accordingly, the cost of collection of 500 qtls of fish, 590 qtls of mutton and 450
qtls of chicken has increased by 22.66 per cent in 1985 as compared to what it was
in 1980. Viewed differently, it indicates that a fixed amount of goods sold at 1985
prices yield 22.66 per cent more revenue than what it did at 1980 prices.
2. It also implies that a fixed amount of goods when purchased at 1985 prices would
cost 22.66 per cent more than what it did at 1980 prices. In this interpretation, the
Laspeyre's Price Index serves as the basis of constructing the cost of living index,
for it tells how much more does it cost to maintain the base period standard of living
at the current period prices.
Laspeyre's Quantity Index, too, has precise interpretations. It reveals the percentage change in
total expenditure in the given (current) period as compared to the base period if varying
amounts of the same basket of goods are sold at the base period prices. When viewed in this
manner, we will be required to spend 12.49 per cent more in 1985 as compared to 1980 if the
quantities of fish, mutton and chicken for 1965 are sold at the base period (1980) prices.
191
Interpretations of Paasche's Index
A careful examination of the Paasche's Price Index will show that this too is amenable to the
following precise interpretations:
1. It compares the cost of collection of a fixed basket of goods selected in the given
period with the cost of collection of the same basket of goods in the base period.
Accordingly, the cost of collection of a fixed basket of goods containing 600 qtls of
fish, 640 qtls of mutton and 500 qtls of chicken in 1985 is about 22.84 per cent more
than the cost of collecting the same basket of goods in 1980. Viewed a little
differently, it indicates that a fixed basket of goods sold at 1985 prices yields 22.84
per cent more revenue than what it would have earned had it been sold at the base
period (1980) prices.
2. It also tells that a fixed amount of goods purchased at 1985 prices will cost 22.84
per cent more than what it would have cost if this fixed amount of goods had been
sold at base period (1980) prices.
Analogously, Paasche's Quantity Index, too, has its own precise meaning. It tells the per cent
change in total expenditure in the given period as compared to the base period if varying
amounts of the same basket of goods are to be sold at given period prices. When so viewed,
we will be required to spend 12.66 per cent more in 1985 as compared to 1980 if the
quantities of fish, mutton and chicken for 1980 are sold at the given period (1985) prices.
Relationship Between Laspeyre’s and Paasche’s Indices
In order to understand the relationship between Laspeyre’s and Paasche’s Indices, the
assumptions on which the two indices are based be borne in mind:
Laspeyre's index is based on the assumption that unless there is a change in tastes and
preferences, people continue to buy a fixed basket of goods irrespective of how high or low
192
the prices are likely to be in the future. Paasche's index, on the other hand, assumes that
people would have bought the same amount of a given basket of goods in the past irrespective
of how high or low were the past prices.
However, the basic contention implied in the assumptions on which the two indices are based
is not true. For, people do make shifts in their purchase pattern and preferences by buying
more of goods that tend to become cheaper and less of those that tend to become costlier. In
view of this, the following two situations that are likely to emerge need consideration:
1. When the prices of goods that enter into the construction of these indices show a
general tendency to rise, those whose prices increase more than the average increase
in prices will have smaller quantities in the given period than the corresponding
quantities in the base period. That is, qi’s will be smaller than q0’s when prices in
general are rising. Consequently, Paasche's index will have relatively smaller weights
than those in the Laspeyre's index and, therefore, the former ( P0Pa
i ) will be smaller
than the latter ( P0La
i ). In other words, Paasche's index will show a relatively smaller
increase when the prices in general tend to rise.
2. On the contrary, when prices in general are falling, goods whose prices show a
relatively smaller fall than the average fall in prices, will have smaller quantities in
the given period than the corresponding quantities in the base period. This means that
qi’s will be smaller than q0’s when prices in general are falling. Consequently,
Paasche's index will have smaller weights than those in the Laspeyre's index and,
La
therefore, the former ( P0Pa
i ) will be smaller than the latter ( P0 i ). In other words,
Paasche's index will show a relatively greater fall when the prices in general tend to
fall.
An important inference based on the above discussion is that the Paasche's index has a
downward bias and the Laspeyre's index an upward bias. This directly follows from the fact
193
that the Paasche's index, relative to the Laspeyre's index, shows a smaller rise when the prices
in general are rising, and a greater fall when the prices in general are falling.
It may, however, be noted that when the quantity demanded increases because of change in
real income, tastes and preferences, advertising, etc., the prices remaining unchanged, the
Paasche's index will show a higher value than the Laspeyre's index. In such situations, the
Paasche's index will overstate, and the Laspeyre's will understate, the changes in prices. The
former now represents the upper limit, and the latter the lower limit, of the range of price
changes.
The relationship between the two indices can be derived more precisely by making use of the
coefficient of linear correlation computed as:
∑ fXY − ⎛⎜ ∑ fX ⎞⎟⎛⎜ ∑ fY ⎞⎟
rxy =
N
⎜
⎝
⎟⎜
⎠⎝
N
N
⎟
⎠
…………(6.15)
SxSy
in which X and Y denote the relative price movements(
movements(
pi
) and relative quantity
p0
qi
) respectively. Sx and Sy are the standard deviations of price and quantity
q0
movements, respectively. While rxy represents the coefficient of correlation between the
relative price and quantity movements; f represents the weights assigned, that is, p0 q0. N is
the sum of frequencies i. e. N = ∑ p 0 q 0 .
Substituting the values of X, Y, f and N in Eq. (6-15), and then rearranging the expression, we
have
rxy S x S y =
∑pq
∑p q
i
i
0
0
−
∑pq x∑p q
∑p q ∑p q
i
0
0
i
0
0
0
0
194
If
∑pq
∑p q
i
i
0
0
= V0i , is the index of value expanded between the base period and the ith period,
then dividing both sides by
rxy S x S y
V0 i
rxy S x S y
V0 i
∑pq
∑p q
= 1−
i
i
0
0
or V0i , we get
∑pq x∑p q
∑p q ∑pq
i
0
0
i
0
0
i
i
= 1 − P0La
i x
1
P0Pa
i
rxy S x S y
P0La
i
= 1−
Pa
V0 i
P0i
…………(6.16)
The relationship in Eq. (6.16) offers the following useful results:
Pa
when either rxy , Sx and Sy is equal to zero. That is, the two indices will
1. P0La
i = P0 i
give the same result either when there is no correlation between the price and
quantity movements, or when the price or quantity movements are in the same ratio
so that Sx or Sy is equal to zero.
2. Since in actual practice rxy will have a negative value between 0 and -1, and as
neither Sx = 0 nor Sy = 0, the right hand side of Eq. (6-16) will be less than 1. This
Pa
means that P0La
i is normally greater than P0 i .
3. Given the overall movement in the index of value ( V0i ) expanded, the greater the
coefficient of correlation (rxy) between price and quantity movements and/or the
greater the degree of dispersion (Sx and Sy) in the price and quantity movements, the
Pa
greater the discrepancy between P0La
i and P0 i .
4. The longer the time interval between the two periods to be compared, the more the
chances for price and quantity movements leading to higher values of Sx and Sy. The
assumption of tastes, habits, and preferences remaining unchanged breaking down
195
over a longer period, people do find enough time to make shifts in their
consumption pattern, buying more of goods that may have become relatively
cheaper and less of those that may have become relatively dearer. All this will end
up with a higher degree of correlation between the price and quantity movement.
Pa
Consequently, P0La
i will diverge from P0 i more in the long run than in the short run.
So long as the periods to be compared are not much apart, P0La
i will be quite close to
P0Pa
i .
Laspeyre’s and Paasche’s Indices Further Considered
The use of different system of weights in these two indices may give an impression as if they
are opposite to each other. Such an impression is not sound because both serve the same
purpose, although they may give different results when applied to the same data.
This raises an important question. Which one of them gives more accurate results and which
one should be preferred over the other? The answer to this question is rather difficult since
both the indices are amenable to precise and useful results.
Despite a very useful and precise difference in interpretation, in actual practice the Laspeyre's
index is used more frequently than the Paasche's index for the simple reason that the latter
requires frequent revision to take into account the yearly changes in weights. No such
revision is required in the case of the Laspeyre's index where once the weights have been
determined, these do not require any change in any subsequent period. It is on this count that
the Laspeyre's index is preferred over the Paasche's index.
However, this does not render the Paasche's index altogether useless. In fact, it supplements
the practical utility of the Laspeyre's index. The fact that the Laspeyre's index has an upward
bias and the Paasche's index downward bias, the two provide the range between which the
index can vary between the base period and the given period. Interestingly, thus, the former
represents the upper limit, and the latter the lower limit.
196
6.6.3.3 Improvements over the Laspeyre’s and Paasche’s Indices
To overcome the difficulty of overstatement of changes in prices by the Laspeyre's index and
understatement by the Paasche's index, different indices have been developed to compromise
and improve upon them. These are particularly useful when the given period and the base
period fall quite apart and result in a greater divergence between Laspeyre's and Paasche's
indices.
Other important Weighted Aggregative Indices are:
1. Marshall-Edgeworth Index
The Marshall-Edgeworth Index uses the average of the base period and given period
quantities/prices as the weights, and is expressed as
P0ME
i
⎛ q0 + qi ⎞
⎟
2 ⎠
=
x100
⎛ q 0 + qi ⎞
∑ p0 ⎜⎝ 2 ⎟⎠
…………(6-17)
Q0ME
i
⎛ p 0 + pi ⎞
⎟
2 ⎠
=
x100
⎛ p 0 + pi ⎞
∑ q0 ⎜⎝ 2 ⎟⎠
…………(6-18)
∑ p ⎜⎝
i
∑ q ⎜⎝
i
2. Dorbish and Bowley Index
The Dorbish and Bowley Index is defined as the arithmetic mean of the Laspeyre’s
and Paasche’s indices.
P0DB
=
i
Pa
P0La
i + P0 i
2
…………(6-19)
Q0DB
i =
Q0Lai + Q0Pai
2
…………(6-20)
3. Fisher’s Ideal Index
197
The Fisher’s Ideal Index is defined as the geometric mean of the Laspeyre’s and
Paasche’s indices.
Pa
P0Fi = P0La
i .P0 i
…………(6-21)
Q0Fi = Q0Lai .Q0Pai
…………(6-22)
6.6.4 INDEX OF WEIGHTED AVERAGE OF PRICE/QUANTITY RELATIVES
An alternative system of assigning weights lies in using value weights. The value weight v for
any single commodity is the product of its price and quantity, that is, v = pq.
If the index of weighted average of price relatives is defined as
⎡ ⎛ pi
P0i =
⎞⎤
x100 ⎟⎟⎥
⎠⎦
⎣ ⎝ 0
∑v
∑ ⎢v⎜⎜ p
…………(6-23)
then v can be obtained either as
(i)
the product of the base period prices and the base period quantities denoted as
v0 that is, v0 = p0 q0 , or
(ii)
the product of the base period prices and the given period quantities denoted
as vi that is, vi = p0 qi
When v is v0 = p0 q0 , the index of weighted average of price relatives, is expressed as
⎡
⎛ pi
⎞⎤
⎜⎜
x100 ⎟⎟⎥
⎝ p0
⎠⎦
⎣
∑ p0 q0
∑⎢p q
0
0
P0i =
0
…………(6-24)
It may be seen that 0 P0i is the same as the Laspeyre’s aggregative price index.
Similarly, When v is vi = p0 qi , the index of weighted average of price relatives, is expressed
as
198
⎡
i
P0i =
⎣
⎞⎤
x100 ⎟⎟⎥
⎠⎦
⎛ pi
∑ ⎢ p q ⎜⎜ p
0
i
⎝
0
∑p q
0
…………(6-25)
i
It may be seen that i P0i is the same as the Paasche’s aggregative price index.
If the index of weighted average of quantity relatives is defined as
⎡ ⎛ qi
Q0 i =
⎞⎤
x100 ⎟⎟⎥
⎠⎦
⎣ ⎝ 0
∑v
∑ ⎢v⎜⎜ q
…………(6-26)
then v can be obtained either as
(i)
the product of the base period quantities and the base period prices denoted as
v0 that is, v0 = q0 p0 , or
(ii)
the product of the base period quantities and the given period prices denoted
as vi that is, vi = q0 pi
When v is v0 = q0 p0 , the index of weighted average of quantity relatives, is expressed as
⎡
⎛ qi
⎞⎤
⎜
⎟⎟⎥
q
p
x
100
⎢
∑ 0 0⎜ q
⎝ 0
⎠⎦
⎣
0 Q0 i =
∑ q0 p0
…………(6-27)
It may be seen that 0 Q0i is the same as the Laspeyre’s aggregative quantity index.
Similarly, When v is vi = vi = q0 pi , the index of weighted average of quantity relatives, is
expressed as
⎡
⎛q
⎞⎤
pi ⎜⎜ i x100 ⎟⎟⎥
⎝ q0
⎠⎦
⎣
∑ q 0 pi
∑ ⎢q
i
Q0 i =
0
…………(6-28)
It may be seen that i Q0i is the same as the Paasche’s aggregative quantity index.
Example 6-5
From the data in Example 6.2 find the:
199
(a)
(b)
Index of Weighted Average of Price Relatives, using
(i)
v0 = p0 q0 as the value weights
(ii)
vi = p0 qi as the value weights
Index of Weighted Average of Quantity Relatives, using
(i)
v0 = q0 p0 as the value weights
(ii)
vi = q0 pi as the value weights
Solution:
Calculations for
Index of Weighted Average of Price Relatives
(Base Year = 1980)
⎞
⎛p
p 0 q1 ⎜⎜ 1 x100 ⎟⎟
⎠
⎝ p0
Item
v0 = p0 q0
Fish
7500
9000
1000000
1200000
Mutton
10620
11520
1357000
1472000
Chicken
9900
11000
1080000
1200000
28020
31520
3437000
3872000
Sum
(a)
⎞
⎛p
p 0 q 0 ⎜⎜ i x100 ⎟⎟
⎠
⎝ p0
→
v1 = p0 q1
Index of Weighted Average of Price Relatives, using
(i)
v0 = p0 q0 as the value weights
⎡
⎛ pi
⎞⎤
⎜⎜
x100 ⎟⎟⎥
⎝ p0
⎠⎦
⎣
∑ p0 q0
∑⎢p q
0
0
P0i =
=
0
3437000
28020
= 122.66
(ii)
vi = p0 qi as the value weights
⎡
i
P0i =
⎛ pi
∑ ⎢ p q ⎜⎜ p
⎣
0
i
⎝
0
∑p q
0
200
⎞⎤
x100 ⎟⎟⎥
⎠⎦
i
=
3872000
31520
= 122.84
Calculations for
Index of Weighted Average of Quantity Relatives
(Base Year = 1980)
⎞
⎛q
q 0 p1 ⎜⎜ 1 x100 ⎟⎟
⎠
⎝ q0
Item
v0 = q0 p0
v1 = q0 p1
Fish
7500
10000
900000
1200000
Mutton
10620
13570
1152000
1472000
Chicken
9900
10800
1100000
1200000
28020
34370
3152000
3872000
Sum
(b)
⎞
⎛q
q 0 p 0 ⎜⎜ 1 x100 ⎟⎟
⎠
⎝ q0
→
Index of Weighted Average of Quantity Relatives, using
(i)
v0 = q0 p0 as the value weights
⎡
⎛q
⎞⎤
p 0 ⎜⎜ i x100 ⎟⎟⎥
⎝ q0
⎠⎦
⎣
∑ q0 p0
∑ ⎢q
0
Q0i =
=
0
3152000
28020
= 112.49
(ii)
vi = q0 pi as the value weights
⎡
⎛q
⎞⎤
pi ⎜⎜ i x100 ⎟⎟⎥
⎝ q0
⎠⎦
⎣
∑ q 0 pi
∑ ⎢q
i
Q0 i =
=
0
3872000
34370
= 112.66
Although the indices of weighted average of price/quantity relatives yield the same results as
the Laspeyre's or Paasche's price/quantity indices, we do construct these indices also in
201
situations when it is necessary and advantageous to do so. Some such situations are as
follows:
(i)
When a group of commodities is to be represented by a single commodity in the
group, the price relative of the latter is weighted by the group as a whole.
(ii)
Where the price/quantity relatives of individual commodities have been computed,
these can be more conveniently utilised in constructing the index.
(iii) Price/quantity relatives serve a useful purpose in splicing two index series having
different base periods.
(iv) Depersonalizing a time series requires construction of a seasonal index, which
also requires the use of relatives.
6.7
TESTS OF ADEQUACY OF INDEX NUMBERS
We have discussed various formulae for the construction of index numbers. None of the
formulae measures the price changes or quantity changes with perfection and has some bias.
The problem is to choose the most appropriate formula in a given situation. As a measure of
the formula error a number of mathematical tests, known as the tests of consistency or tests of
adequacy of index number formulae have been suggested. In this section we will discuss
these tests, which are also sometimes termed as the criteria for a good index number.
1. Unit Test: This test requires that the index number formula should be independent of
the units in which the prices or quantities of various commodities are quoted. All the
formulae discussed in the lesson except the index number based on Simple Aggregate
of Prices/Quantities satisfy this test.
2.
Time Reversal Test: The time reversal test, proposed by Prof Irving Fisher requires
the index number formula to possess time consistency by working both forward and
backward w.r.t. time. In his (Fisher’s) words:
202
“The formula for calculating an index number should be such that it gives the same
ratio between one point of comparison and the other, no matter which of the two is
taken as the base or putting it another way, the index number reckoned forward should
be reciprocal of the one reckoned backward.”
In other words, if the index numbers are computed for the same data relating to two
periods by the same formula but with the bases reversed, then the two index numbers
so obtained should be the reciprocals of each other. Mathematically, we should have
(omitting the factor 100),
Pab xPba = 1
…………(6-29)
or more generally
P01 xP10 = 1
…………(6-29a)
Time reversal test is satisfied by the following index number formulae:
(i)
Marshall-Edgeworth formula
(ii)
Fisher’s Ideal formula
(iii)
Kelly’s fixed weight formula
(iv)
Simple Aggregate index
(v)
Simple Geometric Mean of Price Relatives formula
(vi)
Weighted Geometric Mean of Price Relatives formula with fixed weights
Lespeyre’s and Pasche’s index numbers do not satisfy the time reversal test.
3. Factor Reversal Test: This is the second of the two important tests of consistency
proposed by Prof Irving Fisher. According to him:
“Just as our formula should permit the interchange of two times without giving
inconsistent results, so it ought to permit interchanging the price and quantities without
giving inconsistent results – i.e., the two results multiplied together should give the true
value ratio, except for a constant of proportionality.”
This implies that if the price and quantity indices are obtained for the same data, same
base and current periods and using the same formula, then their product (without the
203
factor 100) should give the true value ratio. Symbolically, we should have (without
factor 100).
P01 xQ01 =
∑pq
∑p q
1 1
0
= V01
…………(6-30)
0
Fisher’s formula satisfies the factor reversal test. In fact fisher’s index is the only
index satisfying this test as none of the formulae discussed in the lesson satisfies this
test.
Remark: Since Fisher’s index is the only index that satisfies both the time reversal
and factor reversal tests, it is termed as Fisher’s Ideal Index.
4. Circular Test: Circular test, first suggested by Westergaard, is an extension of time
reversal test for more than two periods and is based on the shift ability of the base
period. This requires the index to work in a circular manner and this property enables
us to find the index numbers from period to period without referring back to the
original base each time. For three periods a,b,c, the test requires :
Pab xPbc xPca = 1
…………(6-31)
a≠b≠c
In the usual notations Eq. (6-31) can be stated as:
P01 xP12 xP20 = 1
…………(6-31a)
For Instance
P01La xP12La xP21La =
∑p q x∑p q x∑p q
∑p q ∑pq ∑p q
1 0
2 1
0
2
0
1 1
2
2
0
≠1
Hence Laspeyre’s index does not satisfy the circular test. In fact, circular test is not
satisfied by any of the weighted aggregative formulae with changing weights. This
test is satisfied only by the index number formulae based on:
(i)
Simple geometric mean of the price relatives, and
(ii)
Kelly’s fixed base method
204
6.8
SPECIAL ISSUSES IN THE CONSTRUCTION OF INDEX NUMBERS
6.8.1 BASE SHIFTING
The need for shifting the base may arise either
(i)
when the base period of a given index number series is to be made more
recent, or
(ii)
when two index number series with different base periods are to be compared,
or
(iii)
when there is need for splicing two overlapping index number series.
Whatever be the reason, the technique of shifting the base is simple:
New Base Index Number =
Old Index Number of Current Year
x100
Old Index Number of New Base Year
Example 6-6
Reconstruct the following indices using 1997 as base:
Year
:
1991 1992 1993 1994 1995 1996 1997 1998
Index :
100
110
130
150
175
180
200
220
Solution:
Shifting the Base Period
Index Number
Index Number
Year
(1991 = 100)
(1997 = 100)
1991
100
(100/200) x100 = 50.00
1992
110
(110/200) x100 = 55.00
1993
130
(130/200) x100 = 65.00
1994
150
(150/200) x100 = 75.00
1995
175
(175/200) x100 = 87.50
1996
180
(180/200) x100 = 90.00
1997
200
(200/200) x100 = 100.00
1998
220
(220/200) x100 = 110.00
205
6.8.2 SPLICING TWO OVERLAPPING INDEX NUMBER SERIES
Splicing two index number series means reducing two overlapping index series with
different base periods into a single series either at the base period of the old series
(one with an old base year), or at the base period of the new series (one with a recent
base year). This actually amounts to changing the weights of one series into the
weights of the other series.
1. Splicing the New Series to Make it Continuous with the Old Series
Here we reduce the new series into the old series after the base year of the former. As
shown in Table 6.8.2(i), splicing here takes place at the base year (1980) of the new
series. To do this, a ratio of the index for 1980 in the old series (200) to the index of
1980 in the new series (100) is computed and the index for each of the following
years in the new series is multiplied by this ratio.
Table 6.8.2(i)
Splicing the New Series with the Old Series
Price Index
Price Index
(1976 = 100)
(1980 = 100)
(Old Series)
(New Series)
1976
100
--
100
1977
120
--
120
1978
146
--
146
1979
172
--
172
1980
200
100
200
1981
--
110
220
1982
--
116
232
1983
--
125
250
1984
--
140
280
Year
Spliced Index Number
[New Series x (200/100)]
2. Splicing the Old Series to Make it Continuous with the New Series
206
This means reducing the old series into the new series before the base period of the
letter. As shown in Table 6.8.2(ii), splicing here takes place at the base period of the
new series. To do this, a ratio of the index of 1980 of the new series (100) to the index
of 1980 of the old series (200) is computed and the index for each of the preceding
years of the old series are then multiplied by this ratio.
Table 6.8.2(ii)
Splicing the Old Series with the New Series
Price Index
Price Index
(1976 = 100)
(1980 = 100)
(Old Series)
(New Series)
1976
100
--
50
1977
120
--
60
1978
146
--
73.50
1979
172
--
86
1980
200
100
100
1981
--
110
110
1982
--
116
116
1983
--
125
125
1984
--
140
140
Year
Spliced Index Number
[Old Series x (100/200)]
6.8.3 CHAIN BASE INDEX NUMBERS
The various indices discussed so far are fixed base indices in the sense that either the base
year quantities/prices (or the given year quantities/prices) are used as weights. In a dynamic
situation where tastes, preferences, and habits are constantly changing, the weights should be
revised on a continuous basis so that new commodities are included and the old ones deleted
from consideration.
This is all the more necessary in a developing society where new substitutes keep replacing
the old ones, and completely new commodities are entering the market. To take care of such
changes, the base year should be the most recent, that is, the year immediately preceding the
207
given year. This means that as we move forward, the base year should move along the given
year in a chain year after year.
Conversion of Fixed-base Index into Chain-base Index
As shown in Table 6.8.3(i), to convert fixed-base index numbers into chain-base index
numbers, the following procedure is adopted:
¾
The first year's index number is taken equal to 100
¾
For subsequent years, the index number is obtained by following formula:
Current Year' s CBI =
Current Year' s FBI
x 100
Previous Year' s CBI
Table 6.8.3(i)
Conversion of Fixed-base Index into Chain-base Index
Fixed Base Index
Year
Chain Base Index
Number
Conversion
(FBI)
Number
(CBI)
1975
376
--
100
1976
392
(392/376) x100
104.3
1977
408
(408/392) x100
104.1
1978
380
(380/408) x100
93.1
1979
392
(392/380) x100
103.2
1980
400
(400/392) x100
102
Conversion of Chain-base Index into Fixed-base Index
As shown in Table 6.8.3(ii), to convert fixed-base index numbers into chain-base index
numbers, the following procedure is adopted:
¾
The first year's index is taken what the chain base index is; but if it is to form the
base it is taken equal to 100
¾
In subsequent years, the index number is obtained by following formula:
Current Year' s FBI =
Current Year' s CBI x Previous Year' s FBI
100
208
Table 6.8.3(ii)
Conversion of Chain-base Index into Fixed-base Index
Chain Base Index
Year
Fixed Base Index
Number
Conversion
(CBI)
6.9
Number
(FBI)
1978
90
--
90
1979
120
(120 x 90) /100
108
1980
125
(125 x 108) /100
135
1981
110
(110 x 135) /100
148.5
1982
112
(112 x 148.5) /100
166.3
1983
150
(150 x 166.3) /100
249.45
PROBLEMS OF CONSTRUCTING INDEX NUMBERS
The above discussion enables us to identify some of the important problems, which may be
faced in the construction of index numbers:
1. Choice of the Base Period: Choice of the base period is a critical decision because of
its importance in the construction of index numbers. A base period is the reference
period for describing and comparing the changes in prices or quantities in a given
period. The selection of a base year or period does not pose difficult theoretical
questions. To a large extent, the choice of the base year depends on the objective of
the index. A major consideration should be to ensure that the base year is not an
abnormal year. For example, a base period with very low price/quantity will unduly
inflate, while the one with a very high figure will unduly depress, the entire index
number series. An index number series constructed with any such period as the base
may give very misleading results. It is, therefore, necessary that the base period be
selected carefully.
Another important consideration is that the base year should not be too remote in the
past. A more recent year needs to be selected as the base year. The use of a particular
209
year for a prolonged period would distort the changes that it purports to measure. That
is why we find that the base year of major index numbers, such as consumer price
index or index of industrial production, is shifted from time to time.
2. Selection of Weights to be Used: It should be amply clear from the various indices
discussed in the lesson that the choice of the system of weights, which may be used, is
fairly large. Since any system of weights has its own merits and is capable of giving
results amenable to precise interpretations, the weights used should be decided
keeping in view the purpose for which an index is constructed.
It is also worthwhile to bear in, mind that the use of any system of weights should
represent the relative importance of individual commodities that enter into the
construction of an index. The interpretations that are intended to be made from an
index number are also important in deciding the weights. The use of a system of
weights that involves heavy computational work deserves to be avoided.
3. Type of Average to be Used: What type of average should be used is a problem
specific to simple average indices. Theoretically, one can use any of the several
averages that we have, such as mean, median, mode, harmonic mean, and geometric
mean. Besides being locational averages, median and mode are not the appropriate
averages to use especially where the number of years for which an index is to be
computed, is not large.
While the use of harmonic mean and geometric mean has some definite merits over
mean, particularly when the data to be averaged refer to ratios, mean is generally
more frequently used for convenience in computations.
4. Choice of Index: The problem of selection of an appropriate index arises because of
availability of different types of indices giving different results when applied to the
same data. Out of the various indices discussed, the choice should be in favour of one
210
which is capable of giving more accurate and precise results, and which provides
answer to specific questions for which an index is constructed.
While the Fisher's index may be considered ideal for its ability to satisfy the tests of
adequacy, this too suffers from two important drawbacks. First, it involves too
lengthy computations, and second, it is not amenable to easy interpretations as are the
Laspeyre's and Paasche's indices. The use of the term ideal does not, however, mean
that it is the best to use under all types of situations. Other indices are more appropriate under situations where specific answers are needed.
5. Selection of Commodities: Commodities to be included in the construction of an
index should be carefully selected. Only those commodities deserve to be included in
the construction of an index as would make it more representative. This, in fact, is a
problem of sampling, for being related to the selection of commodities to be included
in the sample.
In this context, it is important to note that the selection of commodities must not be
based on random sampling. The reason being that in random sampling every
commodity, including those that are not important and relevant, have equal chance of
being selected, and consequently, the index may not be representative. The choice of
commodities has, therefore, to be deliberate and in keeping with the relevance and importance of each individual commodity to the purpose for which the index is
constructed.
6. Data Collection: Collection of data through a sample is the most important issue in
the construction of index numbers. The data collected are the raw material of an
index. Data quality is the basic factor that determines the usefulness of an index. The
data have to be as accurate, reliable, comparable, representative, and adequate, as
possible.
211
The practical utility of an index also depends on how readily it can be constructed.
Therefore, data should be collected from where these can be easily available. While
the purpose of an index number will indicate what type of data are to be collected, it
also determines the source from where the data can be available.
6.10
SELF-ASSESSMENT QUESTIONS
1.
“Index Numbers are devices for measuring changes in the magnitude of a group of
related variables”. Discuss this statement and point out the important uses of index
numbers.
2.
“Index Numbers are Economic Barometers”. Discuss this statement. What
precautions would you take while constructing index numbers?
3.
4.
5.
6.
(a)
Explain the uses of index numbers.
(b)
What problems are involved in the construction of index numbers?
Describe each of the following:
a.
Base period
b.
Price relatives
c.
Fixed-base index numbers
d.
Chain-base index numbers
Describe briefly the following methods of construction of price index numbers:
a.
Simple Aggregate Method
b.
Simple Average of Price Relatives Method
c.
Weighted Aggregative Method
d.
Weighted Average of Price Relatives
“Laspeyre’s index has an upward bias and the Paasche’s index downward bias”.
Explain this statement.
7.
Discuss the various tests of adequacy of index numbers.
212
8.
State and explain the Fisher’s ideal formula for price index number. Show how it
satisfies the time-reversal and factor- reversal test? Why is it used little in practice?
9.
Briefly explain each of the following:
a.
Base-shifting
b.
Splicing
c.
Deflating
10. From the following data, construct the price index for each year with price of 1995
as base.
Year:
1995 1996 1997 1998 1999 2000
Price of Commodity:
40
50
45
55
65
70
11. From the following data, construct an index number for 2004 taking 2003 as base
year:
Articles:
A
B
C
D
E
Prices (2003):
100
125
50
40
5
Prices (2004):
140
200
80
60
10
12. Find the index number for 1982 and 1983 taking 1981 as base year by the Simple
Average of Price Relatives Method, using (i) Mean, (ii) Median, and (iii) Geometric
Mean:
1981 (Prices)
1982 (Prices)
1983 (Prices)
A
40
55
60
B
50
60
80
C
62
72
93
D
80
88
96
E
20
24
30
Commodities
13. Construct index number of price and index number of quantity from the following
data using:
213
a.
Laspeyre’s formula,
b.
Paasche’s formula,
c.
Dorbish and Bowley’s formula,
d.
Marshall and Edgeworth’s formula, and
e.
Fisher’s Ideal Index formula
Base Year
Current Year
Commodities
Price
Quantity
Price
Quantity
A
2
8
4
6
B
5
10
6
5
C
4
14
5
10
D
2
19
2
13
Which of the formula satisfy
the time reversal test, and
(i)
the factor reversal test?
(ii)
14. Calculate index number using Kelly’s Method of Standard Weights, from the
following data:
Commodities
Quantity
Base Year Price
Current Year Price
A
5
30
40
B
8
20
30
C
10
10
20
15. From the following data, construct price index by using Weighted Average of Price
Relatives Method:
Commodities
Quantity
Base Year Price
Current Year Price
A
6 Qtl
5.00
6.00
B
5 Qtl
5.00
8.00
C
1 Qtl
6.00
9.00
D
4 Kg
8.00
10.00
E
1 Kg
20.00
15.00
214
16. From the information given below, calculate the Cost of Living Index number
for 1985, with 1984 as base year by
a.
Aggregative Expenditure Method, and
b.
Family Budget Method.
Items
Quantity consumed
Unit
Prices in 1984
Prices in 1985
Wheat
2 Qtl
Qtl
75
125
Rice
20 Kg
Kg
12
16
Sugar
10 Kg
Kg
12
16
Ghee
5 Kg
Kg
10
15
Clothing
25 Meter
Meter
4.5
5
Fuel
40 Litre
Litre
10
12
Rent
One house
House
25
40
17. An enquiry into budgets of the middle class families in a city gave the following
information:
Expenses on
Food
Rent
Clothing
Fuel
Miscellaneous
→
40%
10%
20%
10%
20%
Prices(2001)
160
50
60
20
50
Prices(2002)
175
60
75
25
75
What changes in the cost of living figure of 2002 have taken place as compared to
2001?
18. Reconstruct the following indices using 1985 as base:
Year
:
1982 1983 1984 1985 1986 1987
Index
:
100
120
190
200
212
250
19. Given below are two sets of indices one with 1975 as base and the other with 1979 as base:
First set
Year
:
1975 1976 1977 1978 1979
Index Numbers
:
100
110
215
125
180
200
Second Set
Year
:
1979 1980 1981 1982 1983
Index Numbers
:
100
104
110
116
a.
Splice the second set of index numbers from 1975
b.
Splice the first set of index numbers from 1979
124
20. Construct chain index numbers from the following data:
Year :
Price :
1991 1992 1993 1994 1995
25
30
45
60
90
21. Convert into Chain Base Index Number from Fixed Base Index Number
Year
:
1980 1981 1982 1983 1984
Fixed Base Index
:
100
98
102
140
190
22. From the Chain Base Index numbers given below, construct Fixed Base
Index numbers:
Year
:
1993 1994 1995 1996 1997
Chain Base Index
:
100
105
95
115
102
23. From the following data, prepare index number for real wages of workers:
Year
:
1990 1991 1992 1993 1994 1995
Wages (in Rs)
:
300
340
450
460
475
540
Price Index Number :
100
120
220
230
250
300
24. During certain period, the Cost of Living Index number went up from 110 to 200 and
salary of a worker also raised from 325 to 500. State by how much the worker has
gained or lost in real term.
216
6.11
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
217
COURSE:
Course code: MC-106
Author: Dr. B.S. Bodla
Vetter: Karam Pal
Lesson: 7
ANALYSIS OF TIME SERIES
Objective:
This lesson would enable you to understand the meaning, importance,
models, and components of time series along with details of methods
of measuring trends.
Structure
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
7.8.
7.9.
7.10.
Introduction
Objectives of time series analysis
Components of time series
Time series decomposition models
Measurement of secular trend
Seasonal variations
Measurement of cyclical variations
Measurement of irregular variations
Questions
7.1.
INTRODUCTION
A series of observations, on a variable, recorded after successive intervals of time is called a
time series. The successive intervals are usually equal time intervals, e.g., it can be 10 years,
a year, a quarter, a month, a week, a day, and an hour, etc. The data on the population of
India is a time series data where time interval between two successive figures is 10 years.
Similarly figures of national income, agricultural and industrial production, etc., are available
on yearly basis.
7.2
OBJECTIVES OF TIME SERIES ANALYSIS
The analysis of time series implies its decomposition into various factors that affect the value
of its variable in a given period. It is a quantitative and objective evaluation of the effects of
various factors on the activity under consideration.
There are two main objectives of the analysis of any time series data:
(i)
To study the past behaviour of data.
(ii)
To make forecasts for future.
The study of past behaviour is essential because it provides us the knowledge of the effects of
various forces. This can facilitate the process of anticipation of future course of events, and,
thus, forecasting the value of the variable as well as planning for future.
7.3
Components of a Time Series
218
In the typical time-series there are three main components which seem to be independent of
the and seems to be influencing time-series data.
Trend- It is the broad long-term tendency of either upward or downward movement in the
average (or mean) value of the forecast variable y over time. The rate of trend growth usually
varies over time, as shown in fig 7.1(a) and (b).
Cycles- An upward and downward oscillation of uncertain duration and magnitude about the
trend line due to seasonal effect with fairly regular period or long period with irregular
swings is called a cycle. A business cycle may vary in length, usually greater than one year
but less than 5 to 7 years. The movement is through four phases: from peak (prosperity) to
contradiction (recession) to trough (depression) to expansion (recovery or growth) as shown
in Fig. 7.1 (b) and (c).
Seasonal- It is a special case of a cycle component of time series in which the magnitude and
duration of the cycle do not vary but happen at a regular interval each year. For example,
average sales for a retail store may increase greatly during festival seasons.
Irregular- An irregular or erratic (or residual) movements in a time series is caused by shortterm unanticipated and non-recurring factors. These follow no specific pattern.
7.4
TIME SERIES DECOMPOSITION MODELS
The analysis of time series consists of two major steps:
1. Identifying the various forces (influences) or factors which produce the variations in the
time series, and
2.
Isolating, analysing and measuring the effect of these factors separately and
independently, by holding other things constant.
The purpose of decomposition models is to break a time series into its components: Trend
(T), Cyclical (C), Seasonality (S), and Irregularity (I). Decomposition of time series provides
a basis for forecasting. There are many models by which a time series can be analysed; two
models commonly used for decomposition of a time series are discussed below.
7.4.1. Multiplicative Model
This is a most widely used model which assumes that forecast (Y) is the product of the four
components at a particular time period. That is, the effect of four components on the time
series is interdependent.
Y=T x C x S × I Å Multiplicative model
The multiplicative model is appropriate in situations where the effect of S, C, and I is
measured in relative sense and is not in absolute sense. The geometric mean of S, C, and I is
assumed to be less than one. For example, let the actual sales for period 20 be Y20 = 423.36.
Further let, this value be broken down into its components as: let trend component (mean
sales) be 400; effect of current cycle (0.90) is to depress sales by 10 per cent; seasonality of
the series (1.20) boosts sales by 20 per cent. Thus besides the random fluctuation, the
expected value of sales for the period is 400 × 0.90 × 1.20 = 432. If the random factor
depresses sales by 2 per cent in this period, then the actual sales value will be 432 × 0.98 =
423.36.
In this model, it is assumed that the effect of various components can be estimated by adding
the various components of a time-series. It is stated as:
Y=T + C + S + I
Here S, C, and I are absolute quantities and can have positive or negative values. It is
assumed that these four components are independent of each other. However, in real-life time
series data this assumption does not hold good.
219
7.5. MEASUREMENT OF SECULAR TREND
The principal methods of measuring trend fall into following categories:
1. Free Hand Curve methods
2. Method of Averages
3. Method of least squares
The time series methods are concerned with taking some observed historical pattern for some
variable and projecting this pattern into the future using a mathematical formula. These
methods do not attempt to suggest why the variable under study will take some future value.
This limitation of the time series approach is taken care by the application of a causal
method. The causal method tries to identify factors which influence the variable is some way
or cause it to vary in some predictable manner. The two causal methods, regression analysis
and correlation analysis, have already been discussed previously.
A few time series methods such as freehand curves and moving averages simply describe the
given data values, while other methods such as semi-average and least squares help to
identify a trend equation to describe the given data values.
7.5.1. Freehand Method
A freehand curve drawn smoothly through the data values is often an easy and, perhaps,
adequate representation of the data. The forecast can be obtained simply by extending the
trend line. A trend line fitted by the freehand method should conform to the following
conditions:
(i)
The trend line should be smooth- a straight line or mix of long gradual curves.
(ii)
The sum of the vertical deviations of the observations above the trend line should
equal the sum of the vertical deviations of the observations below the trend line.
(iii)
The sum of squares of the vertical deviations of the observations from the trend
line should be as small as possible.
(iv)
The trend line should bisect the cycles so that area above the trend line should be
equal to the area below the trend line, not only for the entire series but as much as
possible for each full cycle.
Example 7.1: Fit a trend line to the following data by using the freehand method.
Year
1991 1992 1993 1994 1995 1996 1997 1998
Sales turnover : 80
90
92
83
94
99
92
104
(Rs. in lakh)
Figure 7.2
freehand
sales
(Rs. in lakh)
1998.
be obtained
extending
line.
110
105
100
Sales
Solution:
presents the
graph of
turnover
from 1991 to
Forecast can
simply by
the trend
95
90
85
80
220 1995 1996 1997 1998
1991 1992 1993 1994
Years
Fig. 7.2: Graph of Sales Turnover
Limitations of freehand method
(i)
This method is highly subjective because the trend line depends on personal
judgement and therefore what happens to be a good-fit for one individual may not
be so for another.
(ii)
The trend line drawn cannot have much value if it is used as a basis for
predictions.
(iii)
It is very time-consuming to construct a freehand trend if a careful and
conscientious job is to be done.
7.5.2. Method of Averages
The objective of smoothing methods into smoothen out the random variations due to irregular
components of the time series and thereby provide us with an overall impression of the
pattern of movement in the data over time. In this section, we shall discuss three smoothing
methods.
(i)
Moving averages
(ii)
Weighted moving averages
(iii)
Semi-averages
The data requirements for the techniques to be discussed in this section are minimal and these
techniques are easy to use and understand.
Moving Averages
If we are observing the movement of some variable values over a period of time and trying to
project this movement into the future, then it is essential to smooth out first the irregular
pattern in the historical values of the variable, and later use this as the basis for a future
projection. This can be done by calculating a series of moving averages.
This method is a subjective method and depends on the length of the period chosen for
calculating moving averages. To remove the effect of cyclical variations, the period chosen
should be an integer value that corresponds to or is a multiple of the estimated average length
of a cycle in the series.
The moving averages which serve as an estimate of the next period’s value of a variable
given a period of length n is expressed as:
Moving average, Mat+1 = ∑{ Dt + Dt −1 + Dt − 2 . + .. + Dt − n +1 }
n
221
where t = current time period
D = actual data which is exchanged each period
n = length of time period
In this method, the term ‘moving’ is used because it is obtained by summing and averaging
the values from a given number of periods, each time deleting the oldest value and adding a
new value.
The limitation of this method is that it is highly subjective and dependent on the length of
period chosen for constructing the averages. Moving averages have the following three
limitations:
(i)
As the size of n (the number of periods averaged) increases, it smoothens the
variations better, but it also makes the method less sensitive to real changes in the
data.
(ii)
Moving averages cannot pick-up trends very well. Since these are averages, it will
always stay within past levels and will not predict a change to either a higher or
lower level.
(iii)
Moving average requires extensive records of past data.
Example 7.2: Using three-yearly moving averages, determine the trend and short-term-error.
Production
Year
Production
Year
(in ‘000 tonnes)
(in ‘000 tonnes)
1987
21
1992
22
1988
22
1993
25
1989
23
1994
26
1990
25
1995
27
1991
24
1996
26
Solution: The moving average calculation for the first 3 years is:
21 + 22 + 23
Moving average (year 1-3) =
= 22
3
Similarly, the moving average calculation for the next 3 years is:
22 + 23 + 25
Moving average (year 2-4) =
= 22.33
3
A complete summary of 3-year moving average calculations is given in Table 7.1
Table 7.1: Calculation of Trend and Short-term Fluctuations
3-yearly Moving
Forecast
Year
3-Year
Production
Average (Trend
Moving
Error (y- ŷ )
Y
Total
values ŷ
1987
21
1988
22
66
22.00
0
1989
23
70
23.33
-0.33
1990
25
72
24.00
1.00
1991
24
71
23.67
0.33
1992
22
71
23.67
-1.67
1993
25
73
24.33
0.67
222
1994
1995
1996
26
27
26
78
79
-
26.00
26.33
-
0
0.67
-
Odd and Even Number of Years
When the chosen period of length n is an odd number, the moving average
at year i is centred on i, the middle year in the consecutive sequence of n
yearly values used to compute i. For instance with n =5, MA3(5) is centred on
the third year, MA4(5) is centred on the fourth year…, and MA9(5) is centred
on the ninth year.
No moving average can be obtained for the first (n-1)/2 years or the last (n1/2) year of the series. Thus for a 5-year moving average, we cannot make
computations for the just two years or the last two years of the series.
When the chosen period of length n is an even numbers, equal parts can
easily be formed and an average of each part is obtained. For example, if n =
4, then the first moving average M3 (placed at period 3) is an average of the
first four data values, and the second moving average M4 (placed at period 4)
is the average of data values 2 through 5). The average of M3 and M4 is
placed at period 3 because it is an average of data values for period 1
through 5.
Example 7.3: Assume a four-yearly cycle and calculate the trend by the method of moving
average from the following data relating to the production of tea in India.
Year
Production (million
Year
Production (million
lbs)
lbs)
464
1992
540
1987
1988
515
1993
557
1989
518
1994
571
1990
467
1995
586
1991
502
1996
612
Solution: The first 4-year moving average is:
464 + 515 + 518+ 467
1964
MA3(4) =
=
= 491.00
4
4
This moving average is centred on the middle value, that is, the third year of the series.
Similarly,
515 + 518 + 467+ 502
2002
223
MA4(4) =
=
= 500.50
4
4
This moving average is centred on the fourth year of the series.
Table 7.2. presents the data along with the computations of 4-year moving averages.
Table 7.2: Calculation of Trend and Short-term Fluctuations
4-Yearly Moving
Year
Production
4-yearly
4-Yearly
Average Centred
(mm lbs)
Moving Totals
Moving
Average
464
1987
515
1988
1964
491.00
518
495.75
1989
2002
500.50
467
503.62
1990
2027
506.75
502
511.62
1991
2066
516.50
540
529.50
1992
2170
542.50
557
553.00
1993
2254
563.50
571
572.00
1994
2326
581.50
586
1995
612
1996
Weighted Moving Averages
In moving averages, each observation is given equal importance (weight). However, different
values may be assigned to calculate a weighted average of the most recent n values. Choice
of weights is somewhat arbitrary because there is no set formula to determine them. In most
cases, the most recent observation receives the most weightage, and the weight decreases for
older data values.
A weighted moving average may be expressed mathematically as
Σ(Weight for period n) (Data value in period n)
Weighted moving average =
ΣWeights
Example 7.4: Vaccum cleaner sales for 12 months is given below. The owner of the
supermarket decides to forecast sales by weighting the past three months as follows:
Weight Applied
Month
3
Last month
2
Two months ago
1
Three months ago
6
Month
: 1
2
4
5
6
7
8
9
10
11
3
Actual sales
: 10 12 13 16 19 23 26 30 28 18
16
(in units)
12
14
Solution: The results of 3-month weighted average are shown in Table 7.3.
224
3 × Sales last month + 2 × Sales two months ago +
1 × Sales three months ago
Forecast for the =
Current month
6
Table 7.3: Weighted Moving Average
Month
Actual Sales
Three-month Weighted
Moving Average
1
10
2
12
3
13
4
16
1
121
[3 × 13) + (2 × 12) + 1 × 10] =
6
6
5
19
1
141
[3 × 16) + (2 × 13) + 1 × 12] =
6
3
6
23
1
[3 × 19) + (2 × 16) + 1 × 13] = 17
6
7
26
1
201
[3 × 23) + (2 × 19) + 1 × 16] =
6
2
8
30
1
235
[3 × 26) + (2 × 23) + 1 × 19] =
6
6
9
28
1
271
[3 × 30) + (2 × 26) + 1 × 23] =
6
2
10
18
1
289
[3 × 28) + (2 × 30) + 1 × 26] =
6
3
11
16
1
231
[3 × 18) + (2 × 28) + 1 × 30] =
6
3
12
14
1
182
[3 × 16) + (2 × 18) + 1 × 28] =
6
3
Example 7.5: A food processor uses a moving average to forecast next month’s demand. Past
actual demand( in units) is shown below:
Month
: 43
44
46
47
48
49
50
51
45
114
121
130
128
137
Actual demand
: 105
106
110 110
(in units)
(a) Compute a simple five-month moving average to forecast demand for month 52.
(b) Compute a weighted three-month moving average where the weights are highest for the
latest months and descend in order of 3, 2, 1.
Solution: Calculation for five-month moving average are shown in Table 7.4.
Month
Actual Demand
5-month Moving
5-month Moving
Total
Average
105
43
106
44
110
545
109.50
45
110
561
112.2
46
114
585
117.0
47
121
603
120.6
48
225
49
50
51
130
128
137
630
-
126.0
-
(a) Five-month average demand for month 52 is
Σx
114 + 121 + 130 + 128 + 137
=
= 126 units
Number of periods
(b)
5
Weighted three-month average as per weights is as follows:
Σ Weight × Data value
MA Wt =
Σ weight
Where
Month Weight × Value = Total
51
50
49
3 × 137
2 × 128
1 × 130
6
=
=
=
141
256
130
797
797
MAWT = 6 = 133 units
Semi-Average Method
The semi-average method permits us to estimate the slope and intercept of the trend the quite
easily if a linear function will adequately described the data. The procedure is simply to
divide the data into two parts and compute their respective arithmetic means. These two
points are plotted corresponding to their midpoint of the class interval covered by the
respective part and then these points are joined by a straight line, which is the required trend
line. The arithmetic mean of the first part is the intercept value, and the slope is determined
by the ratio of the difference in the arithmetic mean of the number of years between them,
that is, the change per unit time. The resultant is a time series of the form : yˆ = a + bx . The
ŷ is the calculated trend value and a and b are the intercept and slope values respectively.
The equation should always be stated completely with reference to the year where x =0 and a
description of the units of x and y.
The semi-average method of developing a trend equation is relatively easy to commute and
may be satisfactory if the trend is linear. If the data deviate much from linearity, the forecast
will be biased and less reliable.
Example 7.6: Fit a trend line to the following data by the method of semi-average and
forecast the sales for the year 2002.
Year
Sales of Firm
Year
Sales of Firm (thousand
(thousand units)
units)
102
1997
108
1993
1994
105
1998
116
1995
114
1999
112
1996
110
226
Solution: Since number of years are odd in number, therefore divide the data into equal parts
(A and B) of 3 years ignoring the middle year (1996). The average of part A and B is
102 + 105 + 114
321
yA
=
=
= 107 units
3
3
108 + 116 + 112
yB
=
336
=
= 112 units
3
2
Part A is centred upon 1994 and part B on 1998. Plot points 107 and 112 against their middle
years, 1994 and 1998. By joining these points, we obtain the required trend line as shown Fig.
7.3. The line can be extended and be used for prediction.
120
Sales
115
110
105
100
1993
1994
1995
1996
1997
1998
1999
Years
Fig. 7.3: Trend Line by the Method of Semi-Average
To calculate the time-series ŷ = a + bx, we need
Δy
Slope b =
=
Δx
Change in sales
Change in year
112 – 107
=
5
=
= 1.25
1998 – 1994 4
Intercept = a = 107 units at 1994
Thus, the trend line is :
ŷ = 107 + 1.25x
Since 2002 is 8 year distant from the origin (1994), therefore we have
ŷ = 107 + 1.25(8) = 117
Exponential Smoothing Methods
Exponential smoothing is a type of moving-average forecasting technique which weighs past
data in an exponential manner so that the most recent data carries more weight in the moving
average. Simple exponential smoothing makes no explicit adjustment for trend effects
whereas adjusted exponential smoothing does take trend effect into account (see next section
for details).
Simple Exponential Smoothing
227
With simple exponential smoothing, the forecast is made up of the last period forecast plus a
portion of the difference between the last period’s actual demand and the last period’s
forecast.
Ft = Ft-1 + α (Dt-1 – Ft-1) = (1-α)Ft-1+ αDt-1
…(7.1)
Where Ft = current period forecast
Ft-1 = last period forecast
α = a weight called smoothing constant (0 ≤ α ≤1)
Dt-1 = last period actual demand
From Eqn. (7.1), we may notice that each forecast is simply the previous forecast plus some
correction for demand in the last period. If demand was above the last period forecast the
correction will be positive, and if below it will negative.
When smoothing constant α is low, more weight is given to past data, and when it is high,
more weight is given to recent data. When α is equal to 0.9, then 99.99 per cent of the
forecast value is determined by the four most recent demands. When α is as low as 0.1, only
34.39 per cent of the average is due to these last 4 periods and the smoothing effect is
equivalent to a 19-period arithmetic moving average.
If α were assigned a value as high as 1, each forecast would reflect total adjustment to the
recent demand and the forecast would simply be last period’s actual demand, that is, Ft =
1.0Dt-1. Since demand fluctuations are typically random, the value of α is generally kept in
the range of 0.005 to 0.30 in order to ‘smooth’ the forecast. The exact value depends upon the
response to demand that is best for the individual firm.
The following table helps illustrate this concept. For example, when α = 0.5, we can see that
the new forecast is based on demand in the last three or four periods. When α = 0.1, the
forecast places little weight on recent demand and takes a 19-period arithmetic moving
average.
Weight Assigned to
2nd Most
3rd Most
Most Recent
4th Most
5th Most
Smoothing
Recent
Period
Recent
Recent
Recent
Constant
Period
Period
Period
Period
(α)
2
3
α(1-α)
α(1-α)
α(1-α)
α(1-α)4
0.1
0.09
0.081
0.073
0.066
α = 0.1
0.5
0.25
0.125
0.063
0.031
α =0.5
Selecting the smoothing constant
The exponential smoothing approach is easy to use and it has been successfully applied by
banks, manufacturing companies, wholesalers, and other organizations. The appropriate value
of the smoothing constant, α, however, can make the difference between an accurate and an
inaccurate forecast. In picking a value for the smoothing constant, the objective is to obtain
the most accurate forecast.
The correct α-value facilitates scheduling by providing a reasonable reaction to demand
without incorporating too much random variation. An approximate value of α which is
equivalent to an arithmetic moving average, in terms of degree of smoothing, can be
estimated as: α = 2 (n +1). The accuracy of a forecasting model can be determined by
comparing the forecasting values with the actual or observed values.
The forecast error is defined as:
Forecast error = Actual values – Forecasted values
One measure of the overall forecast error for a model is the mean absolute deviation (MAD).
This is computed by taking the sum of the absolute values of the individual forecast errors
and dividing by the number of periods n of data.
228
Σ⏐Forecast errors⏐
n
where Standard deviation σ = 1.25 MAD
The exponential smoothing method also facilities continuous updating of the estimate of
Higher values of smoothing constant α make the current MAD more responsive to current
forecast errors.
Example 7.7: A firm uses simple exponential smoothing with α =0.1 to forecast demand.
The forecast for the week of February 1 was 500 units whereas actual demand turned out to
be 450 units.
(a) Forecast the demand for the week of February 8.
(b) Assume the actual demand during the week of February 8 turned out to be 505 units.
Forecast the demand for the week of February 15. Continue forecasting through March 15,
assuming that subsequent demands were actually 516, 488, 467, 554 and 510 units.
Solution: Given Ft-1 = 500, D t-1 = 450, and α = 0.1
(a) Ft = F t-1 – α(Dt-1 - Ft-1) = 500 + 0.1(450-500) = 495 units
(b) Forecast of demand for the week of February 15 is shown in Table 7.5
Table 7.5: Forecast of Demand
Week
Forecast
Correction
New Forecast (Ft)
Old
Error
Forecast
α(Dt-1 -Ft-1)
Ft-1 +α(Dt-1-Ft-1)
(Dt-1 –Ft-1)
Ft-1
495
-5
-50
500
450
Feb. 1
496
1
10
495
505
Feb. 8
498
2
20
496
516
Feb. 15
497
-1
-10
498
488
Feb. 22
494
-3
-30
497
467
Mar. 1
500
6
60
494
554
Mar. 8
501
1
10
500
510
Mar. 15
If no previous forecast value is known, the old forecast starting point may be estimated or
taken to be an average of some preceding periods.
Example 7.8: A hospital has used a 9 month moving average forecasting method to predict
drug and surgical inventory requirements. The actual demand for one item is shown in the
table below. Using the previous moving average data, convert to an exponential smoothing
forecast for month 33.
Month
: 24
25
27
28
29
30
31
32
26
71
80
101
84
60
73
Demand
: 78
65
90
(in units)
Solution: The moving average of a 9-month period is given by
∑Demand (x)
78 + 65 … + 73
MA =
=
= 78
Number of periods
9
2
2
Assume Ft-1 = 78. Therefore, estimated α =
= 0.2
=
n +1 9 +1
Thus,
Ft = Ft-1 + α(Dt-1-Ft-1) = 78 + 0.2 (73 - 78) = 77 units
Demand
Dt-1
229
Methods of least square
The trend project method fits a trend line to a series of historical data points and then projects
the line into the future for medium-to-long range forecasts. Several mathematical trend
equations can be developed (such as exponential and quadratic), depending upon movement
of time-series data.
Reasons to study trend: A few reasons to study trends are as follows:
1. The study of trend allows us to describe a historical pattern so that we may evaluate the
success of previous policy.
2. The study allows us to use trends as an aid in making intermediate and long-range
forecasting projections in the future.
3. The study of trends helps us to isolate and then eliminate its influencing effects on the
time-series model as a guide to short-run (one year or less) forecasting of general business
cycle conditions.
Linear Trend Model
If we decide to develop a linear trend line by a precise statistical method, we can apply the
least squares method. A least squares line is described in terms of its y-intercept (the height at
which it intercepts the y-axis) and its slope (the angle of the line). If we can compute the yintercept and slope, we can express the line with the following equation
yˆ = a + bx
where ŷ = predicted value of the dependent variable
a = y-axis intercept
b = slope of the regression line (or the rate of change in y for a given change in
x)
x = independent variable (which is time in this case)
Least squares is one of the most widely used methods of fitting trends to data because it
yields what is mathematically described as a ‘line of best fit’. This trend line has the
properties that (i) the summation of all vertical deviations about it is zero, that is, Σ(y- ŷ ) = 0,
(ii) the summation f all vertical deviations squared is a minimum, that is, Σ(y- ŷ ) is least, and
(iii) the line goes through the mean values of variables x and y. For linear equations, it is
found by the simultaneous solution for a and b of the two normal equations:
Σy = na + bΣx and Σxy = aΣx + bΣx2
Where the data can be coded so that ∑x = 0, two terms in three equations
drop out and we have Σy = na and Σxy = bΣx2
Coding is easily done with time-series data. For coding the data, we choose
the centre of the time period as x = 0 and have an equal number of plus and
minus periods on each side of the trend line which sum to zero.
Alternately, we can also find the values of constants a
regression line as:
b=
∑ xy − nx y
and a = y − bx
∑ x 2 − n(x ) 2
230
and b
for any
Example 7.9: Below are given the figures of production (in thousand
quintals) of a sugar factory:
Year
Production
:
:
1992
80
1993
90
1994
92
1995
83
1996
94
1997
99
1998
92
(a) Fit a straight line trend to these figures.
(b) Plot these figures on a graph and show the trend line.
(c) Estimate the production in 2001.
Solution: (a) Using normal equations and the sugar production data we can
compute constants a and b as shown in Table 7.6:
Table 7.6: Calculations for Least Squares Equation
Year
Production
(x)
80
x2
xy
1992
Time
Period (x)
1
1
80
Trend
Values y
84
1993
2
90
4
180
86
1994
3
92
9
276
88
1995
4
83
16
332
90
1996
5
94
25
470
92
1997
6
99
36
594
94
1998
7
92
49
644
96
Total
28
630
140
2576
x=
∑ x 28
∑ y 630
=
= 4, y =
=
= 90
n
7
n
7
b=
∑ xy − nx y
2576 − 7(4)(90) 56
=
=
=2
∑ x 2 − n(x ) 2
140 − 7(4) 2
28
a = y − bx = 90 − 2(4) = 82
Therefore, linear trend component for the production of sugar is:
yˆ = a + bx = 82 + 2 x
The slope b = 2 indicates that over the past 7 years, the production of sugar
231
100
Production
95
90
85
80
75
1992
1993
1994
1995
1996
1997
1998
Years
Fig.7.4: Linear Trend for Production of Sugar
(b) Plotting points on the graph paper, we get an actual graph representing
production of sugar over the past 7 years. Join the point a = 82 and b = 2
(corresponds to 1993) on the graph we get a trend line as shown in Fig. 7.4.
(c) The production of sugar for year 2001 will be
ŷ = 82 + 2 (10) = 102 thousand quintals
Parabolic Trend Model
The curvilinear relationship for estimating the value of a dependent variable
y from an independent variable x might take the form
ŷ = a + bx + cx2
This trend line is called the parabola.
For a non-linear equation ŷ = a + bx - cx2, the values of constants a, b, and
c can be determined by solving three normal equations.
Σy = na + bΣx + cΣx2
Σxy = aΣx + bΣx2 + cΣx3
Σx2y = aΣx2 + bΣx3 + cΣx4
232
When the data can be coded so that Σx = 0 and Σx3 = 0, two term in the
above expressions drop out and we have
Σy = na + cΣx2
Σxy = bΣx2
Σx2y = aΣx2 + cΣx4
To find the exact estimated value of the variable y, the values of constants a,
b, and c need to be calculated. The values of these constants can be
calculated by using the following shortest method:
a=
n ∑ x2 y − ∑ x2 ∑ y
∑ y − c ∑ x2
∑ xy
and c =
;b =
n
∑ x2
n ∑ x 4 − (∑ x 2 ) 2
Example 7.10: The prices of a commodity during 1999-2004 are given
below. Fit a parabola to these data. Estimate the price of the commodity for
the year 2005.
Year
Price
Year
Price
1999
100
2002
140
2000
107
2003
181
2001
128
2004
192
Also plot the actual and trend values on a graph.
Solution: To fit a parabola ŷ = a + bx + cx2, the calculations to determine
the values of constants a, b, and c are shown in Table 7.7.
Table 7.7: Calculations for Parabola Trend Line
Year
Price
(y)
x2
x3
x4
xy
x2y
1999
Time
Scale
(x)
-2
100
4
-8
16
-200
400
Trend
Values
( yˆ )
97.72
2000
-1
107
1
-1
1
-107
107
110.34
2001
0
128
0
0
0
0
0
126.68
2002
1
140
1
1
1
140
140
146.50
2003
2
181
4
8
16
362
724
169.88
233
2004
(i)
3
192
9
27
81
576
1728
196.82
3
848
19
27
115
771
3099
847.94
Σy = na- bΣx + cΣx2
848 = 6a + 3b + 19c
(ii) Σxy = aΣx + bΣx2 + cΣx3
771 = 3a + 19b + 27c
(iii) Σx2y = aΣx2 + bΣx2 + cΣx4
3099 = 19a + 27b + 115c
Eliminating a from eqns. (i) and (ii), we get
(iv) 694 = 35b + 35c
Eliminating a from eqns. (ii) and (iii), we get
(v) 5352 = 280b + 168c
Solving eqns. (iv) and (v) for b and c we get b =18.04 and c = 1.78.
Substituting values of b and c in eqn. (i), we get a = 126.68.
Hence, the required non-linear trend line becomes
y = 126.68 +18.04x + 1.78x2
Several trend values as shown in Table 7.7 can be obtained by putting x = 2, -1, 0, 1, 2 and 3 in the trend line. The trend values are plotted on a graph
paper. The graph is shown in Fig. 7.5.
234
600
Years
500
400
300
200
100
0
1994
1995
1996
1997
1998
1999
Price (Rs.)
Fig. 7.5
Exponential Trend Model
When the given values of dependent variable y from approximately a
geometric progression while the corresponding independent variable x values
form an arithmetic progression, the relationship between variables x and y
is given by an exponential function, and the best fitting curve is said to
describe the exponential trend. Data from the fields of biology, banking, and
economics frequently exhibit such a trend. For example, growth of bacteria,
money accumulating at compound interest, sales or earnings over a short
period, and so on, follow exponential growth.
The characteristics property of this law is that the rate of growth, that is, the
rate of change of y with respect to x is proportional to the values of the
function. The following function has this property.
y = abcx, a > 0
The letter b
is a fixed constant, usually either 10 or e, where a
is a
constant to be determined from the data.
To assume that the law of growth will continue is usually unwarranted, so
only short range predictions can be made with any considerable degree or
reliability.
235
If we take logarithms (with base 10) of both sides of the above equation, we
obtain
Log y = log a + (c log b) x
(7.2)
For b =10, log b =1, but for b=e, log b =0.4343 (approx.). In either case, this
equation is of the form y ′ = c + dx
Where y ′ = log y, c = log a, and d = c log b.
Equation (7.2) represents a straight line. A method of fitting an exponential
trend line to a set of observed values of y is to fit a straight trend line to the
logarithms of the y-values.
In order to find out the values of constants a
and b in the exponential
function, the two normal equations to be solved are
Σ log y = n log a + log bΣx
Σx log y = log aΣx + log bΣx2
When the data is coded so that Σx = 0, the two normal equations become
Σ log y = n log a or log a =
and
1
Σlog y
n
Σx log y = log b Σx2 or log b =
∑ x log y
∑ x2
Coding is easily done with time-series data by simply designating the center
of the time period as x =0, and have equal number of plus and minus period
on each side which sum to zero.
Example 7.11: The sales (Rs. In million) of a company for the years 1995 to
1999 are:
Year :
Sales :
1995
1.6
1996
4.5
1997
13.8
1998
40.2
1999
125.0
Find the exponential trend for the given data and estimate the sales for
2002.
236
Solution: The computational time can be reduced by coding the data. For
this consider u = x-3. The necessary computations are shown in Table 7.8.
Table 7.8: Fitting the Exponential Trend Line
Year
u=x-3
u2
Sales y
Log y
u log y
1995
Time
Period x
1
-2
4
1.60
0.2041
-0.4082
1996
2
-1
1
4.50
0.6532
-0.6532
1997
3
0
0
13.80
1.1390
0
1998
4
1
1
40.20
1.6042
1.6042
1999
5
2
4
125.00
2.0969
4.1938
5.6983
4.7366
10
log a =
1
1
Σ log y = (5.6983) = 1.1397
n
5
log b =
Therefore
4.7366
∑ u log y
=
= 0.4737
2
10
∑u
log y = log a + (x+3) log b = 1.1397 + 0.4737x
For sales during 2002, x =3, and we obtain
log y = 1.1397 + 0.4737 (3) = 2.5608
y = antilog (2.5608) = 363.80
Changing the Origin and Scale of Equations
When a moving average or trend value is calculated it is assumed to be
centred in the middle of the month (fifteenth day) or the year (July 1).
Similarly, the forecast value is assumed to be centred in the middle of the
future period. However, the reference point (origin) can be shifted, or the
units of variables x and y are changed to monthly or quarterly values it
desired. The procedure is as follows:
(i)
Shift the origin, simply by adding or subtracting the desired
number of periods from independent variable x
forecasting equation.
237
in the original
(ii)
Change the time units from annual values to monthly values by
dividing independent variable x by 12.
(iii)
Change the y
units from annual to monthly values, the entire
right-hand side of the equation must be divided by 12.
Example 7.12: The following forecasting equation has been derived by a
least-squares method:
ŷ =10.27 + 1.65x (Base year:1992; x = years; y = tonnes/year)
Rewrite the equation by
(a) shifting the origin to 1997.
(b) expressing x units in months, retaining y n tonnes/year.
(c) expressing x units in months and y in tonnes/month.
Solution: (a) Shifting of origin can be done by adding the desired number of
period 5(=1997-1992) to x in the given equation. That is
ŷ =10.27 + 1.65 (x + 5) = 18.52 + 1.65x
where 1997 = 0, x = years, y = tonnes/year
(b) Expressing x units in months
1.65x
ŷ =10.27 +
= 10.27 + 0.14x
12
where July 1, 1992 = 0, x = months, y = tonnes/year
(c) Expressing y in tonnes/month, retaining x months.
1
= (10.27 + 0.14x) = 0.86+0.01x
12
where July 1, 1992 = 0, x = months, y = tones/month
ŷ =
Remarks
1. If both x
and y are to be expressed in months together, then divide
constant ‘a’ by 12 and constant ‘b’ by 24. It is because data are sums of 12
months. Thus monthly trend equation becomes.
Linear trend : yˆ =
a
b
+
x
12 24
238
Parabolic trend : ŷ =
a
b
c
+
x+
x2
12 144
1728
But if data are given as monthly averages per year, then value of ‘a’ remains
unchanged ‘b’ is divided by 12 and ‘c’ by 144.
2. The annual trend equation can be reduced to quarterly trend equation as
:
yˆ =
7.6.
a
b
a b
+
x= +
x
4 4 × 12
4 48
SEASONAL VARIATIONS
If the time series data are in terms of annual figures, the seasonal variations
are absent. These variations are likely to be present in data recorded on
quarterly or monthly or weekly or daily or hourly basis. As discussed earlier,
the seasonal variations are of periodic in nature with period less than or
equal to one year. These variations reflect the annual repetitive pattern of
the economic or business activity of any society. The main objectives of
measuring seasonal variations are:
(i)
To understand their pattern.
(ii)
To use them for short-term forecasting or planning.
(iii)
To compare the pattern of seasonal variations of two or more time
series in a given period or of the same series in different periods.
(iv)
To eliminate the seasonal variations from the data. This process is
known as deseasonalisation of data.
Methods of Measuring Seasonal Variations
The measurement
of seasonal
variation is done by isolating them from
other components of a time series. There are four methods commonly used
for the measurement of seasonal variations. These method are:
1. Method of Simple Averages
2. Ratio to Trend Method
3. Ratio to Moving Average Method
239
4. Method of Line Relatives
Note: In the discussion of the above methods, we shall often assume a
multiplicative model. However, with suitable modifications, these methods
are also applicable to the problems based on additive model.
Method of Simple Averages
This method is used when the time series variable consists of only the
seasonal and random components. The effect of taking average of data
corresponding to the same period (say 1st quarter of each year) is to
eliminate the effect of random component and thus, the resulting averages
consist of only seasonal component. These averages are then converted into
seasonal indices, as explained in the following examples.
Example 7.13.
Assuming that trend and cyclical variations are absent compute the
seasonal index for each month of the following data of sales (in Rs. ‘000) of a
company.
Year
Jan
Feb
Mar Apr
May Jun
Jul
Aug Sep
Oct
Nov
Dec
1987 46
45
44
46
45
47
46
43
40
40
41
45
1988 45
44
43
46
46
45
47
42
43
42
43
44
1989 42
41
40
44
45
45
46
43
41
40
42
45
Solution
Calculation Table
Year
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
1987
46
45
44
46
45
47
46
43
40
40
41
45
1988
45
44
43
46
46
45
47
42
43
42
43
44
1989
42
41
40
44
45
45
46
43
41
40
42
45
Total
133
130
127
136
136
137
139
128
124
122
126
134
At
44.3
43.3
42.3
45.3
45.3
45.7
46.3
42.7
41.3
40.7
42.0
44.7
S.l.
101.4
99.1
96.8
103.7
103.7
104.6
105.9
97.7
94.5
93.1
96.1
102.3
In the above table, A denotes the average and S.I the seasonal index for a
particular month of various years. To calculate the seasonal index, we
240
compute grand average G, given by G =
∑ Ai
523
=
= 43.7 . Then the seasonal
12
12
index for a particular month is given by S.I. =
At
× 100 .
G
Further, ΣS.I.=11998.9≠1200. Thus, we have to adjust these values such
that their total is 1200. This can be done by multiplying each figure by
1200
. The resulting figures are the adjusted seasonal indices, as given
1198.9
below:
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
101.5
99.2
96.9
103.8
103.8
104.7
106.0
97.8
94.6
93.2
96.2
102.3
Remarks: The total equal to 1200, in case of monthly indices and 400, in
case of quarterly indices, indicate that the ups and downs in the time series,
due to seasons, neutralise themselves within that year. It is because of this
that the annual data are free from seasonal component.
Example 7.14
Compute the seasonal index from the following data by the method of simple
averages.
Year
1980
1981
Quarter
I
II
III
IV
I
II
III
IV
Y
106
124
104
90
84
114
107
88
Year
1982
1983
Quarter
I
II
III
IV
I
II
III
IV
Y
90
112
101
85
76
94
91
76
Year
1984
1985
Quarter
I
II
III
IV
I
II
III
IV
Y
80
104
95
83
104
112
102
84
Solution
Calculation of Seasonal Indices
Years
1980
1981
1982
1983
1984
1985
Ist Qr
106
84
90
76
80
104
2nd Qr
124
114
112
94
104
112
241
3rd Qr
104
107
101
91
95
102
4th Qr
90
88
85
76
83
84
Total
Ai
Ai
× 100
G
We have G =
104
90
660
110
600
100
506
84.33
93.67
114.49
104.07
87.77
∑ At
384.33
=
= 96.08 . Further, since the sum of terms in the last
4
4
row of the table is 400, no adjustment is needed. These terms are the
seasonal indices of respective quarters.
Merits and Demerits
This is a simple method of measuring seasonal variations which is based on
the unrealistic assumption that the trend and cyclical variations are absent
from the data. However, we shall see later that this method, being a part of
the other methods of measuring seasonal variations, is very useful.
Ratio to Trend Method
This method is used when cyclical variations are absent from the data, i.e.
the time series variable Y consists of trend, seasonal and random
components.
Using symbols, we can write Y = T.S.R
Various steps in the computation of seasonal indices are:
(i)
Obtain the trend values for each month or quarter, etc. by the
method of least squares.
(ii)
Divide the original values by the corresponding trend values. This
would eliminate trend values from the data. To get figures in
percentages, the quotients are multiplied by 100.
Thus, we have
(iii)
Y
T .S .R
× 100 =
× 100 = S .R.100
T
T
Finally, the random component is eliminated by the method of
simple averages.
242
Example 7.15
Assuming that the trend is linear, calculate seasonal indices by the ratio to
moving average method from the following data:
Quarterly output of coal in 4 years (in thousand tonnes)
Year
I
II
III
IV
1982
65
58
56
61
1983
68
63
63
67
1984
70
59
56
52
1985
60
55
51
58
Solution
By adding the values of all the quarters of a year, we can obtain annual
output for each of the four years. Fit a linear trend to the data and obtain
trend values for each quarter.
Year
Output
X=2(t-1983.5)
XY
X2
1982
240
-3
-720
9
1983
261
-1
-261
1
1984
237
1
237
1
1985
224
3
672
9
Total
962
0
-72
20
From the above table, we get a =
962
− 72
= 240.5 and b =
= −3.6
4
20
Thus, the trend line is Y=240.5 – 3.6X, Origin: Ist January 1984, unit of X:6
months.
The quarterly trend equation is given by
Y=
240.5 3.6
−
X or Y = 60.13-0.45X, Origin: Ist January 1984, unit of X:1
4
8
quarter (i.e., 3 months).
Shifting origin to 15th Feb. 1984, we get
243
Y=60.13-0.45(X+
1
) = 59.9-0.45X, origin I-quarter, unit of X=1 quarter.
2
The table of quarterly values is given by
Year
I
II
III
IV
1982
63.50
63.05
62.50
62.15
1983
61.70
61.25
60.80
60.35
1984
59.90
59.45
59.00
58.55
1985
58.10
57.65
57.20
56.75
The table of Ratio to Trend Values, i.e.
Y
× 100
T
Year
I
II
III
IV
1982
102.36
91.99
89.46
98.15
1983
110.21
102.86
103.62
111.02
1984
116.86
99.24
94.92
88.81
1985
103.27
95.40
89.16
102.20
Total
432.70
389.49
377.16
400.18
Average
108.18
97.37
94.29
100.05
S.I.
108.20
97.40
94.32
100.08
Note : Grand Average, G =
399.89
= 99.97
4
Example 7.16.
Find seasonal variations by the ratio to trend method, from the following
data:
Year
I-Qr
II-Qr
III-Qr
IV-Qr
1995
30
40
36
34
1996
34
52
50
44
1997
40
58
54
48
1998
54
76
68
62
1999
80
92
86
82
Solution
244
First we fit a linear trend to the annual totals.
Year
Annual Totals (Y)
X
XY
X2
1995
140
-2
-280
4
1996
180
-1
-180
1
1997
200
0
0
0
1998
260
1
260
1
1999
340
2
680
4
Total
1120
0
480
10
Now a =
1120
480
= 224 and b =
= 48
5
10
∴Trend equation is Y = 224+48X, origin: Ist July 1997, unit of X = 1 year
The quarterly trend equation is Y=
224 48
+
X=56+3X, origin: Ist July 1997,
4
16
unit of X = 1 quarter.
Shifting the origin to III quarter of 1997, we get
1
Y = 56 + 3 (X+ ) = 57.5 + 3X
2
Table of Quarterly Trend Values
Year
I
II
III
IV
1995
27.5
30.5
33.5
36.5
1996
39.5
42.5
45.5
48.5
1997
51.5
54.5
57.5
60.5
1998
63.5
66.5
69.5
72.5
1999
75.5
78.5
81.5
84.5
Ratio to Trend Values
Year
I
II
III
IV
1995
109.1
131.1
107.5
93.2
1996
86.1
122.4
109.9
90.7
1997
77.7
106.4
93.9
79.3
245
1998
85.0
114.3
97.8
85.5
1999
106.0
117.2
105.5
97.0
Total
463.9
591.4
514.6
445.7
At
92.78
118.28
102.92
89.14
S.I.
92.10
117.35
102.11
88.44
Note that the Grand Average G=
403.12
= 100.78. Also check that the sum of
4
indices is 400.
Remarks: If instead of multiplicative model we have an additive model, then
Y = T + S + R or S + R = Y-T. Thus, the trend values are to be subtracted
from the Y values. Random component is then eliminated by the method of
simple averages.
Merits and Demerits
It is an objective method of measuring seasonal variations. However, it is
very complicated and doesn’t work if cyclical variations are present.
Ratio to Moving Average Method
The ratio to moving average is the most commonly used method of
measuring seasonal variations. This method assumes the presence of all the
four components of a time series. Various steps in the computation of
seasonal indices are as follows:
(i)
Compute the moving averages with period equal to the period of
seasonal variations. This would eliminate the seasonal component
and minimise the effect of random component. The resulting
moving averages would consist of trend, cyclical and random
components.
(ii)
The original values, for each quarter (or month) are divided by the
respective moving average figures and the ratio is expressed as a
246
percentage, i.e.
Y
TCSR
=
= SR' ' , where R´ and R´´ denote the
M . A. TCR '
changed random components.
(iii)
Finally, the random component R´´ is eliminated by the method of
simple averages.
Example 7.17
Given the following quarterly sale figures, in thousand of rupees, for the year
1996-1999, find the specific seasonal indices by the method of moving
averages.
Year
I
II
III
IV
1996
34
33
34
37
1997
37
35
37
39
1998
39
37
38
40
1999
42
41
42
44
Centred
Total
4 Period
M
Y
× 100
M
…
…
279
284
289
294
298
302
305
307
311
318
326
334
…
…
…
…
34.9
35.5
36.1
36.8
37.3
37.8
38.1
38.4
38.9
39.8
40.8
41.8
…
…
…
…
97.4
104.2
102.5
95.1
99.2
103.2
102.4
96.4
97.7
100.5
102.9
98.1
…
…
Solution
Calculation of Ratio of Moving Averages
Year/Quarter
1996
1997
1998
1999
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
IV
Sales
34
33
34
37
37
35
37
39
39
37
38
40
42
41
42
44
4-Period
Moving
Total
138
141
143
146
148
150
152
153
157
161
165
169
Calculation of Seasonal Indices
247
Year
I
II
III
IV
1996
-
-
97.4
104.2
1997
102.5
95.1
99.2
103.2
1998
102.4
96.4
97.7
100.5
1999
102.9
98.1
-
-
Total
307.8
289.6
294.3
307.9
At
102.6
96.5
98.1
102.6
S.I.
102.7
96.5
98.1
102.7
Note that the Grand Average G=
399.8
=99.95. Also check that the sum of
4
indices is 400.
Merits and Demerits
This method assumes that all the four components of a time series are
present and, therefore, widely used for measuring seasonal variations.
However, the seasonal variations are not completely eliminated if the cycles
of these variations are not of regular nature. Further, some information is
always lost at the ends of the time series.
Line Relatives Method
This method is based on the assumption that the trend is linear and cyclical
variations are of uniform pattern. As discussed in earlier chapter, the link
relatives are percentages of the current period (quarter or month) as
compared with previous period. With the computation of link relatives and
their average, the effect of cyclical and random component is minimised.
Further, the trend gets eliminated in the process of adjustment of chained
relatives. The following steps are involved in the computation of seasonal
indices by this method:
(i) Compute the link relative (L.R.) of each period by dividing the figure of
that period with the figure of previous period. For example, link relative of
figure of 3rd quarter
3rd
quarter =
figure of
2nd
× 100
quarter
248
(ii) Obtain the average of link relatives of a given quarter (or month) of
various years. A.M. or Md can be used for this purpose. Theoretically, the
later is preferable because the former gives undue importance to extreme
items.
(iii) These averages are converted into chained relatives by assuming the
chained relative of the first quarter (or month) equal to 100. The chained
relative (C.R.) for the current period (quarter or month)
C.R. of the previous period × L.R. of the current period
=
100
(iv) Compute the C.R. of first quarter (or month) on the basis of the last
quarter (or month). This is given by
C.R. of the last quarter (or month) × L.R. of 1st quarter (or month)
=
100
This value, in general, be different from 100 due to long term trend in the
data. The chained relatives, obtained above, are to be adjusted for the effect
of this trend. The adjustment factor is
d=
1
[New C.R. for Ist quarter-100] for quarterly data
4
and d =
1
[New C.R. for Ist month –100] for monthly data.
12
On the assumption that the trend is linear, d, 2d, 3d, etc. is respectively
subtracted from the 2nd, 3rd, 4th, etc., quarter (or month).
(v) Express the adjusted chained relatives as a percentage of their average to
obtain seasonal indices.
(vi) Make sure that the sum of these indices is 400 for quarterly data and
1200 for monthly data.
Example 7.18
249
Determine the seasonal indices from the following data by the method of link
relatives:
Year
Ist
2nd Qr
3rd Qr
4th Qr
2000
26
19
15
10
2001
36
29
23
22
2002
40
25
20
15
2003
46
26
20
18
2004
42
28
24
21
Solution
Calculation Table
Year
I
II
III
IV
2000
-
73.1
78.9
66.7
2001
360.0
80.5
79.3
95.7
2002
181.8
62.5
80.0
75.0
2003
306.7
56.5
76.9
90.0
2004
233.3
66.7
85.7
87.5
Total
1081.8
339.3
400.8
414.0
Mean
270.5
67.9
80.2
83.0
C.R.
100.0
67.9
54.5
45.2
100.0
62.3
43.3
28.4
S.I.
170.9
106.5
74.0
48.6
The chained relative (C.R.) of the Ist quarter on the basis of C. R. of the 4th
quarter =
270 × 45.2
= 122.3
100
1
The trend adjustment factor d = (122.3 − 100) = 5.6
4
Thus, the adjusted C.R. of 1st quarter = 100
and for 2nd = 67.9 – 5.6 = 62.3
for 3rd = 54.5-2 × 5.6 = 43.3
for 4th = 45.2 – 3 × 5.6 = 28.4
The grand average of adjusted C.R., G =
100 + 62.3 + 43.3 + 28.4
= 58.5
4
250
The seasonal index of a quarter =
G
Merits and Demerits
This method is less complicated than the ratio to moving average and the
ratio to trend methods. However, this method is based upon the assumption
of a linear trend, which may not always hold true.
Deseasonalisation of Data
The deseasonalization of data implies the removal of the effect of seasonal
variations from the time series variable. If Y consists of the sum of various
components, then for its deaseasonalization, we subtract seasonal variations
from it. Similarly, in case of multiplicative model, the deseasonalisation is
done by taking the ratio of Y value to the corresponding seasonal index. A
clue to this is provided by the fact that the sum of seasonal indices is equal
to zero for an additive model while their sum is 400 or 1200 for a
multiplicative model.
It may be pointed out here that the deseasonalization of a data is done
under the assumption that the pattern of seasonal variations, computed on
the basis of past data, is similar to the pattern of seasonal variations in the
year of deseasonalization.
Example 7.19
Deseasonalise the following data on the sales of a company during various
months of 1990 by using their respective seasonal indices. Also interpret the
deseasonalised values.
Month
Jan
Sales
(Rs. ‘000)
16.5
S.I.
109
Feb
21.3
Mar
Month
Jul
Sales
(Rs. ‘000)
36.5
85
105
Aug
44.4
88
27.1
108
Sep
54.9
98
Apr
31.0
102
Oct
62.0
102
May
35.5
100
Nov
67.6
104
251
S.I.
Jun
36.3
89
Dec
78.7
110
Solution
Let Y denote monthly sales and DS denote the deseasonalised sales. Then,
we can write
DS =
Y
× 100
S .I
Computation of Deseasonalised Values
Month
S.I.
DS
Month
Sales (Y)
S.I.
DS
Jan
Sales
(Y)
16.5
109
15.14
Jul
36.5
85
42.94
Feb
21.3
105
20.29
Aug
44.5
88
50.45
Mar
27.1
108
25.09
Sep
54.9
98
56.02
Apr
31.0
102
30.39
Oct
62.0
102
60.78
May
35.5
100
35.50
Nov
67.6
104
65.00
Jun
36.3
89
40.79
Dec
78.7
110
71.55
The deseasonalised figures of sales for each month represent the monthly
sales that would have been in the absence of seasonal variations.
7.7. MEASUREMENT OF CYCLICAL VARIATIONS-RESIDUAL METHOD
As mentioned earlier that a typical time-series has four components: secular
trend (T), seasonal variation (S), cyclical variation (C), and irregular variation
(I). In a multiplicative time-series model, these components are written as:
Y=T×C×S×I
The deseasonalization data can be adjusted for trend analysis these by the
corresponding trend and seasonal variation values. Thus we are left with
only cyclical (C) and irregular (I) variations in the data set as shown below:
Y
T×S
=
T×C×S×I
T×S
252
=C×I
The moving averages of an appropriate period may be used to eliminate or
reduce the effect of irregular variations and thus left behind only the cyclical
variations.
The procedure of identifying cyclical variation is known as the residual
method. Recall that cyclical variations in time-series tend to oscillate above
and below the secular trend line for periods longer than one year. The steps
of residual method are summarized as follows:
(i)
Obtain seasonal indexes and deseasonalized data.
(ii)
Obtain
trend
values
and
expressed
seasonalized
data
as
percentages of the trend values.
(iii)
Divided the original data (Y) by the corresponding trend values (T)
in the time-series to get S × C × I. Further divide S × C × I by S to
get C × I.
(iv)
Smooth out irregular variations by using moving averages of an
appropriate period but of short duration, leaving only the cyclical
variation.
7.8. MEASUREMENT OF IRREGULAR VARIATIONS
Since irregular variations are random in nature, no particular procedure can
be followed to isolate and identify these variations. However, the residual
method can be extended one step further by dividing C × I by the cyclical
component (C) to identify the irregular component (I).
Alternately, trend (T), seasonal (S), and cyclical (C) components of the given
time-series are estimated and then the residual is taken as the irregular
variation. Thus, in the case of multiplicative time-series model, we have
Y
T×C×S
T×C×S×I
=
T×C×S
= I
where S and C are in fractional form and not in percentages.
7.9. QUESTIONS
1. What effect does seasonal variability have on a time-series? What is the
basis for this variability for an economic time-series?
253
2. What is measured by a moving average? Why are 4-quarter and 12month moving averages used to develop a seasonal index?
3. Briefly describe the moving average and least squares methods of
measuring trend in time-series.
4. Distinguish between ratio-to-trend and ratio-to-moving average as
methods of measuring seasonal variations, which is better and why?
5. Why do we deseasonalize data? Explain the ratio-to-moving average
method to compute the seasonal index.
6. Apply the method of link relatives to the following data and calculate
seasonal indexes.
Quarter
1995
1996
1997
1998
1999
I
6.0
5.4
6.8
7.2
6.6
II
6.5
7.9
6.5
5.8
7.3
III
7.8
8.4
9.3
7.5
8.0
IV
8.7
7.3
6.4
8.5
7.1
7. Calculate seasonal index numbers from the following data:
8.
Year
Ist Quarter
2nd Quarter
3rd Quarter
4th Quarter
1991
108
130
107
93
1992
86
120
110
91
1993
92
118
104
88
1994
78
100
94
78
1995
82
110
98
86
1996
106
118
105
98
For what purpose do we apply time series analysis to data collected over a period of
time?
What is the difference between a causal model and a time series model?
9.
10. Explain clearly the different components into which a time series may be
analysed. Explain any method for isolating trend values in a time series.
11.
Explain what you understand by time series. Why is time-series considered to be an
effective tool of forecasting?
254
12.
13.
Explain briefly the additive and multiplicative models of time series. Which of these
models is more popular in practice and why?
A company that manufactures steel observed the production of steel
(in metric tonnes) represented by the time-series:
Year
:
1990
1991 1992 1993
1994
1995
1996
65
80
85
95
Production in steel
:
60
72
75
(a) Find the linear equation that describes the trend in the production of
steel by the company.
(b) Estimate the production of steel in 1997.
14.
The sales (Rs. In lakh) of a company for the years 1990 to 1996 are
given below:
Year
Sales
:
:
1990
32
1991
47
1992
65
1993
88
1994
132
1995
190
1996
275
Find trend values by using the equation Yc = abx and estimate the value for
1997.
15.
A company that specializes in the production of petrol filters has
recorded the following production (in 1000 units) over the last 7 years.
Year
:
1994
1995 1996 1997
1998
1999
75
92
122
Production
:
42
49
62
(a) Develop a second degree estimating equation that best describes these data.
(b) Estimate the production in 2004.
2000
158
1. Spiegel, Murray R.: Theory and Practical of Statistics., London
McGraw Hill Book Company.
2. Yamane, T.: Statistics: An Introductory Analysis, New York, Harpered
Row Publication
3. R.P. Hooda: Statistic for Business and Economic, McMillan India Ltd.
4. G.C. Beri: Statistics for Mgt., TMH.
5. J.K. Sharma: Business Statistics, Pearson Education.
6. S.P. Gupta : Statistical Methods, Sultan Chand and Sons.
255
Course:
Author:
Anil Kumar
Course Code:
MC-106
Vetter:
Dr. Karam Pal
Lesson:
08
PROBABILITY THEORY
Objectives :
The present lesson is an attempt to overview the concept of
probability, thereby enabling the students to appreciate the
relevance of probability theory in decision-making under
conditions of uncertainty. After successful completion of the lesson
the students will be able to understand and use the different
approaches to probability as well as different probability rules for
calculating probabilities in different situations.
Structure
8.1
8.1
Introduction
8.2
Some Basic Concepts
8.3
Approaches to Probability Theory
8.4
Probability Rules
8.5
Bayes’ Theorem
8.6
Some Counting Concepts
8.7
Self-Assessment Questions
8.8
INTRODUCTION
Life is full of uncertainties. ‘Probably’, ‘likely’, ‘possibly’, ‘chance’ etc. is some of the most
commonly used terms in our day-to-day conversation. All these terms more or less convey
the same sense - “the situation under consideration is uncertain and commenting on the
256
future with certainty is impossible”. Decision-making in such areas is facilitated through
formal and precise expressions for the uncertainties involved. For example, product demand
is uncertain but study of demand spelled out in a form amenable for analysis may go a long to
help analyze, and facilitate decisions on sales planning and inventory management.
Intuitively, we see that if there is a high chance of a high demand in the coming year, we may
decide to stock more. We may also take some decisions regarding the price increase, reducing
sales expenses etc. to manage the demand. However, in order to make such decisions, we
need to quantify the chances of different quantities of demand in the coming year. Probability
theory provides us with the ways and means to quantify the uncertainties involved in such
situations.
A probability is a quantitative measure of uncertainty - a number that
conveys the strength of our belief in the occurrence of an uncertain event.
Since uncertainty is an integral part of human life, people have always been interested consciously or unconsciously - in evaluating probabilities.
Having its origin associated with gamblers, the theory of probability today is an indispensable
tool in the analysis of situations involving uncertainty. It forms the basis for inferential
statistics as well as for other fields that require quantitative assessments of chance
occurrences, such as quality control, management decision analysis, and almost all areas in
physics, biology, engineering and economics or social life.
8.2
SOME BASIC CONCEPTS
Probability, in common parlance, refers to the chance of occurrence of an event or happening.
In order that we are able to compute it, a proper understanding of certain basic concepts in
probability theory is required. These concepts are an experiment, a sample space, and an
event.
257
8.2.1
EXPERIMENT
An experiment is a process that leads to one of several possible outcomes. An
outcome of an experiment is some observation or measurement.
The term experiment is used in probability theory in a much broader sense than in physics or
chemistry. Any action, whether it is the drawing a card out of a deck of 52 cards, or reading
the temperature, or measurement of a product's dimension to ascertain quality, or the
launching of a new product in the market, constitute an experiment in the probability theory
terminology.
The experiments in probability theory have three things in common:
¾ there are two or more outcomes of each experiment
¾ it is possible to specify the outcomes in advance
¾ there is uncertainty about the outcomes
For example, the product we are measuring may turn out to be undersize or right size or
oversize, and we are not certain which way it will be when we measure it. Similarly,
launching a new product involves uncertain outcome of meeting with a success or failure in
the market.
A single outcome of an experiment is called a basic outcome or an elementary event. Any
particular card drawn from a deck is a basic outcome.
8.2.2
SAMPLE SPACE
The sample space is the universal set S pertinent to a given experiment. It is
the set of all possible outcomes of an experiment.
So each outcome is visualized as a sample point in the sample space. The sample spaces for
the above experiments are:
Experiment
Sample Space
Drawing a Card
{all 52 cards in the deck}
{all numbers in the range of temperatures}
Measurement of a Product's Dimension
{undersize, outsize, right size}
258
Launching of a New Product
8.2.3
{success, failure}
EVENT
An event, in probability theory, constitutes one or more possible outcomes of an experiment.
An event is a subset of a sample space. It is a set of basic outcomes. We say
that the event occurs if the experiment gives rise to a basic outcome belonging
to the event.
For the experiment of drawing a card, we may obtain different events A, B, and C like:
A
:
The event that card drawn is king of club
B
:
The event that card drawn is red
C
:
The event that card drawn is ace
In the first case, out of the 52 sample points that constitute the sample space, only one sample
point or outcome defines the event, whereas the number of outcomes used in the second and
third case is 13 and 4 respectively.
8.3
APPROACHES TO PROBABILITY THEORY
Three different approaches to the definition and interpretation of probability have evolved,
mainly to cater to the three different types of situations under which probability measures are
normally required. We will study these approaches with the help of examples of distinct types
of experiments.
Consider the following situations marked by three distinct types of experiments. The events
that we are interested in, within these experiments, are also given.
Situation I
Experiment
:
Drawing a Card Out of a Deck of 52 Cards
Event A
:
On any draw, a king is there
Experiment
:
Administering a Taste Test for a New Soup
Event B
:
A consumer likes the taste
Situation II
Situation III
259
Experiment
:
Commissioning a Solar Power Plant
Event C
:
The plant turns out to be a successful venture
Situation I : THE CLASSICAL APPROACH
The first situation is characterized by the fact that for a given experiment we have a sample
space with equally likely basic outcomes. When a card is drawn out of a well-shuffled deck,
every one of the cards (the basic outcomes) is as likely to occur as any other. This type of
situations, marked by the presence of "equally likely" outcomes, gave rise to the Classical
Approach to the probability theory. In the Classical Approach, probability of an event is
defined as the relative size of the event with respect to the size of the sample space. Since
there are 4 kings and there are 52 cards, the size of A is 4 and the size of the sample space is
52. Therefore, the probability of A is equal to 4/52.
The rule we use in computing probabilities, assuming equal likelihood of all basic outcomes,
is as follows:
Probability of the event A:
P(A) =
where
n( A)
N (S )
…………(8-1)
n(A) = the number of outcomes favorable to the event A
n(S) = total number of outcomes
Situation II : THE RELATIVE FREQUENCY APPROACH
If we try to apply the classical definition of probability in the second experiment, we find that
we cannot say that consumers will equally like the taste of the soup. Moreover, we do not
know as to how many persons have been tested. This implies that we should have the past
data on people who were administered the soup and the number that liked the taste. In the
absence of past data, we have to undertake an experiment, where we administer the taste test
on a group of people to check its effect.
260
The Relative Frequency Approach is used to compute probability in such cases. As per this
approach, the probability of occurrence of an event is given by the observed relative
frequency of an event in a very large number of trials. In other words, the probability of
occurrence of an event is the ratio of the number of times the event occurs to the total number
of trials. The probability of the event B:
P(B) =
Where
n
N
…………(8-2)
n = the number of times the event occurs
N = total number of trials
It is appreciated in this approach that, in order to take such a measure, we should have the
soup tested for a large number of people. In other words, the total number of trials in the
experiment should be very large.
Situation III : THE SUBJECTIVE APPROACH
The third situation seems apparently similar to the second one. We may be tempted here to
apply the Relative Frequency Approach. We may calculate the probability of the event that
the venture is a success as the ratio of number of successful ventures to the total number of
such ventures undertaken i.e. the relative frequency of successes will be a measure of the
probability.
However, the calculation here presupposes that either
(a) it is possible to do an experiment with such ventures, or
(b) that past data on such ventures will be available
In practice, a solar power plant being a relatively new development involving the latest
technology, past experiences are not available. Experimentation is also ruled out because of
high cost and time involved, unlike the taste testing situation. In such cases, the only way out
is the Subjective Approach to probability. In this approach, we try to assess the probability
from our own experiences. We may bring in any information to assess this. In the situation
261
cited, we may, perhaps, look into the performance of the commissioning authority in other
new and related technologies.
Therefore the Subjective Approach involves personal judgment, information, intuition, and
other subjective evaluation criteria. A physician assessing the probability of a patient's
recovery and an expert assessing the probability of success of a merger offer are both making
a personal judgment based upon what they know and feel about the situation. The area of
subjective probability - which is relatively new, having been first developed in the 1930s - is
somewhat controversial. One person's subjective probability may very well be different from
another person's subjective probability of the same event. We may note here that since the
assessment is a purely subjective one, it will vary from person to person and, therefore,
subjective probability is also called Personal Probability.
8.3.1
Three Approaches – A Comparative View
As already noted, the different approaches have evolved to cater to different kinds of
situations. So these approaches are not contradictory to one another. In fact, these
complement each other in the sense that where one fails, the other becomes applicable. These
are identical inasmuch as probability is defined as a ratio or a weight assigned to the
occurrence of an event. However, in contrast to the Subjective measure of the third approach,
the first two approaches - Classical and Relative Frequency - provide an objective measure
of probability in the sense that no personal judgment is involved.
We can bring out the commonality between the Classical Approach and the Relative
Frequency Approach with the help of an example. Let us assume that we are interested in
finding out the chances of getting a head in the toss of a coin. By now, you would have come
up with the answer by the Classical Approach, using the argument, that there are two
outcomes, heads and tails, which are equally likely. Hence, given that a head can occur only
once, the probability is ½ : Consider the following alternative line of argument, where the
262
probability can be estimated using the Relative Frequency Approach. If we toss the coin for a
sufficiently large number of times and note down the number of times the head occurs, the
proportion of times that a head occurs will give us the required probability.
Figure 8-1
P(H) = n/N→1/2
as
N→α
Thus, given our definition of the approaches, we find both the arguments to be valid. This
brings out, in a way, the commonality between the Relative Frequency and the Classical
Approach. The difference, however, is that the probability computed by using the Relative
Frequency Approach will be tending to be ½ with a large number of trials; moreover an
experiment is necessary in this case. In comparison, in the Classical Approach, we know
apriori that the chances are ½ , based on our assumption of "equally likely" outcomes.
Example 8-1
A fair coin is tossed twice. Find the probabilities of the following events:
(a)
(b)
B, getting one head and one tail
(c)
C, getting at least one head or one tail
(d)
Solution: Being a Two-Trial Coin Tossing Experiment, it gives rise to the following On = 2n
= 4, possible equally likely outcomes:
HH
HT
TH
263
TT
Thus, for the sample space
N(S) = 4
We can use the Classical Approach to find out the required probabilities.
(a)
For the event A, the number of favourable cases are:
n(A) = 1
{ HH }
So the required probability
P(A) =
=
(b)
n( A)
N (S )
1
4
For the event B, the number of favourable cases are:
n(B) = 2
{ HT, TH }
So the required probability
P(B) =
=
=
(c)
n(B )
N (S )
2
4
1
2
For the event C, the number of favourable cases are:
n(C) = 4
{ HH, HT, TH, TT }
So the required probability
P(C) =
=
n( A)
N (S )
4
4
=1
(d)
For the event D, the number of favourable cases are:
n(D) = 0
264
So the required probability
n(D )
N (S )
P(D) =
=
0
4
=0
It may be noted that the occurrence of C is certainty, whereas D is an impossible event.
Example 8-2
A newspaper boy wants to find out the chances that on any day he will be able to sell more
than 90 copies of The Times of India. From his dairy where he recorded the daily sales of the
last year, he finds out that out of 365 days, on 75 days he had sold 80 copies, on 144 days he
had sold 85 copies, on 62 days he had sold 95 copies and on 84 days he had sold 100 copies
of The Times of India. Find out the required probability for the newspaper boy.
Solution: Taking the Relative Frequency Approach, we find:
Sales(Event)
No. of Days (Frequency)
Relative Frequency
80
75
75/365
85
144
144/365
95
62
62/365
100
84
84/365
Thus, the number of days when his sales were more than 90 = (62 + 84) days = 146 days
So the required probability
P(Sales > 90) =
=
n
N
146
365
= 0.4
265
8.3.2
Probability Axioms
All the three approaches to probability theory share the same basic axioms. These axioms are
fundamental to probability theory and provide us with unified approach to probability.
The axioms are:
(a)
The probability of an event A, written as P(A), must be a number between zero
and one, both values inclusive. Thus
0 ≤ P(A) ≤ 1
(b)
…………(8-3)
The probability of occurrence of one or the other of all possible events is equal
to one. As S denotes the sample space or the set of all possible events, we
write
P(S) = 1.
…………(8-4)
Thus in tossing a coin once; P(a head or a tail) = 1.
(c)
If two events are such that occurrence of one implies that the other cannot
occur, then the probability that either one or the other will occur is equal to the
sum of their individual probabilities. Thus, in a coin-tossing situation, the
occurrence of a head rules out the possibility of occurrence of tail. These
events are called mutually exclusive events. In such cases then, if A and B are
the two events respectively, then
P (A or B) = P (A) + P (B)
It follows from the last two axioms that if two mutually exclusive events form the sample
space of the experiment, then
P(A or B) = P(A) + P(B) = 1; thus P (Head) + P (Tail) = 1
If two or more events together define the total sample space, the events are said to be
collectively exhaustive.
Given the above axioms, we may now define probability as a function, which assigns
probability value P to each sample point of an experiment abiding by the above axioms.
Thus, the axioms themselves define probability.
266
8.3.3
Interpretation of a Probability
From our discussion so far, we can give a general definition of probability:
Probability is a measure of uncertainty. The probability of event A is a
quantitative measure of the likelihood of the event's occurring.
We have also seen that 0 and 1, both values inclusive, sets the range of values that the probability measure may take. In other words 0 ≤ P(A) ≤ 1
When an event cannot occur (impossible event), its probability is zero. The probability of the
empty set is zero: P(Φ) = 0. In a deck where half the cards are red and half are black, the
probability of drawing a green card is zero because the set corresponding to that event is the
empty set: there are no green cards.
Events that are certain to occur have probability 1.00. The probability of the entire sample
space S is equal to 1.00: P(S) = 1.00. If we draw a card out of a deck, 1 of the 52 cards in the
deck will certainly be drawn, and so the probability of the sample space, the set of all 52
cards, is equal to 1.00.
Within the range of values 0 to 1, the greater the probability, the more confidence we have in
the occurrence of the event in question. A probability of 0.95 implies a very high confidence
in the occurrence of the event. A probability of 0.80 implies a high confidence. When the
probability is 0.5, the event is as likely to occur as it is not to occur. When the probability is
0.2, the event is not very likely to occur. When we assign a probability of 0.05, we believe
the event is unlikely to occur, and so on. Figure 8-2 is an informal aid in interpreting
probability.
267
Figure 8-2
Interpretation of a Probability
Note that probability is a measure that goes from 0 to 1. In everyday conversation we often
describe probability in less formal terms. For example, people sometimes talk about odds. If
the odds are 1 to 1, the probability is
1
1
i .e .
3
1 + 2
1
1
i .e .
2
1 + 1
; if the odds are 1 to 2, the probability is
; and so on. Also, people sometimes say, "The probability is 80 percent."
Mathematically, this probability is 0.80.
8.4
PROBABILITY RULES
We have seen how to compute probabilities in certain situations. The nature of the events
were relatively simple, so that direct application of the definition of probability could be used
for computation. Quite often, we are interested in the probability of occurrence of more
complex events. Consider for example, that you want to find the probability that a king or a
club will occur in a draw from a deck of 52 cards. Similarly, on examining couples with two
children, if one child is known as a boy, you may be interested in the probability of the event
of both the children being boys. These two situations, we find, are not as simple as those
discussed in the earlier section. As a sequel to the theoretical development in the field of
probability, certain results are available which help us in computing probabilities in such
situations. Now we will explore these results through examples.
8.4.1
THE UNION RULE
A very important rule in probability theory, the Rule of Unions (also called Addition
Theorem) allows us to write the probability of the union of two events in terms of the
probabilities of the two events and the probability of their intersection.
Consider two events A and B defined over the sample space S, as shone in Figure 8-3
268
Figure 8-3
Two Overlapping Events A and B
We may define
P(A ∪ B) =
n( A ∪ B )
N (S )
=
n( A) + n(B ) − n( A ∩ B )
N (S )
=
n( A)
n(B )
n( A ∩ B )
+
−
N (S )
N (S )
N (S )
= P ( A) + P (B ) − P ( A ∩ B )
Thus, the rule of unions is:
P ( A ∪ B ) = P ( A) + P (B ) − P ( A ∩ B )
…………(8-5)
The probability of the intersection of two events P ( A ∩ B ) is called their joint probability.
The meaning of this rule is very simple and intuitive: When we add the probabilities of A and
B, we are measuring, or counting, the probability of their intersection twice—once when
measuring the relative size of A within the sample space and once when doing this with B.
Since the relative size, or probability, of the intersection of the two sets is counted twice, we
subtract it once so that we are left with the true probability of the union of the two events.
The rule of unions is especially useful when we do not have the sample space for the union of
events but do have the separate probabilities.
Example 8-3
A card is drawn from a well-shuffled pack of playing cards. Find the probability that the card
drawn is either a club or a king.
269
Solution: Let A be the event that a club is drawn and B the event that a king is drawn. Then,
P ( A ∪ B ) = P ( A) + P (B ) − P ( A ∩ B )
= 13/52 + 4/52 – 1/52
= 16/52
= 4/13
Example 8-4
Suppose your chance of being offered a certain job is 0.45, your probability of getting
another job is 0.55, and your probability of being offered both jobs is 0.30. What is the
probability that you will be offered at least one of the two jobs?
Solution: Let A be the event that the first job is offered and B the event that the second job is
offered. Then,
P ( A ) = 0 . 45
P ( B ) = 0 . 55
and P ( A ∩ B ) = 0 . 30
So, the required probability is given as:
P ( A ∪ B ) = P ( A) + P (B ) − P ( A ∩ B )
= 0.45 + 0.55 – 0.30
= 0.70
Mutually Exclusive Events
When the sets corresponding to two events are disjoint (i.e., have no intersection), the two
events are called mutually exclusive (see Figure 8-4).
Figure 8-4
Two Mutually Exclusive Events A and B
270
For mutually exclusive events, the probability of the intersection of the events is zero. This is
so because the intersection of the events is the empty set, and we know that the probability of
the empty set is zero.
For mutually exclusive events A and B:
P(A ∩ B) = 0
…………(8.6)
This fact gives us a special rule for unions of mutually exclusive events. Since the probability
of the intersection of the two events is zero, there is no need to subtract P ( A ∩ B ) when
the probability of the union of the two events is computed. Therefore,
For mutually exclusive events A and B:
P ( A ∪ B ) = P ( A) + P (B )
…………(8.7)
This is not really a new rule since we can always use the rule of unions for the union of two
events: If the events happen to be mutually exclusive, we subtract zero as the probability of
the intersection.
Example 8-5
A card is drawn from a well-shuffled pack of playing cards. Find the probability that the card
drawn is either a king or a queen.
Solution: Let A be the event that a king is drawn and B the event that a queen is drawn. Since
A and B are two mutually exclusive events, we have,
P ( A ∪ B ) = P ( A) + P (B )
= 4/52 + 4/52
= 8/52
= 2/13
We can extend the Rule of Unions to three (or more) events. Let A, B, and C be the three
events defined over the sample space S, as shown in Figure 8-5
Then, the Rule of Unions is
271
P(A ∪ B ∪ C ) =
P ( A ) + P ( B ) + P (C ) − P ( A ∩ B ) − P ( B ∩ C ) − P ( A ∩ C ) + P ( A ∩ B ∩ C )
…………(8.8)
Figure 8-5
Three Overlapping Events A, B and C
When the three events are mutually exclusive (see Figure 8-6), the Rule of Unions is
P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P (C )
Figure 8-6
…………(8.9)
Three Mutually Exclusive Events A, B and C
Example 8-6
A card is drawn from a well-shuffled pack of playing cards. Find the probability that the card
drawn is
(a)
either a heart or an honour or king
(b)
either an ace or a king or a queen
Solution: (a) Let A be the event that a heart is drawn, B the event that an honour is drawn
and C the event that a king is drawn. So we have
n(A) = 13
n(B) = 20
n( A ∩ B ) = 5
and
n(C) = 4
n(B ∩ C ) = 4
n( A ∩ B ∩ C ) = 1
272
n( A ∩ C ) = 1
The required probability (using Eq. (8.8) is
P ( A ∪ B ∪ C ) = 13/52 + 20/52 + 4/52 – 5/52 – 4/52 – 1/52 +1/52
= 28/52
= 7/13
(b)
Let A be the event that an ace is drawn, B the event that a king is drawn and C the
event that a queen is drawn. So we have
n(A) = 4
n(B) = 4
n(C) = 4
Since A, B and C are mutually exclusive events, the required probability (using Eq. (8.9) is
P ( A ∪ B ∪ C ) = 4/52 + 4/52 + 4/52
= 12/52
= 3/13
8.4.2
THE COMPLEMENT RULE
The Rule of Complements defines the probability of the complement of an event in terms of
the probability of the original event. Consider event A defined over the sample space S. The
complement of set A, denoted by A , is a subset, which contains all outcomes, which do not
belong to A (see Figure 8-7).
Figure 8-7
In other words
Complement of an Event
A+ A =S
so
P(A + A ) = P(S)
or
P(A) + P( A ) = 1
or
P( A ) = 1 - P(A)
…………(8.10)
273
Eq. (8.10) is our Rule of Complements. As a simple example, if the probability of rain
tomorrow is 0.3, then the probability of no rain tomorrow must be 1 - 0.3 = 0.7. If the
probability of drawing a king is 4/52, then the probability of the drawn card's not being a
king is 1 - 4/52 = 48/52.
Example 8-7
Find the probability of the event of getting a total of less than 12 in the experiment of
throwing a die twice.
Solution: Let A be the event of getting a total 12.
Then we have,
A = {6,6}
and
P(A) = 1/36
The event of getting a total of less than 12 is the complement of A, so the required probability
is
P( A ) = 1 - P(A)
P( A ) = 1 – 1/36
P( A ) = 35/36
8.4.3
THE CONDITIONAL PROBABILITY RULE
As a measure of uncertainty, probability depends on information. We often face situations
where the probability of an event A is influenced by the information that another event B has
occurred. Thus, the probability we would give the event "Xerox stock price will go up
tomorrow" depends on what we know about the company and its performance; the
probability is conditional upon our information set. If we know much about the company, we
may assign a different probability to the event than if we know little about the company. We
may define the probability of event A conditional upon the occurrence of event B. In this
example, event A may be the event that the stock will go up tomorrow, and event B may be a
favorable quarterly report.
274
Consider two events A and B defined over the sample space S, as shown in Figure 8-8
Figure 8-8
Conditional Probability of Event A
Thus, the probability of event A given the occurrence of event B is
P(A / B) =
n( A ∩ B )
n(B)
n( A ∩ B )
P(A / B) =
P(A / B) =
n(B)
N
N
P(A ∩ B)
P(B)
…………(8.11)
The vertical line in P ( A / B ) is read given, or conditional upon.
Therefore, the probability of event A given the occurrence of event B is defined as the
probability of the intersection of A and B, divided by the probability of event B.
Example 8-8
For an experiment of throwing a die twice, find the probability:
(a)
of the event of getting a total of 9, given that the die has shown up points between
4 and 6 (both inclusive)
(b)
of the event of getting points between 4 and 6 (both inclusive), given that a total
of 9 has already been obtained
Solution: Let getting a total 9 be the event A and the die showing points between 4 and 6
(both inclusive) be the event B
Thus, N(S) = 36 and A = {(3,6) (4,5) (5,4) (6,3)}
275
B = {(4,4) (4,5) (4,6) (5,4) (5,5) (5,6) (6,4) (6,5) (6,6)}
and P ( A ∩ B ) = {(4,5) (5,4)}
So n(A) = 4
n( A ∩ B ) = 2
n(B) = 9
So the required probabilities are
(a)
(b)
8.4.4
P(A / B) =
P(A ∩ B)
P(B)
P(A / B) =
2 / 36
9 / 36
P(A / B) =
2
9
P (B / A) =
P (B ∩ A)
P ( A)
P (B / A) =
2 / 36
4 / 36
P (B / A) =
1
2
THE PRODUCT RULE
The Product Rule (also called Multiplication Theorem) allows us to write the probability of
the simultaneous occurrence of two (or more) events.
In the conditional probability rules
and
P(A / B) =
P(A ∩ B)
P(B)
P (B / A) =
P (B ∩ A)
P ( A)
A ∩ B or B ∩ A is
the event A and B occur simultaneously. So rearranging the
conditional probability rules, we have our Product Rule
P ( A ∩ B ) = P ( A / B ). P ( B )
and
P ( A ∩ B ) = P ( B / A ). P ( A )
…………(8.12)
276
The Product Rule states that the probability that both A and B will occur simultaneously is
equal to the probability that B (or A) will occur multiplied by the conditional probability that
A (or B) will occur, when it is known that B (or A) is certain to occur or has already
occurred.
Example 8-9
A box contains 10 balls out of which 2 are green, 5 are red and 3 are black. If two balls are
drawn at random, one after the other without replacement, from the box. Find the
probabilities that:
(a)
both the balls are of green color
(b)
both the balls are of black color
(c)
both the balls are of red color
(d)
the first ball is red and the second one is black
(e)
the first ball is green and the second one is red
Solution:
(a)
P ( G 1 ∩ G 2 ) = P ( G 2 / G 1 ). P ( G 1 )
1
2
x
9 10
1
=
45
=
(b)
P ( B 1 ∩ B 2 ) = P ( B 2 / B 1 ). P ( B 1 )
2
3
x
9 10
1
=
15
=
(c)
P ( R 1 ∩ R 2 ) = P ( R 2 / R 1 ). P ( R 1 )
4
5
x
9 10
2
=
9
=
(d)
P ( R 1 ∩ B 2 ) = P ( B 2 / R 1 ). P ( R 1 )
277
3
5
x
9 10
1
=
6
=
P ( G 1 ∩ R 2 ) = P ( R 2 / G 1 ). P ( G 1 )
(e)
5
2
x
9 10
1
=
9
=
Example 8-10
A consulting firm is bidding for two jobs, one with each of two large multinational
corporations. The company executives estimate that the probability of obtaining the
consulting job with firm A, event A, is 0.45. The executives also feel that if the company
should get the job with firm A, then there is a 0.90 probability that firm B will also give the
company the consulting job. What are the company's chances of getting both jobs?
Solution: We are given P(A) = 0.45. We also know that P(B / A) = 0.90, and we are looking
for P ( A ∩ B ) , which is the probability that both A and B will occur.
So
P ( A ∩ B ) = P ( B / A ). P ( A )
P ( A ∩ B ) = 0 . 90 x 0 . 45
= 0 . 405
Independent Events
Two events are said to be independent of each other if the occurrence or non-occurrence of
one event in any trial does not affect the occurrence of the other event in any trial. Events A
and B are independent of each other if and only if the following three conditions hold:
Conditions for the independence of two events A and B:
and
P ( A / B ) = P ( A)
…………(8.13a)
P (B / A) = P (B )
…………(8.13b)
P ( A ∩ B ) = P ( A ). P ( B )
…………(8.14)
278
The first two equations have a clear, intuitive appeal. The top equation says that when A and
B are independent of each other, then the probability of A stays the same even when we know
that B has occurred - it is a simple way of saying that knowledge of B tells us nothing about
A when the two events are independent. Similarly, when A and B are independent, then
knowledge that A has occurred gives us absolutely no information about B and its likelihood
of occurring.
The third equation, however, is the most useful in applications. It tells us that when A and B
are independent (and only when they are independent), we can obtain the probability of the
joint occurrence of A and B (i.e. the probability of their intersection) simply by multiplying
the two separate probabilities. This rule is thus called the Product Rule for Independent
Events.
As an example of independent events, consider the following: Suppose I roll a single die.
What is the probability that the number 5 will turn up? The answer is 1/6. Now suppose that I
told you that I just tossed a coin and it turned up heads. What is now the probability that the
die will show the number 5? The answer is unchanged, 1/6, because events of the die and the
coin are independent of each other. We see that P ( 6 / H ) = P ( 6 ) , which is the first rule
above.
The rules for union and intersection of two independent events can be extended to sequences
of more than two events.
Intersection Rule
The probability of the intersection of several independent events A1, A2, ……is just the
product of separate probabilities i.e.
P ( A1 ∩ A 2 ∩ A 3 ) = P ( A1 ). P ( A 2 ). P ( A 3 ).........
Union Rule
279
…………(8.15)
The probability of the union of several independent events A1, A2, ……is given by the
following equation
P ( A1 ∪ A 2 ∪ A 3 ∪ .........) = 1 − P ( A1 ). P ( A 2 ). P ( A 3 ).........
…………(8.16)
The union of several events is the event that at least one of the events happens.
Example 8-11
A problem in mathematics is given to five students A, B,C, D and E. Their chances of solving
it are 1/2, 1/3, 1/3, 1/4 and 1/5 respectively. Find the probability that the problem will
(a)
not be solved
(b)
be solved
Solution: (a) The problem will not be solved when none of the students solve it. So the
required probability is:
P ( problem
will not be solved ) = P ( A ). P ( B ). P ( C ). P ( D ). P ( E )
= (1 − 1 / 2 ).( 1 − 1 / 3 ).( 1 − 1 / 3 ).( 1 − 1 / 4 ).( 1 − 1 / 5 )
= 2 / 15
(b)
The problem will be solved when at least one of the students solve it. So the required
probability is:
P ( A ∪ B ∪ C ∪ D ∪ E ) = 1 − P ( A ). P ( B ). P ( C ). P ( D ). P ( E )
= 1 − 2 / 15
= 13 / 15
8.5
BAYES’ THEOREM
As we have already noted in the introduction, the basic objective behind calculating
probabilities is to help us in making decisions by quantifying the uncertainties involved in the
situations. Quite often, whether it is in our personal life or our work life, decision-making is
an ongoing process. Consider for example, a seller of winter garments, who is interested in
the demand of the product. In deciding on the amount he should stock for this winter, he has
280
computed the probability of selling different quantities and has noted that the chance of
selling a large quantity is very high. Accordingly, he has taken the decision to stock a large
quantity of the product. Suppose, when finally the winter comes and the season ends, he
discovers that he is left with a large quantity of stock. Assuming that he is in this business, he
feels that the earlier probability calculation should be updated given the new experience to
help him decide on the stock for the next winter.
Similar to the situation of the seller of winter garment, situations exist where we are
interested in an event on an ongoing basis. Every time some new information is available, we
do revise our odds mentally. This revision of probability with added information is
formalised in probability theory with the help of famous Bayes' Theorem. The theorem,
discovered in 1761 by the English clergyman Thomas Bayes, has had a profound impact on
the development of statistics and is responsible for the emergence of a new philosophy of
science. Bayes himself is said to have been unsure of his extraordinary result, which was
presented to the Royal Society by a friend in 1763 - after Bayes' death. We will first
understand The Law of Total Probability, which is helpful for derivation of Bayes' Theorem.
8.5.1
The Law of Total Probability
Consider two events A and B. Whatever may be the relation between the two events, we can
always say that the probability of A is equal to the probability of the intersection of A and B,
plus the probability of the intersection of A and the complement of B (event B ).
P ( A) = P ( A ∩ B ) + P ( A ∩ B )
or
P ( A ) = P ( A / B ). P ( B ) + P ( A / B ). P ( B )
…………(8.17)
The sets B and B form a partition of the sample space. A partition of a space is the division
of the sample space into a set of events that are mutually exclusive (disjoint sets) and cover
the whole space. Whatever event B may be, either B or B must occur, but not both. Figure
8-9 demonstrates this situation and the law of total probability.
281
Figure 8-9
Total Probability of Event A
The law of total probability may be extended to more complex situations, where the sample
space X is partitioned into more than two events. Say, we have partition of the space into a
collection of n sets B1, B2,………Bn .The law of total probability in this situation is:
P ( A) =
n
∑
i =1
or
P ( A) =
P ( A ∩ Bi )
n
∑
i =1
P ( A / B i ). P ( B i )
…………(8.18)
Figure 8-10 shows the partition of a sample space into five events B1, B2, B3, B4 and B5 ; and
shows their intersections with set A.
Figure 8-10
Total Probability of Event A
We can demonstrate the rule with a more specific example. Let us define A as the event that
an honour card is drawn out of a deck of 52 cards (the honour cards are the aces, kings,
queens, jacks and 10). Letting H, C, D, and S denote the events that the card drawn is a heart,
club, diamond, or spade, respectively, we find that the probability of an honour card is:
282
Figure 8-11
Total Probability of Event A: An Honour Card
P ( A) = P ( A ∩ H ) + P ( A ∩ C ) + P ( A ∩ D ) + P ( A ∩ S )
= 5/52 + 5/52 + 5/52 + 5/52
= 20/52
= 5/13
which is what we know the probability of an honour card to be just by counting 20 honour
cards out of a total of 52 cards in the deck. The situation is shown in Figure 8-11.
As can be seen from the figure, the event A is the set addition of the intersections of A with
each of the four sets H, D, C, and S.
Example 8-12
A market analyst believes that the stock market has a 0.70 probability of going up in the next
year if the economy should do well, and a 0.20 probability of going up if the economy should
not do well during the year. The analyst believes that there is a 0.80 probability that the
economy will do well in the coming year. What is the probability that stock market will go up
next year?
Solution: Let U be the event that the stock market will go and W is the event that the
economy will do well in the coming year.
Then
P (U ) = P (U / W ). P (W ) + P (U / W ). P (W )
= ( 0 . 70 )( 0 . 80 ) + ( 0 . 20 )( 0 . 20 )
= 0 . 56 + 0 . 04
.
= 0 . 60
BAYES’ THEOREM
We will now develop the Bayes’ theorem. Bayes' theorem is easily derived from the law of
total probability and the definition of conditional probability.
By definition of conditional probability, we have
283
P (B / A) =
P (B ∩ A)
P ( A)
…………(8.19)
By product rule, we have
P ( B ∩ A ) = P ( A ∩ B ) = P ( A / B ). P ( B )
…………(8.20)
Substituting Eq.(8.19) in Eq.(8.20), we have
P (B / A) =
P ( A / B ). P ( B )
P ( A)
…………(8.21)
By the law of total probability, we have
P ( A ) = P ( A / B ). P ( B ) + P ( A / B ). P ( B )
Substituting this expression for P(A) in the denominator of Eq.(8.21), we have the Bayes’
theorem
P (B / A) =
P ( A / B ). P ( B )
P ( A / B ). P ( B ) + P ( A / B ). P ( B )
…………(8.22)
Thus the theorem allows us to reverse the conditionality of events: we can obtain the
probability of B given A from the probability of A given B(and other information).
As we see from the theorem, the probability of B given A is obtained from the probabilities
of B and B and from the conditional probabilities of A given B and A given B .
The probabilities P(B) and P( B ) are called prior probabilities of the events B and B ; the
probability P(B /A) is called the posterior probability of B. It is possible to write Bayes'
theorem in terms of B and A, thus giving the posterior probability of B , P( B /A). Bayes'
theorem may be viewed as a means of transforming our prior probability of an event B into a
posterior probability of the event B - posterior to the known occurrence of event A.
The Bayes' theorem can be extended to a partition of more than two sets. This is done by
using the law of total probability involving a partition in sets B1, B2, ……… Bn .The resulting
form of Bayes' theorem is:
284
P (Bi / A) =
P ( A / B i ). P ( B i )
n
∑
i =1
…………(8.23)
P ( A / B i ). P ( B i )
The theorem gives the probability of one of the sets in the partition Bi, given the occurrence of
event A.
Example 8-13
An Economist believes that during periods of high economic growth, the Indian Rupee
appreciates with probability 0.70; in periods of moderate economic growth, it appreciates
with probability 0.40; and during periods of low economic growth, the Rupee appreciates
with probability 0.20.During any period of time the probability of high economic growth is
0.30; the probability of moderate economic growth is 0.50 and the probability of low
economic growth is 0.20. Suppose the Rupee value has been appreciating during the present
period. What is the probability that we are experiencing the period of (a) high, (b) moderate,
and (c) low, economic growth?
Solution: Our partition consists of three events: high economic growth (event H), moderate
economic growth (event M) and low economic growth (event L). The prior probabilities of
these events are:
P(H) = 0.30
P(M) = 0.50
P(L) = 0.20
Let A be the event that the rupee appreciates. We have the conditional probabilities
P(A / H) = 0.70
P(A / M) = 0.40
P(A / L) = 0.20
By using the Bayes’ theorem we can find out the required probabilities
P(H /A), P(M / A) and P(L / A)
(a)
P(H /A)
285
P ( A / H ). P ( H )
P ( A / H ). P ( H ) + P ( A / M ). P ( M ) + P ( A / L ). P ( L )
( 0 . 70 )( 0 . 30 )
=
( 0 . 70 )( 0 . 30 ) + ( 0 . 40 )( 0 . 50 ) + ( 0 . 20 )( 0 . 20 )
= 0 . 467
P (H / A) =
(b)
P(M /A)
P ( A / M ). P ( M )
P ( A / H ). P ( H ) + P ( A / M ). P ( M ) + P ( A / L ). P ( L )
( 0 . 40 )( 0 . 50 )
=
( 0 . 70 )( 0 . 30 ) + ( 0 . 40 )( 0 . 50 ) + ( 0 . 20 )( 0 . 20 )
= 0 . 444
P (M / A) =
(c)
P(L /A)
P ( A / L ). P ( L )
P ( A / H ). P ( H ) + P ( A / M ). P ( M ) + P ( A / L ). P ( L )
( 0 . 20 )( 0 . 20 )
=
( 0 . 70 )( 0 . 30 ) + ( 0 . 40 )( 0 . 50 ) + ( 0 . 20 )( 0 . 20 )
= 0 . 089
P (L / A) =
8.6
SOME COUNTING CONCEPTS
If there are n events and event i can occur in Ni possible ways, then the number of ways in
which the sequence of n events may occur is
N1. N2. N3 .……….Nn
…………(8.24)
Suppose that a bank has two branches, each branch has two departments, and each
department has four employees. Then there are (2)(2)(4) choices of employees, and the
probability that a particular one will be randomly selected is 1/(2)(2)(4) = 1/16.
We may view the choice as done sequentially: First a branch is randomly chosen, then a
department within the branch, and then the employee within the department. This is
demonstrated in the tree diagram in Figure 8-12.
286
Figure 8-12
Tree Diagram
For any positive integer n, we define n factorial as
n(n- 1)(n- 2) ………1
…………(8.25)
We denote n factorial by n!. The number n! is the number of ways in which n objects can be
ordered. By definition, 0! = 1.
For example, 5! is the number of possible arrangements of five objects. We have 5! = (5) (4)
(3) (2) (1) = 120. Suppose that five applications arrive at a center on the same day, all written
at different times. What is the probability that they will be read in the order in which they
were written? Since there are 120 ways to order five applications, the probability of a
particular order (the order in which the applications were written) is 1/120.
Permutations are the possible ordered selections of r objects out of a total of n objects. The
number of permutations of n objects taken r at a time is denoted by n Pr
n
Pr =
n!
( n − r )!
…………(8.26)
Suppose that 4 people are to be randomly chosen out of 10 people who agreed to be
interviewed in a market survey. The four people are to be assigned to four interviewers. How
many possibilities are there? The first interviewer has 10 choices, the second 9 choices, the
third 8, and the fourth 7. Thus, there are (10)(9)(8)(7) = 5,040 selections. We can see that this
is equal to n(n - l)(n - 2) ……… (n - r + 1), which is equal to n Pr =
287
n!
!.
( n − r )!
If choices are made randomly, the probability of any predetermined assignment of 4 people
out of a group of 10 is 1/5,040.
Combinations are the possible selections of r items from a group of n items regardless of the
order of selection. The number of combinations is denoted by n C r and is read n choose r.
We define the number of combinations of r out of n elements as
n
Cr =
n!
r ! ( n − r )!
…………(8.27)
Suppose that 3 out of the 10 members of the board of directors of a large corporation are to
be randomly selected to serve on a particular task committee. How many possible selections
are there? Using Eq. (8.27), we find that the number of combinations is n C r =
n!
=
r ! ( n − r )!
10!/(3!7!) = 120.
If the committee is chosen in a truly random fashion, what is the probability that the threecommittee members chosen will be the three senior board members? This is 1 combination
out of a total of 120, so the answer is 1/120 = 0.00833.
8.7
SELF-ASSESSMENT QUESTIONS
1.
Explain what do you understand by the term ‘probability’. How is the concept of
probability is relevant to decision making under uncertainty?
2.
What are different approaches to the definition of probability? Are these approaches
contradictory to one another? Which of these approaches you will apply for
calculating the probability that:
3.
(a)
A leap year selected at random, will contain 53 Monday.
(b)
An item, selected at random from a production process, is defective.
(c)
Mr. Bhupinder S. Hooda will win the assembly election from Kiloi.
With the help of an example explain the meaning of the following:
288
4.
(a)
Random experiment, and sample space
(b)
An event as a subset of sample space
(c)
Equally likely events
(d)
Mutually exclusive events.
(e)
Exhaustive events
(f)
Elementary and compound events.
A proofreader is interested in finding the probability that the number of mistakes in a
page will be less than 10. From his past experience he finds that out of 3600 pages he
has proofed, 200 pages contained no errors, 1200 pages contained 5 errors, and 2200
pages contained 11 or more errors. Can you help him in finding the required
probability?
5.
State and develop the Addition Theorem of probability for:
(a)
mutually exclusive events
(b)
overlapping events
(c)
complementary events
5.
Explain the concept of conditional probability with the help of a suitable example.
6.
State and develop the Multiplication Theorem of probability for:
7.
(a)
dependent events
(b)
independent event
State the Bayes’ Theoram of probability. Using an appropriate example, develop the
Bayesian probability rule and generalize it.
8.
What do you understand by permutations and combinations?
(a)
In how many ways we can select three players out of 12 players of the
Indian Cricket team, for playing in the World XI team?
(b)
In how many ways can a sub-committee of 2 out of 6 members of the
executive committee of the employees’ association be constituted?
9.
What is the probability that a non leap year, selected at random, will contain
289
(a)
10.
52 Sundays? (b)
53 Sundays?
(c)
54 Sundays?
A card is drawn at random from well shuffled deck of 52 cards, find the probability
that
11.
(a)
the card is either a club or diamond
(b)
the card is not a king
(c)
the card is either a face card or a club card.
From a well-shuffled deck of 52 cards, two cards are drawn at random.
(a)
If the cards are drawn simultaneously, find the probability that these consists
of (i) both clubs, (ii) a king and a queen, (iii) a face card and a 8.
(b)
If the cards are drawn one after the other with replacement. Find the
probability that these consists of (i) both clubs, (ii) a king and a queen, (iii) a
face card and a 8.
12.
A problem in mathematics is given to four students A, B,C, and D their chances of
solving it are 1/2 , 1/3, 1/4 and 1/5 respectively. Find the probability that the problem
will
13.
(a)
be solved
(b)
not be solved
The odds that A speaks the truth are 3:2 and the odds that B does so are 7:3. In what
percentage of cases are they likely to
14.
(a)
contradict each other on an identical point?
(b)
agree each other on an identical point?
Among the sales staff engaged by a company 60% are males. In terms of their
professional qualifications, 70% of males and 50% of females have a degree in
marketing. Find the probability that a sales person selected at random will be
(a)
a female with degree in marketing
(b)
a male without degree in marketing
290
15.
A and B play for a prize of Rs. 10,000. A is to throw a die first and is to win if he
throws 1: If A fails, B it to throw and is to win if he throws 2 or 1. If B fails, A is to
throw again and to win if he throws 3, 2 or 1: and so on. Find their respective
expectations.
16.
A factory has three units A, B, and C. Unit A produces 50% of its products, and units
B and C each produces 25% of the products. The percentage of defective items
produced by A, B, and C units are 3%, 2% and 1%, respectively. If an item is selected
at random from the total production of the factory is found defective, what is the
probability that it is produced by:
(a)
8.8
Unit A
(b)
Unit B
(c)
Unit C
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
291
Course:
Author:
Anil Kumar
Course Code:
MC-106
Vetter:
Dr. Karam Pal
Lesson:
09
PROBABILITY DISTRIBUTIONS-I
Objectives: The overall objective of this lesson is to discuss the concept of random
variable
and
discrete
probability
distributions.
After
successful
completion of the lesson the students will be able to appreciate the
usefulness of probability distributions in decision-making and also
identify situations where Binomial and Poisson probability distributions
can be applied.
Structure
9.1
9.1
Introduction
9.2
Discrete Probability Distribution
9.3
Bernoulli Random Variable
9.4
The Binomial Distribution
9.5
The Poisson Distribution
9.6
Self-Assessment Questions
9.7
INTRODUCTION
In many situations, our interest does not lie in the outcomes of an experiment as such; we
may find it more useful to describe a particular property or attribute of the outcomes of an
experiment in numerical terms. For example, out of three births; our interest may be in the
292
matter of the probabilities of the number of boys. Consider the sample space of 8 equally
likely sample points.
GGG
GGB
GBG
BGG
GBB
BGB
BBG
BBB
Now look at the variable “the number of boys out of three births”. This number varies
among sample points in the sample space and can take values 0,1,2,3, and it is random –given
to chance.
“A random variable is an uncertain quantity whose value depends on chance.”
A random variable may be…
¾ Discrete if it takes only a countable number of values. For example, number of dots
on two dice, number of heads in three coin tossing, number of defective items,
number of boys in three births and so on.
¾ Continuous if can take on any value in an interval of numbers (i.e. its possible values
are unaccountably infinite). For example, measured data on heights, weights,
temperature, and time and so on.
A random variable has a probability law - a rule that assigns probabilities to different values
of the random variable. This probability law - the probability assignment is called the
probability distribution of the random variable. We usually denote the random variable by X.
In this lesson, we will discuss discrete probability distributions. Continuous probability
distributions will be discussed in the next lesson.
9.2
DISCRETE PROBABILITY DISTRIBUTION
The random variable X denoting “the number of boys out of three births”, we introduced in
the introduction of the lesson, is a discrete random variable; so it will have a discrete
probability distribution. It is easy to visualize that the random variable X is a function of
sample space. We can see the correspondence of sample points with the values of the random
variable as follows:
293
BBB
GGB
GBG
(X=0)
BGG
(X=1)
GBB
BGB
BBG
(X=2)
BBB
(X=3)
The correspondence between sample points and the value of the random variable allows us to
determine the probability distribution of X as follows:
P(X=0) = 1/8
since one out of 8 equally likely points leads to X = 0
P(X=1) = 3/8
since three out of 8 equally likely points leads to X = 1
P(X=2) = 3/8
since three out of 8 equally likely points leads to X = 2
P(X=3) = 1/8
since one out of 8 equally likely points leads to X = 3
The above probability statement constitute the probability distribution of the random variable
X = number of boys in three births. We may appreciate how this probability law is obtained
simply by associating values of X with sets in the sample space. (For example, the set GGB,
GBG, BGG leads to X = 1). We may write down the probability distribution of X in table
format (see Table 9-1) or we may plot it graphically by means of probability Histogram (see
Figure 9-1a) or a Line chart (see Figure 9-1b).
Table 9-1
Probability Distribution of the Number of Boys out of Three Births
No. of Boys X
Probability P(X)
0
1
2
3
1/8
3/8
3/8
1/8
294
(1,
(2,
0.375)
0.375)
(1, 0.375)
P(X)
P(X)
(2, 0.375)
(3, 0.125)
(0, 0.125)
(3, 0.125)
(0, 0.125)
X
X
Figure 9-1
Probability Distribution of the Number of Boys out of Three Births
The probability distribution of a discrete random variable X must satisfy the following two
conditions:
1.
P(X = x) ≥ 0
2.
∑ P( X = x ) = 1
for all values x
all x
These conditions must hold because the P(X = x) values are probabilities. First condition
specifies that all probabilities must be greater than or equal to zero, as we know from Lesson
8.
For the second condition, we note that for each value x, P(x) = P(X = x) is the probability of
the event that the random variable equals x. Since by definition all x means all the values the
random variable X may take, and since X may take on only one value at a time, the
occurrences of these values are mutually exclusive events, and one of them must take place.
Therefore, the sum of all the probabilities P(X = x) must be 1.00.
9.2.1
Cumulative Distribution Function
The probability distribution of a discrete random variable lists the probabilities of occurrence
of different values of the random variable. We may be interested in cumulative probabilities
of the random variable. That is, we may be interested in the probability that the value of the
295
random variable is at most some value x. This is the sum of all the probabilities of the values i
of X that are less than or equal to x.
The cumulative distribution function (also called cumulative probability function)
F(X =
x) of a discrete random variable X is
F(X = x) = P(X ≤ x) =
∑ P(i )
all i ≤ x
For example, to find the probability of at most two boys out of three births, we have
F(X = 2) = P(X ≤ 2) =
∑ P(i )
all i ≤ 2
= P(X = 0)+ P(X = 1)+ P(X = 2)
=1/3 + 3/8 + 3/8
= 7/8
9.2.2
Expected Value and Variance of a Discrete Random Variable
The expected value of a discrete random variable X is equal to the sum of all values of the
random variable, each value multiplied (weighted) by its probability.
μ = E(X) =
∑ x.P(x )
all x
The variance of a discrete random variable is given by
σ2 = V (X) = E [(X - μ)2] =
∑ (x − μ)
2
.P( x)
all x
In the same way we can calculate the other summary measures viz. skewness, kurtosis and
moments.
9.2.3
Probability Distributions are Theoretical Distributions
Consider a random variable X that measures the “number of heads” in a three-trial coin
tossing experiment. The probability distribution of X will be
X
P(X) :
:
0
1
2
3
1/8
3/8
3/8
1/8
296
Now imagine this experiment is repeated 200 times, we may expect ‘no head’ and ‘three
heads’ will each occur 25 times; ‘one head’ and ‘two heads’ each will occur 75 times. Since
these results are what we expect on the basis of theory, the resultant distribution is called a
theoretical or expected distribution.
However, when the experiment is actually performed 200 times, the results, which we may
actually obtain, will normally differ from the theoretically expected results. It is quite
possible that in actual experiment ‘no head’ and ‘three heads’ may occur 20 and 28 times
respectively and ‘one head’ and ‘two heads’ may occur 66 and 86 times respectively. The
distribution so obtained through actual experiment is called the empirical or observed
distribution.
In practice, however, assessing the probability of every possible value of a random variable
through actual experiment can be difficult, even impossible, especially when the probabilities
are very small. But we may be able to find out what type of random variable the one at hand
is by examining the causes that make it random. Knowing the type, we can often approximate
the random variable to a standard one for which convenient formulae are available.
The proper identification of experiments with certain known processes in Probability theory
can help us in writing down the probability distribution function. Two such processes are the
Bernoulli Process and the Poisson Process. The standard discrete probability distributions
that are consequent to these processes are the Binomial and the Poisson distribution. We will
now look into the conditions that characterize these processes, and examine the standard
distributions associated with the processes. This will enable us to identify situations for which
these distributions apply.
Let us first study the Bernoulli random variable, named so in honor of the mathematician
Jakob Bernoulli (1654-1705). It is the building block for other random variables and the
resulting distributions we will study in this lesson.
297
9.3
BERNOULLI RANDOM VARIABLE
Suppose an operator uses a lathe to produce pins, and the lathe is not perfect in the sense that
it does not always produce a good pin. Rather, it has a probability p of producing a good pin
and (1 - p) of producing a defective one. Let us denote a good pin as “success” and a
defective pin as “failure”.
Just after the operator produces one pin, it is inspected; let X denote the "number of good pins
produced” i. e. “the number of successes”.
Now analyzing the trial- “inspecting a pin” and our random variable X-“number of
successes”, we note two important points:
¾ The trial-“inspecting a pin” has only two possible outcomes, which are mutually
exclusive. Such a trial, whose outcome can only be either a success or a failure, is a
Bernoulli trial. In other words, the sample space of a Bernoulli trial is
S = {success, failure}
¾ The random variable, X, that measures number of successes in one Bernoulli trial, is a
Bernoulli random variable. Clearly, X is 1 if the pin is good and 0 if it is defective.
It is easy to derive the probability distribution of Bernoulli random variable
X
:
0
1
P(X)
:
p
1-p
If X is a Bernoulli random variable, we may write
X ~ BER (p)
Where ~ is read as “is distributed as” and BER stands for Bernoulli.
A Bernoulli random variable is too simple to be of immediate practical use. But it forms the
building block of the Binomial random variable, which is quite useful in practice. The
binomial random variable in turn is the basis for many other useful cases, such as Poisson
random variable.
298
9.4
THE BINOMIAL DISTRIBUTION
In the real world we often make several trials, not just one, to achieve one or more successes.
Let us consider such cases of several trials.
Consider n number of identically and independently distributed Bernoulli random variables
X1, X2 ………, Xn. Here, identically means that they all have the same p, and independently
means that the value of one X does not in any way affect the value of another. For example,
the value of X2 does not affect the value of X3 or X8 and so on. Such a sequence of identically
and independently distributed Bernoulli variables is called a Bernoulli Process.
Suppose an operator produces n pins, one by one, on a lathe that has probability p of making
a good pin at each trial, the sequence of numbers (1 or 0) denoting the good and defective
pins produced in each of the n trials is a Bernoulli process. For example, in the sequence of
nine trials denoted by
001011001
the third, fifth, sixth and ninth are good pins, or successes. The rest are failures.
In practice, we are usually interested in the total number of good pins rather than the
sequence of 1's and 0's. In the example above, four out of nine are good. In the general case,
let X denote the total number of good pins produced in n trials. We then have
X = X1 + X2 +………+ Xn
where all Xi ~ BER(p) and are independent.
The random variable that counts the number of successes in many
independent, identical Bernoulli trials is called a Binomial Random
Variable.
9.4.1
Conditions for a Binomial Random Variable
We may appreciate that the condition to be satisfied for a binomial random variable is that
the experiment should be a Bernoulli Process.
299
Any uncertain situation or experiment that is marked by the following three properties is
known as a Bernoulli Process:
¾ There are only two mutually exclusive and collectively exhaustive outcomes in the
experiment i.e. S = {success, failure}
¾ In repeated trials of the experiment, the probabilities of occurrence of these events
remain constant
¾ The outcomes of the trials are independent of one another
The probability distribution of Binomial Random Variable is called the Binomial
Distribution
9.4.2
BINOMIAL PROBABILITY FUNCTION
Now we will develop the distribution of our Binomial random variable. To describe the
distribution of Binomial random variable we need two parameters, n and p we write
X ~ B (n, p)
to indicate that X is Binomially distributed with n number of independent trials and p
probability of success is each trial. The letter B stands for binomial.
Let us analyze the probability that the number of successes X in the n trials is exactly x
(obviously number of failures are n-x) i.e. X = x and x = 0,1,2,…………. n; as n trials are
made, at the best all n can be successes.
Now we know that there are nCx ways of getting x successes out of n trials. We also observe
that each of these nCx possibilities has px(1-p)n-x probability of occurrence corresponding to x
successes and (n-x) failures. Therefore,
P(X = x) = nCx p x (1-p)n-x
for x = 0,1,2,………, n
This equation is the Binomial probability formula. If we denote the probability of failure as q
then the Binomial probability formula is
P(X = x) = nCx p x qn-x
for x = 0,1,2,………, n
300
We may write down the Binomial probability distribution in table format (see Table 9-2)
Table 9-2
Binomial Distribution of X
X=x
0
P(X = x)
n
C0 p0 qn
n
1
…
x
…
…
n
C1 p1 qn-1
…
Cx px qn-x
…
…
n
Cn pn q0
n
Each of the term for x = 0,1,2,………, n correspond to the Binomial expansion of (p + q)n
9.4.3
1.
Characteristics of a Binomial Distribution
Expected Value or Mean
The expected value or the mean, denoted by μ, of a Binomial distribution is computed as
E (X) = μ =
n
∑ x.P( x)
x =0
An evaluation of μ will show that
μ=np
2.
Variance
The variance, denoted by σ2, of a Binomial distribution is computed as
V (X) =σ2 = E [(X - μ)2]
n
=
∑ (x − μ)
2
.P( x)
x =0
An evaluation of σ2 will show that
3.
σ2 = n p q
The rth moment about the origin denoted by mr0 , of a Binomial distribution is computed as:
mr0 =
n
∑x
r
.P( x)
x =0
301
For example, (a) First moment about the origin will be
m10 =
n
∑ x.P( x)
x =0
= np
=μ
(b) Second moment about the origin will be
m20 =
n
∑x
2
.P( x)
x =0
= n(n-1)p2 + np
4.
The rth moment about the mean denoted by mrμ , of a binomial distribution is computed as:
mrμ =
n
∑ ( x − μ ) .P( x)
r
x =0
For example, (a) First moment about the mean will be
m1μ =
n
∑ ( x − μ ) .P( x)
1
x =0
=0
(b) Second moment about the mean will be
m2μ =
n
∑ (x − μ)
2
.P( x)
x =0
= npq
= σ2
(c) Third moment about the mean will be
m3μ =
n
∑ ( x − μ ) .P( x)
3
x =0
= npq(q-p)
(d) Fourth moment about the mean will be
302
n
∑ (x − μ)
m4μ =
4
.P( x)
x =0
= 3(npq)2 + npq(1-6pq)
5.
Skewness
To bring out the skewness of a Binomial distribution we can calculate, moment coefficient of
skewness, γ1
γ1
=
β1
=
(m3μ ) 2
(m2μ ) 3
=
m3μ
(m)
μ
3
2
=
=
Evaluating γ1 =
npq (q − p )
(
npq
)
3
q− p
npq
q− p
npq
we note:
¾ the Binomial distribution is skewed to the right i.e. has positive skewness when γ1 >
0, which is so when p < q
¾ the Binomial distribution is skewed to the left i.e. has negative skewness when
γ1 <
0, which is so when p > q
¾ the Binomial distribution is symmetrical i.e. has no skewness when γ1 = 0, which is
so when p = q
Thus, n being the same, the degree of skewness in a Binomial distribution tends to vanish as
p approaches ½ i.e. as p→ ½
303
¾ for a given value of p, as n increases the Binomial distribution moves to the right,
As n → ∝, γ1 → 0, the distribution tends to be symmetrical.
5.
Kurtosis
A measure of kurtosis of the Binomial distribution is given by the moment coefficient of
kurtosis γ2
γ2
= β2 – 3
=
m4μ
(m )
μ 2
−3
2
=
3n 2 p 2 q 2 + npq(1 − 6 pq )
-3
n2 p 2q 2
=
1 − 6 pq
npq
Evaluating γ2 =
1 − 6 pq
we note
npq
¾ the Binomial distribution is leptokurtic when γ2 > 0, which is so when 6pq <1.
¾ the Binomial distribution is platykurtic when γ2 < 0, which is so when 6pq >1.
¾ the Binomial distribution is mesokurtic when γ2 = 0, which is so when 6pq =1.
6.
Normal approximation of the Binomial distribution
If n is large and if neither of p or q is too close to zero, the Binomial distribution can be
closely approximated by a Normal distribution with standardized variable
Z=
7.
X − np
npq
Poisson approximation of the Binomial distribution
Binomial distribution can reasonably be approximated by the Poisson distribution when n is
infinitely large and p is infinitely small i. e. when
304
n → ∝ and p → 0
Example 9-1
Assuming the probability of male birth as ½, find the probability distribution of number of
boys out of 5 births.
(a)
(b)
Find the probability that a family of 5 children have
(i)
at least one boy
(ii)
at most 3 boys
Out of 960 families with 5 children each find the expected number of families with (i)
and (ii) above
Solution: Let the random variable X measures the number of boys out of 5 births. Clearly X
is a binomial random variable. So we apply the Binomial probability function to calculate the
required probabilities.
X ~ B (5, ½)
P(X = x) = nCx p x qn-x for x = 0, 1, 2, 3, 4, 5
The probability distribution of X is given below
(a)
X=x
:
0
1
P(X = x)
:
1/32
5/32
2
3
4
10/32 10/32 5/32
5
1/32
The required probabilities are
(i)
P(X ≥ 1) = 1- P(X = 0)
= 1- 1/32
= 31/32
(ii)
P(X ≤ 3) = P(X = 0)+ P(X = 1)+ P(X = 2)+ P(X = 3)
= 1/32 + 5/32 + 10/32 + 10/32
= 26/32
305
Out of 960 families with 5 children, the expected number of families with
(b)
(i)
at least one boy = 960 * P(X ≥ 1)
= 960 * 31/32
= 930
(ii)
at most 3 boys = 960 * P(X ≤ 3)
= 960 * 26/32
= 720
9.5
THE POISSON DISTRIBUTION
Poisson Distribution was developed by a French Mathematician Simeon D Poisson (17811840). If a random variable X is said to follow a Poisson Distribution, then its probability
distribution is given by
P(X = x) =
Where
e−μ μ x
x!
x = 0,1,2,………
x is the number of successes
μ is the mean of the Poisson distribution and
e = 2.71828 (the base of natural logarithms)
The random variable X counts the number of successes in Poisson Process. A Poisson
process corresponds to a Bernoulli process under the following conditions:
¾ the number of trials n, is infinitely large i.e. n → ∝
¾ the constant probability of success p, for each trial is infinitely small i.e. p → 0
(obviously q → 1)
¾ np = μ is finite
We can develop the Poisson probability rule from the Binomial probability rule under the
above conditions.
306
Let us consider a Bernoulli process with n trials and probability of success in any trial
p =
μ
n
, where μ ≥ 0. Then, we know that the probability of x successes in n trials is given
by
P (X = x)
⎛μ⎞
= nCx ⎜ ⎟
⎝n⎠
x
⎛ μ⎞
⎜1 − ⎟
n⎠
⎝
n− x
x
n!
⎛μ⎞ ⎛ μ⎞
=
⎜ ⎟ ⎜1 − ⎟
x!(n − x)! ⎝ n ⎠ ⎝
n⎠
n− x
x
n[n − 1][n − 2]............[n − ( x − 1)] ⎛ μ ⎞ ⎛ μ ⎞
=
⎜ ⎟ ⎜1 − ⎟
x!
n⎠
⎝n⎠ ⎝
μ x ⎡n n −1 n − 2
n − ( x − 1) ⎤⎛ μ ⎞
.
.
............
=
⎢
⎥⎦⎜⎝1 − n ⎟⎠
x! ⎣ n n
n
n
=
n− x
n− x
n
μx ⎛
1 ⎞⎛ 2 ⎞
⎛ x − 1 ⎞⎛ μ ⎞ ⎛ μ ⎞
⎟⎜1 − ⎟ ⎜1 − ⎟
⎜1 − ⎟⎜1 − ⎟............⎜1 −
x! ⎝ n ⎠⎝ n ⎠
n ⎠⎝
n⎠ ⎝
n⎠
⎝
−x
⎛ μ⎞
⎛ x −1⎞
⎛ 1⎞ ⎛ 2⎞
Now if n → ∝, then the terms, ⎜1 − ⎟; ⎜1 − ⎟;............; ⎜1 −
⎟ and ⎜1 − ⎟
n ⎠
n⎠
⎝
⎝
⎝ n⎠ ⎝ n⎠
−x
will all be
tending to 1
n
⎛ μ⎞
and ⎜1 − ⎟ → e − μ if n → ∝
n⎠
⎝
Thus we have
P(X = x) =
e−μ μ x
x!
x = 0, 1, 2,………
This equation is the probability distribution function of Poisson distribution.
Thus, we have seen that to describe the distribution of Poisson random variable we need only
one parameter μ, we write
If
X ~ POI (μ)
307
e−μ μ x
Then P(X = x) =
x!
x = 0, 1, 2,………
We may write down the Poisson probability distribution in table format (see Table 9-3)
Table 9-3
X=x
0
1
2
Poisson Distribution of X
P(X = x)
e-μ
μ e-μ or μ P(X = 0)
μ2
2!
e − μ or
…
…
x
μx
x!
…
…
e − μ or
μ
2
…
…
μ
x
…
…
P(X = 1)
P(X = x-1)
Poisson distribution may be expected in situations where the chance of occurrence of any
event is small, and we are interested in the occurrence of the event and not in its nonoccurrence. For example, number of road accidents, number of defective items, number of
deaths in flood or because of snakebite or because of a rare disease etc. In these situations, we
know about the occurrence of an event although its probability is very small, but we do not
know how many times it does not occur. For instance, we can say that two road accidents
took place today, but it is almost impossible to say as to how many times, accident fails to
take place. The reason is that the number of trials is very large here and the nature of event is
of rare type. The Poisson random variable X, counts the number of times a rare event occurs
during a fixed interval of time or space.
9.5.1
1.
Characteristics of a Poisson Distribution
Expected Value or Mean
The expected value or the mean, denoted by μ, of a Poisson distribution is computed as
E (X) = μ =
∑ x.P( x)
all x
308
An evaluation of mean will show that it is always μ itself.
2.
Variance
The variance, denoted by σ2, of a Poisson distribution is computed as
V (X) =σ2 = E [(X - μ)2]
=
∑ (x − μ)
2
.P ( x )
all x
An evaluation of σ2 will show that
σ2 = μ
3.
The rth moments about the origin denoted by mr0 , of a Poisson distribution is computed as:
mr0 =
∑x
r
.P ( x )
all x
For example, (a) First moment about the origin will be
m10 =
∑ x.P( x)
all x
=μ
(b) Second moment about the origin will be
m20 =
∑x
2
.P ( x )
all x
=μ + μ2
4.
The rth moments about the mean denoted by mrμ , of a Poisson distribution is computed as:
mrμ =
∑ ( x − μ ) .P ( x )
r
all x
For example, (a) First moment about the mean will be
309
m1μ =
∑ ( x − μ ) .P ( x )
1
all x
=0
(b) Second moment about the mean will be
m2μ =
∑ (x − μ)
2
.P ( x )
all x
=σ2
=μ
(c) Third moment about the mean will be
m3μ =
∑ ( x − μ ) .P ( x )
3
all x
=μ
(d) Fourth moment about the mean will be
m4μ =
∑ (x − μ)
4
.P ( x )
all x
= 3μ2+μ
5.
Skewness
To bring out the skewness we can calculate, moment coefficient of skewness, γ1
γ1
=
β1
=
(m3μ ) 2
(m2μ ) 3
=
m3μ
(m)
μ
3
2
=
1
μ
310
Evaluating γ1 =
1
we note that Poisson distribution is always skewed to the right i.e. has
μ
positive skewness which is so as it is a distribution of rare events.
The degree of skewness in a Poisson distribution decreases as the value of μ increases.
6.
Kurtosis
A measure of kurtosis of the Poisson distribution is given by the moment coefficient of
kurtosis γ2
γ2
= β2 – 3
=
m4μ
(m )
μ 2
−3
2
=
Evaluating γ2 =
7.
1
μ
1
μ
we note that the Poisson distribution is leptokurtic.
Poisson approximation of the Binomial distribution
Poisson distribution can reasonably approximate Binomial distribution when n is infinitely
large and p is infinitely small i. e. when
n → ∝ and p → 0
Example 9-2
At a parking place the average number of car-arrivals during a specified period of 15 minutes
is 2. If the arrival process is well described by a Poisson process, find the probability that
during a given period of 15 minutes
(c)
no car will arrive
(d)
atleast two cars will arrive
(e)
atmost three cars will arrive
(f)
between 1 and 3 cars will arrive
311
Solution: Let X denote the number of cars arrivals during the specified period of 15 minutes.
So
X ~ POI (μ)
We apply the Poisson probability function P(X = x) =
e−μ μ x
x!
x
=
0,1,2,………
calculate the required probabilities.
(a)
e −2 2 0
=
0!
P(no car will arrive) = P(X = 0)
= 0.1353
(b)
P(atleast two cars will arrive) = P(X ≥ 2)
=1-[ P(X = 0) + P(X = 1)]
= 1-[
e −2 2 0
e −2 21
+
]
0!
1!
= 1-[0.1353 + 0.2707]
= 1 – 0.4060
= 0.5940
(c)
P(atmost three cars will arrive) = P(X ≤ 3)
e −2 2 x
x!
x =0
3
=∑
=P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
= 0.8571
(d)
P(between 1 and 3 cars will arrive) = P(1≤ X ≤ 3)
= P(X ≤ 3) - P(X = 0)
=
e −2 2 x e −2 2 0
∑
x!
0!
x =0
3
= 0.8571 –0.1353
= 0.7218
312
to
9.6
SELF-ASSESSMENT QUESTIONS
1.
Explain what do you understand by random experiment and a random variable.
Briefly explain the following:
2.
a.
Discrete and continuous random variables
b.
Discrete probability distribution.
“Binomial random variable measures the number of successes in a Bernoulli Process”.
Explain this statement. Also develop and generalize Binomial probability rule with
the help of an example.
3.
State the important properties of a Binomial distribution. Give examples of some of
the important area where Binomial distribution is used.
4.
Under what condition can the Poisson distribution approximate Binomial distribution?
Develop the Poisson probability rule from the Binomial probability rule under these
conditions.
5.
List some of the important areas where Poisson distribution is used. Also state the
important properties of a Poisson distribution.
6.
On an average a machine produces 20 % defective item find the probability that a
random sample of 4 items consists of
(a)
none to four defective items
(c)
almost 2 defective items.
(b)
atleast 3 defective items
Out of 200 samples of 4 items, find the expected number of samples with (a), (b), and
(c) above
7.
A gardener knows from his personal experiences that 2% of seedlings fail to service
on transplantation. Find the mean, standard deviation and moment coefficient of
skewness of the distribution of rate of failure to service in a sample of 400 seedlings.
313
8.
If the sum of mean and variance of a binomial distribution of 5 trials is 9/5, find the
binomial distribution.
9.
The mean and variance of a binomial distribution are 2 and 1.5 respectively. Find the
probability of
(a) 2 successes
10.
(b) atleast 2 successes
(c) at most 2 successes.
150 random samples of 4 units each are inspected for number of defective item. The
results are:
Number of defective items
:
0
1
2
3
4
Number of Samples
:
28
62
46
10
4
Fit a binomial distribution to the observed data.
11.
The probability that a particular injection will have reaction to an individual is
0.002. Find the probability that out of 1000 individuals (a) no, (b) 1, (c) at least
1, and (d) almost 2; individuals will have reaction from the injection.
12.
In a razor blades manufacturing factory, there is small chance of 1/500 for any
blade to be defective. The blades are supplied in packets of 10. Find the
approximate number of packets containing (a) no, (b) 1, and (c) 2 defective
blades in a consignment of 10,000 packets.
13.
If P(x = 1) = P(x = 2), for a distribution of Poisson random variable X. Find
the mean of the distribution.
14.
The distribution of typing mistakes committed by a typist is given below:
Number of mistakes (X)
:
0
1
2
3
4
5
Number of pages (f)
:
142
156
69
27
5
1
Fit a Poisson distribution and find the expected frequencies.
314
9.7
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall of
India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New York.
315
Course:
Author:
Anil Kumar
Course Code:
MC-106
Vetter:
Dr. Karam Pal
Lesson:
10
PROBABILITY DISTRIBUTIONS-II
Objectives: The overall objective of the present lesson is to overview the concept of
continuous random variable and Normal distribution. After successful
completion of the lesson the students will be able to appreciate the
usefulness of normal distribution in decision-making and also identify
situations where normal probability distribution can be applied.
Structure
10.1
Introduction
10.2
Continuous Probability Distribution
10.3
The Normal Distribution
10.4
The Standard Normal Distribution
10.5
The Transformation of Normal Random Variables
10.6
Self-Assessment Questions
10.7
10.1 INTRODUCTION
We have learnt that a probability distribution is basically a convenient representation of the
different values a random variable may take, together with their respective probabilities of
occurrence. In the last lesson, we have examined situations involving discrete random
316
variables and the resulting discrete probability distributions. Consider the following random
variables that we have taken up in the last lesson:
1.
Number of Successes (X1) in a Bernoulli’s Process
2.
Number of Successes (X2) in a Poisson Process
In the first case, Binomial random variable X1 could take only finite number of integer values;
0,1,2…n; whereas in the second case, Poisson random variable X2 could take an infinite
number of integer value; 0,1,2,3………… The random variables X1 and X2 are discrete, in the
sense that they could be listed in a sequence, finite or infinite. In contrast to these, let us
consider a situation, where the variable of interest may take any value within a given range.
Suppose we are planning for measuring the variability of an automatic bottling process that
fills ½-liter (500 cm3) bottles with cola. The variable, say X, indicating the deviation of the
actual volume from the normal (average) volume can take any real value - positive or
negative; integer or decimal. This type of random variable, which can take an infinite number
of values in a given range, is called a continuous random variable, and the probability
distribution of such a variable is called a continuous probability distribution. The concepts
and assumption inherent in the treatment of such distributions are quite different from those
used in the context of a discrete distribution. In the present lesson, after understanding the
basic concepts of continuous distributions, we will discuss Normal distribution - an important
continuous distribution that is applicable to many real-life processes.
10.2 CONTINUOUS PROBABILITY DISTRIBUTION
Consider our planning for measuring the variability of the automatic bottling process that fills
½-liter (500cm3) bottles with cola. The random variable X indicates ‘the deviation of the
actual volume from the normal (average) volume.’ Let us, for some time, measure our
random variable X to the nearest one cm3.
317
F
Figure 10-1
Histograms of the Distribution of X as Measurements is refined to
Smaller and Smaller Intervals of Volume, and the Limiting Density Function f(x)
Suppose Figure10-1a represent the histogram of the probability distribution of X. The
probability of each value of X is the area of the rectangle over the value. Since the rectangle
will have the same base, the height of each rectangle is proportional to the probability. The
probabilities also add to 1.00 as required for a probability distribution.
Volume is a continuous random variable; it can take on any value measured on an interval of
numbers. Now let us imagine the process of refining the measurement scale of X to the
nearest 1/2 cm3, the nearest 1/10 cm3… and so on. Obviously, as the process of refining the
measurement scale continues, the number of rectangles in the histogram increases and the
width of each rectangle decreases. The probability of each value is still measured by the area
of the rectangle above it, and the total area of all rectangles remains 1.00. As we keep
refining our measurement scale, the discrete distribution of X tends to a continuous
probability distribution. The step like surface formed by the tops of the rectangles in the
histogram tends to a smooth function. This function is denoted by f(x) and is called the
probability density function of the continuous random variable X. The density function is the
318
limit of the histograms as the number of rectangles approaches infinity and the width of each
rectangle approaches zero. The density function of the limiting continuous variable X is
shown in Figure 10-1 i.e. the values X can assume between the intervals –2.00 to –3.00
approaches infinity. The probability that X assumes a particular value (Say X = 1.5)
approaches zero. Probabilities are still measured as areas under the curve. The probability
that deviation will be between –1.50 and –1.00 is the area under f(x) between the points x = 1.50 and x = -1.00. Let us now make some formal definitions.
A continuous random variable is a random variable that can take on any
value in an interval of numbers.
The probabilities associated with a continuous random variable X are determined by the
probability density function of the random variable. The function, denoted by f(x), has the
following properties:
1.
f(x) = 0 for all x
2.
The probability that X will be between two numbers a and b is equal to the
area under f(x) between a and b.
b
P(a < X < b) =
∫
f ( x ). dx
a
3.
The total area under the entire curve of f(x) is equal to 1.00.
∝
P (− ∝ ≤ X ≤ ∝ ) =
∫
f ( x ). dx = 1 . 00
-∝
When the sample space is continuous, the probability of any single given value is zero. For a
continuous random variable, therefore, the probability of occurrence of any given value is
zero. We see this from property 2, noting that the area under a curve between a point and
itself is the area of a line, which is zero. For a continuous random variable, non-zero
probabilities are associated only with intervals of numbers.
319
We define the cumulative distribution function F(x) for a continuous random variable
similarly to the way we defined it for a discrete random variable: F(x) is the probability that X
is less than (or equal to) x.
Thus, the cumulative distribution function of a continuous random variable:
F(x) = P(X = x) = area under f(x) between the smallest possible value of X (often -∝)
and point x
x
=
∫ f ( x).dx
-∝
The cumulative distribution function F(x) is a smooth, non-decreasing function that increases
from 0 to 1.00.
The expected value of a continuous random variable X, denoted by E(X), and its variance,
denoted by V(X), require the use of calculus for their computation. Thus
∝
E(X ) =
∫ x . f ( x ). dx
-∝
∝
V (X ) =
∫ [x − E ( x ) ] . f ( x ).dx
2
-∝
10.3 THE NORMAL DISTRIBUTION
The Normal Distribution is the most versatile of all the continuous probability distributions.
It is being widely used in all data-based research in the field of agriculture, trade, business
and industry It is found to be useful in characterizing uncertainties in many real-life
processes, in statistical inferences, and in approximating other probability distributions.
A large number of random variables occurring in practice can be approximated to the normal
distribution.
320
A random variable that is affected by many independent causes, and the
effect of each cause is not overwhelmingly large compared to other effects,
The lengths of pins made by an automatic machine; the times taken by an assembly worker to
complete the assigned task repeatedly; the weights of baseballs; the tensile strengths of a
batch of bolts; and the volumes of cola in a particular brand of canned cola - are good
examples of normally distributed random variables. All of these are affected by several
independent causes where the effect of each cause is small. This knowledge helps us in
calculating the probabilities of different events in varied situations, which in turn is useful for
decision-making.
In many real life situations, we face the problem of making statistical inferences about
processes based on limited data. Limited data is basically a sample from the full body of data
on the process. Irrespective of how the full body of data is distributed, it has been found that
the Normal Distribution can be used to characterize the sampling distribution of many of the
sample statistics. (we will see it in next few lessons). This helps considerably in Statistical
Inferences.
Finally, the Normal Distribution can be used to approximate certain probability
distributions. This helps considerably in simplifying the probability calculations.
10.3.1
Probability Density Function
If X is normally distributed with mean μ and variance σ 2, we write
X ~ N (μ, σ2)
and the probability density function ƒ(x) is given by the formula
f ( x) =
1 ⎛ x−μ ⎞ 2
⎟
σ ⎠
− ⎜
1
e 2⎝
2πσ
-∝ < x < +∝
321
In the equation e is the base of natural logarithm, equal to 2.71828.... By substituting desired
values for μ and σ , we can get any desired density function. For example, a distribution
with mean 100 and standard deviation 5 will have the density function.
f (x) =
1 ⎛ x −100 ⎞ 2
⎟
5
⎠
− ⎜
1
e 2⎝
2π 5
-∝ < x < +∝
This function when plotted (see Figure 10-2) will give the famous bell-shaped mesokurtic
normal curve.
Figure 10-1
Figure 10-2
Normal Curve
A Normal Distribution with μ = 100 and σ = 5
Many mathematicians have worked on the mathematics behind the normal distribution and
have made many independent discoveries. In the initial stages, the normal distribution was
developed by Abraham De Moivre (1667-1754). His work was later taken up by Pierre S
Laplace (1949-1827). But the discovery of equation for the normal density function is
attributed to Carl Friedrich Gauss (1777-1855),who did much work with the formula. In
science books, this distribution is often called the Gaussian distribution.
We will now examine the properties of the Normal distribution.
10.3.2
Properties of Normal Distribution
1. The normal curve is not a single curve representing only one continuous distribution.
Obviously, it represents a family of normal curves; since for each different value of μ
322
and σ , there is a specific normal curve different in its positioning on the X-axis and
the extent of spread around the mean. Figure 10-3 shows three different normal
distributions – with different shapes and positions.
Figure 10-3
Three Different Normal Distribution
2. The normal curve is bell-shaped and perfectly symmetric about its mean. As a result
50% of the area lies to the right of mean and balance 50% to the left of mean. Perfect
symmetry, obviously, implies that mean, median and mode coincide in case of a
normal distribution. The normal curve gradually tapers off in height as it moves in
either direction away from the mean, and gets closer to the X-axis.
3. The normal curve has a (relative) kurtosis of 0, which means it has average
peakedness and is mesokurtic.
4. Theoretically, the normal curve never touches the horizontal axis and extends to
infinity on both sides. That is the curve is asymptotic to X-axis.
5. If several independent random variables are normally distributed, then their sum will
also be normally distributed. The mean of the sum will be the sum of all the
individual means, and by virtue of the independence, the variance of the sum will be
the sum of all the individual variances.
If X1, X2,………… Xn are independent normal variables, the their sum S will also be a
normal variable with
323
E(S) = E(X1) + E(X2) +…………E(Xn)
and
V(S) = V(X1) + V(X2) +…………V(Xn)
6. If a normal variable X under goes a linear change in scale such as Y = aX + b, where a
and b are constants and a ≠ 0; the resultant Y variable will also be normally distributed
with mean = a E(X) + b and Variance = a2 V(X)
We can combine the above two properties.
If X1, X2,………… Xn are independent random variables that are normally distributed, then the
random variable Q defined as
Q = a1X1+ a2X2 + … anXn + b will also be normally distributed with
E(Q) = a1E(X1) + a2E(X2) +…………anE(Xn) + b
and
V(Q) = a12 V(X1) + a 22 V(X2) +………… a n2 V(Xn)
Let us see the application of this result with the help of an example.
Example 10-1
A cost accountant needs to forecast the unit cost of a product for the next year. He notes that
each unit of the product requires 10 labor hours and 5 kg of raw material. In addition, each
unit of the product is assigned an overhead cost of Rs 200. He estimates that the cost of a
labor hour next year will be normally distributed with an expected value of Rs 45 and a
standard deviation of Rs 2; the cost of raw material will be normally distributed with an
expected value of Rs 60 and a standard deviation of Rs 3. Find the distribution of the unit
cost of the product. Find its expected value and variance.
Solution: Since the cost of labor L may not influence the cost of raw material M, we can
assume that the two are independent. This makes the unit cost of the product Q a random
variable. So if
L ~ N (45, 22)
M ~ N (60, 32)
and
Then, Q = 10L + 5M + 200 will follow normal distribution with
324
Mean = E(Q) = 10E(L) + 5E(M) + 200
= 10(45) + 5(60) + 200
= 950
Variance = V(Q) = 102V(L) + 52V(M)
= 100(4) + 25(9)
= 625
So
Q ~ N (950, 252)
7. Same important area relationships under normal curse are
Area between μ - 1σ and μ + 1σ is about 0.6826
Area between μ - 2σ and μ + 2σ is about 0.9544
Area between μ - 3σ and μ + 3σ is about 0.9974
Area between μ – 1.96σ and μ + 1.96σ is 0.95
Area between μ – 2.58σ and μ + 2.58σ is 0.99
10.4 THE STANDARD NORMAL DISTRIBUTION
There are infinitely many possible normal random variables and the resulting normal curves
for different values of μ and σ2. So the range probability P(a < X < b) will be different for
different normal curves. We can make use of integral calculus to compute the required range
probability
b
P(a < X < b) =
∫
f ( x ). dx
a
It may be appreciated that we can simplify this process of computing range probabilities to a
great extent by tabulating the range probabilities. Since it is not practicable and indeed
impossible to have separate probability tables for each of the infinitely many possible normal
curves, we select one normal curve to serve as a standard. Probabilities associated with the
range of values of this standard normal random variable are tabulated. A special
325
transformation then allows us to apply the tabulated probabilities to any normal random
variable. The standard normal random variable is denoted by a special name, Z (rather than
the general name X we use for other random variables).
We define the standard normal random variable Z as the normal random
variable with mean = 0 and standard deviation = 1.
We say
Z ~ N (0,12)
10.4.1
Standard Area Tables
The probabilities associated with standard normal distribution are tabulated in two ways – say
Type I and Type II tables, as shown in Figure 10-4. Type I Tables give the area between μ =
0 and any other z value, as shown by vertical hatched area in Figure 10-4a. The hatched area
shown in figure is P (0 < Z < z).
P(0 < Z < z)
Figure 10-4
P(Z > z)
Standard Area Tables
Type II Tables give the area towards the tail–end of the standard normal curve beyond the
ordinate at any particular z value. The hatched area shown in Figure 10-4b is
P (Z >
z).
As the normal curve is perfectly symmetrical, the areas given by Type 1 Tables when
subtracted from 0.5 will provide the same areas as given by Type II Tables and vice-versa.
i.e
P (0 < Z < z) = 0.5 - P (Z > z).
326
10.4.2
Finding Probabilities of the Standard Normal Distribution
We will now illustrate the use of standard normal area tables for calculating the range
probabilities. Probability of intervals is areas under the density curve ƒ(z) over the intervals in
question.
Example 10-2
Find the probability that the value of the standard normal random variable will be…
(a)
between 0 and 1.74
(b)
less than -1.47
(c)
between 1.3 and 2
(d)
between -1 and 2
Solution: (a) P(Z is between 0 and 1.74)
That is, we want P(0 < Z < 1.74). In Figure 10-4a, substitute 1.74 for the point z on the graph.
We are looking for the table area in the row labeled 1.7 and the column labeled 0.04. In the
table, we find the probability 0.4591.Thus
P (0 < Z < 1.74) = 0.4591
(b)
P(Z is less than -1.47)
That is, we want P(Z < -1.47). By the symmetry of the normal curve, the area to the left of 1.47 is exactly equal to the area to the right of 1.47. We find
P(Z < -1.47) = P(Z >1.47)
= 0.5000 - 0.4292
= 0.0808
(c)
P(Z is between 1.3 and 2)
That is, we want P(1.3 < Z < 2). The required probability is the area under the curve between
the two points 1.3 and 2. The table gives us the area under the curve between 0 and 1.3, and
the area under the curve between 0 and 2. Areas are additive; therefore,
P(1.30 < Z< 2) = TA(for 2.00) - TA(for 1.30)
= P(0 < Z < 2) - P(0 < Z < 1.3)
327
= 0.4772 - 0.4032
= 0.0740
(d)
P(Z is between -1 and 2)
That is, we want P(-1< Z < 2). The required probability is the area under the curve between
the two points -1 and 2. The table gives us the area under the curve between 0 and 1, and the
area under the curve between 0 and 2. Areas are additive; therefore,
P(-1 < Z< 2) = P(-1 < Z < 0) + P(0 < Z < 2)
= P(0 < Z < 1) + 0.4772
= 0.3413 + 0.4772
= 0.8185
In cases, where we need probabilities based on values with greater than second-decimal
accuracy, we may use a linear interpolation between two probabilities obtained from the
table.
Example 10-3
Find
P(0 ≤ Z ≤ 1.645)
Solution: P(0 ≤ Z ≤ 1.645) is found as the midpoint between the two probabilities P(0 ≤ Z ≤
1.64) and P(0 ≤ Z ≤ 1.65). So
P(0 ≤ Z ≤ 1.645) = ½[P(0 ≤ Z ≤ 1.64) + P(0 ≤ Z ≤ 1.65)]
= ½[0.4495 + 0.4505]
= 0.45
10.4.3
Finding Values of Z Given a Probability
In many situations, instead of finding the probability that a standard normal random variable
will be within a given interval; we may be interested in the reverse: finding an interval with a
given probability. Consider the following examples.
Example 10-4
328
Find a value z of the standard normal random variable such that the probability that the
random variable will have a value between 0 and z is 0.40.
Solution: We look inside the table for the value closest to 0.40. The closest value we find to
0.40 is the table area 0.3997. This value corresponds to 1.28 (row 1.2 and column .08).
So for P(0 < Z < z) = 0.40; z = 1.28
Example 10-5
Find the value of the standard normal random variable that cuts off an area of 0.90 to its left.
Solution: Since the area to the left of the given point z is greater than 0.50, z must be on the
right side of 0. Furthermore, the area to the left of 0 all the way to -∝ is equal to 0.50.
Therefore, TA = 0.90 - 0.50 = 0.40. We need to find the point z such that TA = 0.40.
We find that for TA = 0.40; z =1.28.
Thus z =1.28 cuts off an area of 0.90 to the left of standard normal curve.
Example 10-6
Find a 0.99 probability interval, symmetric about 0, for the standard normal random variable.
Solution: The required area between the two z values that are equidistant from 0 on either
side is 0.99. Therefore, the area under the curve between 0 and the positive z value is TA =
0.99/2 = 0.495. We now look in our normal probability table for the area closest to 0.495. The
area 0.495 lies exactly between the two areas 0.4949 and 0.4951, corresponding to z = 2.57
and z = 2.58, Therefore, a simple linear interpolation between the two values gives us z =
2.575. The answer, therefore, is z = ± 2.575.
So for P(-z < Z< z) = 0.99;
z = 2.575
10.5 THE TRANSFORMATION OF NORMAL RANDOM VARIABLES
The importance of the standard normal distribution derives from the fact that any normal
random variable may be transformed to the standard normal random variable. If we want to
329
transform X, where X ~ N (μ, σ 2), into the standard normal random variable Z ~ N (0, 12), we
can do this as follows:
Z=
X −μ
σ
We move the distribution from its center of μ to a center of 0. This is done by subtracting μ
from all the values of X. Thus, we shift the distribution μ units back so that its new center is
0. To make the standard deviation of the distribution equal to 1, we divide the random
variable by its standard deviation σ. The area under the curve adjusts so that the total remains
the same. All probabilities (areas under the curve) adjust accordingly. Thus, the
transformation from X to Z is achieved by first subtracting μ from X and then dividing the
result by σ.
Example 10-7
If X ~ N (50, 10 2), find the probability that the value of the random variable X will be greater
than 60
Solution:
P(X > 60) = P(
X −μ
σ
= P( Z >
>
60 − μ
)
10
60 − 50
)
10
= P( Z >1)
= P( Z > 0) - P(0 < Z <1)
= 0.5000 - 0.3413
= 0.1587
Example 10-8
The weekly wage of 2000 workmen is normally distribution with mean wage of Rs 70 and
wage standard deviation of Rs 5. Estimate the number of workers whose weekly wages are
330
(a)
between Rs 70 and Rs 71
(b)
between Rs 69 and Rs 73
(c)
more than Rs 72
(d)
less than Rs 65
Solution: Let X be the weekly wage in Rs, then
X ~ N (70, 5 2)
(a)
So
The required probability to be calculated is P(70 < X < 71)
P(70 < X < 71) = P(
= P(
70 − μ
σ
<
X −μ
σ
<
71 − μ
σ
)
70 − 70
71 − 70
<Z<
)
5
5
= P(0 < Z < 0.2)
= 0.0793
So the number of workers whose weekly wages are between Rs 70 and Rs 71
= 2000 x 0.0793
= 159
(b)
So
The required probability to be calculated is P(69 < X < 73)
P(69 < X < 73) = P(
= P(
69 − μ
σ
<
X −μ
σ
<
73 − μ
σ
)
69 − 70
73 − 70
<Z<
)
5
5
= P(-0.2 < Z < 0.6)
= P(-0.2 < Z < 0)+ P(0 < Z < 0.6)
= P(0 < Z < 0.2)+ P(0 < Z < 0.6)
= 0.0793 + 0.2257
= 0.3050
So the number of workers whose weekly wages are between Rs 69 and Rs 73
= 2000 x 0.3050
= 610
331
(c)
So
The required probability to be calculated is P(X > 72)
P(X > 72) = P(
X −μ
>
σ
72 − μ
σ
)
72 − 70
)
5
= P(Z >
= P(Z > 0.4)
= 0.5 - P(0 < Z < 0.4)
= 0.5 – 0.1554
= 0.3446
So the number of workers whose weekly wages are more than Rs 72
= 2000 x 0.3446
= 689
(d)
So
The required probability to be calculated is P(X < 65)
P( X < 65) = P(
X −μ
σ
<
= P( Z <
65 − μ
σ
)
65 − 70
)
5
= P(Z < -1.0)
= P(Z >1.0)
= P(Z >0) - P(0 < Z < 1.0)
= 0.5 - 0.3413
= 0.1567
So the number of workers whose weekly wages are less than Rs 65
= 2000 x 0.1567
= 313
332
10.5.1
The Inverse Transformation
The transformation Z =
X −μ
σ
takes us from a random variable X with mean μ, and standard
deviation σ to the standard normal random variable. We also have an opposite, or inverse,
transformation, which takes us from the standard normal random variable Z to the random
variable X with mean μ and standard deviation σ. The inverse transformation is given as
X = μ + Zσ
We use the inverse transformation when we want to get from a given probability, the value or
values of a normal random variable X.
Example 10-9
The amount of fuel consumed by the engines of a jetliner on a flight between two cities is a
normally distributed random variable X with mean μ = 5.7 tons and standard derivation σ =
0.5 tons. Carrying too much fuel is inefficient as it slows the plans. If, however, too little fuel
is loaded on the plane, an emergency landing may be necessary. What should be the amount
of fuel to load so that there is 0.99 probability that the plane will arrive at its destination
without emergency landing?
X ~ N (5.7, 0.5 2),
Solution: Given that
We have to find the value x such that
P(X < x) = 0.99
or
or
P(
X −μ
σ
< z) = 0.99
P(Z < z) = 0.99
= 0.5 + 0.49
= 0.5 + P(0 < Z < z)
From the table, value of z is 2.33
So
x = μ + zσ
333
x = 5.7 + 2.33 x0.5
x = 6.865
Therefore, the plane should be loaded with 6.865 tons of fuel to give 0.99 probability that the
fuel will last throughout the flight.
Example 10-10
Monthly sale of beer at a bar is believed to be approximately normally distributed with mean
2450 units and standard 400 units. To determine the level of orders and stock, the
management wants to find two values symmetrically on either side of mean, such that the
probability that sales of beer during the month will be between the two values is
(a)
0.95
(b)
0.99
Find the required values.
Solution: Let X be the monthly sale of beer, then
X ~ N (2450, 400 2),
(a)
We have to find the values x1 and x2 such that
P(x1 < X < x2) = 0.95
or
P(
x1 − μ
σ
<
X −μ
σ
<
x2 − μ
σ
) =0.95
P( z1 < Z < z 2 ) =0.95
or
We know
P(-1.96 < Z < 1.96) = 0.95
So
z1 = -1.96
and
z2 = 1.96
and
x 2 = μ + z 2σ
Using the inverse transformation,
x1 = μ + z1σ
x1 = 2450 + (− 1.96 )400
x 2 = 2450 + (1.96 )400
x1 = 1666
x 2 = 3234
334
Therefore, the management may be 95% sure that sales in any given month will be between
1666 and 3234 units.
(b)
We have to find the values x1 and x2 such that
P(x1 < X < x2) = 0.99
or
P(
x1 − μ
σ
<
X −μ
σ
<
x2 − μ
σ
) =0.99
P( z1 < Z < z 2 ) =0.99
or
We know
P(-2.58 < Z < 2.58) = 0.99
So
z1 = -2.58
and
z2 = 2.58
and
x 2 = μ + z 2σ
Using the inverse transformation,
x1 = μ + z1σ
x1 = 2450 + (− 2.58)400
x 2 = 2450 + (2.58)400
x1 = 1418
x 2 = 3482
Therefore, the management may be 99% sure that sales in any given month will be between
1418 and 3482 units.
We can summarize the procedure of obtaining values of a normal random variable, given a
probability, as:
¾ draw a picture of the normal distribution in question and the standard normal
distribution
¾ in the picture, shade in the area corresponding to the probability
¾ use the table to find the z value (or values) that gives the required probability
¾ use the transformation from Z to X to get the appropriate value (or values) of the
original normal random variable
335
10.6 SELF-ASSESSMENT QUESTIONS
1.
Define continuous probability distribution. State the properties of the probability
density function of a continuous random variable.
2.
(a)
Define normal random variable. State the probability density function of a
normal random variable.
(b)
List down important properties of a normal curve.
3.
Discus the role of normal distribution in statistical theory.
4.
What do you mean by standard normal variable? Bring out the need for having a
standard normal curve.
5.
Find the probability that a standard normal variable will have a value
(a)
6.
less than –10
(b)
between -0.01 and 0.05
A sensitive measuring device is calibrated so that errors in the measurements it
provides are normally distributed with mean 0 and variance 1.00. Find the probability
that a given error will be between -2 and 2.
7.
The deviation of a magnetic needle from the magnetic pole in a certain area in
northern Canada is a normally distributed random variable with mean 0 and standard
deviation 1.00. What is the probability that the absolute value of the deviation from
the north pole at a given moment will be more than 2.4?
8.
9.
Find two values of the standard normal random variable, z and -z, such that
(a)
the two corresponding "tail areas" of the distribution add to 0.01.
(b)
each tail have an area of 0.05
Let X be a normally distributed random variable with mean μ = 16 and standard
deviation σ = 3. Find
(a)
P(10 < X< 18)
(b)
P(16 < X< 18)
336
(c)
P(X > 14)
10.
For a normally distributed random variable with mean -44 and standard deviation 16,
find the probability that the value of the random variable will be
(a)
11.
above 0
(b)
-10
(c)
below 0
A normal random variable has mean 0 and standard deviation 4. Find the probability
that the random variable will be…
(a)
12.
above 2.5
(b)
between 2 and 3
(c)
below 1
The time it takes an international telephone operator to place an overseas phone call is
normally distributed with mean 45 seconds and standard deviation 10 seconds.
(a)
What is the probability that my call will go through in less than 1 minute?
(b)
What is the probability that my call will get through in less than 40 seconds?
(c)
What is the probability that I will have to wait more than 70 seconds for my
call to go through?
13.
The number of votes cast in favor of a controversial proposition is believed to be
approximately normally distributed with mean 8,000 and standard deviation 1,000.
The proposition needs at least 9,322 votes in order to pass. What is the probability
that the proposition will pass? (Assume numbers are on a continuous scale.)
14.
A manufacturing company regularly consumes a special type of glue purchased from
a foreign supplier. From past experience, the materials manager notes that the
company’s demand for glue during the uncertain lead-time is normally distributed
with a mean of 187.6 gallons and a standard deviation of 12.4 gallons. The company
follows a policy of placing the order when the glue stock falls to a predetermined
value, called “re-order point”. It the demand during lead-time exceeds the reorder
level, the glue would go ‘stock-out’ and production process would have to stop.
(a)
If the re-order point is kept at 187.6 gallons, what is the probability that a
stock-out condition would occur?
337
(b)
If the reorder point is kept at 200 gallons, what is the probability that a stockout condition would occur?
(c)
If the company wants to be 95% confident that the stock-out condition will
not occur, what should be the reorder point? The reorder point minus the
mean demand during lead-time is known as the "safety stock." What is the
safety stock in this case?
(d)
If the company wants to be 99% confident that the stock-out condition will not
occur, what should be the reorder point? What is the safety stock in this case?
15.
If X is a normally distributed random variable with mean 125 and standard deviation
44, find a value x such that the probability that X will be less than x is 0.66.
16.
For a normal random variable with mean 10.5 and standard deviation 0.4, find a point
of the distribution such that there is a 0.95 probability that the value of the random
variable will be above it.
17.
For a normal random variable with mean 29,500 and standard deviation 410, find a
point of the distribution such that the probability that the random variable will exceed
this value is
(a)
18.
0.03
(b)
0.25
Find two values of the normal random variable with mean 80 and standard deviation 5
lying symmetrically on either side of the mean and covering an area of 0.98 between
them.
19.
For X~ N(32, 72), find two values x1 and x2, symmetrically lying on each side of the
mean, with
(a)
20.
P(x1 < X< x2) = 0.99
(b)
P(x1 < X < x2) = 0.95
The results of a given selection test exercise are summarized as
(i)
cleared with distinction = 10%
338
(ii)
cleared without distinction = 60%
(iii)
those who failed = 30%.
A candidate gets failed if he/she obtains less than 40% marks, while one must obtain
at least 75% marks to pass with distinction. Determine the mean and standard
deviation of the distribution of marks, assuming the same to be normal.
21.
The demand for gasoline at a service station is normally distributed with mean 27,009
gallons per day and standard deviation 4,530. Find two values that will give a
symmetric 0.95 probability interval for the amount of gasoline demanded daily.
22.
The percentage of protein in a certain brand of dog food is a normally distributed
random variable with mean 11.2 % and standard deviation 0.6 %. The manufacturer
would like to state on the package that the product has a protein content of at least x1
% and no more than x %. He wants the statement to be true for 99% of the packages
sold. Determine the values x1 and x2.
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
339
AUTHOR: DR. PARDEEP GUPTA
COURSE CODE: MC-106
VETTER: DR. B.S. BODLA
LESSON: 11
Sampling and sampling methods
Objective: After going through this chapter, you will be able to understand: various terms
associated with sampling; various methods of probability and non-probability
sampling and how to determine sample size.
Structure
11.1. Census Vs. sampling method
11.2. Definitions
11.3. Probability samples vs. non-probability samples
11.4. Probability sampling methods
11.5. Non-probability sampling methods
11.6. Determination of sample size
11.7. Self-test questions
11.1. Census Vs. sampling method
Sample is a part of the population from which it is selected. The process of selecting a sample
is known as sampling. Thus, the sampling theory is a study of relationship that exists between
the population and the samples drawn from the population. The complete enumeration,
popularly known as census, may not be feasible either due to non-availability of time or
because of high cost involved. Therefore, it becomes essential to draw inferences for the
population on the basis of sample information. Thus, sampling helps us to get as much
information as possible of the whole universe. The sampling also helps us in determining the
reliability of the estimates. This can be done by drawing samples from the same parent
population and comparing the results obtained from different samples.
In a survey of the entire population, data is collected from every elementary unit of the
population. Suppose, one is studying the wage structure of the coal mining industry in the
country, then one approach is to collect the data on wages of every worker in the coal
industry. From this data, one can calculate the various characteristics of the population, such
as average wage, the range and the variance, etc. This is referred as census survey. The
advantages of the census approach are
every unit of the population is considered and the respective data on the
various characteristics are compiled,
the analysis made on the basis of census data is very accurate and reliable,
and
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in one time studies of special importance, only census method is adopted in
order to get accurate and reliable data. The data collected by this
method becomes a data base for all future studies. This is one of the
reasons why population data are collected once in a decade by the
census method.
Although there are many advantages with the census method, the cost, effort and the time
required to conduct census survey is very large, unless the population is very small, and in
many cases it is so prohibitive that one rarely uses this method in surveys.
Sampling involves an examination of a small portion of the elementary units in a population.
Although, a census operation gives a more reliable data, sampling method is more desired
when
1.1.
the population is very large, i.e., infinite and it would be impossible to
conduct census surveys;
2.1.
when quick results are required it would be appropriate to conduct
sample surveys rather than census surveys;
3.1.
in studies involving destruction of the elementary units under study,
it would only be appropriate to go for sample testing. Items such as
light bulbs and ammunition often must be destroyed as a part of
testing process;
4.1.
cost of conducting surveys would be very prohibitive in census
method, and therefore, it is advisable to carry out a sample survey,
and lastly; and
5.1.
some times accuracy may be lost because of the large size of the
population. Sampling involves a small portion of the population and
therefore, would involve very few people for conducting surveys and
for data collection and compilation. This would not be so in the
census method and the chances of committing errors would increase.
As the sampling involves less time and money, it would be possible to give attention to
different characteristics of the elementary units. A sample using same money and time can
produce a detailed study of lesser number of units. The process of sampling involves
selecting a sample, collecting all relevant information, and finally drawing conclusions about
the population from which the sample has been drawn.
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11.2. Definitions
The surveys are concerned with the attributes of certain entities, such as business enterprises,
human beings, etc. The attributes that are the object of the study are known as characteristics
and the units possessing them are called the elementary units.
The aggregate of elementary units to which the conclusions of the study apply is termed as
population/universe, and the units that form the basis of the sampling process are called
sampling units. The sampling unit may be an elementary unit.
The sample is defined as an aggregate of sampling units actually chosen in obtaining a
representative subset from which inferences about the population are drawn. The frame— a
list or directory, defines all the sampling units in the universe to be covered. This frame is
either constructed for the purpose of a particular survey or may consist of previously
available description of the population; the latter is the commonly used method. For example,
telephone directory can be used as a frame for conducting opinion surveys in a city or
locality.
In order that, sampling results reflect the characteristics of the population, it is necessary that
the sample selected for study should be
1.1.
Truly representative, i.e., the selected sample truly represent the
universe so that the results can be generalised;
2.1.
Adequate, i.e., the size of the sample or the sample size should be
adequate enough to represent the various characteristics of the
universe;
3.1.
Independent, i.e. the elementary units selected should be independent
of one another and all units of the population should have the same
chance of being selected in the sample; and lastly
4.1.
Homogeneous, i.e., there should not be any basic difference between
the characteristics of the units in the sample and that of the
population. This means that if two or more samples are drawn from
the same population, the results should be more or less identical.
11.3. Probability samples vs. non-probability samples
A probability sample is one for which the inclusion or exclusion of any individual element of
the population depends upon the application of probability methods and not on a personal
judgement. It is so designed and drawn that the probability of inclusion of an element is known.
The essential feature of drawing such a sample is the randomness. As against the probability
sample, we have a variety of other samples, termed as judgement samples, purposive samples,
quota samples, etc. These samples have one common distinguishing feature: personal
judgement rather than the random procedure to determine the composition of what is to be
taken as a representative sample. The judgement affects the choice of the individual elements.
All such samples are non-random, and no objective measure of precision may be attached to the
results arrived at.
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In a probability sampling, it is possible to estimate the error in the estimates and they can be
minimized also. It is also possible to evaluate the relative efficiency of the various probability
sampling designs. Probability sampling does not depend upon the detailed information about
population for its effectiveness. However, probability sampling requires a high level of skill
and experience for its use. It also requires sufficient time and money to execute.
Non-probability sampling is a procedure of selecting a sample without the use of probability
or randomisation. It is based on convenience, judgement, etc. The major difference between
the two approaches is that it is possible to estimate the sampling variability in the case of
probability sampling while it is not possible to estimate the same in the non-probability
sampling. The classification of various probability and non-probability methods are shown in
Fig. 11.1.
POPULATION
SAMPLING
Probability Samples
•Simple Random Sampling
•Stratified Random Sampling
•Cluster Sampling or
Multistage Sampling
•Systematic Sampling
Non probability Samples
•Convenience Sampling
•Quota Sampling
•Judgement Sampling
Fig. 11.1. Classification of sampling schemes
11.4. Probability Sampling Methods
The various probability sampling methods are described as under:
(a) Simple random sampling method
In simple random sampling, drawing of elements from the population is random and the
choice of an element is made in such a way that every element has the same probability of
being chosen. When the sample is so selected, every possible set of elements has the same
chance of being drawn. With N, population size, fairly large, the number of such possible sets
of size n is of course very large. This number is given by NCn. Of course, it is unnecessary in
a specific case to compute the number of possible sets of stated size that might be drawn from
a given population, but the process of sample selection should be such that the probability of
selection is the same for every such set.
The objective is to achieve randomness in drawing the individual elements of a sample for
ensuring that all possible samples have the same chance of being selected. If we are to draw
from a population containing N elementary units, the elementary unit also being a sampling
unit, it is necessary that each of the N units should be individually numbered or otherwise
distinctively designed. One of the approaches for drawing random sample of size n from a
population of N units is to draw n cards from N cards which are numbered from 1 to N and
mixed thoroughly. The sample size n, thus drawn, would constitute a simple random sample
(SRS). Another popular method of selecting a random sample is by lottery method. In this
method all the elements are named or numbered on a small slip of paper of identical shape
and size. These slips are folded identically and mixed up well in a container. Number of slips
of desired sample size is selected blindly from this container. Thus, the selection of
elementary units depends purely on chance and no personal bias exists. We shall illustrate
this method of selection of a sample with the following example: Suppose the warden of a
student’s hostel with 200 occupants wants to constitute a welfare committee with the
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members randomly selected. The lottery method of selecting these five members from a
group of 200 would be first to prepare 200 slips of identical shape and size and write the
name of each student on a slip. Fold these 200 slips identically and mix them well in a
container. Then select five folded slips, from the container at random. The five students so
selected would constitute a welfare committee of the hostel.
There are, however, some difficulties in these procedures. For, if N is large, the task becomes
physically difficult. So it is desirable to use better methods for ensuring randomness. One
such method is the use of random number tables.
Use of random number tables
If the N elements of a total population are numbered serially from 1 to N, a random sample
may be most readily and reliably drawn by using a table of random numbers. Such tables
enable us to select n numbers at random from the full list of serial numbers from 1 to N. In a
random number table, digits in each column are in random order and so are the digits in each
row. As the arrangement is random in all directions, it makes no difference where we begin in
our selection of random numbers from such a table. However, the column arrangement is
generally found more convenient for references.
Several random number tables are available for use. These numbers have been adequately
tested for randomness. Among them, the most popular ones are:
1.1.
Tippett’s (1927) 10,400 sets of four-digited random numbers;
2.1.
Fisher and Yates (1938) table of random numbers with 1,500 sets of
ten-digited random numbers; and
3.1.
Rand Corporation (1955) table of random numbers of 2,00,000 sets of
five-digited random numbers.
Tippet’s table of random numbers is most popularly used in practice. Given below are the
first forty sets from Tippet’s table as an illustration of the general appearance of random
numbers:
2952
6641
3992
9792
7969
5911
3170
5624
4167
9524
1545
1396
7203
5356
1300
2693
2670
7483
3408
2762
3563
1089
6913
7691
0560
5246
1112
6107
6008
8125
4233
8776
2754
9143
1405
9025
7002
6111
8816
6446
Tippett’s numbers have been subjected to numerous tests and used in many investigations
and their randomness has been well established for all practical purposes. An example to
illustrate how Tippett’s table of random numbers may be used is given below.
Suppose ten numbers from out of 0 and 80 are required. We start anywhere in the table and
write down the numbers in pairs. The table can be read horizontally, vertically, diagonally or
in any methodical way. Starting with the first and reading horizontally first we obtain 29, 52,
66, 41, 39, 92, 97, 92, 79, 69, 59, 11, 31, 70, 56, 24, 41, 67 and so on. Ignoring the numbers
greater than 80, we obtain for one purpose ten random numbers, namely 29, 52, 66, 41, 39,
79, 69, 59, 11 and 31.
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The sampling procedure described above is quite satisfactory for a small population. With a
large population, the process of identification of numbers to each elementary sampling unit
becomes very prohibitive with respect to both time and money. Moreover, the population is
often geographically spread out or composed of clearly identified strata possessing unique
characteristics. Whenever any of the above situations arise, alternative sampling schemes that
are sophisticated combinations of simple random sampling provide significantly better results
for the same expenditure and time. As a result, the simple random sampling method is not
very frequently used in practice. However, the simple random sampling scheme is the basis
of any other probabilistic sampling schemes.
(b)
Stratified random sampling method
In simple random sampling, the population to be sampled is treated as homogeneous and the
individual elements are drawn at random from the whole universe. However, it is often
possible and desirable to classify the population into distinctive classes or strata and then
obtain a sample by drawing at random the specified number of sampling units from each of
the classes thus constructed. This may be desirable because of our interest in the distinct
classes of the universe as a whole.
In stratified random sampling, the population is sub-divided into strata before the sample is
drawn. Strata are so designed that they should not overlap. A sample of specified size is
drawn at random from the sampling units that make up each stratum. If a given stratum is of
our interest, the corresponding sub-sample provides the basis for estimates concerning the
attributes of the population stratum, or sub-universe from which it is drawn. The total of subsamples constitutes the aggregate sample on which estimates of attributes of the entire
population are based.
Stratified samples may be either proportional or non-proportional. In a proportional stratified
sampling, the number of elements to be drawn from each stratum is proportional to the size of
that stratum compared with the population. For example, if a sample size of 500 elementary
units have to be drawn from a population with 10,000 units divided in four strata in the
following way:
Population size
Sample size
Stratum I =
2000
500 × 0.2 = 100
Stratum II =
3000
500 × 0.3 = 150
Stratum III =
4000
500 × 0.4 = 200
Stratum IV =
1000
500 × 0.1 = 50
Total 10000
500
Thus, the elements to be drawn from each stratum would be 100, 150, 200 and 50 respectively.
Proportional stratification yields a sample that represents the population with respect to the
proportion in each stratum in the population. Proportional stratified sampling yields satisfactory
results if the dispersion in the various strata is of proportionately the same magnitude. If there is a
significant difference in dispersion from stratum to stratum, sample estimates will be much more
efficient if non-proportional stratified random sampling is used. Here, equal numbers of elements
are selected from each stratum regardless of how the stratum is represented in the population.
Thus, in the earlier example, an equal number, i.e., 125, of elementary units will be drawn to
constitute the sample.
A sample drawn by stratified random sampling scheme ensures a representative sample as the
population is first divided into various strata and then a sample is drawn from each stratum.
Stratified random sampling also ensures greater accuracy and it is maximum if each stratum
is formed in such a way that it consists of uniform or homogeneous items. Compared with a
simple random sample, a stratified sample can be more concentrated geographically, i.e., the
elementary units from different strata may be selected in such a way that all of them are
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located in one geographical area. This would also reduce both time and cost involved in data
collection. However, care should be exercised in dividing the population into various strata.
Each stratum must contain, as far as possible, homogeneous units, as otherwise the reliability
of the results would be lost.
In conclusion, stratification is an effective sampling device to the extent that it creates classes
that are more homogeneous than the total. When this can be done, the classes are
distinguished that differ among themselves in respect of a stated characteristic. Stratification
may be futile if classes do not differ among themselves. Thus, there should be homogeneity
within classes and heterogeneity between classes.
(c) Cluster sampling or multistage sampling
Under this method, the random selection is made of primary, intermediate and final (or the
ultimate) units from a given population or stratum. There are several stages in which the
sampling process is carried out. At first, the first stage units are sampled by some suitable
method, such as simple random sampling. Then, a sample of second stage unit is selected
from each of the selected first stage units, again by some suitable method which may be same
as or different from the method employed for the first stage units. Further stages may be
added as required. The procedure may be illustrated as follows:
Suppose we want to take a sample of 5,000 households from the State of Haryana. At the first
stage, the state may be divided into a number of districts and a few districts are selected at
random. At the second stage, each district may be sub-divided into a number of villages and a
sample of villages may be taken at random. At the third stage, a number of households may
be selected from each of the villages selected at second stage. To take another example
supposes in a particular survey, we wish to take a sample of 10,000 students from a
University. We may take colleges at the first stage, then draw departments at the second
stage, and choose students as the third and last stage.
Merits: Multi-stage sampling introduces flexibility in the sampling method which is lacking
in the other methods. It enables existing divisions and sub-divisions of the population to be
used as units at various stages, and permits the field work to be concentrated and yet large
area to be covered.
Another advantage of this method is that sub-division into second stage units need be carried
out for only those first stage units which are included in the sample. It is, therefore,
particularly valuable in surveys of under-developed areas where no frame is generally
sufficiently detailed and accurate for subdivision of the material into reasonably small
sampling units.
Limitations: However, a multi-stage sample is in general less accurate than a sample
containing the same number of final stage units which have been selected by some suitable
single stage process.
(d) Systematic sampling
Another sampling form, simple in design and execution, may be employed when the
members of population to be sampled are arranged in order, the order corresponding to
consecutive numbers. The arrangements of names in a telephone directory or income-tax
returns in the income tax department are the illustrations of such orderings. A sample of
suitable size is obtained by taking every unit say, seventh unit of the population, one of the
first seven units in this ordered arrangement is chosen at random and the sample is
completely by selecting every seventh unit from the rest of the list. If the first unit selected is
the fifth, the researcher will include in his sample 12th, 19th, 26th, 33rd, etc. We can generalize
the approach as follows: if the requirements of the survey call for the inclusion of one unit
out of every m units in the population, a unit is chosen at random from the first m units,
thereafter, every mth unit in the population when arranged in order, is included in the sample.
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This mode of selection is called systematic sampling, m is generally referred to as the
N
sampling ratio, i.e., the ratio of the population size to the sample size. Symbolically m = n .
where N is the population size and n is the sample size. While calculating the value of m, we
may get a fractional value. In such cases, it is rounded off to the nearest digit.
Which sampling scheme to select
In sampling, one scheme is said to be more efficient than another when the sample estimates
developed by the scheme tend to cluster more closely around the population parameter being
estimated. An estimator of the population parameter should possess the following
characteristics:
1.1.
It should be unbiased: An estimator is unbiased when the expected
(average) value of the sample statistic is equal to the population
parameter being estimated.
2.1.
It should be efficient: Efficiency is with respect to sample size and it
means that the sample estimates should be clustered as closely
possible to the population parameter being estimated for a given
sample
size.
For
example,
when
the
population
is
normally
distributed, both the sample mean and the median are unbiased
estimators of the population mean. However, for any given sample
size, the sample means cluster more closely around the population
mean than do the sample medians. Thus, both mean and the median
are the unbiased estimators of the population mean. However, the
sample mean is the unbiased efficient estimator of the population
mean.
In stratified random sampling, where stratification is meaningful, a stratified random sample
will be more efficient than a simple random sample of the same size. A sampling design is
considered efficient with respect to cost if the sample estimates cluster more closely around
the population parameter being estimated than they would for any alternative sampling
scheme involving equivalent rupee expenditure.
It should be consistent: An estimator is considered to be consistent if the
sample
estimates cluster
more
and
more
closely around
the
population parameter being estimated as the sample size increases.
11.5. Non-Probability Sampling Methods
There are three methods of sampling in this category. These are explained as follows:
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1. Convenience sampling
In this scheme, a sample is obtained by selecting ‘convenient’ population elements. For
example, a sample selected from the readily available sources or lists such as telephone
directory or a register of the small scale industrial units, etc. will give us a convenient sample.
In these cases, even if a random approach is used for identifying the units, the scheme will
not be considered as simple random sampling. For example, if one studies the wage structure
in a close by textile industry by interviewing a few selected workers, then the scheme adopted
here is convenient sampling. The results obtained by convenience sampling method can
hardly be said to be representative of the population parameters. Therefore, the results
obtained are generally biased and unsatisfactory. However, convenient sampling approach is
generally used for making pilot studies, particularly for testing a questionnaire and to obtain
2. Quota sampling
In this method of sampling, the basic parameters which describe the population are identified
first. Then the sample is selected which conform to these parameters. Thus, in a quota
sample, quotas are fixed according to these parameters, and each field investigator is assigned
with quotas of the number of units to be interviewed. Within the preassigned quotas, the
selection of the sample elements depends on the personal judgement. For example, if one is
studying the consumer preferences for ice creams among children and college going students
and supposes it is fixed to interview 250 individuals from each category. If the city has five
colleges, one decides to fix up a quota of 50 students to be interviewed from each college. It
entirely depends upon the interviewer who will constitute this sub-sample of 50 students in a
college— they may be the first 50 students who visit the ice cream parlour or may be the 50
students who visit the parlour between 4 p.m. and 6 p.m., etc.
Quota sampling method has the advantage that the sample will conform to the selected
parameters of the population. The cost and time involved in getting information from the
sample will be relatively less for a quota sample but there are many weaknesses too. Some of
these are:
1.1.
It is difficult to validate the information gathered on the elementary
units,
2.1.
It may be difficult to specify the characteristics of the population and
therefore it may be hard to identify it,
3.1.
Even when the sample does conform to the characteristics used in the
quotas, the sample may be distorted on other factors of importance in
the study. For example, interviewing first 50 students or the last 50
students visiting the ice cream parlour can make a lot of difference
completely distort the results.
Quota sampling method is generally used in public opinion studies, election forecast polls, as
there is not sufficient time to adopt a probability sampling scheme.
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3. Judgement sampling
Judgement sampling method can also be called as sampling by opinion. In this method,
someone who is well acquainted with the population decides which members (elementary
units) in his or her judgement would constitute a proper cross-section representing the
parameters of relevance to the study. This method of sampling is generally used in studies
involving performance of personnel. For example, if one is studying the performance of sales
staff in a marketing organisation, the people here are classified into top grade, medium grade
and low grade performers. Having specified qualities that are important in the study, the
expert (possibly here the Vice-President-sales) indicates the people who, in his or her
knowledge, would be representative of each of the three categories mentioned earlier. This, of
course, is not a scientific method, but in the absence of better evidence, such a judgement
method may have to be used.
11.6. Determination of sample size
We prefer samples to complete enumeration because of convenience and reduced cost of data
collection. However, in sampling, there is a likelihood of missing some useful information
about the population. For a high level of precision, we need to take a larger sample. How
large should be the sample and what should be the level of precision? In specifying a sample
size, care should be taken such that (i) neither so few are selected so as to render the risk of
sampling error intolerably large, nor (ii) too many units are included, which would raise the
cost of the study to make it inefficient. It is, therefore, necessary to make a trade-off between
(i) increasing sample size, which would reduce the sampling error but increase the cost, and
(ii) decreasing the sample size, which might increase the sampling error while decreasing the
cost. Therefore, one has to make a compromise between obtaining data with greater precision
and with that of lower cost of data collection. Several factors need to be considered before
determining the sample size.
The first and the foremost is the size of the error that would be tolerable for the purposes of
decision-making. The second consideration would be the degree of confidence with the
results of the study, i.e., if one wants to be 100 per cent confident of the results, the entire
population must be studied. However, this is generally too impractical and costly. Therefore,
one must accept something less than 100 per cent confidence. In practice, the confidence
limits most often used are 99 per cent, 95 per cent and 90 per cent. Most commonly used
confidence limit is 95 per cent. This means that there is a 5 per cent risk that the true
population statistic is outside the range of possible error specified by the confidence interval.
This 5 per cent risk appears to be acceptable in most of the decisions. Thus, for 95 per cent
level of confidence, Z value is 1.96. The Z value can be obtained from normal probability
distribution for a specified level of confidence. For determining the sample size, we make use
of the following relationship:
σ x = standard error of the estimate = σ
n
σ x can be calculated if we know the upper and lower confidence limits. Let these limits be
Y, then
Z σx = Y
Where Z is the value of the normal variate for a given confidence level. The procedure has
been explained using the illustration given below:
Illustration 11.1. A state cooperative department is performing a survey to determine the
annual salary earned by managers numbering 3000 in the cooperative sector within the state.
How large a sample size it should take in order to estimate the mean annual earnings within
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plus and minus 1,000 and at 95 per cent confidence level? The standard deviation of annual
earnings of the entire population is known to be Rs. 3,000.
Solution. As the desired upper and lower limit is Rs. 1,000, i.e., we want to estimate the
annual earnings within plus and minus Rs. 1,000.
∴
z σ x = 1,000
As the level of confidence is 95 per cent, the Z value is 1.96
∴
1.96 σ x = 1,000
σ x = 1‚000 = 510.20
1.96
The standard error σ x is given by σ/ n where σ is the population standard deviation
σ
= 510.20
∴
n
3000
= 510.20
i.e.,
n
3000
n = 510.2 = 5.88
i.e.,
This gives n = 34.57
Therefore, the desired sample size is about 35.
11.6.1. Sample size for stratified sampling
Once the strata have been established, we are interested in the size of the stratified random
sample. The size will depend upon whether the proportional or disproportional (optimal)
sample is being taken.
A proportional stratified sample is one in which the sample units in a given stratum are
allocated in proportion to the relative size of the stratum. The following formula is used for
calculation of the proportional sample for each stratum
Ni
ni = N × n
Where ni = number of sample units from stratum i, N = the total number of units in the
population, Ni = the total number of units in the stratum i, n = sample size desired.
The standard error of mean is
k
σx =
∑wi2σi2/ni
i=1
where wi = the weight of stratum i = Ni/N, σi = the standard deviation of the ith stratum, k =
the total number of strata. In case of disproportionate stratified sampling, the proportion of
units in the sample stratum is not equal to the proportion of the population. The formula for
sample allocation in this case is
wiσin
ni =
k
∑wiσi
1
Thus, the disproportional stratified sample is more desirable if standard deviation (σi) of each
stratum is known. The standard error of the mean of a disproportionate stratified sample is
350
k
∑(wiσi)2
1
σx =
∑ni
It may be observed that the standard error for stratified sample is smaller than for simple
random sample, i.e., much smaller samples may be utilized when the population has been
stratified.
Illustration 11.2. In a market area, shops are divided into two categories, viz., those that
have daily turnover of more than Rs. 2000 and those that have daily turnover of less than Rs.
2000 for the study of estimating the total sales in the area. The total number of shops in the
first stratum are 420 and in the second stratum 180. A sample of 50 was selected, the standard
deviation has been found to be 70 for first stratum and 95 for second stratum. What size of
stratified random sample should be taken under proportional and disproportional stratified
sampling?
Solution. Under the proportional stratified sampling, the sample size is given by
Ni
ni = N × n
420
and, therefore n1 = 600 × 50 = 35
180
and
n2 = 600 × 50 = 15
The standard error (σ x ) =
∑w n
i
2 σi
2
i
(70)2 (0.3)2 × (95)2
(0.7)2 × 35 +
15
122.75 = 11.079
=
For disproportionate sampling, the sample size is given by:
wiσin
ni =
∑wiσi
0.7 × 70 × 50
2450
∴ n1 = 0.7 × 70 + 0.3 × 95 = 77.5 = 32.0
0.3 × 95 × 50
1425
and n2 = 0.7 × 70 + 0.3 × 95 = 77.5 = 18.0
The standard error is given by
k
∑(wiσi)2
1
(0.7 × 70 + 0.3 × 95)2
σx =
=
50
∑ni
=
120.125 = 10.96
=
11.6.2.
Cost as a factor in the determination of the sample size
Another consideration in determining the sample size is the cost. Management may reduce
the level of confidence in an attempt to reduce the cost of sampling. An illustration will
clarify how cost of sampling can be reduced by reducing the sample size.
Illustration 11.3. In a market area there are 600 shops. A researcher wishes to estimate
number of customers visiting these shops per day. The researcher wants to estimate the
sampling error in the number of customers visiting is no larger than ± 10 with probability of
0.95. The previous studies indicated that the standard deviation is 85 customers. If the cost
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per interview is Rs. 20 (this includes field work, supervision of interviewers, coding, editing
and tabulation of results and report writing, etc.), calculate the total cost involved. Researcher
is willing to sacrifice some accuracy in order to reduce cost. If he settles for an estimate with
0.90 probability, how much reduction in cost can be achieved?
Solution. For 95 per cent confidence levels,
Z σx = Y
i.e., 1.96 σ x = 10.0
10
∴ σ x = 1.96
Now, σ x is given by σ/ n and therefore, the sample size will be determined by the equation
10
σ
= 1.96
n
Since σ = 85, we have
85
10
= 1.96
n
∴ n = 277.6
Thus, if the sample is taken as 278, the total cost involved will be 278 × 20 = Rs. 5560. As
this cost is considered to be on the higher side by the researcher and in order to reduce the
cost, the researcher has now settled to 90 per cent confidence level. At 90 per cent confidence
level, the sample size can be calculated as follows:
Z σ x = 10
1.65 σ x =
10
1.65 σ x =
10
10
or σ x = 1.65
10
σ
∴
= 1.65
n
10
85
=
i.e.,
1.65
n
n = 196.7
The cost of survey for this sample size will be 197 × 20 = Rs. 3940. Thus, we have observed
that by reducing the confidence level from 95 per cent to 90 per cent, the researcher would
reduce the cost from Rs. 5560 to Rs. 3940. The researcher may not like to reduce the
confidence level further and so further cost reduction may not be desirable.
11.7. Self-Test Questions
1.
Describe the various methods of drawing a sample. Which one would
you suggest and why?
2.
Describe the importance of sampling. Critically examine the merits of probability
sampling and non-probability sampling methods.
3.
Specify and explain the factors that make sampling preferable to a complete census in
a statistical investigation.
352
4.
How would you determine the sample size for stratified sampling?
Explain with the help of a suitable example.
5.
To determine the effectiveness of the advertising campaign of a new
VCR, management would like to know what percentage of the
household are aware of the new brand. The advertising agency thinks
that this figure is as high as 70 per cent. The management would like
a 95% confidence interval and a margin of error no greater than plus
or minus 2 per cent. (a) What sample size should be used for this
study? (b) Suppose that management wanted to be 99 per cent
confident but could tolerate an error of plus or minus 3 per cent. How
would the sample size change?
1.
Statistical Methods by S.P. Gupta. Sultan Chand and Sons, New
Delhi.
2.
Statistics for MBA by T.R. Jain and Dr. S.C. Aggarwal. VK (India)
Enterprises, New Delhi. First edition.
3.
Business Statistics by Shenoy and Shenoy.
353
Course:
Author:
Anil Kumar
Course Code:
MC-106
Vetter:
Dr. B. S. Bodla
Lesson:
12
SAMPLING DISTRIBUTIONS
Objectives: The present lesson is an attempt to overview the concept of sampling
distributions. After successful completion of the lesson the students will be
able to understand the meaning and the need of studying sampling
distribution of a sample statistic.
Structure
12.1
Introduction
12.2
Sampling Distribution of the Mean
12.3
Central Limit Theorem
12.4
Sampling Distribution of the Proportion
12.5
Sampling Distribution of the Difference of Sample Means
12.6
Sampling Distribution of the Difference of Sample Proportions
12.7
Small Sampling Distributions
12.8
Sampling Distribution of the Variance
12.9
F Distribution
12.10
t-Distribution
12.11
Self-Assessment Questions
12.12
12.1 INTRODUCTION
Having discussed the various methods available for picking up a sample from a population,
we would naturally be interested in drawing statistical inferences - making generalizations
354
about the population on the basis of a sample drawn from it. The generalizations to be made
about the population are usually either by way of
¾ estimating the unknown population parameters, or
¾ testing appropriate hypotheses stated in relation to population parameters in the light
of sample data
These generalizations, together with the measurement of their reliability, are made in terms of
the relationship between the values of any sample statistic and those of the corresponding
population parameters. Population parameter is any number computed (or estimated) for the
entire population viz. population mean, population median, population proportion, population
variance and so on. Population parameter is unknown but fixed, whose value is to be
estimated from the sample statistic that is known but random. Sample Statistic is any
numbers computed from our sample data viz. sample mean, sample median, sample
proportion, sample variance and so on.
It may be appreciated that no single value of the sample statistic is likely to be equal to the
corresponding population parameter. This owes to the fact that the sample statistic being
random, assumes different values in different samples of the same size drawn from the same
population.
Referring to our earlier discussion on the concept of a random variable in the lessons on
Probability Distributions, it is not difficult to see that any sample statistics is a random
variable and, therefore, has a probability distribution better known as the Sampling
Distribution of the statistic.
The sampling distribution of a statistic is the probability distribution of all
possible values the statistic may take when computed from random samples
of the same size drawn from a specified population.
355
Figure 12-1
Sampling Distribution of a Statistic
In reality, of course we do not have all possible samples and all possible values of the
statistic. We have only one sample and one value of the statistic. This value is interpreted
with respect to all other outcomes that might have happened, as represented by the sampling
distribution of the statistic. In this lesson, we will refer to the sampling distributions of only
the commonly used sample statistics like sample mean, sample proportion, sample variance
etc., which have a role in making inferences about the population.
Why We Study Sampling Distributions?
Sample statistics form the basis of all inferences drawn about populations. Thus, sampling
distributions are of great value in inferential statistics. The sampling distribution of a sample
statistic possess well-defined properties which help lay down rules for making
generalizations about a population on the basis of a single sample drawn from it. The
variations in the value of sample statistic not only determine the shape of its sampling
distribution, but also account for the element of error in statistical inference. If we know the
probability distribution of the sample statistic, then we can calculate risks (error due to
356
chance) involved in making generalizations about the population. With the help of the
properties of sampling distribution of a sample statistic, we can calculate the probability that
the sample statistic assumes a particular value (if it is a discrete random variable) or has a
value in a given interval. This ability to calculate the probability that the sample statistic lies
in a particular interval is the most important factor in all statistical inferences. We will
demonstrate this by an example.
Suppose we know that 40% of the population of all users of hair oil prefers our brand to the
next competing brand. A "new improved" version of our brand has been developed and given
to a random sample of 100 users for use. If 55 of these prefer our "new improved" version to
the next competing brand, what should we conclude? For an answer, we would like to know
the probability that the sample proportion in a sample of size 100 is as large as 55% or higher
when the true population proportion is only 40%, i.e. assuming that the new version is no
better than the old. If this probability is quite large, say 0.5, we might conclude that the high
sample proportion viz. 55% is perhaps because of sampling errors and the new version is not
really superior to the old. On the other hand, if this probability works out to a very small
figure, say 0.001, then rather than concluding that we have observed a rare event we might
conclude that the true population proportion is higher than 40%, i.e. the new version is
actually superior to the old one as perceived by members of the population. To calculate this
probability, we need to know the probability distribution of sample proportion i.e. the
sampling distribution of the proportion.
12.2 SAMPLING DISTRIBUTION OF THE MEAN
Suppose we have a simple random sample of size n, picked up from a population of size N.
We take measurements on each sample member in the characteristic of our interest and
denote the observation as x1 , x 2 ,......x n respectively. The sample mean for this sample is
defined as:
357
X =
x1 + x 2 + ...... + x n
n
If we pick up another sample of size n from the same population, we might end up with a
totally different set of sample values and so a different sample mean. Therefore, there are
many (perhaps infinite) possible values of the sample mean and the particular value that we
obtain, if we pick up only one sample, is determined only by chance. In other words, the
sample mean is a random variable. The possible values of this random variable depends on
the possible values of the elements in the random sample from which sample mean is to be
computed. The random sample, in turn, depends on the distribution of the population from
which it is drawn. As a random variable, X has a probability distribution. This probability
distribution is the sampling distribution of X .
The sampling distribution of X is the probability distribution of all possible
values the random variable X may take when a sample of size n is taken
from a specified population.
To observe the distribution of X empirically, we have to take many samples of size n and
determine the value of X for each sample. Then, looking at the various observed values of
X , it might be possible to get an idea of the nature of the distribution. We will derive the
distribution of X in three cases:
(a)
Sampling from infinite populations
(b)
Sampling with replacement from finite populations
(c)
Sampling without replacement from finite populations
12.2.1
Sampling from Infinite Populations
Let us assume we have a population, with mean μ and variance σ 2 , which is infinitely large.
If we take a sample of size n with individual values x1 , x 2 ,......x n , then
358
( )
Sample Mean X =
x1 + x 2 + ...... + x n
n
here x1 representing the first observed values in the sample, is a random variable since it may
take any of the population values. Similarly x 2 , representing the second observed value in
sample is also a random variable since it may take any of the population values. In other
words, we can say that xi , representing the ith observed value in the sample is a random
variable.
Now when the population is infinitely large, whatever is the value of x1 , the distribution of
x 2 is not affected by it. This is true for any other pair of random variables as well. In other
words; x1 , x 2 ,......x n are independent random variables and all are picked up from the same
population.
So
E (xi ) = μ
and
Var (xi ) = σ2
for
i =1, 2,3,………n
Finally, we have
( )
μx = E X
⎛ x + x 2 + ...... + x n ⎞
= E⎜ 1
⎟
n
⎠
⎝
⎛x
⎛x ⎞
= E ⎜ 1 ⎟ + E⎜ 2
⎝ n
⎝n⎠
⎛x ⎞
⎞
⎟ + ...... + E ⎜ n ⎟
⎠
⎝ n ⎠
[as E(A + B) = E(A) + E(B)]
1
1
1
= E (x1 ) + E (x 2 ) + ...... + E ( x 2 ) [as E(nA) = n E(A)]
n
n
n
=
1
1
1
μ + μ +.…..+ μ
n
n
n
=μ
and
⎛ x1 + x 2 + ...... + x n ⎞
⎟
n
⎝
⎠
σ x2 = Var( X ) = Var ⎜
⎛x ⎞
⎛x ⎞
⎛x ⎞
= Var ⎜ 1 ⎟ + Var ⎜ 2 ⎟ + ...... + Var ⎜ n ⎟
⎝ n⎠
⎝n⎠
⎝ n ⎠
[as Var(A + B) = Var (A) + Var (B)]
359
=
1
1
1
Var ( x1 ) + 2 Var ( x 2 ) + ...... + 2 Var ( x n )
2
n
n
n
=
1 2 1 2
1
σ + 2 σ + ...... + 2 σ 2
2
n
n
n
=
σ2
n
So, σ x = SD( X ) =
12.2.2
[as Var(nA) = n2 Var(A)]
σ
n
Sampling With Replacement from Finite Populations
The above results have been obtained under the assumption that the random variables
x1 , x 2 ,......x n are independent. This assumption is valid when the population is infinitely
large. It is also valid when the sampling is done with replacement, so that the population is
back to the same form before the next sample member is picked up. Hence, if the sampling is
done with replacement, we would again have:
( )
μx = E X = μ
12.2.3
σ x2 = Var( X ) =
and
σ2
or
n
σ x = SD( X ) =
σ
n
Sampling Without Replacement from Finite Populations
When sampling without replacement from a finite population, the probability distribution of
the second random variable depends on what has been the outcome of the first pick and so on.
In other words, the n random variables representing the n sample members do not remain
independent, the expression for the variance of X changes. The results in this case will be:
( )
μx = E X = μ
and
σ x2 = Var( X ) =
σ2 N −n
n
.
N −1
or
σ x = S.D( X ) =
σ
n
.
N −n
N −1
By comparing these expressions with the ones derived above we find that the variance of
X is the same but further multiplied by a factor
N −n
. This factor is, therefore, known as
N −1
the finite population multiplier or the correction factor.
360
In practice, almost all the samples are picked up without replacement. Also, most populations
are finite although they may be very large and so the variance of the mean should
theoretically be found by using the expression given above. However, if the population size
(N) is large and consequently the sampling ratio (n/N) small, then the finite population
multiplier is close to 1 and is not used, thus treating large finite populations as if they were
infinitely large. For example, if N = 100,000 and n = 100, the finite population multiplier will
be 0.9995, which is very close to 1 and the variance of the mean would, for all practical
purposes, be the same whether the population is treated as finite or infinite. As a rule of that,
the finite population multiplier may not be used if the sampling ratio (n/N) is smaller than
0.05.
Above discussion on the sampling distribution of mean, presents two very important results,
which we shall be using very often in statistical estimation and hypotheses testing. We have
seen that the expected value of the sample mean is the same as the population mean.
Similarly, that the variance of the sample mean is the variance of the population divided by
the sample size (and multiplied by the correction factor when appropriate).
The fact that the sampling distribution of X has mean μ is very important. It means that, on
the average, the sample mean is equal to the population mean. The distribution of the statistic
is centered on the parameter to be estimated, and this makes the statistic X a good estimator
of μ. This fact will become clearer in the next lesson, where we will discuss estimators and
their properties. The fact that the standard deviation of X is σ
n means that as the sample
size increases, the standard deviation of X decreases, making X more likely to be close to
μ. This is another desirable property of a good estimator, to be discussed in the next lesson.
If we take a large number of samples of size n, then the average value of the sample means
tends to be close to the true population mean. On the other hand, if the sample size is
361
increased then the variance of X gets reduced and by selecting an appropriately large value
of n, the variance of X can be made as small as desired.
The standard deviation of X is also called the standard error of the mean. It indicates the
extent to which the observed value of sample mean can be away from the true value, due to
sampling errors. For example, if the standard error of the mean is small, we may be
reasonably confident that whatever sample mean value we have observed cannot be very far
away from the true value.
Before discussing the shape of the sampling distribution of mean, let us verify the above
results empirically, with the help of a simple example.
Consider a discrete uniform population consisting of the values 1, 2, and 3. If the random
variable X represents these population values, its mean is
μ=
∑X
i
=
N
6
=2
3
and variance is
2
σ =
(a)
∑ (X
i
− μ)
N
2
2
2
(1 − 2) 2 + (2 − 2) 2 + (3 − 2)
=
=
3
3
Sampling with Replacement
If random samples of size n = 2 are drawn with replacement from this population, we will
have Nn = 32 = 9 possible samples. These are shown in Box 12-1 along with the
corresponding sample mean values, which vary from 1 to 3.
The resulting distribution of X is given below:
X
:
1
1.5
2
2.5
3
P( X )
:
1/9
2/9
3/9
2/9
1/9
Box 12-1
Sample No. 1
Sample No. 2
362
Sample No. 3
(1,1)
X=1
Sample No. 4
(2,1)
X = 1.5
Sample No. 7
(3,1)
X=2
(1,2)
X = 1.5
Sample No. 5
(2,2)
X=2
Sample No. 8
(3,2)
X = 2.5
(1,3)
X=2
Sample No. 6
(2,3)
X = 2.5
Sample No. 9
(3,3)
X=3
Now we can find out the mean and variance of the sampling distribution, the necessary
calculations are given in Table 12-1.
Table 12-1
Calculations for μ x and σ x2
( )
( )
( )
PX
X .P X
P X .[ X − E X ] 2
1
1/9
1/9
1/9
1.5
2/9
3/9
2/36
2
3/9
6/9
0
2.5
2/9
5/9
2/36
3
1/9
3/9
1/9
∑ P(X ) = 1
∑ X .P(X ) = 2
∑ P(X ).[ X − E (X )]
So the mean of the sampling distribution,
( )
μx = E X
=
∑ X .P(X ) = 2 = μ
and the variance of the sampling distribution,
σ x2 = Var( X )
=
(b)
( )
X
∑ P(X ).[ X − E (X )]
2
=1/3 =
Sampling without Replacement
363
2/3 σ 2
=
n
2
2
= 1/ 3
If random samples of size n = 2 are drawn without replacement from this population, we will
have
N
Pn = 3P2 = 6 possible samples. These are shown in Box 12-2 along with the
corresponding sample mean values, which vary from 1.5 to 2.5.
Box 12-2
Sample No. 1
(1,2)
X = 1.5
Sample No. 4
(2,3)
X = 2.5
Sample No. 2
(1,3)
X=2
Sample No. 5
(3,1)
X=2
Sample No. 3
(2,1)
X = 1.5
Sample No. 6
(3,2)
X = 2.5
The resulting distribution of X is given below:
X
:
1.5
2
2.5
P( X )
:
2/6
2/6
2/6
Now we can find out the mean and variance of the sampling distribution, the necessary
calculations are given in Table 12-2.
Table 12-2
Calculations for μ x and σ x2
( )
( )
( )
( )
X
PX
X .P X
P X .[ X − E X ] 2
1.5
2/6
3/6
2/24
2
2/6
4/6
0
2.5
2/6
5/6
2/24
∑ P(X ) = 1
∑ X .P(X ) = 2
So the mean of the sampling distribution,
( )
μx = E X
=
∑ X .P(X ) = 2 = μ
and the variance of the sampling distribution,
σ x2 = Var( X )
364
∑ P(X ).[ X − E (X )]
2
= 1/ 6
( )
( )
2/3 3− 2 σ 2 N − n
.
= ∑ P X .[ X − E X ] = 1/6 =
=
.
n N −1
2 3 −1
2
Now if we compare the shapes of the parent population and the resulting sampling
distribution of mean, we find that although our parent population is uniformly distributed, the
sampling distribution of mean is symmetrically distributed as shown in Figure 12-2.
If we increase the sample size n we observe an interesting and important fact. As n increases
¾ the possible values X can assume increases, so the number of rectangles increases
¾ the probability that X assumes a particular value decreases i.e. the width of rectangles
Probability
decreases
Parent
Population
Figure 12-2
Sampling Distribution of Mean
n=2
n=5
Parent Population and Sampling Distribution of Mean
for n = 2 and n = 5
In the limiting case when the sample size n increases infinitely, the particular values X can
assume approaches infinity and the probability that X assumes a particular value approaches
to zero. In other words, the limiting distribution of X is normal distribution.
Thus as n → ∝
X ~ N (μ,
2
σ2 n )
f (X )
f( X )
Total
Area
P(a < X < b)
Aaaa
Aaaa
Aaaaaaaaaa
aa
365
Figure 12-3
Limiting Distribution of X
12.3 THE CENTRAL LIMIT THEOREM
The result we just stated - the limiting distribution of X is the normal distribution - is one of
the most important results in statistics. It is popularly known as the central limit theorem.
When sampling is done from a population with mean μ and standard
deviation σ, the sampling distribution of the sample mean X tends to a normal distribution with mean μ and standard deviation σ
n as the sample
size n increases.
For "Large Enough" n:
X ~ N (μ,
2
σ2 n )
The central limit theorem is remarkable because it states that the distribution of the sample
mean X tends to a normal distribution regardless of the distribution of the population from
which the random sample is drawn. The theorem allows us to make probability statements
about the possible range of values the sample mean may take. It allows us to compute
probabilities of how far away X may be from the population mean it estimates. We will
extensively use the central limit theorem in the next two lessons about statistical estimation
and testing of hypotheses.
366
Figure 12-4
Sampling Distributions of X for different Sample Sizes
The central limit theorem says that, in the limit, as n goes to infinity (n → ∝), the distribution
of X becomes a normal distribution (regardless of the distribution of the population). The
rate at which the distribution approaches a normal distribution does depend, however, on the
shape of the distribution of the parent population:
¾ if the population itself is normally distributed, the distribution of X is normal for any
sample size n
¾ if the population distributions are very different from a normal distribution, a
relatively large sample size is required to achieve a good normal approximation for
the distribution of X
Figure 12-4 shows several parent population distributions and the resulting sampling
distributions of X for different sample sizes.
Since we often do not know the shape of the population distribution, it would be useful to
have some general rule of thumb telling us when a sample is “Large Enough” that we may
apply the central limit theorem:
In general, a sample of 30 or more elements is considered “Large Enough”
for the central limit theorem to be applicable.
367
We emphasize that this is a general, and somewhat arbitrary, rule. A larger minimum sample
size may be required for a good normal approximation when the population distribution is
very different from a normal distribution. By the same reason, a smaller minimum sample
size may suffice for a good normal approximation when the population distribution is close to
a normal distribution.
Figure 12-5
Population Distribution and the Sampling Distribution of X
Figure 12-5 should help clarify the distinction between the population distribution and the
sampling distribution of X . The figure emphasizes the three aspects of the central limit
theorem:
1.
When the sample size is large enough, the sampling distribution of X is
normal
2.
The expected value of X is μ
3.
The standard deviation of X is σ
n
The last statement is the key to the important fact that as the sample size increases, the
(take a larger sample), our uncertainty (measured by the standard deviation) about the
parameter being estimated decreases.
368
The History of the Central Limit Theorem
What we call the central limit theorem actually comprises several theorems developed over
the years. The first such theorem was the discovery of the normal curve by Abraham De
Moivre in 1733, when he discovered the normal distribution as the limit of the binomial
distribution. The fact that the normal distribution appears as a limit of the binomial distribution as n increases is a form of the central limit theorem. Around the turn of the twentieth
century, Liapunov gave a more general form of the central limit theorem, and in 1922
Lindeberg gave the final form we use in applied statistics. In 1935, W Feller gave the proof
of the necessary condition of the theorem.
Let us now look at an example of the use of the central limit theorem.
Example 12-1
ABC Tool Company makes Laser XR; a special engine used in speedboats. The company’s
engineers believe that the engine delivers an average power of 220 horsepower, and that the
standard deviation of power delivered is 15 horsepower. A potential buyer intends to sample
100 engines (each engine to be run a single time). What is the probability that the sample
mean X will be less than 217 horsepower?
Solution: Given that:
Population mean
μ = 220 horsepower
Population standard deviation
σ = 15 horsepower
Sample size
n = 100
Here our random variable X is normal (or at least approximately so, by the central limit
theorem as our sample size is large).
369
X ~ N (μ,
or
2
σ2 n )
2
X ~ N (220, 15 2 100 )
So we can use the standard normal variable Z =
P( X < 217)
= P(Z <
X −μ
to find the required probability,
σ n
217 − 220
)
15 100
= P(Z < -2)
= 0.0228
So there is a small probability that the potential buyer’s tests will result in a sample mean less
than 217 horsepower.
12.4 SAMPLING DISTRIBUTION OF THE PROPORTION
Let us assume we have a binomial population, with a proportion p of the population possesses
a particular attribute that is of interest to us. This also implies that a proportion q (=1-p) of the
population does not possess the attribute of interest. If we pick up a sample of size n with
replacement and found x successes in the sample, the sample proportion of success ( p ) is
given by
p =
x
n
x is a binomial random variable, the possible value of this random variable depends on the
composition of the random sample from which p is computed. The probability of x
successes in the sample of size n is given by a binomial probability distribution, viz.
P( x) = nCx p x qn-x
Since p =
x
and n is fixed (determined before the sampling) the distribution of the number
n
of successes (x) leads to the distribution of p .
370
The sampling distribution of p is the probability distribution of all possible
values the random variable p may take when a sample of size n is taken
from a specified population.
The expected value and the variance of x i.e. number of successes in a sample of size n is
known to be:
E(x) = n p
Var (x) = n p q
Finally we have mean and variance of the sampling distribution of p
()
⎛ x⎞
⎝n⎠
μp = E p = E⎜ ⎟
=
and
1
1
E(x) = .n p = p
n
n
()
⎛ x⎞
⎝n⎠
σ 2p = Var p = Var ⎜ ⎟
=
()
σ p = SD p =
1
1
pq
. Var(x) = 2 . n p q =
2
n
n
n
pq
n
When sampling is without replacement, we can use the finite population correction factor, so
sampling distribution of p has its
Mean
μp = p
Variance
σ 2p =
and standard deviation
pq ⎛ N − n ⎞
.⎜
⎟
n ⎝ N −1 ⎠
σp =
pq N − n
.
n N −1
371
As the sample size n increases, the central limit theorem applies here as well. The rate at
which the distribution approaches a normal distribution does depend, however, on the shape
of the distribution of the parent population.
¾ if the population is symmetrically distributed, the distribution of p approaches the
normal distribution relatively fast
¾ if the population distributions are very different from a symmetrical distribution, a
relatively large sample size is required to achieve a good normal approximation for
the distribution of p
In order to use the normal approximation for the sampling distribution of p , the sample size
needs to be large. A commonly used rule of thumb says that the normal approximation to the
distribution of p may be used only if both n p and n q are greater than 5.
We now state the central limit theorem when sampling for the population proportion p .
When sampling is done from a population with proportion p, the sampling
distribution of the sample proportion p approaches to a normal distribution
with proportion p and standard deviation pq n as the sample size n
increases.
For "Large Enough" n:
p ~ N (p,
2
pq n )
The estimated standard deviation of p is also called its standard error. We demonstrate the
use of the theorem in Example 12-2
Example 12-2
A manufacturer of screws has noticed that on an average 0.02 proportion of screws produced
are defective. A random sample of 400 screws is examined for the proportion of defective
372
screws. Find the probability that the proportion of the defective screws ( p ) in the sample is
between 0.01 and 0.03?
Solution: Given that:
Population proportion
p = 0.02
So
q = 0.08 (= 1-0.02)
Sample size
n = 400
Since the population is infinite and also the sample size is large, the central limit theorem
applies. So
p ~ N (p,
2
pq n )
p ~ N (0.02,
2
(0.02)(0.08) 400 )
⎛ p− p ⎞
⎟
We can find the required probability using standard normal variable Z = ⎜
⎜ pq / n ⎟
⎠
⎝
⎛
⎜
0.01 − 0.02
<Z<
P(0.01 < p < 0.03) = P ⎜⎜
(0.02)(0.08)
⎜
400
⎝
⎞
⎟
0.03 − 0.02 ⎟
(0.02)(0.08) ⎟
⎟
400
⎠
0.01 ⎞
⎛ − 0.01
= P⎜
<Z<
⎟
0.007 ⎠
⎝ 0.007
= P(-1.43 < Z < 1.43)
= 2 P(0 < Z < 1.43)
= 0.8472
So there is a very high probability that the sample will result in a proportion between 0.01
and 0.03.
12.5 SAMPLING DISTRIBUTION OF THE DIFFERENCE OF SAMPLE MEANS
In order to bring out the sampling distribution of the difference of sample means, let us
assume we have two populations labeled as 1 and 2. So that
μ1 and μ2 denote the two population means.
σ1 and σ2 denote the two population standard deviations
n1 and n2 denote the two sample sizes
373
X 1 and X 2 denote the two sample means
Let us consider independent random sampling from the populations so that the sample sizes
need not be same for both populations.
Since X 1 and X 2 are random variables so is their difference X 1 - X 2 . As a random variable,
X 1 - X 2 has a probability distribution. This probability distribution is the sampling
distribution of X 1 - X 2 .
The sampling distribution of X 1 - X 2 is the probability distribution of all
possible values the random variable X 1 - X 2 may take when independent
samples of size n1 and n2 are taken from two specified populations.
Mean and Variance of X 1 - X 2
(
μX −X = E X1 - X 2
1
2
)
( ) ( )
= E X1 − E X 2
= μ1 - μ2
and
(
σ X2 − X = Var X 1 - X 2
1
2
)
( )
( )
= Var X 1 + Var X 2
=
σ 12
=
σ 12 ⎛ N 1 − n1 ⎞ σ 22 ⎛ N 2 − n2 ⎞
n1
+
σ 22
n2
; when sampling is with replacement
⎟+
⎟ ; when sampling is without
.⎜
.⎜
n1 ⎜⎝ N 1 − 1 ⎟⎠ n 2 ⎜⎝ N 2 − 1 ⎟⎠
replacement
As the sample sizes n1 and n2 increases, the central limit theorem applies here as well. So we
state the central limit theorem when sampling for the difference of population means X 1 - X 2
When sampling is done from two populations with means μ1 and μ2 and
standard deviations σ1 and σ2 respectively, the sampling distribution of the
difference of sample means X 1 - X 2 approaches to a normal distribution
374
with mean μ1 - μ2 and standard deviation
σ 12
n1
+
σ 22
n2
as the sample sizes n1
and n2 increases.
For "Large Enough" n1 and n2:
X 1 - X 2 ~ N (μ1 - μ2,
σ 12
n1
+
σ 22
n2
2
)
The estimated standard deviation of X 1 - X 2 is also called its standard error. We
demonstrate the use of the theorem in Example 12-3.
Example 12-3
The makers of Duracell batteries claims that the size AA battery lasts on an average of 45
minutes longer than Duracell’s main competitor, the Energizer. Two independent random
samples of 100 batteries of each kind are selected. Assuming σ 1 = 84 minutes and
σ 2 = 67 minutes, find the probability that the difference in the average lives of Duracell and
Energizer batteries based on samples does not exceed 54 minutes.
Solution: Given that:
μ1 - μ2 = 45
σ1 = 84 and σ2 = 67
n1 =100 and n2 = 100
Let X 1 and X 2 denote the two sample average lives of Duracell and Energizer batteries
respectively. Since the population is infinite and also the sample sizes are large, the central
limit theorem applies.
i.e
X 1 - X 2 ~ N (μ1 - μ2,
σ 12
n1
+
σ 22
2
n2
2
X 1 - X 2 ~ N (45,
84 2 67 2
+
)
100 100
375
)
So we can find the required probability using standard normal variable
Z =
(X
1
)
− X 2 − (μ 1 − μ 2 )
σ 12
n1
So
P( X 1 - X 2 < 54) = P(Z<
+
σ 22
n2
54 − 45
84 2 67 2
+
100 100
)
= P(Z < 0.84)
= 1- 0.20045
= 0.79955
So there is a very high probability that the difference in the average lives of Duracell and
Energizer batteries based on samples does not exceed 54 minutes.
12.6 SAMPLING DISTRIBUTION OF THE DIFFERENCE OF SAMPLE
PROPORTIONS
Let us assume we have two binomial populations labeled as 1 and 2. So that
p1 and p2 denote the two population proportions
n1 and n2 denote the two sample sizes
p1 and p 2 denote the two sample proportions
Let us consider independent random sampling from the populations so that the sample sizes
need not be same for both populations.
Since p1 and p 2 are random variables so is their difference p1 - p 2 . As a random variable,
p1 - p 2 has a probability distribution. This probability distribution is the sampling
distribution of p1 - p 2 .
The sampling distribution of p1 - p 2 is the probability distribution of all
possible values the random variable p1 - p 2 may take when independent
samples of size n1 and n2 are taken from two specified binomial populations.
376
Mean and Variance of p1 - p 2
(
μ p − p = E p1 - p 2
1
2
)
( ) ( )
= E p1 − E p 2
= p1 - p2
and
(
σ p2 − p = Var p1 - p 2
1
2
)
( )
( )
= Var p1 + Var p 2
=
=
p1 q1 p 2 q 2
; when sampling is with replacement
+
n1
n2
p1 q1 ⎛ N 1 − n1 ⎞ p 2 q 2 ⎛ N 2 − n 2
⎟+
.⎜
.⎜
n1 ⎜⎝ N 1 − 1 ⎟⎠
n 2 ⎜⎝ N 2 − 1
⎞
⎟⎟ ; when sampling is
⎠
without replacement
As the sample sizes n1 and n2 increases, the central limit theorem applies here as well. So we
state the central limit theorem when sampling for the difference of population proportions
p1 - p 2
When sampling is done from two populations with proportions p1 and p2
respectively, the sampling distribution of the difference of sample proportions
p1 - p 2 approaches to a normal distribution with mean p1 - p2 and standard
deviation
p1 q1 p 2 q 2
as the sample sizes n1 and n2 increases.
+
n1
n2
2
For "Large Enough" n1 and n2:
p1 - p 2 ~ N (p1 - p2,
p1 q1 p 2 q 2
+
)
n1
n2
The estimated standard deviation of p1 - p 2 is also called its standard error. We demonstrate
the use of the theorem in Example 12-4.
Example 12-4
It has been experienced that proportions of defaulters (in tax payments) belonging to business
class and professional class are 0.20 and 0.15 respectively. The results of a sample survey
are:
377
Professional class
Sample size:
n1 = 400
n2 = 420
Proportion of defaulters:
p1 = 0.21
p 2 = 0.14
Find the probability of drawing two samples with a difference in the two sample proportions
larger than what is observed.
Solution: Given that:
p1 = 0.20
p2 = 0.15
q1 = 1-0.20 = 0.80
q2 = 1-0.15 = 0.85
n1 = 400
n2 = 420
p1 = 0.21
p 2 = 0.14
Since the population is infinite and also the sample sizes are large, the central limit theorem
applies. i.e.
2
p1 - p 2 ~ N (p1 - p2,
p1 q1 p 2 q 2
+
)
n1
n2
2
(0.20)(0.80) (0.15)(0.85)
p1 - p 2 ~ N (0.05,
+
)
400
420
So we can find the required probability using standard normal variable
Z =
(p
1
)
− p 2 − ( p1 − p 2 )
p1 q1
p q
+ 2 2
n1
n2
P( p1 - p 2 > 0.07) = P(Z >
0 . 07 − 0 . 05
)
( 0 . 20 )( 0 . 80 ) ( 0 . 15 )( 0 . 85 )
+
400
400
= P(Z > 0.76)
= 0.22363
So there is a low probability of drawing two samples with a difference in the two sample
proportions larger than what is observed.
378
12.7 SMALL SAMPLING DISTRIBUTIONS
Up to now we were discussing the large sampling distributions in the sense that the various
sampling distributions can be well approximated by a normal distribution for “Large Enough”
sample sizes. In other words, the Z-statistic is used in statistical inference when sample size is
large. It may, however, be appreciated that the sample size may be prohibited from being
large either due to physical limitations or due to practical difficulties of sampling costs being
too high.
Consequently, for our statistical inferences, we may often have to contend
ourselves with a small sample size and limited information. The consequences of the sample
being small; n < 30; are that
¾ the central limit theorem ceases to operate, and
¾ the sample variance S2 fails to serve as an unbiased estimator of σ 2
Thus, the basic difference which the sample size makes is that while the sampling
distributions based on large samples are approximately normal and sample variance S2 is an
unbiased estimator of σ 2 , the same does not occur when the sample is small.
It may be appreciated that the small sampling distributions are also known as exact sampling
distributions, as the statistical inferences based on them are not subject to approximation.
However, the assumption of population being normal is the basic qualification underlying the
application of small sampling distributions.
In the category of small sampling distributions, the Binomial and Poisson distributions were
already discussed in lesson 9. Now we will discuss three more important small sampling
distributions – the chi-square, the F and the student t-distribution. The purpose of discussing
these distributions at this stage is limited only to understanding the variables, which define
them and their essential properties. The application of these distributions will be highlighted
in the next two lessons.
379
The small sampling distributions are defined in terms of the concept of degrees of freedom.
We will discuss this before concept proceeding further.
Degrees of Freedom (df)
The concept of degrees of freedom (df) is important for many statistical calculations and
probability distributions. We may define df associated with a sample statistic as the number
of observations contained in a set of sample data which can be freely chosen. It refer to the
number of independent variables which vary freely without being influenced by the
restrictions imposed by the sample statistic(s) to be computed.
Let x1 , x 2 ......x n be n observations comprising a sample whose mean x =
1 n
∑ xi is a value
n i =1
known to us. Obviously, we are free to assign any value to n-1 observation out of n
observations. Once the value are freely assigned to n-1observations, freedom to do the same
for the nth observation is lost and its value is automatically determined as
nth observation = n x - sum of n-1 observations
n −1
= n x − ∑ xi
i =1
As the value of nth observation must satisfy the restriction
n
∑x
i =1
i
= nx
We say that one degree of freedom, df is lost and the sum n x of n observations has n-1 df
associated with it.
For example, if the sum of four observations is 10, we are free to assign any value to three
observations only, say, x1 = 2, x 2 = 1 and x3 = 4 . Given these values, the value of fourth
observation is automatically determined as
4
x 4 = ∑ xi − (x1 + x 2 + x3 )
i =1
380
x 4 = 10 − (2 + 1 + 4)
x4 = 3
Sampling essentially consists of defining various sample statistics and to make use them in
estimating the corresponding population parameters. In this respect, degrees of freedom may
be defined as the number of n independent observations contained in a sample less the
number of parameters m to be estimated on the basis of that sample information, i.e. df =
n-m.
For example, when the population variance σ2 is not known, it is to be estimated by a
particular value of its estimator S2; the sample variance. The number of observations in the
sample being n,
df = n-m = n-1 because σ2 is the only parameter (i.e. m =1) to be
estimated by the sample variance.
12.8 SAMPLING DISTRIBUTION OF THE VARIANCE
We will now discuss the sampling distribution of the variance. We will first introduce
the concept of the sample variance as an unbiased estimator of population variance
and then present the chi-square distribution, which helps us in working out
probabilities for the sample variance.
12.8.1 THE SAMPLE VARIANCE
By now it is implicitly clear that we use the sample mean to estimate the population mean
and sample proportion to estimate the population proportion, when those parameters are
unknown. Similarly, we use a sample statistic called the sample variance to estimate the
population variance.
As will see in the next lesson on Statistical Estimation a sample statistic is an unbiased
estimator of the population parameter when the expected value of sample statistic is equal to
the corresponding population parameter it estimates.
Thus, if we use the sample variance S2 as an unbiased estimator of population varianceσ2
381
E(S2) = σ2
Then
However, it can be shown empirically that while calculating S2 if we divide the sum of square
∑ (x − x )
n
of deviations from mean (SSD) i.e.
2
by n, it will not be an unbiased estimator of σ2
i =1
(
⎛ n
⎜∑ x−x
E ⎜ i =1
⎜
n
⎜
⎝
and
∑ (x − x )
)
2
⎞
⎟
⎟
⎟
⎟
⎠
=
n −1 2
σ
n
σ2 −
=
σ2
n
2
i.e.
n
will underestimate the population variance σ by the factor
2
σ2
n
compensate for this downward bias we divide
∑ (x − x )
2
by n-1, so that S 2 =
To
∑ (x − x )
n
n
.
2
i =1
i =1
n −1
is
an unbiased estimator of population variance σ2 and we have:
(
⎛ n
⎜∑ x−x
E ⎜ i =1
⎜ n −1
⎜
⎝
)
2
⎞
⎟
⎟ = σ2
⎟
⎟
⎠
In other words to get the unbiased estimator of population variance σ2, we divide the
n
(
)
sum ∑ x − x by the degree of freedom n-1
i =1
2
12.8.2 THE CHI-SQUARE DISTRIBUTION
Let X be a random variable representing N values of a population, which is normally
distributed with mean μ and varianceσ2, i. e.
X = {X 1 , X 2 ...... X N }
We may draw a random sample of size n comprising x1 , x 2 ......x n values from this population.
As brought out in section 12.2, each of the n sample values x1 , x 2 ......x n can be treated as an
independent normal random variable with mean μ and variance σ2. In other words
xi ~ N (μ, σ2)
where i = 1, 2, ………n
382
Thus each of these n normally distribution random variable may be standardized so that
Zi =
xi − μ
σ
~ N (0, 12)
where i = 1, 2, ………n
A sample statistic U may, then, be defined as
U = Z 12 + Z 22 + ......... + Z n2
n
U = ∑ Z i2
i =1
⎛x −μ⎞
U = ∑⎜ i
⎟
σ ⎠
i =1 ⎝
n
2
Which will take different values in repeated random sampling. Obviously, U is a random
variable. It is called chi-square variable, denoted by χ2. Thus the chi-square random
variable is the sum of several independent, squared standard normal random variables.
The chi-square distribution is the probability distribution of chi-square variable. So
The chi-square distribution is the probability distribution of the sum of
several independent, squared standard normal random variables.
The chi-square distribution is defined as
1
− χ2
2
n
−1
2 2
f (χ ) =Ce (χ ) dχ2
2
for
χ2 ≥ 0
where e is the base of natural logarithm, n denotes the sample size (or the number of
independent normal random variables).C is a constant to be so determined that the total area
under the χ2 distribution is unity. χ2 values are determined in terms of degrees of freedom, df
=n
Properties of χ2 Distribution
1.
A χ2 distribution is completely defined by the number of degrees of freedom, df = n.
So there are many χ2 distributions each with its own df.
383
2.
χ2 is a sample statistic having no corresponding parameter, which makes χ2distribution
a non-parametric distribution.
3.
As a sum of squares the χ2 random variable cannot be negative and is, therefore,
bounded on the left by zero.
Figure 12-6
4.
χ2 Distribution with Different Numbers of df
The mean of a χ2 distribution is equal to the degrees of freedom df. The variance of
the distribution is equal to twice the number of degrees of freedom df .
E(χ2) = n
5.
Var (χ2 ) = 2n
Unless the df is large, a χ2 distribution is skewed to the right. As df increases, the
χ2 distribution looks more and more like a normal. Thus for large df
χ2 ~ N (n,
2
2n )
Figure 12-6 shows several χ2 distributions with different numbers of df.
In general, for n ≥ 30, the probability of χ2 taking a value greater than or less than a
particular value can be approximated by using the normal area tables.
6.
If χ 12 , χ 22 , χ 32 ,.........χ k2 are k independent χ2 random variables, with degrees of
freedom n1 , n2 , n3 ,.........nk . Then their sum χ 12 + χ 22 + χ 32 + ......... + χ k2 also possesses
a χ2 distribution with df = n1 + n2 + n3 + ......... + nk .
12.8.3 The χ2Distribution in terms of Sample Variance S2
We can write
384
2
1
⎛ xi − μ ⎞
⎜
⎟ = 2
∑
σ ⎠ σ
i =1 ⎝
n
=
=
=
⎡
⎢since
⎣
1
σ
2
1
σ2
∑ [( x − x) + ( x − μ )]
2
n
i
i =1
∑ [( x
n
i =1
i
− x) 2 + ( x − μ ) 2 + 2( xi − x)( x − μ )
n
∑ ( xi − x ) 2 +
i =1
(n − 1) S 2
σ2
n
n
1
σ2
⎛ x−μ ⎞
⎟
+ ⎜⎜
⎟
σ
/
n
⎝
⎠
∑ (x − μ)2 +
i =1
σ2
n
( x − μ )∑ ( x i − x )
i =1
2
n
∑ ( xi − x) 2 = (n − 1)S 2 ; ∑ ( x − μ ) = n( x − μ ) and
i =1
2
]
i =1
n
∑ (x
i =1
i
⎤
− x ) = 0⎥
⎦
Now, we know that the LHS of the above equation is a random variable which has chi-square
distribution, with df = n
We also know that if
x ~ N (μ,
2
σ2 n )
2
⎛ x−μ ⎞
⎟ will have a chi-square distribution with df = 1
Then ⎜⎜
⎟
σ
n
⎠
⎝
Since the two terms on the RHS are independent,
(n − 1)S 2
σ2
will also has a chi-square
distribution with df = n-1. One degree of freedom is lost because all the deviations are
measured from x and not from μ..
Expected Value and Variance of S2
In practice, therefore, we work with the distribution of
(n − 1)S 2
of S2 directly.
Since
(n − 1)S 2
σ2
has a chi-square distribution with df = n-1
385
σ2
and not with the distribution
⎡ (n − 1)S 2 ⎤
E⎢
⎥ = n −1
2
⎣ σ
⎦
So
n −1
σ
2
E (S 2 ) = n − 1
E (S 2 ) = σ 2
⎡ (n − 1) S 2 ⎤
Also Var ⎢
⎥ = 2(n − 1)
2
⎣ σ
⎦
Using the definition of variance, we get
⎡ (n − 1)S 2
⎛ (n − 1) S 2
E⎢
− E ⎜⎜
2
2
⎝ σ
⎣ σ
2
⎞⎤
⎟⎟⎥ = 2(n − 1)
⎠⎦
⎡ (n − 1)S 2
⎤
E⎢
− (n − 1)⎥ = 2(n − 1)
2
⎣ σ
⎦
2
or
⎡ (n − 1)2 S 4
(n − 1)S 2 ⎤ = 2(n − 1)
2
E⎢
(
1
)
2
(
1
)
+
n
−
−
n
−
⎥
4
σ2 ⎦
⎣ σ
2
or
or
(n − 1)2 E [S 4 + σ 4 − 2S 2σ 2 ]2 = 2(n − 1)
σ4
or
(n − 1)2 E ( S 2 − σ 2 ) 2 = 2(n − 1)
or
E (S 2 − σ 2 ) 2 =
So
Var ( S 2 ) =
σ4
2(n − 1) 4
σ
(n − 1) 2
2σ 4
n −1
It may be noted that the conditions necessary for the central limit theorem to be operative in
the case of sample variance S2 are quite restrictive. For the sampling distribution of S2 to be
approximately normal requires not only that the parent population is normal, but also that the
sample is at least as large as 100.
Example 12-5
386
In an automated process, a machine fills cans of coffee. The variance of the filling process is
known to be 30. In order to keep the process in control, from time to time regular checks of
the variance of the filling process are made. This is done by randomly sampling filled cans,
measuring their amounts and computing the sample variance. A random sample of 101 cans
is selected for the purpose. What is the probability that the sample variance is between 21.28
and 38.72?
Solution: We have
Population variance σ2 = 30
n = 101
We can find the required probability by using the chi-square distribution
χ2 =
So
(n − 1)S 2
σ2
(101 − 1)38.72 ⎞
⎛ (101 − 1)21.28
P(21.28 < S2 < 38.72) = P⎜
< χ2 <
⎟
30
30
⎠
⎝
= P(70.93 < χ2 < 129.06)
= P(χ2 > 70.93) - P(χ2 > 129.06)
≈ 0.990 – 0.025
= 0.965
Since our population is normal and also sample size is quite large, we can also estimate the
required probability using normal distribution.
2
We have
So
S ~ (σ ,
2
2
2σ 4
)
n −1
⎛
⎜
38.72 − σ 2
⎜ 21.28 − σ 2
2
P(21.28 < S < 38.72) = P⎜
<Z<
4
σ
2
2σ 4
⎜
⎜
n −1
n −1
⎝
387
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
21.28 − 30
38.72 − 30
<Z<
= P⎜
⎜ 2 x30 x30
2 x30 x30
⎜
101 − 1
⎝ 101 − 1
⎞
⎟
⎟
⎟
⎟
⎠
8.72 ⎞
⎛ − 8.72
= P⎜
<Z<
⎟
4.36 ⎠
⎝ 4.36
= P(− 2 < Z < 2 )
= 2 P(0 < Z < 2 )
= 2x0.4772
= 0.9544
Which is approximately the same as calculated above using χ2distribution
12.9 THE F -DISTRIBUTION
Let us assume two normal population with variances σ 12 and σ 22 repetitively. For a random
sample of size n1 drawn from the first population, we have the chi-square variable
(
n1 − 1)S12
χ =
2
2
1
σ1
which process a χ2 distribution with ν1 = n1 -1 df
Similarly, for a random sample of size n2 drawn from the second population, we have the chisquare variable
(
n2 − 1)S 22
χ =
2
2
2
σ2
which process a χ2 distribution with ν2 = n2 -1 df
A new sample statistic defined as
χ 12
F=
χ
2
2
v1
v2
is a random variable known as F statistic, named in honor of the English statistician Sir
Ronald A Fisher.
388
Being a random variable it has a probability distribution, which is known as F distribution.
The F distribution is the distribution of the ratio of two chi-square random
variables that are independent of each other, each of which is divided by its
own degrees of freedom.
Properties of F- Distribution
1.
The F distribution is defined by two kinds of degrees of freedom – the degrees of
freedom of the numerator always listed as the first item in the parentheses and the
degrees of freedom of the denominator always listed as the second item in the
parentheses. So there are a large number of F distributions for each pair of v1 and v2.
Figure 12-7 shows several F distributions with different v1 and v2.
2.
As a ratio of two squared quantities, the F random variable cannot be negative and is,
therefore, bounded on the left by zero.
Figure 12-7 F- Distribution with different v1 and v2
3.
The F( v1 ,v2 ) has no mean for v2 ≤ 2 and no variance for v2 ≤ 4.
However, for
v2
>2, the mean and for v2 > 4, the variance is given as
E( F( v1 ,v2 ) ) =
4.
v2
v2 − 2
Var( F( v1 ,v2 ) ) =
2v 22 (v1 + v 2 − 2)
v1 (v 2 − 2) 2 (v 2 − 4)
Unless the v2 is large, a F distribution is skewed to the right. As v2 increases, the F
distribution looks more and more like a normal. In general, for v2 ≥ 30, the probability
389
of F taking a value greater than or less than a particular value can be approximated by
using the normal area tables.
5.
The F distributions defined as F( v1 ,v2 ) and as F( v2 ,v1 ) are reciprocal of each other.
i.e.
1
F( v1 ,v2 ) =
F( v2 ,v1 )
12.10 THE t-DISTRIBUTION
Let us assume a normal population with mean μ and variance σ 2 . If xi represent the n values
of a sample drawn from this population. Then
Zi =
xi − μ
σ
~ N (0, 12)
where i = 1, 2, ………n
∑ (x
and
2
⎛x −μ⎞
U = ∑⎜ i
⎟ =
σ ⎠
i =1 ⎝
n
i =1
)
2
n
i
−x
σ2
~ χ2 (n-1 df)
where i = 1, 2, ………n
A new sample statistic T may, then, be defined as
xi − μ
σ
T=
∑ (x
1
n −1
T=
i =1
)
2
n
i
−x
σ2
xi − μ
∑ (x
i =1
)
2
n
i
−x
n −1
T=
xi − μ
S
This statistic - the ratio of the standard normal variable Z to the square root of the χ2
variable divided by its degree of freedom - is known as ‘t’ statistic or student ‘t’ statistic,
named after the pen name of Sir W S Gosset, who discovered the distribution of the quantity.
390
The random variable
xi − μ
follows t-distribution with n-1 degrees of freedom.
S
xi − μ
~ t (n-1 df)
S
12.10.1
The t-distribution in terms of Sampling Distribution of Sample Mean
We know
X −μ
σ
X −μ
σ
2
σ2 n )
X ~ N (μ,
So
Putting
where i = 1, 2, ………n
n
for
xi − μ
σ
~ N (0, 12 )
n
xi − μ
σ
in T =
∑ (x
1
n −1
i =1
)
2
n
−x
i
, we get
σ2
X −μ
σ
T=
n
∑ (x
1
n −1
i =1
)
2
n
i
−x
σ2
(X − μ )
or
σ
T=
∑ (x
1
i =1
σ
or
−x
X −μ
T=
1
n
T=
i
n(n − 1)
∑ (x
i =1
)
2
n
or
)
2
n
i
−x
n −1
X −μ
S
n
When defined as above, T again follows t-distribution with n-1 degrees of freedom.
391
X −μ
~ t (n-1 df)
S
n
where i = 1, 2, ………n
Properties of t- Distribution
1. The t-distribution like Z distribution, is unimodal, symmetric about mean 0, and the tvariable varies from -∝ and∝
2. The t-distribution is defined by the degrees of freedom v = n-1, the df associated with
the distribution are the df associated with the sample standard deviation.
3. The t-distribution has no mean for n = 2 i.e. for v = 1 and no variance for n ≤ 3 i.e. for v
≤ 2. However, for v >1, the mean and for v > 2, the variance is given as
E(T) = 0
Figure 12-8
4. The variance
Var(T) =
v
v−2
t-Distribution with different df
v
of the t-distribution must always be greater than 1, so it is more
v−2
variable as against Z distribution which has variance 1. This follows from the fact what
while Z values vary from sample to sample owing to the change in the X alone, the
variation in T values are due to changes in both X and S.
5. The variance of t-distribution approaches 1 as the sample size n tends to increase. In
general, for n ≥ 30, the variance of t-distribution is approximately the same as that of Z
distribution. In other words the t-distribution is approximately normal for
392
n ≥ 30.
12.11 SELF-ASSESSMENT QUESTIONS
1.
What is a sampling distribution, and what are the uses of sampling distributions?
2.
How does the size of population and the kind of random sampling determine the
shape of the sampling distributions?
3.
(a)
A sample of size n = 5 is selected from a population. Under what conditions is
the sampling distribution of X normal?
(b)
Suppose the population mean is μ = 125 and the population standard deviation
is 20. What are the expected value and the standard deviation of X ?
4.
What is the most significant aspect of the central limit theorem? Discuss the practical
utility of central limit theorem in applied statistics.
5.
Under what conditions is the central limit theorem most useful in sampling for
making statistical inferences about the population mean?
6.
If the population mean is 1,247, the population variance is 10,000, and the sample size
is 100, what is the probability that X will be less than 1,230?
7.
When sampling is from a population with standard deviation σ = 55, using a sample of
size n = 150, what is the probability that X will be at least 8 units away from the
population mean μ?
8.
The Colosseum, once the most popular monument in Rome, dates from about AD 70.
Since then, earthquakes have caused considerable damage to the huge structure, and
engineers are currently trying to make sure the building will survive future shocks.
The Colosseum can be divided into several thousand small sections. Suppose that the
average section can withstand a quake measuring 3.4 on the Richter scale with a
standard deviation of 1.5. A random sample of 100 sections is selected and tested for
the maximum earthquake force they can withstand. What is the probability that the
393
average section in the sample can withstand an earthquake measuring at least 3.6 on
the Richter scale?
9.
On June 10, 1997, the average price per share on the Big Board Composite Index in
New York rose 15 cents. Assume the population standard deviation that day was 5
cents. If a random sample of 50 stocks is selected that day, what is the probability that
the average price change in this sample was a rise between 14 and 16 cents?
10. An economist wishes to estimate the average family income in a certain population.
The population standard deviation is known to be Rs 4,000, and the economist uses a
random sample of size n = 225. What is the probability that the sample mean will fall
within Rs 750 of the population mean?
11. When sampling is done from a population with population proportion p = 0.2, using a
sample size n = 15, what is the sampling distribution of p ? Is it reasonable to use a
normal approximation for this sampling distribution? Explain.
12. When sampling is done for the proportion of defective items in a large shipment,
where the population proportion is 0.18 and the sample size is 200, what is the
probability that the sample proportion will be at least 0.20?
13. A study of the investment industry claims that 55% of all mutual funds outperformed
the stock market as a whole last year. An analyst wants to test this claim and obtains a
random sample of 280 mutual funds. The analyst finds that only 128 of the funds
outperformed the market during the year. Determine the probability that another
random sample would lead to a sample proportion as low as or lower than the one
obtained by the analyst, assuming the proportion of all mutual funds that outperformed the market is indeed 0.55.
14. In recent years, convertible sport coupes have become very popular in Japan. Toyota
is currently shipping Celicas to Los Angeles, where a customizer does a roof lift and
394
ships them back to Japan. Suppose that 25% of all Japanese in a given income and
lifestyle category are interested in buying Celica convertibles. A random sample of
100 Japanese consumers in the category of interest is to be selected. What is the probability that at least 20% of those in the sample will express an interest in a Celica convertible?
15. What are the limitations of small samples?
16. What do you understand by small sampling distributions? Why are the small sampling
distributions called exact distributions?
17. What do you understand by the concept of degrees of freedom?
18. Define the χ2 statistic. What are important properties of χ2 distribution?
19. Define the F statistic. What are important properties of F distribution?
20. Define the t statistic. What are important properties of t-distribution? How does t
statistic differ from Z statistic?
1. Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers and
Distributors (P) Ltd., Agra.
2. Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing Company
Ltd., New Delhi.
3. Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4. Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd., New
Delhi.
5. Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6. Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7. Statistics for Management by Richard I. Levin and David S. Rubin. Prentice Hall
of India Pvt. Ltd., New Delhi.
8. Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
395
COURSE:
Author: Dr. B.S. Bodla
Course code: MC-106
Vetter: Karam Pal
Lesson: 13
STATISTICAL ESTIMATION
Objectives
Having studied this chapter, you should be able to•
Understand the term statistical estimation and types of estimates;
•
Construct and interpret confidence interval estimates to know the
precision of the estimate of a population mean and proportion; and
•
Determine the sample size to ensure that the margin of error will be within acceptable limits.
Structure
13.1 Introduction
13.2 Types of Estimates
13.3 Criteria of a good estimator
13.4 Method of Maximum Likelihood
13.5 Point Estimation
13.6 Interval Estimation
13.7 Sample size Determination
13.8 Summary
13.9 Questions
13.1. INTRODUCTION
The sampling process is used to draw statistical inference about the characteristics of a
population or process of interest. On many occasions we do not have enough information to
calculate an exact value of population parameters (such as μ, σ and p) and therefore make the
best estimate of this value from the corresponding sample statistics (such as x , s, and P ).
The need to use the sample statistic to draw conclusions about the population characteristic is
one of the fundamental applications of statistical inference in business and economics. A few
applications of statistical estimation are given below :
•
A production manager needs to determine the proportion of items being
manufactured that do not match with quality standards.
•
A mobile phone service company may be interested to know the average length of
a long distance telephone call and its standard deviation.
396
•
A bank needs to understand consumer awareness of its services and credit
schemes.
•
Any service centre needs to determine the average amount of time a customer
spends in queue.
In all such cases, a decision-maker needs to examine the following two concepts that are
useful for drawing statistical inference about an unknown population or process parameters
based upon random samples:
(i)
Estimation– a sample statistic to estimate an unknown parameter value
(ii)
Hypothesis testing– a claim or belief about an unknown parameter value.
In this lesson we shall discuss methods to estimate unknown population parameters and then
to determine the range of values (confidence interval) likely to contain the parameter value.
13.2 TYPES OF ESTIMATES
Let us first know the concept of ‘estimate’ as used in Statistics. According to some
dictionaries, an estimate is a valuation based on opinion or roughly made from imperfect or
incomplete data. This definition may apply, for example, when an individual who has an
opinion about the competence of one of his colleagues. But, in Statistics the term estimate is
not used in this sense. In Statistics too the estimates are made when the information available
is incomplete or imperfect. However, such estimates are made only when they are based on
sound judgement or experience and when the samples are scientifically selected.
There are two types of estimates that we can make about a population : a point estimate and
an interval estimate. A point estimate is a single number, which is used to estimate an
unknown population parameter. Although a point estimate may be the most common way of
expressing an estimate, it suffers from a major limitation since it fails to indicate how close it
is to the quantity it is supposed to estimate. In other words, a point estimate does not give any
idea about the reliability of precision of the method of estimation used. For instance, if
someone claims that 40 percent of all children in a certain town do not go to the school and
are devoid of education, it would not be very helpful if this claim is based on a small number
of households, say, 20. However, as the number of households interviewed for this purpose
increases from 20 to 100, 500 or even 5,000, the claim that 40 percent of children have no
school education would become more and more meaningful and reliable. This makes it clear
that a point estimate should always be accompanied by some relevant information so that it is
possible to judge how far it is reliable.
The second type of estimate is known as the interval estimate. It is a range of values used to
estimate an unknown population parameter. In case of an interval estimate, the error is
indicated in two ways: first by the extent of its range; and second, by the probability of the
true population parameter lying within that range. Taking our previous example of 40 percent
children not having a school education, the statistician may say that actual percentage of such
children in that town may lie between 35 percent and 45 percent. Thus, he will have a better
idea of the reliability of such an estimate as compared to the point estimate of 40 percent.
Estimator and Estimate
When we make an estimate of a population parameter, we use a sample statistic. This sample
statistic is an estimator.
397
n
For example, the samples mean x =
∑x
i =1
i
n
x is a point estimator of the population mean μ. The value obtained by the estimator is
known as an estimate. Many different Statistics can be used to estimate the same parameter.
For example, we may use the sample mean or the sample median or even the range to
estimate the population mean. The question here is: how can we evaluate the properties of
these estimates, compare then with one another, and finally, decide which the ‘best’ is? The
answer to this question is possible only when we have certain criteria that a good estimator
must satisfy. These criteria are briefly discussed below.
13.3 CRITERIA OF A GOOD ESTIMATOR
There are four criteria by which we can evaluate the quality of a statistic as an estimator.
These are: unbiasedness, efficiency, consistency and sufficiency.
Unbiasedness
This is a very important property that an estimator should possess. If we take all possible
samples of the same size from a population and calculate their means, the mean μ x of all
these means will be equal to the mean μ of the population. This means that the sample mean
x is an unbiased estimator of the population mean μ. When the expected value (or mean) of
a sample statistic is equal to the value of the corresponding population parameter, the sample
statistic is said to be an unbiased estimator.
Suppose we take the smallest sample observation as an estimator of the population mean μ, it
can be easily shown that this estimator is biased. Since the smallest observation must be less
than the mean, its expected value must be less than μ. Symbolically, E(Xs) < μ, where Xs
stands for the smallest item and E stands for the expected value. Thus, this estimator is biased
downwards. The extent of bias is the difference between the expected value of the estimator
and the value of the parameter. In this case, bias is equal to E(Xs)- μ. In contrast, the biases
for the sample mean x is zero.
Consistency
Another important characteristic that an estimator should possess is consistency. Let us take
the case of the standard deviation of the sampling distribution of x . The standard deviation
of the sampling distribution of sample mean is computed by following formula :
σx =
σ
n
The formula states that the standard deviation of the sampling distribution of x decreases as
the sample size increases and vice versa. When the sample size n increases, the population
standard deviation σ is to be divided by a higher denominator. This results in the reduced
value of sample standard deviation σξ. Let us take an example.
Illustration 13.1: A company has 4,000 employees whose average monthly wage comes to
Rs.4,800 with a standard deviation of Rs.1,200. Let x be the mean monthly wage for a
random sample of certain employees selected from this company. Find the mean and
standard deviation of ξ for a sample size of (a) 81, (b) 100 and (c) 180.
Solution
From the given information, for the population of all employees, N = 4,000 μ = Rs.4,800 σ =
Rs.1,200.
398
(a)
The mean μξ of the sampling distribution of the ξ is μξ = μ = Rs.4,800.
As n = 81 and N = 4,000, which gives n/N = 0.01. At this value is less than 0.05, the standard
deviation of ξ is obtained by using the formula. Substituting the values.
1,200 1,200
σ
= Rs.133.33
σx =
or, σx =
=
81
9
n
(b)
In this case, n = 100 and n/N = 100/4,000 = 0.025, which is also less than 0.05. The
mean and the standard deviation ξ are
μξ = μ = Rs.4,800
σ
1,200 1,200
= Rs.120
σx =
or, σ x =
=
10
n
100
(c)
In this case, n = 180 and n/N = 180/4,000 = 0.045, which again is less than 0.05. The
mean and the standard deviation ξ are
μx = μ = Rs.4,800
σ
1,200 1,200
= Rs.89.42
σx =
or, σ x =
=
n
180 13.42
From the above three sets of calculation, it becomes clear that the mean of the sampling
distribution of x is always equal to the mean of the population regardless of the sample size.
But, in case of the standard deviation, we find the change. In the given example, we find that
standard deviation of x decreased from Rs.189.87 to Rs.120 and then to Rs.133.33 as the
sample size increased from 40 to 100 and then to 180.
Efficiency
Another desirable property of a good estimator is that it should be efficient. Efficiency is
measured in terms of size of the standard error of the statistic. Since an estimator is a random
variable, it is necessarily characterised by a certain amount of variability. This means that
some estimates may be more variable than others. Just as bias is related to the expected value
of the estimator, so efficiency can be defined in terms of the variance. In large samples, for
example, the variance of the sample mean is V( x )=σ2/n. As the sample size n increases, the
variance of the sample mean (V x ) becomes smaller, so the estimator becomes more
efficient. This criterion, when applied to large samples, gives better estimates as compared to
the small ones.
The efficiency of one estimator in relation to another estimator can be judged by comparing
their sampling variances. Thus, efficiency relates to the size of the standard error. Given the
same sample size, the statistic that has a smaller standard error is preferable as it is efficient
in relation to another statistic that has a larger standard error. The sampling distribution of the
mean and the median have the same mean, that is, the population mean. However, the
variance of the sampling distribution of the means is smaller than the variance of the
sampling distribution of the medians. As such, the sample mean is an efficient estimator of
the population mean, while the sample median is an inefficient estimator.
Sufficiency
The fourth property of a good estimator is that it should be sufficient. A sufficient statistic
utilises all the information a sample contains about the parameter to be estimated. ξ, for
example, is a sufficient estimator of the population mean μ. It implies that no other estimator
of μ, such as the sample median, can provide any additional information about the parameter
μ. Likewise, we can say that the sample proportion π.
399
Having looked into properties of a good estimator briefly, a pertinent question arises: how
can we find estimators with these desirable properties? This brings us to the method of
maximum likelihood.
13.4 METHOD OF MAXIMUM LIKELIHOOD (ML)
The maximum likelihood method provides estimators with the desirable properties such as
efficiency, consistency and sufficiency, which we have just discussed. It usually does not
give an unbiased estimate. Let us take an example to explain this method.
Example: Suppose we want to estimate the average grade μ of a large number of
students. A random sample of size n = 64 is taken and the sample mean x is found
to be 90 marks. Now, the assumption on which we have to base our reasoning is
that the random sample of n = 64 is representative of the population. We saw how
samples that were similar to the population had greater probability of being
selected.
Let us now reverse this reasoning as follows: we have before us a random sample size n = 64
and x = 90 marks. From which population did it most probably come-a population with μ =
85, 90 or 95? According to our earlier approach, we would think that it most probably came
from a population with μ =90 marks. Thus, it can be concluded that the population mean μ,
based on our sample, is most likely to be μ=90 marks.
A point worth noting is that the population mean μ is either 90 or not; it has only one value.
Hence, we have used the term likely instead of probably.
This technique to find the estimators was first used and developed by Sir R.A. Fisher in 1922,
who called it the maximum likelihood method.
13.5 POINT ESTIMATION
In point estimation, a single sample statistic (such as x , s, and p ) is calculated from the
sample to provide a best estimate of the true value of the corresponding population parameter
(such as μ, σ and p ). Such a single relevant statistic is termed as point estimator, and the
value of the statistic is termed as point estimate. For example, we may calculate that 10 per
cent of the items in a random sample taken from a day’s production are defective. The result
‘10 per cent’ is a point estimate of the percentage of items in the whole lot that are defective.
Thus, until the next sample of items is not drawn and examined, we may proceed on
manufacturing on the assumption that any day’s production contains 10 per cent defective
items.
13.6 INTERVAL ESTIMATION
Generally, a point estimate does not provide information about ‘how close is the estimate’ to
the population parameter unless accompanied by a statement of possible sampling errors
involved based on the sampling distribution of the statistic. It is therefore important to know
the precision of an estimate before relying on it to make a decision. Thus, decision-makers
prefer to use an interval estimate that is likely to contain the population parameter value.
However, it is also important to state ‘how confident’ he is that the interval estimate actually
contains the parameter value. Hence an interval estimate of a population parameter is
therefore a confidence interval with a statement of confidence that the interval contains the
parameter value.
400
The confidence interval estimate of a population parameter is obtained by applying the
formula :
Point estimate + Margin of error
Where Margin of error = zc × Standard error of a particular statistic
zc = critical value of standard normal variable that represents confidence
level (probability of being correct) such as 0.90,0.95, and so on.
13.6.1. Interval estimation of population mean (σknown)
Suppose the population mean μ is unknown and the true population standard deviation σ is
known. Then for a large sample size (n=>30), the interval estimation of population mean μ is
given by
x ± z α/2 σξ or, x ± z α/2
or
x - z α/2
σ
σ
n
≤ μ ≤ x + z α/2
σ
n
n
where za/2 is the z-value representing an area a/2 in the right and left tails of the standard
normal probability distribution, and (1-α) is the level of confidence as shown in Fig.13.1.
Fig. 13.1 : Sampling Distribution of Mean
For example, if a 95 per cent level of confidence is desired to estimate the mean, then 95 per
cent of the area under the normal curve would be divided equally, leaving an area equal to
47.5 per cent between each limit and population mean μ as shown in Fig.13.2.
Fig. 13.2: Sampling Distribution of Mean ξ
401
If n = 100 and σ = 25, then σξ = σ/ n = 25/ 100 = 2.5. Using a table of
areas for the standard normal probability distribution 95 per cent of the
values of a normally distributed population are within ±1.96σξ or 1.96 (2.5) =
± 4.90 range. Hence 95 per cent of the sample means will be within ± 4.90 of
the population mean μ. In other words, there is a 0.95 probability that the
sample mean will provide a sampling error equal to |ξ - μ| = 4.90 or less.
The value 0.95 is called confidence coefficient and the interval estimate ξ ±
4.90 is called a 95 per cent confidence interval.
In general, a 95 per cent confidence interval estimate implies that if all
possible samples of the same size were drawn, then 95 per cent of them
would include the true population mean somewhere within the interval
around their sample mean and only 5 per cent of them would not. The
values for z α/2 for the most commonly-used as well as the other confidence
levels can be seen from standard normal probability table as shown in Table
13.1.
Table 13.1 : Values of Standard Normal Probability zα/2
Confidence
Level, (1-α) (%)
Acceptable
Error Level, α
α/2
zα/2
90%
0.10
0.05
1.645
95%
0.05
0.025
1.960
99%
0.01
0.005
2.576
Illustration 13.2: The average monthly electricity consumption for a sample
of 100 families is 1250 units. Assuming the standard deviation of electric
consumption of all families is 150 units, construct a 95 per cent confidence
interval estimate of the actual mean electric consumption.
Solution: The information given is:
x =12.50,
σ =150, n= 100 and
confidence level (1-α) = 95 per cent. Using the ‘Standard Normal Curve’ we
find that the half of 0.95 yields a confidence coefficient z α/2 = 1.96. Thus
confidence limits with
α/2
= ± 1.96 for 95 per cent confidence are given by
402
ξ ± z α/2
σ
150
= 1250 ± 29.40 units
n
100
Thus for 95 per cent level of confidence, the population mean μ is likely to fall between
1220.60 units and 1279.40 units, that is, 1220.60 ≤ μ ≤ 1279.40.
Illustrator 13.3: The quality control manager at a factory manufacturing light bulbs is
interested to estimate the average life of a large shipment of light bulbs. The standard
deviation is known to be 100 hours. A random sample of 50 light bulbs gave a sample
average life of 350 hours.
(a)
Setup a 95 per cent confidence interval estimate of the true average life of light bulbs
in the shipment.
(b)
Does the population of light bulb life have to be normally distributed? Explain.
Solution: The following information is given :
x = 350, σ = 100, n =50, and confidence level, (1-α) = 95 per cent.
(a)
Using the ‘Standard Normal Curve’, we have z α/2 = ± 1.96 for 95 per cent confidence
level. Thus confidence limits are given by
100
σ
= 350 ± 1.96
= 350 ± 27.72
x ± z α/2
n
50
Hence for 95 per cent level of confidence the population mean μ is likely to fall between
322.28 hours to 377.72 hours, that is, 322.28 ≤ μ ≤ 377.72.
(b)
No, since σ is known and n = 50, from the central limit theorem we may assume that
x is normally distributed.
13.6.2 Interval Estimation for Difference of Two Means
If all possible samples of large size n1 and n2 are drawn from two different populations, then
sampling distribution of the difference between two means ξ1 and ξ2 is approximately normal
with mean (μ1-μ2) and standard deviation:
=12.50 ± 1.96
σ x −x =
1
2
σ12
n1
+
σ 22
n2
For a desired confidence level, the confidence interval limits for the population mean (μ1-μ2)
are given by
( x1 - x 2 ) ± z α/2 σ x1 − x2
Illustration 13.4: The strength of the wire produced by company A has a mean of 4,500 kg
and a standard deviation of 200 kg. Company B has a mean of 4000 kg and a standard
deviation of 300 kg. A sample of 50 wires of company A and 100 wires of company B are
selected at random for testing the strength. Find 99 per cent confidence limits on the
difference in the average strength of the populations of wires produced by the two companies.
Solution: The following information is given:
Company A: x1 = 4500, σ1= 200, n1 = 50
Company B: x 2 = 4000, σ2= 300, n2 = 100
Therefore μ x1 − x2 = 4500 - 4000 = 500 and z α/2 = 2.576
σ 12
σ 22
40,000 90,000
+
= 41.23
n1
n2
50
100
The required 99 per cent confidence interval limits are given by
( x1 − x 2 ) + za/2 σ x1 − x2 = 500 + 2.576 (41.23) = 500 ± 106.20
σ x −x =
1
2
+
=
Hence, the 99 percent confidence limits on the difference in the average strength of wires
produced by the two companies are likely to fall in the interval 393.80 ≤ μ ≤ 606.20.
403
13.6.3
Interval estimation of population mean (σknown)
In practice, the standard deviation of a population σ, is not likely to be
known. Thus in the large sample case, the sample standard deviation σ, and
we use a z-table to compute za/2 for providing an area of a/2 in the right tail
of the standard normal probability distribution curve. Hence the interval
estimate of a population mean for a large sample case (n > 30) with
confidence coefficient 1-α is given by
x + zα/2 s x = x ± zα/2
s
n
When the population standard deviation is not known and the sample size is small, the
procedure of interval estimation of population mean is based on a probability distribution
known as the t-distribution. This distribution is very similar to the normal distribution.
However, the t-distribution has more area in the tails and less in the center than doe’s normal
distribution. The t-distribution depends on a parameter known as degree of freedom. As the
number of degrees of freedom increases, t-distribution gradually approaches the normal
distribution, and the sample standard deviation s becomes a better estimate of population
standard deviation σ.
The interval estimate of a population mean when the sample size is small (n <30) with
confidence coefficient (1-α), is given by
s
s
s
x +tα/2
≤ μ ≤ x + zα/2
or x -tα/2
n
n
n
where tα/2 is the critical value of t-test statistic providing an area α/2 in the
right tail of the t-distribution with n-1 degrees of freedom, and
s=
∑( xi − x ) 2
n −1
The critical values of t for the given degrees of freedom can be obtained from
the table of t-distribution (see appendix).
The procedure of the confidence interval estimation of population mean μ
when population standard deviation is unknown and sample size is large or
small, is summarised in Table 13.2.
Table 13.2: Confidence Interval for μ
Sample size
Large
•
Interval Estimate of
Population Mean μ
σ assumed known
x +zα/2
404
σ1
n
•
x +zα/2
σ estimated by s
Small
•
σ assumed known
•
x +zα/2
σ estimated by s
x +tα/2
s
n
σ
n
s
n
Illustration 13.5: A random sample of 64 sales invoices was taken from a
large population of sales invoices. The average value was found to be
Rs.2000 with a standard deviation of Rs.540. Find a 90 per cent confidence
interval for the true mean value of all the sales.
Solution: The information given is: x1 = 2000, s = 540, n = 64, and α = 10 per
cent. Therefore
sx =
s
n
=
540
64
= 67.50 and zα/2 = 1.64 (from Normal table)
The required confidence interval of population mean μ is given by
x ± zα / 2
s
= 2000 + 1.64 (67.50) = 2000 + 110.70
n
Thus the mean of the sales invoices for the whole population is likely to fall
between Rs.1889.30 and Rs.2110.70, that is, 1889.30 ≤ μ ≤ 2110.70.
Illustration 13.6: The personnel department of an organization would like
to estimate the family dental expenses of its employees to determine the
feasibility of providing a dental insurance plan. A random sample of 10
employees reveals the following family dental expenses (in thousand Rs.) in
the previous year: 11, 37, 25, 62, 51, 21, 18, 43, 32, 20.
Setup a 99 per cent confidence interval of the average family dental
expenses for the employees of this organization.
Solution: The calculations for sample mean x and standard deviation are
shown in Table 13.3.
Table 13.3 : Calculations for x and s
405
Variable, x
( x − x ) = ( x − 32)
11
37
25
62
51
21
18
43
32
20
320
−21
05
−07
30
19
−11
−14
11
−
−12
0
From the data in Table 13.3, the sample mean x =
standard deviation s =
∑( x − x ) 2 / n − 1 =
(x − x)2
441
25
49
900
361
121
196
121
−
144
2358
∑ x 320
= Rs.32, and the sample
=
n
10
2358
= Rs.5.11. Using this information and tα/2
9
= 1.833 at df = 9, we have
5.11
s
x ± tα/2
= 32 + 1.833
= 32 + 2.962
n
10
Hence the mean expenses per family are likely to fall between Rs.29.038 and Rs.34.962, that
is, 29.038 ≤ μ ≤ 34.962.
13.6.4. INTERVAL ESTIMATION FOR POPULATION PROPORTION
You know that normal distribution as an approximation of the sampling
distribution of sample proportion p = x / n is based on the large sample
conditions: np >5 and nq=n (1-p) >5, where p is the population proportion as
shown in Fig.13.3. The confidence interval estimate for a population
proportion at 1-α confidence coefficient is given by
p ± zα/2 σ p = p ± zα/2
p (1 − p )
or p -zα/2 σ p ≤ p ≤ p + zα/2 σ p
n
where zα/2 is the z-value providing an area of α/2 in the right tail of the
standard normal probability distribution and the quantity zα/2 σ p is the
margin of error.
406
Fig. 13.3 : Sampling Distribution of Proportion p ; np > 5, and nq > 5
Illustration 13.7: Suppose we want to estimate the proportion of families in
a town, which have two or more children. A random sample of 144 families
shows that 48 families have two or more children. Setup a 95 per cent
confidence interval estimate of the population proportion of families having
two or more children.
Solution: The sample proportion is : p =
Using the information, n = 144, p =
x 48 1
=
=
n 144 3
1
and zα/2 = 1.96 at 95 per cent
3
confidence coefficient, we have
p ± zα / 2
p (1 − p ) 1
( 13 )( 23 )
= ± 1.96
= 0.333 ± 0.077
3
n
144
Hence the population proportion of families who have two or more children is likely
to be between 25.6 to 41 per cent, that is, 0.256 ≤ p ≤ 0.410.
13.7
SAMPLE SIZE DETERMINATION
From previous sections we understand that standard error σ x = σ / n and
σ p = σ / pq / n of sampling distribution of sample statistic x and p are both
inversely proportional to the sample size n, which is also related to the width
of the confidence intervals x ± z a/2 σ x and p ± z a/2 σ p . Obviously, the width or
range of the confidence interval can be decreased by increasing the sample
size n. The decision regarding the appropriate size of the sample, however,
depends on (i) deciding in advance how good an estimate is required, and (ii)
the availability of funds, time, and ease of sample selection. For example, an
insurance company wants to estimate the proportion of claims settled within
2 months of the receipt of claim. For this purpose, the company must decide
how much error it is willing to allow in estimating the population proportion
407
of claims settled in a particular financial year. This means, whether
accuracy is required to be within ± 80 claims, ± 100 claims, and so on. Also,
the company needs to determine in advance the level of confidence for
estimating the true population parameter. Hence for determining the sample
size for estimating population mean or proportion, such requirements must
be kept in mind along with information regarding standard deviation.
13.7.1 Sample Size for Estimating Population Mean
When the distribution of sample mean x is normal, the standard normal variable z is given as
σ
x−μ
or x − μ =
z=
σ/ n
n
The value of z can be seen from ‘standard normal table’ for a specified confidence coefficient
1-α. The value of z in the above equation will be positive or negative, depending or whether
the sample mean x is larger or smaller than, population mean μ as shown in Fig.13.4. This
difference between x and mean μ is called the sampling error or margin of error E. Thus for
estimating the population mean μ with a condition that the error in its estimation should not
exceed a fixed value, say E, we require that the sample mean x should fall within the range, μ
± E with a specified probability. Thus the margin of error acceptable (i.e. maximum tolerable
difference between unknown population mean μ and the sample estimate at a particular level
of confidence) can be written as:
x − μ = zα / 2
n=
or
σ
n
or E = zα / 2
σ
n
zα / 2σ
(z )2σ 2
, i.e., n = α / 2 2
Ε
Ε
Fig.13.4
This formula for sampling size n will provide the tolerable margin of error E, at the chosen
confidence level 1-α (which determines the critical value of z from the normal table) with
known or estimated population standard deviation σ.
Note: If population standard deviation σ is not known, then sample standard deviation s can
be used to determine the sample size n.
Illustration 13.8: Suppose the sample standard deviation of P/E ratios for stocks listed on the
Mumbai Stock Exchange (BSE) is s = 7.8. Assume that we are interested in estimating the
population mean of P/E ratio for all stocks listed on BSE with 95 per cent confidence. How
many stocks should be included in the sample if we desire a margin of error of 2?
Solution: The information given is: E=2, s =7.8, zα/2 = 1.96 at 95 per cent level of
confidence.
Using the formula for n and substituting the given values, we have
408
n=
( za / 2) 2 σ 2
Ε2
=
(1.96) 2 (7.8) 2
=
(2) 2
3.84 × 60.84
= 59 approx.
4
Thus a sample size n = 59 should be chosen to estimate the population mean of P/E ratio for
all stocks on the BSE.
Note: The general rule used in determining sample size is to always round off to the nearest
integer value in order to slightly over-satisfy the desire of estimation.
13.7.2 Sample Size for Estimating Population Proportion
The method for determining a sample size for estimating the population proportion is similar
to that used in the previous section. We require that the sample proportion p should fall
within the range p ± Ε, with a specified probability
Ε = zα / 2σ p = zα / 2
Ε = ( zα / 2 ) 2
Or
pq
;q = 1− p
n
( z ) 2 pq
pq
, i.e., n =
n
α /2
Ε2
The value of z can be calculated from ‘Standard normal table’ for a specified confidence
coefficient. This formula for n will provide the desired margin of error E at the chosen
confidence level 1-α (which determines the critical value of z) with known or estimated
population proportion p.
Illustration 13.9: A car manufacturing company received a shipment of
petrol filters. These filters are to be sampled to estimate the proportion that
is unusable. From past experience, the proportion of unusable filter is
estimated to be 10 per cent. How large a random sample should be taken to
estimate the true proportion of unusable filters to within 0.07 with 99 per
cent confidence.
Solution: The information given is : E=0.07, p = 0.10, and zα/2 = 2.576 at 99
per cent confidence level.
Using the formula for n and substituting the given values, we have
n=
( zα / 2 ) 2 pq
Ε2
=
(2.576) 2 (0.10 × 0.90)
(0.07) 2
= 121.88
Therefore a slightly larger sample size of n = 122 filters should be taken
13.7.3 Sample Size Determination for Finite Population
When samples are drawn without replacement from a finite population of size N, the use of
finite population correction factor reduces the standard error by a value equal to
( N − n) /( N − 1) . For example, for deciding sample size n for estimating the population
mean μ, the desired margin of error is given by
409
zα / 2σ
n
Ε=
N −n
N −l
Similarly, when estimating the proportion, the desired margin of error is
given by
σ p or E = zα/2 pq N − n
n N −1
Let n0 be the size for estimating population mean without using correction factor. Then
(z ) 2σ 2
n0 =
α /2
Ε2
The revised sample size, taking into consideration the size of the population,
is given by
n=
n0 N
n0 + ( N − 1)
Illustration 13.10: For a population of 1000, what should be the sampling
size necessary to estimate the population mean at 95 per cent confidence
with a sampling error of 5 and the standard deviation equal to 20?
Solution: We have E =5, σ=20, zα/2 = 1.96 at 95 per cent confidence level, and N=1000.
Thus
( z ) 2 σ 2 (1.96) 2 (20) 2
n=
α /2
Ε2
=
(5) 2
= 61.456
Since the population size is finite, the revised sample size obtained by using
the correction factor
n0 N
(61.456)(1000)
61456
=
=
= 57.952
n0 + ( N − 1) 61.456 + (1000 − 1) 1060.456
Thus a sample size of n = 58 should be taken
n=
13.8
SUMMARY
There are two types of estimates that we can make about a population: a point estimate and
an interval estimate. A point estimate is a single number, which is used to estimate an
unknown population parameter. Although a point estimate may be the most common way of
expressing an estimate, it suffers from a major limitation since it fails to indicate how close it
is to the quantity it is supposed to estimate.
The second type of estimate is known as the interval estimate. It is a range of values used to
estimate an unknown population parameter. In case of an interval estimate, the error is
indicated in two ways: first by the extent of its range; and second, by the probability of the
true population parameter lying within that range.
410
There are four criteria by which we can evaluate the quality of a statistic as an
estimator. These are: unbiasedness, efficiency, consistency and sufficiency.
13.9.
CONCEPTUAL QUESTIONS
1. Distinguish between the point estimation and interval estimation. Explain how an interval
estimate is better than a point estimate.
2. Explain the concept of ‘margin of error’ in deciding the size of a sample.
3. Prove that the mean of a simple random sample from a given population is an unbiased
estimator of the population mean.
4. Under what circumstances can the normal distribution be used to construct a confidence
interval estimate of the population mean?
5. What are the properties of a good estimator? Explain, how these properties are essential
for estimating the population characteristic of interest.
6. Distinguish between statistic and parameter and explain the meaning of confidence
interval of a population parameter.
7. Explain the following terms with an example
(a) Point estimate
(b) Interval estimate
(c) Confidence interval
(d) Confidence limits
8. Describe the effect of sample size on the margin of sampling error of point estimate of the
proportion mean. Doest this error depends on the sample size in the same way?
9.
In an effort to estimate the mean amount spend per customer for diner at a city
hotel, data were collected for a sample of 49 customers. Assume a population
standard deviation of Rs.25.
(a) At 95 per cent confidence, what is the margin of error?
(b) If the sample mean is Rs.124, what is the 95 per cent confidence interval for
the population mean?
10. The following data have been collected for a sample from a normal population: 5,
10, 8, 11, 12, 6, 15, 13
(a) What is the point estimate of population mean and standard deviation?
(b) What is the confidence interval for population mean at 95 per cent confidence interval?
11.
A machine is producing ball bearing with a diameter of 0.5 inches. It
is known that the standard deviation of the ball bearings is 0.005 inch. A
sample of 100 ball bearings is selected and their average diameter is found
to be 0.48 inch. Determine the 99 per cent confidence interval.
12.
Suppose a wholesaler of paints wants to estimate the actual amount
of paint contained in 10 kg cans purchased from a paint manufacturing
company. It is known from the manufacture’s specifications that the
411
standard deviation of the amount of paint is equal to 0.02 kg. A random
sample of 50 cans is selected, and the average amount of paint per 10 kg
can is 0.995 kg. Setup a 99 per cent confidence interval estimate of the true
population average amount of paint included in a 10 kg can. Based on your
results, do you think that the wholesaler has a right to complaint to the
manufacture? Why?
13.
A survey of 672 audited tax returns showed that 448 resulted in
additional payments. Construct a 95 per cent confidence interval for the
true percentage of all audited tax returns that resulted in additional
payments.
14.
In a survey carried out in a large city, 170 households out of a
random sample of 250 owned at least one pet. Find the 95 per cent
confidence interval for the percentage of households in the city who own at
least one pet. Does the result support a pet food manufacturer’s claim that
75 per cent of all households have at least one pet?
15. A cigarette manufacturer wishes to use random sampling to estimate the average nicotine
content. The sampling error should not be more than one milligram above or below the
true mean with a 99 per cent confidence coefficient. The population standard deviation
is 4 milligrams. What sample size should the company use in order to satisfy these
requirements?
16. An agency responsible for electricity distribution would like to estimate the average
electric bills for a particular month for single-family homes in a large city. Based on
studies conducted in other cities, the standard deviation is assumed to be Rs.40. The
agency would like to estimate the average bill for that month to within Rs.10 of the true
average. If 95 per cent confidence is desired, then what sample size is necessary?
412
1. Spiegel, Murray R.: Theory and Practical of Statistics., London
McGraw Hill Book Company.
2. Yamane, T.: Statistics: An Introductory Analysis, New York, Harpered
Row Publication
3. R.P. Hooda: Statistic for Economic and Management McMillan India
Ltd.
4. G.C. Beri: Statistics for Mgt., TMA
5. J.K. Sharma: Business Statistics, Pearson Education
6. S.P. Gupta : Statistical Methods, Sultan Chand and Sons.
413
Course:
Author:
Anil Kumar
Course Code:
MC-106
Vetter:
Dr. Karam Pal
Lesson:
14
TESTING OF HYPOTHESES
Objectives: The present lesson is an attempt to overview the concept of hypotheses
testing. After successful completion of the lesson the students will be able
to specify the most appropriate test of hypothesis in a given situation,
apply the procedure and make inferences from the results.
Structure
14.1
Introduction
14.2
The Null and the Alternative Hypothesis
14.3
Some Basic Concepts
14.4
Critical Region in Terms of Test Statistic
14.5
General Testing Procedure
14.6
Tests of Hypotheses about Population Means
14.7
Tests of Hypotheses about Population Proportions
14.8
Tests of Hypotheses about Population Variances
14.9
The Comparison of Two Populations
14.10 Solved Problems
14.11 Self-Assessment Questions
14.1 INTRODUCTION
Closely related to Statistical Estimation discussed in the preceding lesson, Testing of
Hypotheses is one of the most important aspects of the theory of decision-making. In the
present lesson, we will study a class of problems where the decision made by a decision
414
maker depends primarily on the strength of the evidence thrown up by a random sample
drawn from a population. We can elaborate this by an example where the operations manager
of a cola company has to decide whether the bottling operation is under statistical control or it
has gone out of control (and needs some corrective action). Imagine that the company sells
cola in bottles labeled 1-liter, filled by an automatic bottling machine. The implied claim that
on the average each bottle contains 1,000 cm3 of cola may or may not be true.
¾ If the claim is true, the process is said to be under statistical control. It is in the
interest of the company to continue the bottling process
¾ If the claim is not true i.e. the average is either more than or less than 1,000 cm3, the
process is said to be gone out of control. It is in the interest of the company to halt the
bottling process and set right the error
Therefore, to decide about the status of the bottling operation, the operations manager needs a
tool, which allows him to test such a claim.
Testing of Hypotheses provides such a tool to the decision maker. If the operations manager
were to use this tool, he would collect a sample of filled bottles from the on-going bottling
process. The sample of bottles will be evaluated and based on the strength of the evidence
produced by the sample; the operations manager will accept or reject the implied claim and
accordingly make the decision. The implied claim (μ = 1,000 cm3) is a hypothesis that needs
to be tested and the statistical procedure, which allows us to perform such a test, is called
Hypothesis Testing or Testing of Hypotheses.
What is a Hypothesis?
A thesis is some thing that has been proven to be true. A hypothesis is something that has not
yet been proven to be true. It is some statement about a population parameter or about a
population distribution. Our hypothesis for the example of bottling process could be:
“The average amount of cola in the bottles is equal to 1,000 cm3”
415
This statement is tentative as it implies some assumption, which may or may not be found
valid on verification. Hypothesis testing is the process of determining whether or not a given
hypothesis is true.
If the population is large, there is no way of analyzing the population or of testing the
hypothesis directly. Instead, the hypothesis is tested on the basis of the outcome of a random
sample.
14.2
THE NULL AND THE ALTERNATIVE HYPOTHESIS
As stated earlier, a hypothesis is a statement about a population parameter or about a
population distribution. In any testing of hypotheses problem, we are faced with a pair of
hypotheses such that one and only one of them is always true. One of this pair is called the
null hypothesis and the other one the alternative hypothesis.
A null hypothesis is an assertion about the value of a population parameter. It
is an assertion that we hold as true unless we have sufficient statistical
evidence to conclude otherwise.
For example, a null hypothesis might assert that the population mean is equal to 1,000.
Unless we obtain sufficient evidence that it is not 1,000, we will accept it as 1,000.
We write the null hypothesis compactly as:
H0:
μ =1,000
Where the symbol H0 denotes the null hypothesis.
The alternative hypothesis is the negation of the null hypothesis.
For the null hypothesis H0:
μ =1,000, the alternative hypothesis is μ ≠ 1000. We will write
it as
H1:
μ ≠ 1,000
We use the symbol H1 (or Ha) to denote the alternative hypothesis.
416
The null and alternative hypotheses assert exactly opposite statements. Obviously, both H0
and H1 cannot be true and one of them will always be true. Thus, rejecting one is equivalent
to accepting the other. At the end of our testing procedure, if we come to the conclusion that
H0 should be rejected, this also amounts to saying that H1 should be accepted and vice versa.
It is not difficult to identify the pair of hypotheses relevant in any decision situation. Can any
one of the two be called the null hypothesis? The answer is a big NO — because the roles of
H0 and H1 are not symmetrical.
The possible outcomes of a test can be summarized as:
Either:
Accept H0
-a weak conclusion without any evidence in as a
reasonable possibility support of H0
or:
Reject H0 and
Accept H1
-a strong conclusion with strong evidence
against H0
To better understand the role of null and alternative hypotheses, we can compare the process
of hypothesis testing with the process by which an accused person is judged to be innocent or
guilty. The person before the bar is assumed to be “innocent until proven guilty” So using
the language of hypothesis testing, we have:
H0:
The person is innocent
H1:
The person is guilty
The outcomes of the trial process may result
¾ Accepting H0 of innocence: when there was not enough evidence to convict.
However, it does not prove that the person is truly innocent
¾ Rejecting H0 and accepting H1 of guilt: when there is enough evidence to rule out
innocence as a possibility and to strongly establish guilt
417
The jury acquitted Michael Jackson, on June 13, of all charges against him in the child
molestation case. In other words, using the language of hypothesis testing the jury had to
accept the null hypothesis
H0:
Michael Jackson is innocent
because the prosecution could not prove their case against H0 of innocence.
In a trial case we do not have to rule out guilt in order to find someone innocent, but we do
have to rule out innocence in order to find someone guilty. On the similar lines, we do not
have to rule out H1 in order to accept H0; but we do have to rule out H0 in order to accept H1.
Thus, it is clear that the two hypotheses - null and alternative - are not interchangeable; each
one plays a different, a special role. So it becomes more important to be clear about what the
null and alternative hypotheses should be in a given situation, or else the test is meaningless.
One can conceptualize the whole procedure of testing of hypothesis as trying to answer one
basic question: Is the sample evidence strong enough to enable us to reject H0? This means
that H0 will be rejected only when there is strong sample evidence against it. However, if the
sample evidence is not strong enough, we shall conclude that we cannot reject H0 and so we
accept H0 by default. Thus, H0 is accepted even without any evidence in support of it whereas
it can be rejected only when there is overwhelming evidence against it. In other words, the
decision maker is somewhat biased towards the null hypothesis and he does not mind
accepting the null hypothesis. However, he would reject the null hypothesis only when the
sample evidence against the null hypothesis is too strong to be ignored.
The null hypothesis is called by this name because in many situations, acceptance of this
hypothesis would lead to null action. Thus, one way to ensure what the null hypothesis
should be is to note that…
…if the null hypothesis is true, then no corrective action would be necessary. If
the alternative hypothesis is true, then some corrective action would be necessary.
418
Recall our example of the cola-company in which an automatic bottling machine fills 1-liter
bottles with cola. Now consider three different situations:
The operations manager wants to test the average amount filled, in
Situation I:
order to know whether the process is under statistical control.
In this situation, the operations manager will have to take corrective action when the average
is either more than or less than 1,000 cm3. Only when the average equals 1,000 cm3, no
corrective action is necessary. So we have
H0:
μ = 1,000 cm3
H1:
μ ≠ 1,000 cm3
Situation II: A consumer advocate suspects that the average amount of cola is less than
1,000 cm3 and wants to test it.
In this situation, if the average amount of cola is greater than or equal to 1,000 cm3, no
corrective action is needed, but if the average amount is less than 1,000 cm3, the company
has to halt the bottling process and set right the error. So, in this case, we have
H0:
μ ≥ 1,000 cm3
H1:
μ < 1,000 cm3
Situation III: The owner of the company suspects that the machine is wasting cola by
filling more than 1,000 cm3 on the average and wants to test it.
From the owner's point of view, no corrective action is necessary if the average is less than
or equal to 1,000 cm3. And, therefore, in this case we have
H0:
μ ≤ 1,000 cm3
H1:
μ > 1,000 cm3
As the bottling example indicates, there are three possible cases for the null hypothesis,
involving ≥, ≤ and = relationships. The exact null hypothesis should be finalized before any
evidence is gathered, or the test will not be valid. Data snooping - formulating the null and
419
alternative hypotheses at one's convenience after collecting and looking at the evidence - is
unethical.
14.3 SOME BASIC CONCEPTS
We will now discuss some concepts, which are essential for setting up a procedure for testing
of hypotheses.
14.3.1
TYPE I AND TYPE II ERRORS
After the null and alternative hypotheses are spelled out, the next step is to gather evidence
from a random sample of the population. An important limitation of making interferences
from the sample data is that we cannot be 100% confident about it. Since variations from
one sample to another can never be eliminated until the sample is as large as the population
itself, it is possible that the conclusion drawn is incorrect which leads to an error. As shown
in Table 14-1 below, there can be two types of errors.
Table 14-1
Type I and Type II Errors of Hypothesis Testing
Decision based on Sample
Accept H0
Reject H0
States of Population
H0 True
H0 False
Correct decision
(No Error)
Wrong Decision
(Type I Error)
Wrong Decision
(Type II Error)
Correct Decision
(No Error)
Type I Error
In the context of statistical testing, the wrong decision of rejecting a true null hypothesis is
know as Type I Error. If the operations manager reject H0 and conclude that the process has
gone out of control, when in reality it is under control, he would be making a type I error.
Type II Error
420
The wrong decision of accepting (not rejecting, to be more accurate) a false null hypothesis is
known as Type II Error. If the operations manager do not reject H0 and conclude that the
process is under control, when in reality it has gone out of control, he would be making a type
II error.
Both the type I and type II errors are undesirable and should be reduced to the minimum. Let
us analyse how we can minimize the chances of type I and type II errors. It may be easily
realized that it is possible, even with imperfect sample evidence, to reduce the probability of
type I error all the way down to zero. Just accept the null hypothesis; no matter what the
evidence is. Since we will never reject any null hypothesis, we will never reject a true null
hypothesis and thus we will never commit a type I error! However, it is obvious that this
would be foolish. If we always accept a null hypothesis, then given a false null hypothesis, no
matter how wrong it is, we are sure to accept it. In other words, our probability of committing
a type II error will be 1. Similarly, we find it foolish to reduce the probability of type II error
all the way down to zero by always rejecting a null hypothesis, for we would then reject
every true null hypothesis, no matter how right it is. Our probability of type I error will be 1.
Therefore, we cannot and should not try to completely avoid either type of error. We should
plan, organize, and settle for some small, optimal probability of each type of error. Before we
discuss this issue, we need to learn a few more concepts.
14.3.2
TEST STATISTIC AND THE p-VALUE
Consider the case of owner’s suspicion related to our bottling process example. The null and
alternative hypotheses in this case are:
H0:
μ ≤ 1,000
H1:
μ > 1,000
Suppose the population variance is 25 and a random sample of size 100 yields a sample
mean of 1,000.5. Because the sample mean is more than 1,000, the evidence goes against
the null hypothesis (H0). Can we reject H0 based on this evidence?
421
¾ if we reject it, there is some chance that we might be committing a type I error, and
¾ if we accept it, there is some chance that we might be committing a type II error.
Then what can we do? We should ask a natural question at this situation- “What is the
probability that H0 can still be true despite the evidence?” The question asks for the
"credibility" of H0 in light of unfavorable evidence. However, due to mathematical
complexities, it is not possible to compute the probability that H0 is true. We, therefore, settle
for a question that comes very close.
“When the actual μ
= 1,000, and with sample size 100, what is the
probability of getting a sample mean that is more than or equal to 1000.5?”
The answer to this question is then taken, as the "credibility rating" of H0. Analyzing the
question carefully, we note an important aspect:
The condition assumed is μ = 1,000; although H0 states μ ≤ 1,000 . The reason for
assuming μ = 1,000 is that it gives the most benefit of doubt to H0. If we assume μ
= 999, for instance, the probability of the sample mean being more than or equal to
1,000.5 will only be smaller, and H0 will only have less credibility. Thus the
assumption μ = 1,000 gives the maximum credibility to H0.
Now using our knowledge of sampling distribution of sample mean, we can easily answer our
question.
Since population variance is known and sample size is large enough, the Central Limit
Theorem is applicable here i. e.
2
⎛ ⎛
⎞
⎞
σ
⎜
X ~ N ⎜ μ, ⎜
⎟ ⎟⎟
n⎠ ⎠
⎝ ⎝
and the standard normal variable Z =
(
X −μ
σ
n
)
probability P X ≥ 1,000.5
422
is to be used to calculate the required
(
)
So P X ≥ 1,000.5
⎛
⎞
⎜
1,000.5 − 1,000 ⎟
= P⎜ Z ≥
⎟
5
⎜
⎟
100 ⎠
⎝
= P(Z ≥ 1.00)
= 0.1587
≈ 0.16
So the answer to our question is 16%. That is, there is a 16% chance for a sample of size 100
to yield a sample mean more than or equal to 1000.5 when the actual µ = 1,000. Statisticians
call this 16% the p-value. In other words p-value-the probability of observing a sample
statistic as extreme as the one observed if the null hypothesis is true-.
is a kind of "credibility rating" of H0 in light of the evidence. A p-value of zero means H0 is
certainly false and a p-value of 1 means that H0 is certainly true. A p-value of 16% means
that there is roughly 16% probability that H0 is true, despite the evidence. Conversely, we can
be roughly 84% confident that H0 is false in light of the evidence. The implication is that if
we reject H0, then there is about an 84% chance that we are doing the right thing, and about a
16% chance that we are committing a type I error. The formal definition of the p-value
follows:
Given a null hypothesis and sample evidence with sample size n, the
p-
value is the probability of getting a sample evidence with the same n that is
equally or more unfavorable to the null hypothesis while the null hypothesis is
actually true. The p-value is calculated giving the null hypothesis the
maximum benefit of doubt.
The random variable, as Z in this case, used to calculate the p-value is called test statistic. The
formal definition of the test statistic follows:
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A test statistic is a random variable calculated from the sample evidence,
which follows a well-known distribution and thus can be used to calculate the
p-value.
Most of the time, the test statistic we use will be Z, t, χ2, or F. The distributions of these
random variables are well known and we can calculate the p-value.
Up to this point it is very much clear that statistical hypothesis is always stated with reference
to a population parameter (mean, proportion or variance). The appropriate random variable
calculated from the sample evidence acts as a test statistic and provide the means to decide
whether statistical hypothesis is to be rejected or accepted.
14.3.3
THE SIGNIFICANCE LEVEL-α
From our discussion on p-value, it becomes clear that the p-value of a test i.e. the credibility
of the null hypothesis varies with actual observed value of the sample statistic. This fact
necessitates having a policy for rejecting H0 based on p-value.
The most common policy in statistical hypothesis testing is to establish a significance level,
denoted by α, and to reject H0 when the p-value falls below it. When this policy is followed,
one can be sure that the maximum probability of type I error is α.
Policy: When the p-value is less than α, reject H0
In other words, we can say that the rejection region for H0 is the area under the curve where
the p-value is less than α. This region is also called critical region
The standard values for α are 10%, 5%, and 1%. Suppose α is set at 5%. In the preceding example, for a sample mean of 1,000.5 the p-value was 16%, and H0 will not be rejected. For a
sample mean of 1001 the p-value will be 2.28%, which is below α = 5%. Hence H0 will be
rejected.
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Let us analyze in some detail the implications of using a significance level α for rejecting a
null hypothesis.
¾ The first thing to note is that if we do not reject H0, this does not prove that H0 is true.
For example, if α = 5% and the p-value = 6%, we will not reject H0. But there is only
about 6% chance that H0 is true, which is hardly proof that H0 is true. It may be
possible that H0 is false and by not rejecting it, we are committing a type II error. For
this reason, we should say "We cannot reject H0 at an α of 5%" rather than "We
accept H0."
¾ The second thing to note is that α is the maximum probability of type I error we set
for ourselves. Since α is the maximum p-value at which we reject H0, it is the maximum probability of committing a type I error. In other words, setting α = 5% means
that we are willing to put up with up to 5% chance of committing a type I error.
¾ The third thing to note is that the selected value of α indirectly determines the
probability of type II error as well. In general, other things remaining the same,
increasing the value of α will decrease the probability of type II error. This should be
intuitively obvious. For example, increasing α from 5% to 10% means that in those
instances with p-value in the range 5% to 10% the H0 that would not have been
rejected before would now be rejected. Thus, some cases of false H0 that escaped
rejection before may not escape now. As a result, the probability of type II error will
decrease
¾ The fourth thing to note about α is the meaning of (1 - α). If we set α = 5%, then (1 -
α) = 95% is the minimum confidence level that we set in order to reject H0. In other
words, we want to be at least 95% confident that H0 is false before we reject it.
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14.3.3.1
One-Tailed and Two-Tailed Tests
Consider the null and alternative hypotheses:
H0:
μ ≥ 1,000
H1:
μ < 1,000
In this case, we will reject H0 only when X is significantly less than 1,000 or only when Z
falls significantly below zero. Thus the rejection occurs only when Z takes a significantly low
value in the left tail of its distribution.
Such a case where rejection occurs in the left tail of the distribution of the test statistic is
called a left-tailed test, as seen in Figure 14-1.
Figure 14-1
A Left-tailed Test
In the case of a left-tailed test, the p-value is the area to the left of the calculated value of the
test statistic.
Now consider the case where the null and alternative hypotheses are:
H0:
μ ≤ 1,000
H1:
μ > 1,000
In this case, we will reject H0 only when X is significantly more than 1,000 or only when Z is
significantly greater than zero. Thus the rejection occurs only when Z takes a significantly
high value in the right tail of its distribution.
Such a case where rejection occurs in the right tail of the distribution of the test statistic is
called a right-tailed test, as seen in Figure 14-2.
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Figure 14-2
A Right-tailed Test
In the case of a right-tailed test, the p-value is the area to the right of the calculated value of
the test statistic.
In left-tailed and right-tailed tests, rejection occurs only on one tail. Hence each of them is
called a one-tailed test.
Finally, consider the case where the null and alternative hypotheses are:
H0:
μ = 1,000
H1:
μ ≠ 1,000
In this case, we have to reject H0 in both cases, that is, whether X is significantly less than or
greater than 1,000. Thus, rejection occurs when Z is significantly less than or greater than
zero, which is to say that rejection occurs on both tails. Therefore, this case is called a twotailed test. See Figure 14-3, where the shaded areas are the rejection regions.
Figure 14-3
A Two-tailed Test
In the case of a two-tailed test, the p-value is twice the tail area. If the calculated value of the
test statistic falls on the left tail, then we take the area to the left of the calculated value and
multiply it by 2. If the calculated value of the test statistic falls on the right tail, then we take
the area to the right of the calculated value and multiply it by 2. For example, if the calculated
427
Z = +1.75, the area to the right of it is 0.0401. Multiplying that by 2, we get the p-value as
0.0802.
14.3.3.2
Selecting Optimal α
All tests of hypotheses hinge upon this concept of the significance level and it is possible that
a null hypothesis can be rejected at α = 5% whereas the same evidence is not strong enough
to reject the null hypothesis at α = 1%. In other words, the inference drawn can be sensitive
to the significance level used. We should note that selecting a value for α is a question of
compromise between type I and type II error probabilities. In practice, the significance level
is supposed to be arrived at after considering the cost consequences of type I error and type II
error. However, most of the time the costs are difficult to estimate since they depend, among
other things, on the unknown actual value of the parameter being tested. Thus, arriving at a
"calculated" optimal value for α is impractical. Instead, we follow an intuitive approach of
assigning one of the three standard values, 1%, 5%, and 10%, to α.
In the intuitive approach, we try to estimate the relative costs of the two types of errors. For
example, suppose we are testing the average tensile strength of a large batch of bolts
produced by a machine to see if it is above the minimum specified. Here type I error will
result in rejecting a good batch of bolts and the cost of the error is roughly equal to the cost of
the batch of bolts. Type II error will result in accepting a bad batch of bolts and its cost can
be high or low depending on how the bolts are used.
If the bolts are used to hold together a structure, then the cost is high because defective bolts
can result in the collapse of the structure, causing great damage. In this case, we should strive
to reduce the probability of type II error more than that of type I error. In such cases where
type II error is more costly, we keep a large value for α, namely, 10%.
428
On the other hand, if the bolts are used to secure the lids on trash cans, then the cost of type II
error is not high and we should strive to reduce the probability of type I error more than that
of type II error. In such cases where type I error is more costly, we keep a small value for α,
namely, 1%.
Then there are cases where we are not able to determine which type of error is more costly.
If the costs are roughly equal, or if we have not much knowledge about the relative costs of
the two types of errors, then we keep α = 5%.
14.3.3.3
β and Power of the Test
Denoted by β, Type II error is committed when a wrong decision is taken in accepting a false
null hypothesis. It is the probability of accepting H0 when it should have rejected for being
false. It should be noted that β depends on the actual value of the parameter being tested, the
sample size, and α. Let us see exactly how it depends.
Consider the null and alternative hypotheses
Figure 14-4
H0:
μ ≤ 1,000
H1:
μ > 1,000
Type II Error: H0:
μ ≤ 1,000 and actual μ = 1,002
Suppose the actual value of μ = μ1 (say 1,002), such that μ1 > 1,000. Obviously, H0 is false.
The cross-hatched area under the normal curve centered at μ1 in Figure 14-4 is then the
probability of accepting H0 when it is false. This area - in the acceptance region of the normal
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curve centered at μ0 = 1,000; represents the probability that the observed sample mean X
falls in the acceptance region when μ = μ1 (1,002), that is when H0 is false.
Given the acceptance region (1 - α) for the normal curve centered at μ = μ0 = 1,000, a careful
analysis of figure reveals the following.
¾ The value of β decreases as μ1 move away from μ0, displaying the entire normal curve
centered at μ1 farther and farther away from the normal curve centered at μ0.
¾ The value of β tends to increase as μ1 moves nearer to μ0. A limit is reached when μ1
coincides with μ0, and the entire acceptance region (1- α) for μ = μ0 will represent the
value of β. This is important conclusion in the sense that when H0 is true for μ = μ0,
the entire acceptance region is Type II error. Hence when H0 is true, β = 1 - α and α
=1-β.
¾ The un-shaded area under the normal curve centered at μ1, which falls outside the
acceptance region for μ = μ0, represents the probability of rejecting H0 when it is false
for μ = μ1. This complement of β; (1-β) is known as the power of the test.
The power of a test is the probability that a false null hypothesis will be
detected by the test.
¾ A change in the level of significance α means a change in the acceptance region (1-
α), which obviously implies a change in the cross hatched area i.e. β. In other words,
the smaller the α, the larger the β and vice-versa. Type I and type II errors are,
therefore negatively related.
Type I error and the power of the test (1-β) are, however, positively related. Thus, the
smaller the probability (α) of rejecting H0 when it is true, the smaller is the
probability (1-β) of rejecting H0 when it is false.
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14.3.3.4
Sample Size
In the discussion above we said that we can keep a α low or aβ low depending on which type
of error is more costly. What if both types of error are costly and we want to have low α as
well as low β? The only way to do this is to make our evidence more reliable, which can he
done only by increasing the sample size. If the sample size increases, then the evidence
becomes more reliable and the probability of any error will decrease.
Figure 14-5 shows the relationship between α and β for various values of sample size n. As n
increases, the curve shifts downwards reducing both α and β. Thus, when the costs of both
types of error are high, the best policy is to have a large sample and a low α, such as 1%.
Figure 14-5
β versus α for Various Values of n
After understanding the basic concepts of testing of hypotheses, we are now, able to
develop tests concerning different population parameters. Under different conditions the
test procedures have to be developed differently and different test statistics are used for
testing. Before proceeding further let us define the critical region in terms of test statistic,
which is often more helpful in many situations.
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14.4
CRITICAL REGION IN TERMS OF TEST STATISTIC
We have seen that the most common policy in statistical hypothesis testing is to establish a
significance level-α. We decide to reject or not to reject the null hypothesis H0 by comparing
the p-value with the significance level. We define the critical or rejection region as:
Critical Region:
p-value < α
But in many situations we find it more useful to define the critical region in terms of test
statistic. We, then, decide to reject or not to reject the null hypothesis H0 by comparing the
observed value of the test statistic with the cut-off value or the critical value of the test
statistic.
Z-test
When in the testing of hypotheses, we use the random variable Z for calculating the p-value
and for defining the critical region of the test; we call the test as Z-test. The critical region in
terms of Z are summarized in Table 14-2
Table 14-2
Critical Region of Z-test
Test
Critical Region
Z< -Zα
Left-tailed
Z > Zα
Right-tailed
Z > Zα/2
Two-tailed
and
Z< -Zα/2
t-test
When in the testing of hypotheses, we use the random variable t for calculating the p-value
and for defining the critical region of the test; we call the test as t-test. The critical region in
terms of t are summarized in Table 14-3
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Table 14-3
Critical Region of t-test
Test
Critical Region
Left-tailed
t< -tα
Right-tailed
t > tα
t > tα/2
Two-tailed
t< -tα/2
and
χ2-test
When in the testing of hypotheses, we use the random variable χ2 for calculating the p-value
and for defining the critical region of the test; we call the test as χ2-test. The critical region in
terms of χ2 are summarized in Table 14-4
Table 14-4
Critical Region of χ2-test
Test
Critical Region
Left-tailed
χ2< χ21-α
Right-tailed
χ2 > χ2 α
χ2> χ2α/2
Two-tailed
and
χ2< χ21-α/2
F-test
When in the testing of hypotheses, we use the random variable F for calculating the p-value
and for defining the critical region of the test; we call the test as F-test. The critical region in
terms of F are summarized in Table 14-5
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Table 14-5
Critical Region of F-test
Test
Critical Region
Left-tailed
F< F1−α ( n1 −1, n2 −1 )
i.e. F< Fα ( n2 −1, n1 −1 )
Right-tailed
F> Fα (n1 −1, n2 −1 )
F> Fα / 2 ( n1 −1, n2 −1 )
Two-tailed
and
F< F1−α / 2 ( n1 −1, n2 −1 )
i.e. F< Fα / 2 ( n2 −1, n1 −1 )
14.5 GENERAL TESTING PROCEDURE
We have learnt a number of important concepts about hypothesis testing. We are now in a
position to lay down a general testing procedure in a more systematic way. By now it should
be clear that there are basically two phases in testing of hypothesis - in the first phase, we
design the test and set up the conditions under which we shall reject the null hypothesis. In
the second phase, we use the sample evidence and draw our conclusion as to whether the null
hypothesis can be rejected. The detailed steps involved are as follows:
Step 1:
State the Null and the Alternate Hypotheses. i.e. H0 and H1
Step 2:
Specify a level of significance α
Step 3:
Choose the test statistic and define the critical region in terms of the test
statistic
Step 4:
Make necessary computations
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Step 5:
¾
calculate the observed value of the test statistic
¾
find the p- value of the test
Decide to accept or reject the null hypothesis either
¾
by comparing the p- value with α or
¾
by comparing the observed value of the test statistic with the cut- off
value or the critical value of the test statistic.
14.6 TESTS OF HYPOTHESES ABOUT POPULATION MEANS
When the null hypothesis is about a population mean, the test statistic can be either Z or t. If
we use μ0 to denote the claimed population mean the null hypothesis can be any of the three
usual forms:
H0:
μ = μ0
two-tailed test
H0:
μ ≥ μ0
left-tailed test
H0:
μ ≤ μ0
right-tailed test
Cases in Which the Test Statistic is Z
1. The population standard deviation, σ, is known and the population is normal.
2. The population standard deviation, σ, is known and the sample size, n, is at least 30
(The population need not be normal).
The formula for calculating the test statistic Z in both these cases is
Z=
X − μ0
σ
n
3. The population is normal and the population standard deviation, σ, is unknown, but
the sample standard deviation, S, is known and the sample size, n, is large enough.
The formula for calculating the test statistic Z in this case is
Z=
X − μ0
S
n
Cases in Which the Test Statistic is t
435
1. The population is normal and the population standard deviation, σ, is unknown, but
the sample standard deviation, S, is known and the sample size, n, is small.
2. The population is not normal and the population standard deviation, σ, is unknown,
but the sample standard deviation, S, is known and the sample size, n, large enough.
The formula for calculating the test statistic t in both these cases is
t=
X − μ0
S
n
The degrees of freedom for this t is (n-1)
14.7 TESTS OF HYPOTHESES ABOUT POPULATION ROPORTIONS
When the null hypothesis is about a population proportion, the test statistic can be either the
Binomial random variable or its Poisson or Normal approximation. If we use p0 to denote the
claimed population proportion the null hypothesis can be any of the three usual forms:
H0:
p = p0
two-tailed test
H0:
p ≥ p0
left-tailed test
H0:
p ≤ p0
right-tailed test
Cases in which the Test Statistic is Binomial Random Variable X
The Binomial distribution can be used whenever we are able to calculate the necessary
binomial probabilities. When the Binomial distribution is used, the number of successes X
serves as the test statistic. It is conveniently applicable to problems where sample size, n, is
small and p0 is neither very close to 0 nor to 1.
Cases in which the Test Statistic is Poisson Random Variable X
The Poisson approximation of Binomial distribution is conveniently applicable to problems
where sample size, n, is large and p0 is either very close to 0 or to 1. When the Poisson
distribution is used, the number of successes X serves as the test statistic.
436
Note that the Binomial random variable or its Poisson approximation X follows a discrete
distribution, and recall that the p-value is the probability of the test statistic being equally or
more unfavorable to H0 than the value obtained from the evidence. For example, for a righttailed test with H0: p ≤ 0.5, the p-value = P(X ≥ observed number of successes).
Cases in Which the Normal Approximation is to be used
The Normal approximation of Binomial distribution is conveniently applicable to problems
where sample size, n, is large and p0 is neither very close to 0 nor to 1. When the normal
distribution is used, the test statistic Z is calculated as:
p − p0
Z=
p0 (1 − p0 )
n
14.8 TESTS OF HYPOTHESES ABOUT POPULATION VARIANCES
When the null hypothesis is about a population variance, the test statistic is χ2. If we use σ0 to
denote the claimed population proportion the null hypothesis can be any of the three usual
forms:
H0:
σ = σ0
two-tailed test
H0:
σ ≥ σ0
left-tailed test
H0:
σ ≤ σ0
right-tailed test
The formula for calculating the test statistic χ2 is:
χ2
=
(n − 1)S 2
σ 02
The degrees of freedom for this χ2 is (n - 1).
14.9 THE COMPARISON OF TWO POPULATIONS
Almost daily we compare products, services, investment opportunities, management styles
and so on. In all such situations we are interested in the comparisons of two populations with
respect to some population parameter - the population mean, the population proportion, or the
437
population variance. Now we will learn how to conduct such comparisons in an objective and
meaningful way.
14.9.1 TESTING FOR DIFFERENCE BETWEEN MEANS
When we want to arrive at same conclusion about the difference between two population
means, we draw one sample from each of the population. The samples drawn may be
dependent on each other or these may be independent of each other.
14.9.1.1
Dependent Samples- Paired Observations
In many situations, we can design our test in such a way that the samples drawn are
dependent on each other and our observations come from two populations and are paired in
some way. In general, when possible, it is often advisable to pair the observations, as this
makes the experiment more precise. We can see the advantage of pairing observations with
the helps of an example.
Consider a sales manager who wants to know if display at point of purchase helps in
increasing the sales of his product. He may design the experiment in two ways:
Design I: He picks up a sample of, say 12, retail shops with no display at point of purchase.
Similarly he picks up a sample of, say 10, retail shops with display at point of purchase. He
will note his observations from both samples independently of each other.
Design II: He picks-up a random sample, of say 11, retail shops and note down the
observations about weekly sale in each of these shops. Next he introduces display at point of
purchase at each of these shops and again observes the weekly sales in them.
Obviously design II much better, as this tends to remove much of the extraneous variations in
sales – the variation in the location of the soap, experimental conditions and other extraneous
factors. Now after eliminating the effect of all other major factors, we can attribute the
difference only to the ‘treatment’ we are studying -the display at point of purchase.
438
Let us label the two populations as 1 and 2. Under the situation of paired observations, it is
easy to see that the variable in which we are interested is the differences between the two
observations i.e. d = x1 − x 2 . In other words our two-population comparison test is reduced
to a hypothesis test about one parameter - the difference between the means of two
populations’ i.e. μ d = μ1 − μ 2
Thus the null hypothesis can be any of the three usual forms:
H0:
μ1 − μ 2 = μ d 0
or
μd = μd 0
two-tailed test
H0:
μ1 − μ 2 ≥ μ d 0
or
μd ≥ μd 0
left-tailed test
H0:
μ1 − μ 2 ≤ μ d 0
or
μd ≤ μd 0
right-tailed test
The test statistic can be either t or Z.
Cases in Which the Test Statistic is t
The population standard deviation of the difference, σd, is not known and the sample
size, n, is small.
The formula for calculating the test statistic t is
t=
d − μ d0
Sd
n
The degrees of freedom for this t is (n-1)
Cases in Which the Test Statistic is Z
The sample size, n, is large and/or we happen to know the population standard
deviation of the difference, σd.
The formula for calculating the test statistic t is
Z=
d − μ d0
Sd
n
439
Z=
or
14.9.1.2
d − μ d0
σd
n
Independent Samples
When independent random sample are taken, the sample size need not be same for both
populations. Let us label the two populations as 1 and 2. So that
μ1 and μ2 denote the two population means.
σ1 and σ2 denote the two population standard deviations
n1 and n2 denote the two sample sizes
X 1 and X 2 denote the two sample means
S1 and S2 denote the two sample standard deviations
If we use (μ1 − μ 2 )0 to denote the claimed difference between the two population means,
then the null hypothesis can be any of the three usual forms:
H0:
μ1 − μ 2 = (μ1 − μ 2 )0
two-tailed test
H0:
μ1 − μ 2 ≥ (μ1 − μ 2 )0
left-tailed test
H0:
μ1 − μ 2 ≤ (μ1 − μ 2 )0
right-tailed test
The test statistic can be either Z or t.
Cases in Which the Test Statistic is Z
1. The population standard deviations; σ1 and σ2; are known and both the populations are
normal.
2. The population standard deviations; σ1 and σ2; are known and the sample sizes; n1 and
n2; are both at least 30 (The population need not be normal).
The formula for calculating the test statistic Z in both these cases is
Z =
(X
1
)
− X 2 − (μ 1 − μ 2 )0
σ 12
n1
440
+
σ 22
n2
Cases in Which the Test Statistic is t
The populations are normal; the population standard deviations; σ1 and σ2; are
unknown, but the sample standard deviations; S1 and S2; are known.
The formula for calculating the test statistic t depends on two sub cases:
Subcase I:
σ1 and σ2 are believed to be equal (although unknown)
t =
(X
1
)
− X 2 − (μ 1 − μ 2 )0
S p2 ⎛⎜ 1 + 1 ⎞⎟
n2 ⎠
⎝ n1
Where S P2 is the pooled variance of the two samples, which serves as the estimator of the
common population variance.
S =
2
p
(n1 − 1)S12 + (n 2 − 1)S 22
n1 + n 2 − 2
The degrees of freedom for this t is (n1 +n2 -2).
Subcase II:
σ1 and σ2 are believed to be unequal (although unknown)
t =
(X
1
− X
2
) − (μ
1
− μ 2 )0
⎛ S 12
⎞
S2
+ 2
⎜
n1
n 2 ⎟⎠
⎝
The degrees of freedom for this t is given by:
2
⎛ S 12
S 22 ⎞
+
⎜ n
n 2 ⎟⎠
1
⎝
df =
⎛ ⎛ S 2 ⎞ 2 ⎞⎛ ⎛ S 2 ⎞ 2
⎜ ⎜ 1 ⎟ ⎟⎜ ⎜ 2 ⎟
⎜ ⎝ n1 ⎠ ⎟⎜ ⎝ n 2 ⎠
⎜ n − 1 ⎟⎜ n − 1
2
⎜ 1
⎟⎜
⎜
⎟⎜
⎝
⎠⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎠
14.9.2 TESTING FOR DIFFERENCE BETWEEN POPULATION PROPORTIONS
We will consider the large-sample tests for the difference between population proportions.
For ‘large enough’ sample sizes the distribution of the two sample proportions and also the
distribution of the difference between the two sample proportions is approximated well by a
441
normal distribution. This gives rise to Z-test for comparing the two population proportions.
Let us assume independent random sampling from the two populations, labeled as 1 and 2, so
that
p1 and p2 denote the two population proportions
n1 and n2 denote the two sample sizes
p 1 and p 2 denote the two sample proportions
We will use
( p1 − p 2 )0
to denote the claimed difference between the two population
proportions. Then the null hypothesis can be any of the three usual forms:
H0:
p1 − p 2 = ( p1 − p 2 )0
two-tailed test
H0:
p1 − p 2 ≥ ( p1 − p 2 )0
left-tailed test
H0:
p1 − p 2 ≤ ( p1 − p 2 )0
right-tailed test
The formula for calculating the test statistic Z depends on two cases.
Case I:
When ( p1 − p 2 )0 = 0 i.e. the claimed difference between the two population
proportions is zero
Z =
(
(p
1
)
− p2 − 0
)
p 1 − p ⎛⎜ 1 + 1 ⎞⎟
n2 ⎠
⎝ n1
Where p is the combined sample proportion in both the samples
p=
Case II:
x1 + x 2
n1 + n 2
When ( p1 − p 2 )0 ≠ 0 i.e. the claimed difference between the two population
proportions is some number other than zero
Z =
(p
)
p (1 − p ) p (1 − p )
+
1
− p 2 − ( p1 − p 2 )0
1
1
2
n1
2
n2
442
14.9.3 TESTING FOR EQUALITY OF TWO POPULATION VARIANCES
Many a times, we may be interested in comparing the degree of variability or dispersion of
two different populations. Here the problem essentially involves testing the equality of two
population variances. Let us assume independent random sampling from the two populations,
labeled as 1 and 2, so that
σ 12 and σ 22 denote the two population variances
n1 and n2 denote the two sample sizes
S 12 and S 22 denote the two sample variances
Then the null hypothesis can be any of the three usual forms:
H0:
σ 12 = σ 22
two-tailed test
H0:
σ 12 ≥ σ 22
left-tailed test
H0:
σ 12 ≤ σ 22
right-tailed test
The formula for calculating the test statistic F is:
F(n1 −1, n2 −1)
S 12
= 2
S2
The degrees of freedom for this F is (n1-1, n2-1)
14.10 SOLVED PROBLEMS
Now we will solve some problems relating to testing the hypotheses stated about different
population parameters, under different conditions.
Example 14-1
An automatic bottling machine fills oil into 2-liter (2,000 cm3) bottles. A consumer advocate
wants to test the null hypothesis that the average amount filled by the machine into a bottle is
at least 2,000 cm3. A random sample of 40 bottles coming out of the machine was selected
and the exact contents of the selected bottles are recorded. The sample mean was 1,999.6
cm3. The population standard deviation is known from past experience to be 1.30 cm3.
443
(a)
Test the null hypothesis at an α of 5%.
(b)
Assume that the population is normally distributed with the same standard
deviation of 1.30 cm3. Assume that the sample size is only 20 but the sample
mean is the same 1,999.6 cm3. Conduct the test once again at an α of 5%.
(c) If there is a difference in the two test results, explain the reason for the difference.
Solution: (a) 1.
The null and alternative hypotheses:
H0:
μ ≥ 2,000
H1:
μ < 2,000
The test is a left-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
Z; as the population standard deviation is known and
sample size is greater than 30
4.
Critical region:
5.
Computations:
Z < -Z0.05 Where Z0.05=1.645
X =1,999.6
n = 40
X − μ0
Z=
Z =
σ = 1.30
σ
n
1,999 . 6 − 2 , 000
1 . 30
40
Z = −1.95
6.
Conclusion:
We reject the null hypothesis at α =0.05 since
= -1.95 < -Z0.05 = -1.645
(b)
Since the population is normally distributed, the test statistic is once again Z
Computations:
X =1,999.6
σ = 1.30
444
n = 20
Z
Z=
Z=
X − μ0
σ
n
1,999.6 − 2,000
1.30
20
Z = −1.38
Conclusion:
We do not reject the null hypothesis at α=0.05 since
Z=
-1.38 > -Z0.05 = -1.645
(c)
In the first case we could reject the null hypothesis but in the second we could not,
although in both cases the sample mean was the same. The reason is that in the first case the
sample size was larger and therefore the evidence against the null hypothesis was more
reliable. This produced a smaller p-value in the first case.
Example 14-2
An automobile manufacturer substitutes a different engine in cars that were known to have an
average miles-per-gallon rating of 31.5 on the highway. The manufacturer wants to test
whether the new engine changes the miles-per-gallon rating of the automobile model. A
random sample of 100 trial runs gives X = 29.8 miles per gallon and S = 6.6 miles per
gallon. Using the 0.05 level of significance, is the average miles-per-gallon rating on the
highway for cars using the new engine different from the rating for cars using the old engine?
Solution: 1.
The null and alternative hypotheses:
H0:
μ = 31.5
H1:
μ ≠ 31.5
The test is a two-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
Z; as the sample standard deviation is known and sample size
is greater than 30
445
4.
Critical region:
5.
Computations:
Z0.025 < Z < -Z0.025 Where Z0.025 =1.96
X =29.8
S = 6.6
Z=
Z=
n = 100
X − μ0
S
n
29.8 − 31.5
6.6
100
Z = −2.57
6.
Conclusion:
We reject the null hypothesis at α = 0.05 since
Z=-
2.57 < -Z0.025 = -1.96. So we conclude that the average miles-per-gallon rating on the
highway for cars using the new engine is different from the rating for cars using the
old engine.
Example 14-3
Sixteen oil tins are taken at random from an automatic filling machine. The mean weight of
the tins is 14.2 kg, with a standard deviation of 0.40 kg. Can we conclude that the filling
machine is wasting oil by filling more than the intended weight of 14 kg, at a significance
level of 5%?
Solution: 1.
The null and alternative hypotheses:
H0:
μ ≤ 14.2
H1:
μ > 14.2
The test is a right-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
t; as the sample standard deviation is known and sample size is
small.
4.
Critical region:
t > t0.05 Where t0.05 for 15 df =1.7530
446
5.
Computations:
X =14.2
t=
t=
S = 0.40
n = 16
X − μ0
S
n
14.2 − 14
0.40
16
t =2
6.
We reject the null hypothesis at α = 0.05 since
Conclusion:
t =2
> t0.05 =1.7530. So we conclude that the filling machine is wasting oil by filling more
than the intended weight of 14 kg.
Example 14-4
A coin is to be tested for fairness. It is tossed 15 times and only 8 heads are observed. Test if
the coin is fair at α = 5%.
Solution: 1.
The null and alternative hypotheses:
H0:
p = 0.5
H1:
p ≠ 0.5
The test is a two-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
Binomial random variable X
4.
Critical region:
p-value < α
5.
Computations:
p 0 = 0.5
n = 15
p-value = 2*P(X ≤ 8)
447
⎛ 8
n− X ⎞
= 2 * ⎜ ∑ nC X p X (1 − p) ⎟
⎝ X =0
⎠
⎛ 8
n− X ⎞
= 2 * ⎜ ∑ 15C X 0.5 X (1 − 0.5) ⎟
⎝ X =0
⎠
= 0.5034
6.
We cannot reject the null hypothesis at α = 0.05 since
Conclusion:
p-
value > α. So we accept that the coin is fair.
Example 14-5
A wholesaler received a shipment of goods, which is reported to be containing at most 2%
defective items. He will accept the shipment if the claim is found true and reject if the
percentage of defective items is more. To verity this claim, he draws a sample of 200 items
and finds that 10 items are defective. What should be his decision at 5% level of significance?
Solution:1.
The null and alternative hypotheses:
H0:
p ≤ 0.02
H1:
p > 0.02
The test is a right-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
Poisson random variable X since p0 is very small and the
sample size is large enough to use poisson approximation of binomial distribution.
4.
Critical region:
5.
Computations:
p-value < α
p 0 = 0.02
n = 200
μ=4
p-value = P(X ≥ 10)
= 1- P (X ≤ 9)
448
9
⎛ e−μ μ X
= 1 − ∑⎜⎜
X!
X =0 ⎝
⎞
⎟⎟
⎠
= 1- 0.9919
= 0.0081
6.
We reject the null hypothesis at α =0.05 since p-value < α. So
Conclusion:
the wholesaler will not accept the shipment.
Example 14-6
SBI claims that more than 55% of the saving accounts in Haryana are at SBI. A sample
survey of 400 account holders revealed that only 180 account holders have account at SBI.
Verify, using 5% level of significance, if the sample results underestimate the claim of SBI.
Solution: 1.
The null and alternative hypotheses:
H0:
p ≥ 0.55
H1:
p < 0.55
The test is a left-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
Z; since p0 is not too close to 0 or 1 and the sample size is
large enough to use normal approximation of binomial distribution.
4.
Critical region:
5.
Computations:
Z < -Z0.05 Where Z0.05 =1.645
p0 = 0.55
Z=
Z=
p = 180/400 = 0.45
p − p0
p0 (1− p0 )
n
0.45 − 0.55
0.55(1 − 0.55)
449
400
n = 400
Z=
− 20000
4975
Z = −4.02
6.
We reject the null hypothesis at α = 0.05 since
Conclusion:
Z=-
4.02 < -Z0.05 = -1.645. So the sample results underestimate the claim of SBI.
Example 14-7
A manufacturer of golf balls claims that the company controls the weights of the golf balls
accurately so that the variance of the weights is not more than 1 mg2. A random sample of 31
golf balls yields a sample variance of 1.62 mg2.Is that sufficient evidence to reject the claim
at an α of 5%?
Solution: 1.
The null and alternative hypotheses:
H0:
σ2 ≤ 1
H1:
σ2 > 1
The test is a right-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
χ2
4.
Critical region:
χ2 > χ20.05 Where χ20.05 for 30 df = 43.7729
5.
Computations:
σ 02 =1
S 2 = 1.62
χ
2
n = 31
(
n − 1)S 2
=
σ 02
=
30x1.62
1
= 48.6
450
6.
We reject the null hypothesis at α =0.05 since
Conclusion:
χ2=48.6 >
χ20.05 = 43.7729. So we conclude that there is sufficient evidence to reject the claim of
the company.
Example 14-8
A sales manager wants to know if display at point of purchase helps in increasing the sales of
his product. He note the following observations:
Shop No.
1
2
3
4
5
6
7
8
9
10
11
Sales before
4500 5275 7235 6844 5991 6672 4943 7615 6148 5623 5154
display
Sales
after
4834 5010 7562 6957 6401 6423 5334 8004 6729 6277 5769
display
Difference(d) -334 265 -327 -113 -410 249 -391 -389 -581 -654 -615
d = -300
Sd = 314.53
Is there sufficient evidence to conclude that display at point of purchase helps in increasing
the sales of his product?
Solution: 1.
The null and alternative hypotheses:
H0:
μd ≥ 0
H1:
μ<0
The test is a left-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
t; as the population standard deviation of the difference, σd, is
not known and the sample size, n, is small.
4.
Critical region:
5.
Computations:
t < -t0.05 Where t0.05 for 10 df =1.812
d =-300
S = 314.53
451
n = 11
t=
t=
d − μ d0
Sd
n
− 300− 0
314.53
11
t = −3.16
6.
Conclusion:
We reject the null hypothesis at α =0.05 since
t=-
3.16 < t0.05 = -1.812. So the sales manager has sufficient evidence to conclude that
display at point of purchase helps in increasing the sales.
Example 14-9
The makers of Duracell batteries want to demonstrate that their size AA battery lasts on an
average of at least 45 minutes longer than Duracell’s main competitor, the Energizer. Two
independent random samples of 100 batteries of each kind are selected. The sample average
lives for Duracell and Energizer batteries are found to be X 1 = 308 minutes and X 2 = 254
minutes respectively. Assume σ 1 = 84 minutes and σ 2 = 67 minutes. Is there evidence to
substantiate Duracell’s claim that its batteries last, on an average, at least 45 minutes longer
than Energizer of the same size?
Solution: 1.
The null and alternative hypotheses:
H0:
μ1 − μ 2 ≤ 45
H1:
μ1 − μ 2 > 45
The test is a right-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
Z
4.
Critical region:
Z > Z0.05 Where Z0.05 =1.645
5.
Computations:
452
X 1 =308
σ 1 = 84
X 2 =254
Z =
(X
Z =
1
σ 2 = 67
n1 = n2 = 100
)
− X 2 − (μ 1 − μ 2 )0
σ 12
n1
+
σ 22
n2
308 − 254 − 45
84 2
100
+
67
2
100
Z = 0.838
6.
Conclusion:
We cannot reject the null hypothesis at α = 0.05 since Z=0.838
< Z0.05=1.645. In fact the observed value of the test statistic falls in the non-rejection
region of our right-tailed test at any conventional level of significance. So we must
conclude that there is insufficient evidence to support Duracell’s claim.
Example 14-10
The following information relate to the prices (in Rs) of a product in two cities A and B.
City A
City B
Mean price
22
17
Standard deviation
5
6
The observations related to prices are made for 9 months in city A and for 11 months in city
B. Test at 0.01 level whether there is any significant difference between prices in two cities,
assuming (a) σ 12 = σ 22 (b) σ 12 =/ σ 22
Solution: 1.
The null and alternative hypotheses:
H0:
μ1 − μ 2 = 0
H1:
μ1 − μ 2 ≠ 0
The test is a two-tailed test
2.
Level of significance:α =1% or 0.01
453
3.
Test statistic:
t; since the population standard deviations, σ1 and σ2, are
unknown, but the sample standard deviations, S1 and S2, are known and sample sizes
are small.
4.
Critical region:
5.
Computations:
t0.005 < t < -t0.005
X 1 =22
X 2 =17
n1 = 9
n2 =11
S1 = 5
S2 = 6
(a) σ 12 = σ 22
t =
t =
t =
t =
(X
− X
1
2
) − (μ
1
− μ 2 )0
S p2 ⎛⎜ 1 + 1 ⎞⎟
n2 ⎠
⎝ n1
(X
1
− X
2
) − (μ
1
⎛ (n 1 − 1 )S 12 + (n 2 − 1 )S 22
⎜⎜
n1 + n 2 − 2
⎝
(
⎛ 8 x 25 + 10 x 36 ⎞ 1
⎜
⎟ 9 + 1 11
18
⎝
⎠
5
2 . 51
The degrees of freedom for this t are n1 + n2 –2 i.e. 9 +11-2 =18
t0.005 =2.88
(b) σ 12 =/ σ 22
t=
⎞⎛ 1
⎟⎟ ⎜
+ 1 ⎞⎟
n1
n2 ⎠
⎝
⎠
22 − 17
t = 1 . 99
For 18 df ,
− μ 2 )0
(X
1
)
− X 2 − (μ 1 − μ 2 )0
2
⎛ S 12
⎞
+ S2
⎜
n1
n 2 ⎟⎠
⎝
454
)
t =
t =
22 − 17 − 0
25 + 36
9
11
(
)
5
2 . 46
t = 2.03
The degrees of freedom for this t are given by
2
⎛ S 12
S2 ⎞
+ 2
⎜
n1
n 2 ⎟⎠
⎝
df =
⎛ ⎛ S 2 ⎞ 2 ⎞⎛ ⎛ S 2 ⎞ 2
⎜⎜ 1
⎟⎜ 2
n1 ⎟⎠ ⎟ ⎜ ⎜⎝
n 2 ⎟⎠
⎜⎝
⎜ n − 1 ⎟⎜ n − 1
1
2
⎜
⎟⎜
⎜
⎟⎜
⎝
⎠⎝
⎛ 25
⎞
+ 36
⎜
⎟
9
11
⎝
⎠
df =
2
⎞ ⎛ 36
⎛ 25
⎜
9 ⎟⎜
11
⎟⎜
⎜
10
8
⎟⎜
⎜
⎠⎝
⎝
(
)
(
⎞
⎟
⎟
⎟
⎟
⎟
⎠
2
)
2
⎞
⎟
⎟
⎟
⎠
= 18
Against which, t0.005 =2.88
6.
Conclusion:
(a) We cannot reject the null hypothesis at α = 0.01,
when σ 12 = σ 22 since t=1.99 < t0.005 =2.88.
(b) We cannot reject the null hypothesis at α = 0.01, when σ 12 =/ σ 22 since t=2.03 <
t0.005 =2.88.
Example 14-11
A sample survey of tax-payers belonging to business class and professional class yielded the
following results:
Professional Class
Sample size
n1 = 400
n2 = 420
Defaulters in tax payment
x1 = 80
x2 = 65
455
Given these sample data, test the hypothesis at α = 5% that
(a)
the defaulters rate is the same for the two classes of tax-payers
(b)
the defaulters rate in the case of business class is more than that in the case of
professional class by 0.07.
Solution: (a) 1.
The null and alternative hypotheses:
H0:
p1 − p 2 = 0
H1:
p1 − p 2 ≠ 0
The test is a two-tailed test
2.
Level of significance:α =1% or 0.01
3.
Test statistic:
Z; since the sample sizes are large enough.
4.
Critical region:
Z0.005 < Z < -Z0.005 Where Z0.005 = 2.58
5.
Computations:
p1 =
x1
80
=
= 0 . 20
n1
400
p =
Z =
Z =
p2 =
x2
65
=
= 0 . 15
n2
420
x1 + x 2
80 + 65
=
= 0 . 177
n1 + n 2
400 + 420
(
(p
1
)
− p2 − 0
)
p 1 − p ⎛⎜ 1 + 1 ⎞⎟
n2 ⎠
⎝ n1
0 . 20 − 0 . 15
(0 .177 x 0 .823 ) 1 400 + 1 420
(
)
Z = 1.87
6.
We cannot reject the null hypothesis at α = 0.05 since Z
Conclusion:
=1.87 < Z0.005 =2.58
(b)
1.
The null and alternative hypotheses:
H0:
p1 − p 2 = 0.07
456
H1:
p1 − p 2 ≠ 0.07
The test is a two-tailed test
2.
Level of significance:α =1% or 0.01
3.
Test statistic:
Z; since the sample sizes are large enough.
4.
Critical region:
Z0.005 < Z < -Z0.005 Where Z0.005 = 2.58
5.
Computations:
p1 =
x1
80
=
= 0 . 20
n1
400
Z =
p2 =
(p
)
p (1 − p ) p (1 − p )
+
1
− p 2 − ( p1 − p 2 )0
1
1
n1
Z =
x2
65
=
= 0 . 15
n2
420
2
2
n2
(0 . 20
− 0 . 15 ) − (0 . 07 )
0 . 20 x 0 . 80
0 . 15 x 0 . 85
+
400
420
Z = −0.76
6.
We cannot reject the null hypothesis at α = 0.05 since Z
Conclusion:
= -0.76 > -Z0.01 = -2.58
Example 14-12
Use the data of Problem 14-10: n1 = 9, n2 =11 and S1 = 5 , S 2 = 6 to test the assumption of
equal population variances.
Solution: 1.
The null and alternative hypotheses:
H0:
σ 12 = σ 22
H1:
σ 12 ≠ σ 22
The test is a two-tailed test
2.
Level of significance:α = 5% or 0.05
3.
Test statistic:
F
457
4.
Critical region:
F(n1 −1,n2 −1 ) > Fα / 2 (n1 −1, n2 −1 )
and F( n1 −1, n2 −1 ) < F1−α / 2 ( n1 −1, n2 −1 ) i.e. F(8,10) > F0.025(8,10) = 3.85 and
F(8,10) < F0.095(8,10) = 0.23
4. Computations:
S1 = 5
S2 = 6
n1 = 9
n2 =11
F(n1 −1, n2 −1 ) =
S 12
S 22
F(8 ,10 ) =
25
36
= 0.694
6.
Conclusion:
We cannot reject the null hypothesis at α = 0.05 since F(8,10) <
F0.025(8,10) =3.85 and F(8,10) > F0.095(8,10) =0.23. So the sample evidence supports the
view that the two populations do not have different variances.
14.11 SELF-ASSESSMENT QUESTIONS
1.
What is a Hypothesis? Explain how Hypothesis Testing is useful to management?
2.
What are Null and Alternative hypotheses? How you will set up null and alternative
hypotheses under following conditions:
(a)
A pharmaceutical company claims that four out of five doctors prescribe the
pain medicine it produces. You wish to test this claim.
(b)
A manufacturer of golf balls claims that the variance of the weights of the
company's golf balls is controlled within 0.0028 oz2. You wish to test this
claim.
(c)
A medicine is effective only if the concentration of a certain chemical in it is
at least 200 parts per million (ppm). At the same time the medicine would
produce an undesirable side effect if the concentration of the same chemical
458
exceeds 200 parts per million (ppm). You wish to test the concentration of the
chemical in the medicine.
3.
What are Type I and Type II Errors in hypothesis testing? Explain the relationship
between the two types of errors.
4.
What is a Test Statistic? Why do we have to know the distribution of the test statistic?
What are the commonly used test statistics in hypotheses testing?
5.
Distinguish between a One-tailed and Two-tailed test, give a diagram and an example
in each case.
6.
What is the p-value of a test? How it is calculated? Find the p-value of a (a) lefttailed, (b) right-tailed, and (c) two-tailed test if
(i)
In the test, the test statistic Z = -1.86. In which of these three cases will H0 be
rejected at an α of 5%?
(ii)
In the test, the test statistic Z = 1.75. In which of these three cases will H0 be
rejected at an α of 5%?
7.
What do you mean by Level of Significance of a test? “Level of significance should
be specified after due consideration to the costs associated with Type I and Type II
errors”. Explain this statement.
8.
What do you mean by Critical Regain and Acceptance Region of a test?
9.
What is the Power of a hypothesis test? Why is it important? How is the power of a
hypothesis test related to
10.
(a)
the significance level?
(b)
the sample size?
(c)
the actual value of the parameter?
Consider the use of metal detectors in airports to test people for concealed weapons.
In essence, this is a form of hypothesis testing.
(a)
What are the null and alternative hypotheses?
459
(b)
What are type I and type II errors in this case?
(c)
Which type of error is more costly?
(d)
Based on your answer to part (c), what value of α would you
recommend
for this test?
(e)
If the sensitivity of the metal detector is increased, how would the
probabilities of type I and type II errors be affected?
(f)
If α is to be increased, should the sensitivity of the metal detector be increased
or decreased?
11.
When planning a hypothesis test, what should be done if the probabilities of both type
I and type II errors are to be small?
12.
“Not – rejecting a Null hypothesis” is a more precise term rather than “Accepting a
Null hypothesis”. Do you agree with this statement? Explain.
13.
What step are involved in statistical testing of a hypothesis?
14.
A company is engaged in the packaging of a superior quality tea in jars of 500gm
each. The company is of the view that as long as the jars contains 500gm of tea, the
process is under control. The standard deviation of the process is 50gm. A sample of
225 jars is taken at random and the sample average is found to be 510 gm. Has the
process gone out of control?
15.
A sample of size 400 was drawn and the sample mean found to be 99. Test, at 5%
level of significance, whether this sample could have come form normal population
with mean 100 and variance 64.
16.
A manufacturer of a new motorcycle claims for it an average mileage of 60 km/liter
under city conditions. However, the average mileage in 16 trials is found to be 57 km,
with a standard deviation of 2 km. Is the manufacturer’s claim justified?
460
17.
In a big city, 450 men out of a sample of 850 men were found to be smokers. Does
this information, at 5% level of significance, supports the view that the majority of
men in this city are smokers?
18.
A stock-broker claims that she can predict with 85% accuracy whether a stock’s
market value will rise or fall during the coming month. Test the stock-broker’s claim
at 5% level of significance if, as a test, she predict the outcome of 6 stocks and is
correct in 5 of the predictions.
19.
A company engaged in manufacturing of radio tubes, finds that the life of its tubes has
a variance of 0.7 years. As a result of some qualitative improvement brought about in
the product, the company claims that the variance of the life of its tubes has reduced.
If the sample variance, S2, on observation of 9 tubes is observed 0.55 years at test the
claim of the company (a) 5% level of significance (b) 1% level of significance.
20.
Seven persons were appointed in officer cadre in an organisation. Their performance
was evaluated by giving a test and the marks were recorded out of 100. They were
given two-month training and another test was held and marks were recorded out of
100.
Officer:
a
b
c
d
e
f
g
Score Before Training:
80
76
92
60
70
56
74
Score After Training:
84
70
96
80
70
52
84
Can it be concluded that the training has benefited the employees? Use 5% level of
significance.
21.
The makers of Philips bulb want to demonstrate that their bulb lasts on an average of
at least 100 hours longer than Philips’ main competitor, Surya. Two independent
random samples of 100 bulbs of each kind are selected. The sample average lives for
Philips and Surya bulbs are found to be X 1 = 1232 hours and X 2 = 1016 hours
461
respectively. Assume σ 1 = 84 hours and σ 2 = 67 hours. Is there evidence to
substantiate Philips’ claim that its bulbs last, on an average, at least 180 hours longer
than Surya bulb of the same size?
22.
Consider the following data:
Sample A
Sample B
Sample Mean
100
105
Standard Deviation
16
24
Sample Size
800
1600
Test, at 5% level of significance, the difference between means of two populations
from which samples are taken.
23.
The following information relate to the wages (in Rs) of mill workers in two cities A
and B.
City A
City B
Mean wage
40
34
Standard deviation
5
6
The observations related to wages are for 8 workers in city A and for 10 workers in
city B. Test at 0.01 level whether there is any significant difference between wages in
two cities, assuming (a) σ 12 = σ 22 (b) σ 12 =/ σ 22
24.
Test market result of two advertisements A and B, yielded the following results:
Who tried the Product
A
B
n1 = 200
n2 = 220
x1 = 40
x2 = 35
Given the data, test the hypotheses at α = 5% that
462
Effectiveness of the advertisements are measured as proportion of viewers who tried
the product.
25.
Use the data of Problem 22: n1 = 8, n2 =10 and S1 = 5 , S 2 = 6 to test the assumption
of equal population variances.
1.
Statistics (Theory & Practice) by Dr. B.N. Gupta. Sahitya Bhawan Publishers
and Distributors (P) Ltd., Agra.
2.
Statistics for Management by G.C. Beri. Tata McGraw Hills Publishing
Company Ltd., New Delhi.
3.
Business Statistics by Amir D. Aczel and J. Sounderpandian. Tata McGraw Hill
Publishing Company Ltd., New Delhi.
4.
Statistics for Business and Economics by R.P. Hooda. MacMillan India Ltd.,
New Delhi.
5.
Business Statistics by S.P. Gupta and M.P. Gupta. Sultan Chand and Sons., New
Delhi.
6.
Statistical Method by S.P. Gupta. Sultan Chand and Sons., New Delhi.
7.
Statistics for Management by Richard I. Levin and David S. Rubin. Prentice
Hall of India Pvt. Ltd., New Delhi.
8.
Statistics for Business and Economics by Kohlar Heinz. Harper Collins., New
York.
463
COURSE:
Author: Dr. B.S. Bodla
Course code: MC-106
Vetter: Karam Pal
Lesson: 15
NON-PARAMETRIC TESTS
Objective:
This lesson would enable you to differentiate between parametric and
nonparametric tests; understand the relevance of non-parametric test in data
analysis; understand the procedure involved in carrying out non-parametric
tests; and design and conduct some selected non-parametric tests.
Structure
15.1. Introduction
15.2.
15.3.
15.4.
15.5.
15.6.
15.7.
15.8.
15.9.
15.10.
15.11.
15.12.
Sign tests
The two-sample and K-sample Median Tests
Wilcoxon matched-pairs test (or Signed Rank Test)
The Mann-Whitney U Test
The Kruskal-Wallis Test
The spearman's rank correlation test
Tests of Randomness: Runs Above and Below the Median
Kolmogorov-Smirnov One-sample Test
Summary
Questions
15.1. Introduction
In contrast to parametric tests, non-parametric tests do not require any assumptions about the
parameters or about the nature of population. It is because of this that these methods are
sometimes referred to as the distribution free methods. Most of these methods, however, are
based upon the weaker assumptions that observations are independent and that the variable
under study is continuous with approximately symmetrical distribution. In addition to this,
these methods do not require measurements as strong as that required by parametric methods.
Most of the non-parametric tests are applicable to data measured in an ordinal or nominal
scale. As opposed to this, the parametric tests are based on data measured at least in an
interval scale. The measurements obtained on interval and ratio scale are also known as high
level measurements.
Level of measurement
1.
Nominal scale: This scale uses numbers or other symbols to identify the groups or
classes to which various objects belong. These numbers or symbols constitute a
nominal or classifying scale. For example, classification of individuals on the basis of
sex (male, female) or on the basis of level of education (matric, senior secondary,
graduate, post graduate), etc. This scale is the weakest of all the measurements.
464
2.
Ordinal scale: This scale uses numbers to represent some kind of ordering or ranking
of objects. However, the differences of numbers, used for ranking, don’t have any
meaning. For example, the top 4 students of class can be ranked as 1, 2, 3, 4,
according to their marks in an examination.
3.
Interval scale: This scale also uses numbers such that these can be ordered and their
differences have a meaningful interpretation.
4.
Ratio scale: A scale possessing all the properties of an interval scale along with a true
zero point is called a ratio scale. It may be pointed out that a zero point in an interval
scale is arbitrary. For example, freezing point of water is defined at 0° Celsius or 32°
Fahrenheit, implying thereby that the zero on either scale is arbitrary and doesn’t
represent total absence of heat. In contrast to this, the measurement of distance, say in
metres, is done on a ratio scale. The term ratio is used here because ratio comparisons
are meaningful. For example, 100 kms of distance is four times larger than a distance
of 25 kms while 100°F may not mean that it is twice as hot as 50°F.
It should be noted here that a test that can be performed on high level measurements can
always be performed on ordinal or nominal measurements but not vice-versa. However, if
along with the high level measurements the conditions of a parametric test are also met, the
parametric test should invariably be used because this test is most powerful in the given
circumstances.
From the above, we conclude that a non-parametric test should be used when either the
conditions about the parent population are not met or the level of measurements is inadequate
for a parametric test.
The non-parametric tests have gained popularity in recent years because of their usefulness in
certain circumstances. Some advantages of non-parametric tests are mentioned below:
1.
Non-parametric tests require less restrictive assumptions vis-à-vis a comparable
parametric test.
2.
These tests often require very few arithmetic computations.
3.
There is no alternative to using a non-parametric test if the data are available in
ordinal or nominal scale.
4.
None of the parametric tests can handle data made up of samples from several
populations without making unrealistic assumptions. However, there are suitable nonparametric tests available to handle such data.
1.
2.
It is often said that non-parametric tests are less efficient than the parametric tests
because they tend to ignore a greater part of the information contained in the sample.
In spite of this, it is argued that although the non-parametric tests are less efficient, a
researcher using them has more confidence in using his methodology than he does if
he must adhere to the unsubstantiable assumptions inherent in parametric tests.
The non-parametric tests and their accompanying tables of significant values are
widely scattered in various publications. As a result of this, the choice of most
suitable method, in a given situation, may become a difficult task.
15.2. Sign tests
One of the easiest non-parametric tests is the sign test. The test is known as the sign test as it
is based on the direction of the plus or minus signs of observations in a sample instead of
their numerical values. There are two types of sign tests: (a) One-sample sign test, and (b)
Two-sample sign test.
465
One-sample sign test
The one-sample sign test is a very simple non-parametric test applicable on the assumption
that we are dealing with a population having a continuous symmetrical distribution. As such,
the probability of getting a value less than the mean is 0.5. Likewise, the probability of
getting a value greater than the mean is also 0.5. To test the null hypothesis µ = µ0 against an
appropriate alternative, each sample value greater than µ0 is replaced by plus (+) sign and
each sample value less than µ0 with a minus (-) sign. Having done this, we can test the null
hypothesis that the probabilities of getting both plus and minus signs are 0.5. It may be noted
that if a sample value happens to be equal to µ0, it is simply discarded.
To perform the actual test, we use either of the two methods. When the sample is small, the
test is performed by computing the binomial probabilities or by referring to the binomial
probabilities table. When the sample is large, the normal distribution is used as an
approximation of the binomial distribution. Let us take an example to show how the onesample sign test is applied.
Example 1: We are required to test the hypothesis that the mean value µ of a continuous
distribution is 20 against the alternative hypothesis µ ≠ 20. Fifteen observations were taken
and the following results were obtained:
18, 19, 25, 21, 16, 15, 19, 22, 24, 21, 18, 17, 15, 26 and 24.
We may use α = 0.05 level of significance.
Solution: Replacing each value greater than 20 with a plus (+) sign and each value less than
20 with a minus (-) sign, we get
--++---+++---++
Now, the question before us is whether 7 plus signs observed in 15 trials support the null
hypothesis p = 0.5 or the alternative hypothesis p ≠ 0.5. Using the binomial probability tables
or binomial probabilities, we find that the probability of 7 or more successes is 0.196 + 0.196
+ 0.153 + 0.092 + 0.042 + 0.014 + 0.003 = 0.696* and p = 0.5 and since this value is greater
than α/2 = 0.025, we find that the null hypothesis will have to be accepted. We can also use
normal approximation to the binomial distribution when np ≥ 5. As here p = ½, the condition
for the normal approximation to the binomial distribution is satisfied as n > 10. As such, we
can use the Z statistic for which the following formula is to be used.
X − np
X − (np )
Z=
=
npq
n
4
=
7 − (15 / 2)
14 − 15
−0.5
2
=
=
= -0.26
1.9365
1.9365
15
4
Since calculated Z = -0.26 lies between Z = - 1.96 and Z = 1.96 (the critical value of Z at 0.05
level of significance), the null hypothesis is accepted.
The two-sample sign test
The sign test can be applied to problems that deal with paired data. In such problems, each
pair can be replaced with a plus sign if the first value is greater than the second or a minus
sign if the first value is smaller than the second. In case the two values in the pair turn out to
be equal, these are discarded. These are essentially two kinds of situations: (a) the data are
actually given as pairs and (b) the data comprise two independent samples that are randomly
paired.
466
Example 2: Suppose we have the following table indicating the ratings assigned to
two brands of cold drink X and Y by 12 consumers. Each respondent was asked to
taste the two brands of cold drink and then rate them.
Table 15.1. Ratings of brands X and Y cold drinks
Brand X
26
30
44
23
18
50
34
16
25
49
37
20
Brand Y
22
27
39
7
11
56
30
14
18
51
33
16
Sign
+
+
+
+
+
-
+
+
+
-
+
+
We have to apply the two-sample sign test. H0 being both brands enjoy equal preference.
Solution: Row three of Table 15.1 shows + and – signs. When X’s rating is higher than that
of Y, then the third row shows the ‘+’ sign. As against this, when X’s rating is lower than that
of Y, then it shows the ‘-‘ sign. The table shows 10 plus signs and 2 minus signs. Now, we
have to examine whether ’10 successes in 12 trials’ supports the null hypothesis p = ½ or the
alternative hypothesis p > ½. The null hypothesis implies that both the brands enjoy equal
preferences and none is better than the other. The alternative hypothesis is that the brand X is
better than brand Y. Referring to the binomial probabilities table, we find that for n = 12 and
p = ½ the probability of ’10 or more successes’ is 0.016 + 0.003 = 0.019. It follows that the
null hypothesis can be rejected at α = 0.05 level of significance. We can, therefore, conclude
that brand X is a preferred brand as compared to brand Y.
Example 3: To illustrate the second case, which relates to two independent samples, let us
consider the following data pertaining to the downtimes (periods in which computers were
inoperative on account of failures, in minutes of two different computers. We have to apply
the two-sample sign test.
Computer
A
Computer
B
58
60
42
62
65
59
60
52
50
75
59
52
57
30
46
66
40
78
55
52
58
44
32
48
50
41
45
40
43
43
70
60
80
45
36
56
40
70
50
53
50
30
42
45
Solution: These data are shown in Table 15.2 along with + or – sign as may be applicable in
case of each pair of values. A plus sign is assigned when the downtime for computer A is
greater than that for computer B and a minus sign is given when the downtime for computer
B is greater than that for computer A.
Table 15.2: Downtime of computers A and B (Minutes)
Computer A 58 60 42 62 65 59 60 52 50 75 59
Computer B
32
48
50
41
45
40
43
43
70
60
80
Sign
+
+
-
+
+
+
+
+
-
+
-
Computer A
52
57
30
46
66
40
78
55
52
58
44
Computer B
45
36
56
40
70
50
53
50
30
42
45
Sign
+
+
-
+
_
_
+
+
+
+
-
It will be seen that there are 15 plus signs and 7 minus signs. Thus, we have to ascertain
whether ‘15 successes in 22 trials’ support the null hypothesis p = ½. The null hypothesis
implies that the true average downtime is the same for both the computers A and B. The
467
alternative hypothesis is p ≠ ½. The null hypothesis implies that the true average downtime is
the same for both the computers A and B. The alternative hypothesis is p = ½.
Let us use in this case the normal approximation of the binomial distribution. This can be
done since np and n (1 – p) are both equal to 11 in this example. Substituting n = 22 and p =
½ into the formulas for the mean and the standard deviation of the binomial distribution, we
get µ = np = 22 (½) = 11 and
σ = np (1 - p) = 22.½.½ = 2.345
Hence, Z = (X – µ)/σ = (15 – 11)/2.345 = 1.71
Since this value of 1.71 falls between – Z0.025 = - 1.96 and Z0.025 = 1.96, we find that the null
hypothesis cannot be rejected. This means that the downtime in the two computers is the
same.
This seems to be surprising as we find that there are substantial differences. The two sample
means, for example, are 55.5 for A and 48.6 for B. This example illustrates the point that at
times the sign test can be quite a waste of information. It may also be noted that had the
continuity correction been used, we would have obtained:
Z = 3.5/2.345 = 1.49
This would not have changed our earlier conclusion.
15.3. Median test for two independent samples
In order to perform this test, let us use our previous example, which pertains to the
downtimes of the two computers. The median of the combined data is 52, which can easily be
checked. There are 5 values below 52 and 15 values above it, in case of computer A. As
regards computer B, the corresponding figures are 16 and 6. All this information is
summarised in Table 15.3, which also indicates the totals of the rows and columns.
Table 15.3. Classification of downtime for computers A and B
Below median
Above median
Total
Computer A
5
15
20
Computer B
16
6
22
Total
21
21
42
Our null hypothesis H0 is that there is no difference in the median downtime for the two
computers. The alternative hypothesis H1 is that there is difference in the downtime of the
two computers.
We now calculate the expected frequencies by the formula (Rowi × Columni)/Grand total.
Thus, Table 15.4 shows both the observed and the expected frequencies. Of course, we could
have obtained these results by arguing that half the values in each sample can be expected to
fall above the median and the other half below it.
Table 15.4. Calculation of chi-square
Observed
Expected
O–E
(O – E)2
(O – E)2/E
freque
freque
5ncies
10
ncies
-5
25
2.50
15 (O)
10 (E)
5
25
2.50
16
11
5
25
2.27
6
11
-5
25
2.27
Total
9.54
468
(O i − E i )2
= 9.54
χ =∑
Ei
2
The critical value of χ2 at 0.05 level of significance for (2 – 1) (2 – 1) = 1 degree of freedom
is 3.841 (χ2-Table). Since the calculated value of χ2 exceeds the critical value, the null
hypothesis has to be rejected. In other words, there is no evidence to suggest that the
downtime is the same in case of the two computers.
It may be recalled that in the previous example having the same data, the null hypothesis
could not be rejected. In contrast, we find here that the two-sample median test has led to the
rejection of the null hypothesis. This may be construed as evidence that the median test is not
quite as wasteful of the information as the sign test. However, in general, it is very difficult to
make a meaningful comparison of the merits of two or more non-parametric tests, which can
be used for the same purpose.
15.3.1. The K-sample median test
The median test can easily be generalised so that it can be applied to K-samples. In
accordance with the earlier procedure, first find the median of the combined data. We then
determine how many of the values in each sample fall above or below the median. Finally,
we analyse the resulting contingency table by the method of chi-square. Let us take an
example.
Example 4: Suppose that we are given the following data relating to marks obtained by
students in Statistics in the three different sections of a MBA class in G.J.U. Hisar. The
maximum marks were 100.
Section A
46
60
58
80
66
39
56
61
81
70
Section B
Section C
75
48
43
64
57
59
87
50
73
62
60
55
82
70
46
63
88
69
61
43
76
54
58
65
73
52
74
67
37
80
72
92
19
52
70
40
83
76
68
21
90
74
49
70
65
58
Test whether the differences among the three sample means are significant.
Solution: In case of such problems, analysis of variance is ordinarily performed. However,
here we find that the data for Section C have much more variability as compared to the data
for the other two sections. In view of this, it would be wrong to assume that the three
population standard deviations are the same. This means that the method of one-way analysis
of variance cannot be used.
In order to perform a median test, we should first determine the median of the combined data.
This comes out to 63.5, as can easily be checked. Then we count how many of the marks in
each sample fall below or above the median. Thus, the results obtained are shown in Table
15.5.
Table 15.5. Worksheet for calculating chi-square
Below median
Above median
Section A
12
8
Section B
9
7
Section C
7
13
469
Since the corresponding expected frequencies for Section A are 10 and 10, for Section B are
8 and 8, and for Section C 10 and 10, we can obtain the value of chi-square. These
calculations are shown below:
(12-10)2 (8-10)2 (9-8)2 (7-8)2 (7-10)2 (13-10)2
2
χ = 10 + 10 + 8 + 8 + 10 + 10
= 0.4 + 0.4 + 0.125 + 0.125 + 0.9 + 0.9 = 2.85
Now, we have to compare this value with the critical value of χ2 at 5 per cent level of
significance. This value is 5.991 for 2 (K – 1 = 3 – 1) degrees of freedom (Chi-square Table).
As the calculated value of χ2 is less than the critical value, the null hypothesis cannot be
rejected. In other words, we cannot conclude that there is a difference in the true average
(median) marks obtained by the students in Statistics test from the three sections.
15.4. Wilcoxon
Wilcoxon matched-pairs test is an important non-parametric test, which can be used in
various situations in the context of two related samples such as a study where husband and
wife are matched or when the output of two similar machines are compared. In such cases we
can determine both direction and magnitude of difference between matched values, using
Wilcoxon matched-pairs test.
The procedure involved in using this test is simple. To begin with, the difference (d) between
each pair of values is obtained. These differences are assigned ranks from the smallest to the
largest, ignoring signs. The actual signs of differences are then put to corresponding ranks
and the test statistic T is calculated, which happens to be the smaller of the two sums, namely,
the sum of the negative ranks and the sum of the positive ranks.
There may arise two types of situations while using this test. One situation may arise when
the two values of some matched-pair(s) is/are equal as a result the difference (d) between the
values is zero. In such a case, we do not consider the pair(s) in the calculations. The other
situation may arise when we get the same difference (d) in two or more pairs. In such a case,
ranks are assigned to such pairs by averaging their rank positions. For instance, if two pairs
have rank score of 8, then each pair is assigned 8.5 rank [(8 + 9)/2 = 8.5] and the next largest
pair is assigned the rank 10.
After omitting the number of tied pairs, if the given number or matched pairs is equal to or
less than 25, then the table of critical value T is used for testing the null hypothesis. When the
calculated value of T is equal to or smaller than the table (i.e. critical) value at a desired level
of significance, then the null hypothesis is rejected. In case the number exceeds 25, the
sampling distribution of T is taken as approximately normal with mean µT = n (n + 1)/µ and
standard deviation
σT = n (n + 1) (2n + 1)/24
where n is taken as the number of given matched pairs- number of tied pairs omitted, if any.
In such a situation, the test Z statistic is worked out as follows:
Z = (T – µr)/σr
Let us now take an example to illustrate the application of Wilcoxon matched-pairs test.
Example 5: The management of the Punjab National Bank wants to test the effectiveness of
an advertising company that is intending to enhance the awareness of the bank’s service
measure the awareness of services offered. After the advertising campaign, the bank
administered the same questionnaire to the same group of people. Both the before and after
advertising campaign scores are given in the following table.
Consumer awareness of bank services offered
Consu
1
2
3
4
5
6
7
8
9
10
m
er
470
Before
82
81
89
74
68
80
77
66
77
75
87
84
84
76
78
81
79
81
81
83
a
After d
a
c
Using Wilcoxon
matched-pairs test, test the hypothesis that there is no difference in
d
a
awareness of services offered after the advertising campaign.
c
m
Solution:
Table 15.6.
a
p Application of Wilcoxon matched-pairs test
Rank of Rank (-) Rank (+)
Diff. di
m
ai
di
er
ca
si
si
p
g
1
87 m
82 ca
5
6.5
6.5 gn
gn
ai
n
2
84 pn
81 m
3
4
4
g
3
84 .
89 pn
-5
6.5
-6.5
n
4
76
74 .
2
2.5
2.5
5
78
68
10
9
9
6
81
80
1
1
1
7
79
77
2
2.5
2.5
8
81
66
15
10
10
9
81
77
4
5
5
10
83
75
8
8
8
Total
-6.5
+48.5
Null hypothesis H0: There is no difference in the awareness of bank services after the ad
campaign. Alternative hypothesis H1: There is a difference in the awareness of bank services
Computed ‘T’ value is 6.5. The critical value of T for n = 10 at 5 per cent level of
significance is 8 (Area Table). Since the computed T value is less than the critical T value,
the null hypothesis is rejected. We can conclude that after the ad campaign there is difference
in the consumer awareness of the bank’s services needs some explanation. Had there been no
difference in the awareness before and after the ad campaigns, the sum of positive and
negative ranks would have been almost equal. However, if the difference between the two
series being compared is larger, then the value of T will tend to be smaller as it is defined as
smaller of ranks. This is the case we find in this problem. It may be noted that with this test
the calculated value of T must be smaller than the critical value in order to reject the null
hypothesis.
15.5. The Mann-Whitney U Test
One of the most common and best known distribution-free tests is the Mann-Whitney test for
two independent samples. The logical basis of this test is particularly easy to understand.
Suppose we have two independent treatment groups, with n1 observations in Group 1 and n2
observations in Group 2. Now, we assume that the population from which Group 1 scores
have been sampled contained generally lower values than the population from which Group 2
471
scores were drawn. If we were to rank these scores disregarding the group to which they
belong then the lower ranks would generally fall to Group 1 scores and the higher ranks
would generally fall to Group 2 scores. Proceeding one step further, if we were to add
together the ranks assigned to each group, the sum of the ranks in Group 1 would be expected
to be considerably smaller than the sum of the ranks in Group 2. This would result in the
rejection of the null hypothesis.
Let us now take another situation where the null hypothesis is true and the scores for the two
groups are sampled from identical populations. If we were to rank all N scores regardless of
the group, we would expect a mix of low and high ranks in each group. Thus, the sum of the
ranks assigned to Group 1 would be broadly equal to the sum of the ranks assigned to Group
2.
The Mann-Whitney test is based on the logic just described, using the sum of the ranks in one
of the groups as the test statistic. In case that sum turns out to be too small as compared to the
other sum, the null hypothesis is rejected. The common practice is to take the sum of the
ranks assigned to the smaller group, or if n1 = n2, the smaller of the two sums as the test
statistic. This value is then compared with the critical value that can be obtained from the
table of the Mann-Whitney statistic (Ws) to test the null hypothesis.
Let us take an example to illustrate the application of this test.
Example 6: The following data indicate the lifetime (in hours) of samples of two kinds of
light bulbs in continuous use:
Brand A
603
625
641
622
585
593
660
600
633
580
615
648
Brand B
620
640
646
620
652
639
590
646
631
669
610
619
We are required to use the Mann-Whitney test to compare the lifetimes of brands A and B
light bulbs.
Solution: The first step for performing the Mann-Whitney test is to rank the given data
jointly (as if they were one sample) in an increasing or decreasing order of magnitude. For
our data, we thus obtain the following array where we use the letters A and B to denote
whether the light bulb was from brand A or brand B.
Table 15.7. Ranking of light bulbs of brands A and B
Sample score
Group
Rank
Sample
Group
Rank
sco
580
A
1
625re
A
13
585
A
2
631
B
14
590
B
3
633
A
15
593
A
4
639
B
16
600
A
5
640
B
17
603
A
6
641
A
18
610
B
7
646
B
19.5
615
A
8
646
B
19.5
619
B
9
648
A
21
620
B
10.5
652
B
22
620
B
10.5
660
A
23
472
622
A
12
669
B
24
As both the samples come from identical populations, it is reasonable to assume that the
means of the ranks assigned to the values of the two samples are more or less the same. As
such, our null hypothesis is:
H0: Means of ranks assigned to the values in the two groups are the same.
H1: Means are not the same.
However, instead of using the means of the ranks, we shall use rank sums for which the
following formula will be used.
U = n1n2 + [n1(n1 + 1)]/2 – R1
Where n1 and n2 are the sample sizes of Group 1 and Group 2, respectively, and R1 is the sum
of the ranks assigned to the values of the first sample. In our example, we have n1 = 12, n2 =
12 and R1 = 1 + 2 + 4 + 5 + 6 + 8 + 12 + 13 + 15 + 18 + 21 + 23 = 128. Substituting these
values in the above formula,
U = (12) (12) + [12 (12 + 1)]/2 – 128
= 144 + 78 – 128
= 94
From Appendix Table 9 for n1 and n2, each equal to 12, and for 0.05 level of significance is
37. Since the critical value is smaller than the calculated value of 94, we accept the null
hypothesis and conclude that there is no difference in the average lifetimes of the two brands
of light bulbs.
The test statistic we have just applied is suitable when n1 and n2 are less than or equal to 25.
For larger values of n1 and/or n2, we can make use of the fact that the distribution of Ws
approaches a normal distribution as sample sizes increase. We can then use the Z test to test
the hypothesis.
The normal approximation
Although our observations are limited, we may apply the normal approximation to this
problem. For this, we have to use the Z statistic.
1.
Mean = µu = [(N1N2/2] = [(12 × 12)/2] = 72
2.
Standard error =
n1n2 (n1 + n2 + 1)
12
12 × 12 (12 + 12 + 1)
12
= 300 = 17.3
3.
(Statistic – Mean)/Standard deviation
= (94 – 72)/17.3 = 1.27
The critical value of Z at 0.05 level of significance is 1.64. Since the calculated value of Z =
1.27 is smaller than 1.64, the null hypothesis is accepted. This shows that there is no
difference in average lifetimes of brands A and B bulbs. The Z test is more dependable as
compared to the earlier one. It may be noted that Mann-Whitney test required fewer
assumptions than the corresponding standard test. In fact, the only assumption required is that
the populations from which samples have been drawn are continuous. In actual practice, even
when this assumption turns out to be wrong, this is not regarded serious.
=
15.6. The Kruskal-Wallis test
This test is used to determine whether k independent samples can be regarded to have been
obtained from identical populations with respect to their means. The Kruskal-Wallis Test is
the non-parametric counter part of the one-way analysis of variance. The assumption of the
473
F-test, used in analysis of variance, was that each of the k populations should be normal with
equal variance. In contrast to this, the Kruskal-Wallis test only assumes that the k populations
are continuous and have the same pattern (symmetrical or skewed) of distribution. The null
and the alternative hypotheses of the Kruskal-Wallis test are:
H0: m1 = m2 = … = mk (i.e., means of the k populations are equal)
Ha: Not all mi’s are equal.
The Test Statistic: The computation of the test statistic follows a procedure that is very
similar to the Mann-Whitney Wilcoxon test.
(i) Rank all the n1 + n2 + … + nk = n observations, arrayed in ascending order.
(ii) Find R1, R2, … Rk, where Ri is the sum of ranks of the ith sample.
The test statistic, denoted by H, is given by
2
2
Rk2⎞
12 ⎛R1 R2
H = n(n+1) ⎜ n + n + … + n ⎟ - 3 (n + 1).
⎝ 1
2
k⎠
It can be shown that the distribution of H is c2 with k – 1 d.f., when size of each sample is at
least 5. Thus, if H > χ 2k −1 , H0 is rejected.
Example 7: To compare the effectiveness of three types of weight-reducing diets, a
homogeneous groups of 22 women was divided into three sub-groups and each sub-group
followed one of these diet plans for a period of two months. The weight reductions, in kgs,
were noted as given below:
I
II
III
Diet Plans
4.3
5.3
1.4
3.2
7.4
2.1
2.7
8.3
2.7
6.2
5.5
3.1
5.0
6.7
1.5
3.9
7.2
0.7
8.5
4.3
3.5
0.3
Use the Kruskal-Wallis test to test the hypothesis that the effectiveness of the three weight
reducing diet plans is same at 5% level of significance.
Solution:
It is given that n1 = 6, n2 = 7 and n3 = 9.
The total number of observations is 6 + 7 + 9 = 22. These are ranked in their ascending order
as given below:
Diet
Pla
I
12.5 9
6.5
17
14
11
70
II
15
20
21
16
18
19
22
II
3
5
6.5
8
4
2
12.5 10
131
1
52
From the
nsabove table, we get R1 = 70, R2 = 131 and R3 = 52.
⎛702 1312 522⎞
12
∴ H = 22 × 23 ⎜ 6 + 7 + 9 ⎟ - 3 × 23 = 15.63
⎝
⎠
2
The tabulated value of χ at 2 d.f. and 5% level of significance is 5.99. Since H is greater than
this value, H0 is rejected at 5% level of significance.
15.7. The spearman’s rank correlation test
6∑di2
The Spearman’s Rank Correlation rs = 1 – n (n2 - 1), can be used to test the significance of
correlation in population. We can write H0: rs = 0, where rs is the coefficient of correlation in
population.
474
The test statistic: It can be shown that for n ≥ 10, the distribution of rs, under H0, is
approximately normal with mean 0 and standard error
1
. Thus, z = rs n-1 is a standard normal variate.
n-1
Example 8: Twelve entries in a painting competition were ranked by two judges, as shown
below:
Entry: A
B
C
D
E
F
G
H
I
J
K
L
Judge I:
5
2
3
5
1
6
8
7
10
9
12
11
Judge II:
4
5
2
1
6
7
10
9
11
12
3
8
Test the hypothesis that coefficient of rank correlation in population is positive.
Solution: We have to test H0:σs ≤ 0 against Ha:σs > 0.
From the given data, we can find d1 = Ru – R2i and then ∑di2 = 154.
6 × 154
∴ rs = 1 – 12 × 143 = 0.46 and z = 0.46 11 = 1.53.
Since the value of z is less than 1.645, there is no evidence against H0 at 5% level of
significance. Hence, the correlation in population cannot be regarded as positive.
15.8. The median test for randomness
Any sample comprising numerical observations can be treated in the same manner by using
the letters a and b to denote, respectively, values above the median and values below the
median of the sample. In case an observation is equal to the median, it is omitted. The
resulting series of as and bs (representing the data in their original order) can be tested for
randomness on the basis of the total number of runs above and below the median,
respectively. Let us take an example.
Example 9: Suppose we have the following series of 29 college students. After performing a
set of study exercises, increases in their pulse rate were recorded as follows:
22, 23, 21, 25, 33, 32, 25, 30, 17, 20, 26, 12, 21, 20, 27, 24, 28, 14, 29, 23, 22, 36, 25, 21, 23,
19, 17, 26 and 26.
We have to test the randomness of these data.
Solution: First, we have to calculate the median of this series. If we arrange these values in
an ascending order, we find that the size of (n +1)/2th item, that is, 15th item is 24. Thus, the
median is 24. As there is one value, which is 24 we omit it and get the following arrangement
of as and bs where a stands for an item greater than (or above) the median and b stands for an
item lower than (or below) the median:
bbb aaaaa bb a bbb aa b a bb aa bbbb aa
On the basis of this arrangement, we find that n1, (i.e. a) = 13, n2, (i.e. bs) = 15, and u = 12,
we get
µr = [(2n1n2)/(n1 + n2)] + 1
= [(2 × 13 × 15)/(13 + 15)] + 1 = (390/28) + 1 = 14.93
2n1n2 (2n1n2 - n1 - n2)
σu =
(n1n2)2 (n1 + n2 - 1)
σu =
=
(2 × 13 × 15) ( 2 × 13 × 15 - 13 - 15)
(13 + 15)2 ( 13 + 15 - 1)
390 × 362
(28)2 (27)
475
141180
21168 = 6.6695 = 2.58
Z = (u – µr)/σu = (12 – 14.93)/2.58 = -2.93/2.58 = -1.14
Since Z = -1.14 falls between –Z0.025 = -1.96 and Z0.025 = 1.96, the null hypothesis cannot be
rejected at the level of significance α = 0.05. We can, therefore, conclude that the
randomness of the original sample cannot be questioned.
It may be noted that this test is particularly useful in detecting trends and cyclic patterns in a
series. If there is a trend, there would be first mostly as and later mostly bs or vice versa. In
case of a repeated cyclic pattern, there would be a systematic alternation of as and bs and
probably, too many runs.
=
15.9. Kolmogorov-Smirnov one-sample test
This test is concerned with the degrees of agreement between a set of observed values and the
values specified by the null hypothesis. It is similar to the chi-square test of goodness-of-fit.
It is used when one is interested in comparing a set of values on an ordinal scale. Let us take
an example.
Example 10: Suppose that a company has conducted a field survey covering 200
respondents. Apart from other questions, it asked the respondents to indicate on a 5-point
scale how much the durability of a particular product is important to them. The respondents
indicated as follows:
Very important
50
Somewhat important
Neither
60
important
nor
20
Somewhat
unimportant
unimportant
40
Very unimportant
30
Total respondents
200
We have been asked to use the Kolmogorov-Smirnov test to test the hypothesis that there is
no difference in importance ratings for durability among the respondents.
Solution: In order to apply the Kolmogorov-Smirnov test to the above data, first of all we
should have the cumulative frequency distribution from the sample. Second, we have to
establish the cumulative frequency distribution, which would be expected on the basis of the
null hypothesis. Third, we have to determine the largest absolute deviation between the two
distributions mentioned above. Finally, this value is to be compared with the critical value to
ascertain its significance.
Table 15.8 shows the calculations.
Table 15.8: Worksheet for the Kolmogorov-Smirnov D
Importance of
Observed
Observed
Observed
n
p
c
p
c
di
u
r
u
r
u
ff
m
o
m
o
m
er
Very important
50 b
0.25 p
0.25 ul
0.2 p
0.2 ul
0.05 e
Somewhat important
60 er
0.30 o
0.55 at
0.2 o
0.4 at
0.15 n
rt
iv
rt
iv
c
io
e
io
e
e
n 476
p
n
p
o
r
r
b
o
o
s
durability
Null
Null
Absolute
Neither important nor
20
0.10
0.65
0.2
0.6
0.05
Somewhat
40
0.20
0.85
0.2
0.8
0.05
Very unimportant
unimportant
30
0.15
1.00
0.2
1.0
0.00
unimportant
From Table 15.8, we find that the largest absolute difference is 0.15, which is known as the
Kolmogorov-Smirnov D value. For a sample size of more than 35, the critical value of D at
an α = 0.05 is 1.36/ n. As sample size in this example is 200, D = 1.36/ 200 = 0.096. As the
calculated D exceeds the critical value of 0.096, the null hypothesis that there is no difference
in importance ratings for durability among the respondents is rejected.
Although there are a number of non-parametric tests, we have presented some of the more
frequently used tests in this chapter. While using these tests, we must know that the
advantages we derive by limiting our assumptions may be offset by the loss in the power of
such tests. However, when basic assumptions as required for parametric tests are valid, the
use of non-parametric tests may lead to a false hypothesis and thus we may commit a Type II
error. We have to consider this aspect very carefully before deciding in favour of nonparametric tests. It may be reiterated that such tests are more suitable in case of ranked,
scaled or rated data.
15.10. Summary
Non-parametric tests: Tests that rely less on parameter estimation and/or assumptions about
the shape of a population distribution.
One-Sample Runs test: A non-parametric test used for determining whether the items in a
sample have been selected randomly.
Run: A sequence of identical occurrences that may be preceded and followed by different
occurrences. At times, they may not be preceded or followed by any occurrences.
Sign test: A non-parametric test that takes into account the difference between paired
observations where plus (+) and minus (-) signs are substituted for quantitative values.
Theory of runs: A theory concerned with the testing of samples for the randomness of the
order in which they have been selected.
Wilcoxon Matched-pairs Test (or Signed Rank Test): A non-parametric test that can be used
in various situations in the context of two related samples.
Kolmogorov-Smirnov test: A non-parametric test that is concerned with the degrees of
agreement between a set of observed ranks (sample values) and a theoretical frequency
distribution.
Kurskal-Wallis test: A non-parametric method for testing the null hypothesis that K
independent random samples come from identical populations. It is a direct generalisation of
the Mann-Whitney test.
Mann-Whitney U test: A non-parametric test that is used to determine whether two different
samples come from identical populations or whether these populations have different means.
15.11. Questions
1.
2.
3.
4.
What do you understand by non-parametric or distribution free methods?
What are the major advantages of non-parametric methods over parametric methods?
What are the main limitations of non-parametric tests?
Enumerate the different non-parametric tests and explain any two of them.
477
5.
The sequence of occurrence of ‘zeros’ and ‘ones’ in a message sent in a digital code is
shown below. Test at 5 per cent whether the sequence of ‘0’ and ‘1’ is random 00110
11011 00001 11100 00110 11001 11110 00011 00100 11000 11100 00011 00111
11100 00000 11111 10001 11000 10001 01110.
6.
The proprietor of a small business computed his average earnings per day over a period
of 12 days. For each day, an L was recorded if the earnings were less than the average,
otherwise an M was recorded. These data are given below:
LLLLMMLLLLMM
7.
In a metropolitan city, a city bus service was scheduled to reach a major bus stop at 11
a.m. each day. If the bus reached that stop within 5 minutes of 11 a.m. it was
considered to be on time. Over a 15-day period, an A was recorded if the bus was on
time, otherwise a B was recorded. The picture that emerged after ten days was as
follows:
AABABBABAABBBAA
8.
The following data show employees’ rate of substandard performance before and after
a new incentive scheme. Determine whether the introduction of the new incentive
scheme has reduced the substandard performance at 0.05 level of significance.
Befo 7
8
5
9
10
6
5
9
6
8
After r 5
6
7
6
8
7
6
6
5
7
9.
Aecompany manufacturing electronic toys toys has recently been taken over by another
company. Prior to the takeover of the company, certain workers were approached to
ascertain their satisfaction levels. The same workers were again approached to know
their satisfaction level after the takeover of the company. The two sets of data are given
below.
Befo 69
73
58
76
82
65
75
64
87
70
After r 65
75
63
75
82
68
71
65
85
68
Using
e an appropriate test, find out whether there has been an improvement in the
satisfaction level of workers after the takeover of their company by a new company
10. The following data relate to the costs of building comparable lots in the two Resons A
and B (in million rupees):
Resort
30.9
32.5
44.3
39.5
35.0
48.9
ResortA
53.9
61.0
36.0
42.5
40.9
47.9
The
B company owning the resort area A claimed that the median price of building lots
was less in area A as compared to resort area B. You are asked to test this claim, using
a nonparametric test with a 1 per cent level of significance.
11. On 15 different days, A had to wait for the city bus to reach his office as shown below:
17, 12, 18, 20, 25, 30, 10, 15, 7, 10, 9, 11, 5, 11 and 20 minutes.
Use the sign test at 5 per cent level of significance to test the bus company’s claim that
on an average A should not have to wait for more than 15 minutes.
12. A company used three different methods of advertising its product in three cities. It
later found the increased sales (in thousand rupees) in identical retail outlets in the three
cities as follows:
City
70
58
60
45
55
62
80
72
City A 65
57
48
55
75
68
45
52
City B 53
59
71
70
63
60
58
75
C
478
63
Use Kruskal-Wallis method to test the hypothesis that the mean increase in sales on
account of three different methods of advertising was the same in the retail outlets in A,
B and C cities. Use 5 per cent level of significance.
1. Spiegel, Murray R.: Theory and Practical of Statistics., London
McGraw Hill Book Company.
2. Yamane, T.: Statiscs: An Introductory Analysis, New York, Harpered
Row Publication
3. R.P. Hooda: Statistic for Economic and Management McMillan India
Ltd.
4. G.C. Beri: Statistics for Mgt., TMA
5. J.K. Sharma: Business Statistics, Pearson Education
6. S.P. Gupta : Statistical Methods, Sultan Chand and Sons.
479
AUTHOR: DR. PARDEEP GUPTA
COURSE CODE: MC-106
VETTER: DR. B.K. PUNIA
LESSON: 16
Statistical quality control
Objective:
After going through this chapter, you will be able to understand: the
concept and importance of quality control; set up different types of
control charts to keep the process under control; and the concepts of
acceptance sampling, single, double and multiple sampling plans and
select the most appropriate sampling plan.
Structure
16.1. Introduction
16.2. Statistical Quality Control
16.3. Control charts
16.4. x Charts: Control charts for process means
16.5. R-Charts: Control charts for process variability
16.6. Control chart for C (Number of defects per unit)
16.7. p-charts: Control charts for attributes
16.8. Benefits of Statistical Quality Control
16.9. Limitations of statistical quality control
16.10. Acceptance sampling
16.11. Self-test questions
16.1. Introduction
The subject of ‘quality control’ has assumed considerable importance in recent years in the
wake of globalisation of economies world over. As a result, there has been tremendous
increase in competition amongst business enterprises both within and outside the country.
Quality has been defined in different ways by different experts but almost all those
definitions emphasis that quality must meet the requirements of the customer. While quality
is very vital for providing satisfaction to the customer, it goes far beyond this. For industrial
and commercial organisations, quality is not only central to profitability but crucial to
business survival. This aspect has assumed considerable importance in today’s tough and
challenging business environment. If quality is ignored or overlooked by these organisations
then their continued existence is in danger.
The main factor that affects quality is variability in the process. This variability does not
allow a factory to provide consistently a standard quality product. Prior to mass production,
480
an individual worker or a few of them produced by hand, checking frequently if the product
manufactured is coming out as they had conceived it. If it was distorted, they would again
check where the fault laid, measure, and rework on it. However, when goods began to be
manufactured on a mass scale, it became apparent that individual items could not be identical.
It is almost impossible to eliminate variability completely. Such a situation poses a major
problem in that the parts that are supposed to fit together would not fit. This shows that
variability is the cause of poor quality.
The various causes of variation in the product may be classified into two categories:
(a) Scientific and identifiable
(b) Random and Chance
The first category comprises such causes as the use of defective raw material, poor
equipment, poor workmanship, and so on. While the second category contains causes that do
not have any bearing on the production process. The main purpose of our quality control
exercise is to segregate specific and identifiable causes from the chance or random causes.
In the early days of mass production, inspection of the product and sorting out the defective
ones was the chief method used for quality control. It was thought that the rejection of
defective items would not cost much as the marginal cost of each unit was small. But
gradually it became apparent that the costs of defective items were much higher than
supposed earlier. This is because a number of people had to be employed to inspect the
product besides losing the goodwill of the customers.
This realisation laid emphasis on doing things right at the very first time, focussing on the
concept of zero defects. This means that efforts must be made to prevent defects at each stage
of manufacturing a product or delivering a service. In order to achieve this, workers engaged
in the production are given the responsibility to check their output rather than to pass it on for
a final inspection. One major benefit of this approach is that workers feel a sense of pride and
satisfaction for the responsibility given to them.
16.2. Statistical Quality Control
Statistical Quality Control (SQC) is the application of appropriate statistical tools to
processes to ensure continuous improvement in quality of products, services and productivity
in the workforce. As far back as in 1920, Walter A. Shewart created a system for tracking
variation and identifying its causes. His system of SQC was further developed by W.
Edwards Deming, a one-time colleague of Shewart. It is nothing else but a differentiation of
the causes of variation during the operation of any process. The basic approach to statistical
quality control is to identify a parameter that is easy to measure and is relevant to ascertain
whether the quality is being maintained. For this purpose, control charts are used.
16.3. Control charts
Control charts show a step-by-step approach to statistical quality control. These are ‘roadmaps’ that are very helpful in solving the problems pertaining to quality. The underlying
feature of such a chart is that there are certain SQC techniques that are most appropriate in each
step.
Figure 16.1 gives a schematic control chart. It will be seen from Fig. 16.1 that the control
chart has three zones. These are: stable zone, warning zone and action zone.
3
Action zone
Upper control limit
2
Warning zone
ariable or attribute
Upper warning limit
1
Stable
481
Central line
Zone
1
Lower warning limit
2
Warning zone
3
Action zone
Time
Lower control limit
Fig. 16.1. A specimen of control chart
The action required depends on the zones in which the results fall. The possibilities are:
Nothing needs to be done in case of stable zone wherein variation occurs
due to common causes only.
In respect of warning zone, there seem to be special causes of variation.
There is a need for collecting more information and having a watchful
eye on the process.
Action zone suggests that special causes of variation in the process are
present. The situation demands further investigation and where
appropriate the process needs to be adjusted.
These three situations can be compared to traffic lights, which signal ‘stop’, ‘caution’ or ‘go’.
Let us examine in some more detail major parts of a control chart.
16.3.1. Major parts of a control chart
A control chart generally includes the following four major parts which are shown in Fig.
16.2.
Quality
Scale
Out of control
(Upper control limit)
Sample (sub-group) Number
Fig. 16.2. Major parts of a control chart
(b)
(c)
(Central line)
Quality scale- This is a vertical scale, which is marked as per the chosen quality
Average
characteristic (either in variables or attributes) of each sample.
Plotted samples- The control chart does not show the qualities of individual items of
a sample. Instead, the quality of the entire sample represented by a single value (a
(Lower control limit)
statistic) is shown. The single value plotted on the chart is in the
form of a dot (or
LCL
sometimes a small circle or a cross).
of controlin SQC, on
Sample numbers- The samples, which are also referred to as Out
sub-groups
a control chart are numbered individually and are shown on a horizontal line. The line
is usually shown at the bottom of the chart. It may be noted that the utility of the
2 3 depends
4 5 to
6 a great
7 8 extent
9 10
11 proper
12 13 grouping of items into
control chart1technique
on the
samples. The grouping should be such that variation in quality among items within the
3 Signals
(a)
3 Signals
UCL
482
(d)
same sample is small, while variation between one sample and another is large. Such
a sample is regarded as ‘rational sub-group’.
The horizontal lines- The central line represents the average quality of the samples
plotted on the chart. The lines above the central line shows the upper control limit
(UCL), which is commonly obtained by adding 3 sigmas (σ) to the average that is,
mean +3 standard deviation. Similarly, the lower control limit (LCL) is given below
the central line. It is obtained by subtracting 3 sigmas (σ) from the average, that is,
mean –3 standard deviation. The upper and lower control limits are usually drawn as
dotted lines.
16.3.2. Why 3-Sigma Limits?
We have just said that upper and lower control limits are set at 3σ limits. One may ask the
reason for this approach. It may be noted that the 3σ limits were first proposed by Shewart
for his control charts. On the basis of probability consideration, if variable X is normally
distributed, the probability that a random observation on the variable will lie between µ ± 3σ
(where µ is the mean and σ the standard deviation of X) is 0.997, which is extremely high. It
may be recalled the area of the normal curve between µ ± 3σ is 99.73 per cent. This means
that the probability of a random observation going beyond these limits is nearly 0.003. This
means that the variable quality characteristic is assumed to be normally distributed and that
the probability of a sample point going outside 3σ limits when the process is in control is
very small. If a sample point goes beyond this limit, it is highly likely that the normality
assumption of the process is not applicable.
In order to set up a sound quality control mechanism, the concerned organisation must be
keenly interested. It must take the following steps.
First, it must select the quality characteristics, which need to be kept under control. Besides,
both their upper and lower limits within which variation can be tolerated, should be fixed up.
Second, the production process must be analysed so that the possible causes of variation can
be determined. Finally, it must lay down as to how the inspection data will be collected and
recorded as also how they will be subdivided. Depending on the type of inspection data
available, any one of the following types of control charts can be used.
4.1.
Control charts for x
2.
Control chart for σ or R alone
3.
Control chart for C
4.
Control chart for p or pn
16.4. x Charts: Control charts for process means
In order to ascertain whether the process is in control or out of control, x -charts are
constructed. In regard to the process output, there is an assumption of normality where µ and
σ are known, though in many situations this assumption may not hold good. We know that
the sample means have a sampling distribution with µ x = µ and σ x = σ/√n.
483
The construction of x -chart needs the values of µ and σ and also a sample size n. There are
three lines in a x control chart, viz. the centre line indicating µ x , the upper control limit
(UCL), with value µ x + 3σ and the lower control limit (LCL), with value µ x – 3σ. In
addition to the control limits, there are warning limits, which are determined by 1.96 σ on
either side of the centre line. Thus, the upper warning limit (UWL) = µ + (1.96)/√n and the
lower warning limit (LWL) = µ - (1.96)/√n. Figure 16.1 shows these two warning limits.
However, the control charts do not normally show the warning limits.
Let us take an example to illustrate the procedure used in construction x control charts.
Example 16.1. A company is engaged in the manufacture of battery cells in its plant. The
process is said to be under control if the mean life of battery cells is 1,200 hrs with a standard
deviation of 75 hrs. Considering these values to be the process average and process
dispersion, you are required to determine the 3-sigma control limits for x -chart for samples
of size 16.
Solution- Given are µ = 1,200 hrs, σ = 75 hrs and n = 16.
As the estimates of process average and process dispersion are based on a large sample, the
desired control limits can be obtained by the following formula:
µ ± 3 σ/√n
Substituting the values in the above formula,
UCL = µ + 3(75/√16)
= 1,200 + 56.25
= 1,256.25
LCL = µ - 3(75/√16)
= 1,200–56.25
= 1,143.75
16.4.1. x Chart when µ and σ are not known
The preceding discussion has given us some basic ideas on x -chart. The question is that
when population mean and population standard deviation are not known to us, then how to
construct x -charts. In such cases, we use sample information to estimate unknown
parameters. Let us take first the estimation of µ. This can be done by taking the mean of the
sample mean ( x ). This can be calculated by the following formula
x = Σx/n × k = Σ x /k
Where, n = number of observations in each sample
k = number of samples taken
In respect of control charts, it has become customary to use R as an estimate of σ. R
signifies the average of the sample ranges. It is a biased estimator of σ, and d is the correction
factor. The value for d2 is given in question. Thus, the upper and lower control limits (UCL
and LCL) for an x -chart are computed with the following formulas:
3R
UCL = x +
d2√n
3R
LCL = x –
d2√n
In the above formula, d2 stands for control chart factor. These limits are often calculated as x
± A2 R where A2 = 3/(d2√n).
484
By using these formulas, we can now plot the three lines— CL (central line), UCL (upper
control line) and LCL lower control line). Let us take an example to show how these
formulas can be used.
Example 16.2. Suppose we are given the following information:
n = 20, x = 75, d2 = 3.735 and R = 15. We are asked to find the CL, UCL and LCL for a x
control chart.
Solution. It is obvious that CL is the grand mean, that is, 75.
3R
UCL =
x +
d2√n
3(15)
=
75 +
3.735×√20
45
=
75 + 16.70
= 77.69
3R
UCL =
x –
d2√n
3(15)
= 75 –
3.735×√20
45
= 75 – 16.70
= 72.31
Example 16.3. A company manufactures tyres. A quality control engineer is responsible to
ensure that the tyres turned out are fit for use up to 40,000 km. He monitors the life of the
output from the production process. From each of the 10 batches of 900 tyres, he has tested 5
tyres and recorded the following data, with x and R measured in thousands of km.
Batch 1
2
3
4
5
6
7
8
9
10
40.2
43.1
42.4
39.8
43.1
41.5
40.7
39.2
38.9
41.9
x
1.3
1.5
1.8
0.6
2.1
1.4
1.6
1.1
1.3
1.5
R
Construct an x -chart using the above data. Do you think that the production process is in
control? Explain. (Value of d2 = 2.326)
Solution.
∑ x 410.8
x =
k = 10 = 41.08
∑R 14.2
R =
k = 10 = 1.42
CL =
41.08
3R
UCL =
x +
d2√n
3(1.42)
=
41.08 +
2.326×√5
4.26
=
41.08 + 2.326×2.24
=
41.08 + 0.82 = 41.9
3R
LCL =
x –
d2√n
485
3(1.42)
2.326×√5
4.26
=
41.08 – 2.326×2.24
=
41.08 – 0.82 = 40.26
The production process is in control in respect of only 3 batches as is indicated in Fig. 16.3.
The production process is in respect of batches 1, 4, 8 and 9 has gone out of control so also
batch numbers 2, 3 and 5.
=
41.08 –
44
43
42
UCL
41
CL
40
LCL
39
38
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Batch Number
Fig. 16.3. x -Chart for the data given in example 16.3
16.5. R-Charts: Control charts for process variability
In this chart, the value of the sample range for each of the samples is plotted. The central line
for R-charts is placed at R . Now, we have to decide the control limits for which we need
some additional information regarding the sampling distribution of R, in particular its
standard deviation σR. For this purpose, the following formula is used.
σR = d3σ
where
σ = population standard deviation
d3 = another factor depending on n
The values of d3 are given in question.
Control Limits for an R-Chart
3d3 R
⎛ 3d3⎞
UCL =
R + d = R ⎜1+ d ⎟
⎝
2
2⎠
3d3 R
⎛ 3d3⎞
R – d = R ⎜1– d ⎟
LCL =
⎝
2
2⎠
It may be noted that these limits are often calculated as:
3d3
UCL = R D4, where D4 = 1 + d
2
486
3d3
LCL = R D3, where D3 = 1 – d
2
The values of D3 and D4 can also be found from Table of control charts.
Example 16.4. We have to determine the UCL and the LCL by applying the above formulae
to the data given in Example 16.3.
Solution. The UCL and the LCL are calculated as follows:
⎛ 3d3⎞
UCL =
R ⎜1+ d ⎟
⎝
2⎠
⎛ 3(0.864)⎞
=
1.42 ⎜1+ 2.326 ⎟
⎠
⎝
=
1.42 (1 + 1.11) = 2.996 or 3 approx.
⎛ 3d3⎞
R ⎜1– d ⎟
LCL =
⎝
2⎠
⎛ 3(0.864)⎞
=
1.42 ⎜1– 2.326 ⎟
⎠
⎝
=
1.42 × -0.11 = -0.156 (to be taken as zero)
Some explanation is needed for the zero value of LCL. A sample range is always a nonnegative number (because it is the difference between the largest and smallest observations in
the sample). However, when n ≤ 6, the LCL computed by the above equation will be
negative. Although in this case n is 10, yet the calculation shows a negative value. As such,
we set the value of LCL at zero.
A major limitation of R-chart arises from the characteristic of range itself. As we know that
the range considers only the highest and the lowest values in a distribution, it may ignore the
nature of variation in the remaining observations. Further, it is influenced by extreme values,
which may significantly differ from one sample to the other. In view of these limitations, Rchart is only a convenient device for examining variability of the process.
16.6. Control chart for C (Number of defects per unit)
So far we have consider the control charts for attributes in those cases wherein a random
sample of definite size is selected and examined in some way. However, there are certain
situations where the number of events, defects, errors can be counted, but there is no
information about the number of events, defects or errors that are not present. Each item is
classified in one of the two categories- defective or non-defective. In such cases, we know the
number of defects, say, number of holes in a fabric but we do not know the number of nondefects present. In such cases, the Poisson distribution is to be applied.
The central lines of the control chart for C is C and the 3-sigma control limits are
UCL = C + 3 C
LCL = C – 3 C
This formula is based on a normal curve approximation to the Poisson distribution. The use
of the C-chart is appropriate if the occasions for a defect in each production unit are infinite,
but the probability of a defect at any point is very small and is constant.
Example 16.5. Fifteen pieces of cloth from different rolls contained respectively 1, 5, 3, 2, 7,
6, 3, 2, 6, 5, 4, 3, 5, 6, and 3 imperfections. Draw a control chart using these data and state
whether the process is in a state of statistical control.
Solution.
C = (1 + 5 + 3 + 2 + 7 + 6 + 3 + 2 + 6 + 5 + 4 + 3 + 4 + 6 + 3)/15
= 60/15 = 4
UCL =
C +3 C
487
=
4 + 3√4 = 4 + 6 = 10
LCL = C – 3 C
= 4 – 3√4 = 4 – 6 = –2
Since the number of defectives cannot be negative, the lower control limit will be taken as
zero. Figure 16.4 shows both the control limits. The chart clearly shows that all the
imperfections in cloth are within the control limits, that is, no point lies outside the control
limits. This suggests that the process is in a state of statistical control.
Number of imperfections
12
UCL
10
8
6
CL
4
2
LCL
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Rolls of Cloth
Fig. 16.4. Control chart for C
16.7. p-charts: Control charts for attributes
The control chart for attributes is known as the p-chart. Such a chart is used to control the
proportion or percentage of defectives per sample. It may be noted that there is an assumption
that the items are produced by Bernoulli process, which implies the following three
assumptions: (i) There are only two outcomes– acceptable or defective. (ii) The outcomes
occur randomly. (iii) There is no change in the probability of either outcome for each trial.
As we have seen earlier that the C-chart is concerned with the number of defectives, it can be
easily converted into proportion by dividing the number of defectives by the sample size.
Thus, we can use the p-chart in place of the C-chart. In order to draw the p-chart, we have to
5.1.
Calculate the average fraction defective ( p ) by dividing the number of
defective units by the total number of units inspected.
6.1.
The value of p is now used to draw a horizontal line.
488
7.1.
The upper and lower control limits are to be obtained by using the
following formulas:
UCL = p + 3
pq
n
pq
where q = (1 – p )
n
Any sample point falling outside the UCL and the LCL indicates that the process is not in
control. It is preferable to set up the chart to express ‘percent defective’ to ‘fraction
defective’.
Example 16.6. The following figures give the number of defects in 10 samples, each
containing 200 items: 40, 44, 22, 34, 24, 32, 28, 32, 34 and 30. Calculate the values for
central line and the upper and lower control limits of p-chart. Draw the p-chart and comment
if the process can be regarded in control.
Solution.
Table 1. Worksheet for calculating the values for p-chart
Sample No.
No. of defectives
Fraction defectives
1
40
0.20
2
44
0.22
3
22
0.11
4
34
0.17
5
24
0.12
6
32
0.16
7
28
0.14
8
32
0.16
9
34
0.17
10
30
0.15
Total
320
No. of units defective
320
p = Total no. of units inspected = 2000 = 0.16
p(1 − p )
0.16(1 − 0.16)
= 0.16 + 3
UCL = p + 3
n
200
= 0.16 + 0.07776 = 0.2378
p(1 − p )
0.16(1 − 0.16)
LCL = p – 3
= 0.16 – 3
n
200
= 0.16 – 0.07776 = 0.0822
It will be seen from Fig. 16.6 that all the units fall within the upper and lower control limits.
On the basis of this chart, we can say that the process is well under control. It may be noted
that we have plotted the percentage defective instead of fraction defective in the above chart.
LCL = p – 3
489
45
Percentage defective
40
35
30
UCL
25
20
CL
15
10
5
LCL
0
0
1
2
3
4
5
6
7
8
9
10
11
Number of samples
Fig. 16.6. p-Chart for data given in example 16.6
16.8. Benefits of Statistical Quality Control
There are several benefits of SQC approach and these include:
8.1.
SQC can be applied to any type of problem selected and process
originally tackled will result into improvement.
9.1.
This approach eliminates the ‘emotion’ factor and the decisions are
based on facts rather than on opinions.
10.1. As the workers are directly involved in the improvement process, their
‘quality awareness’ increases.
11.1. The knowledge and experience potential of those involved in the
process is released in a systematic way through the investigative
approach. They increasingly realise that their role in problem solving
is collecting and communicating the relevant facts on which decisions
12.1. Managers and supervisors solve problems methodically instead of in a
haphazard manner. Thus, the approach to the problem becomes
unified in place of an individual approach earlier.
490
13.1. In case of any inquiry from the government or any other appropriate
authority, the quality can be defended on the basis of statistical
process control.
14.1. Since the firm strictly adheres to the SQC, the users of the product
may rely on it and may not resort to check the quality themselves.
16.9. Limitations of statistical quality control
Despite the above mentioned advantages of the SQC, it may be noted that it is unable to solve
all the problems arising in quality improvement. There are several highly complex problems
where SQC may not be in a position to contribute much towards reduction of variability. This
apart, at times, managers use SQC mechanically and construct control charts without going
into the depth of the problem. As a result, statistical methods have been criticised at times. It
has been argued that continuous improvement in quality can be attained by studying all parts
of an organisation and not merely one part viz. production process.
16.10. Acceptance sampling
Acceptance sampling involves sampling inspection by a purchaser who has to decide whether
to accept a shipment of product. Thus, the objective of acceptance sampling is either to
accept or to reject the product. It does not attempt to control the quality during the
manufacturing process. This is altogether a different approach from what has been followed
in control charts discussed earlier.
A major advantage of acceptance sampling is that it can motivate suppliers to improve the
quality of their items. Suppose a company receives a batch of components from its supplier
and finds that 10 percent of the supply is defective. Although 90 per cent of it is free from
defects, but the company may decide to reject the entire lot to ensure its qualitative output.
This decision of the company would result into heavy loss to the supplier. He has to suffer
even though a small proportion of his equipment was defective. In order to avoid such an
eventuality, the supplier would be very particular from the very beginning to ensure his
supplies are free from defects. In contrast, if the company rejects only 10 percent of defective
equipment, it amounts to imposing a high cost on itself and a low cost on the supplier. The
various kinds of sampling plans used by purchasers to accept or reject a lot are discussed as
under:
16.10.1. Single-sampling plan
When the decision on whether to accept or reject a plot is based on only one sample, the
acceptance plan is said to be on a single-sampling plan. There are three things that need to be
specified in a single-sample plan. These are: (a) number of items N in the lot from which the
sample is chosen, (b) number of articles n drawn by random sample from the given lot, and
(c) the acceptance number c, which specifies the maximum number of defective articles
allowable in the sample. In case the number of defective articles crosses this limit in the
sample drawn, the entire lot is to be rejected.
16.10.2. Double-sampling plan
Double-sampling plan is obviously more complicated than the single-sampling plan. In this
case, a lot is immediately accepted or rejected depending on the condition of the first sample.
At times, the management finds that the first sample is neither good enough nor bad enough
so as to take a decision one way or the other. In such a situation, it defers its decision until a
second sample is drawn. On the basis of the evidence from both the first and the second
491
samples, a decision is finally taken whether to accept or reject the lot. A double-sampling
plan depends on five specified numbers (besides N): n1, c1, n2, n1+n2 and c2 (> c1), which are
used as follows:
First, a sample of size n1 is taken. Let b1 denote the number of defective pieces in the first
sample and c1 denote the number of defective pieces acceptable in the lot, then
(a) accept the lot if b1 ≤ c1
(b) reject the lot if b1 > c1
(c) an additional n2 units are sampled if c1 < b1 ≤ c2
Let b2 be the total number of defective pieces in the combined sample of n1+n2 units:
(d) accept the lot if b2 ≤ c2
(e) reject the lot if b2 > c2
As mentioned earlier, double-sampling plans are more complicated than the single-sampling
plan. But, as they are more powerful, they are more frequently used in quality control
problems.
16.10.3. Multiple or sequential sampling plan
We have seen earlier that when a single-sampling plan is unable to give us a clear decision,
we take recourse to the double-sampling plan. It may just be possible that even a doublesampling plan may not give us a clear decision. In such a case, we may go on to have another
sample before we reach a definite decision. Thus, three or more sampling plans can be used.
This is known as multiple or sequential sampling. Since such plans are extremely
complicated, they are seldom used in practice.
16.11. Self-test questions
1.
What is statistical quality control? How is it useful to industry?
2.
What is a control chart? Describe how it is constructed and used?
3.
Describe briefly the working of the p-chart.
4.
Write a detailed note on ‘Acceptance Sampling’.
1. Business Statistics by Shenoy and Shenoy.
2. Statistical Methods by S.P. Gupta.
3. Statistics for Business and Economics by R.P. Hooda.
492
AUTHOR: DR. PARDEEP GUPTA
COURSE CODE: MC-106
VETTER: DR. V.K. BISHNOI
LESSON: 17
Indian Statistics: An Overview
Objective:
After going through this chapter, you will be able to understand: the
statistical system in India; various aspects of labour statistics; statistics
of internal and external trade; Index numbers as a statistical device and
role of various non-governmental agencies in collecting statistics in the
country.
Structure
17.1. Indian statistical system
17.2. Statistical organisations at the Centre
17.3. Important publications
17.4. Statistical Organisations in the State
17.5. Non-Governmental Agencies
17.6. Labour statistics
17.7. Trade union and other miscellaneous statistics
17.8. Industrial Statistics
17.10. Index numbers
17.11. Self-test questions
17.1. Indian statistical system
The statistical system in India at present is a decentralised one, in which the task of collection
of statistics is divided between the Central Government and the State Government on the
subject-wise basis. The Central Statistical Organization has the responsibility for
coordination at the national level for all the activities of the state and central statistical
agencies. The State Statistical Bureaus have the responsibility of the similar task at the state
level. The subject-wise division between the Central and State level Bureaus is as follows:
The items like foreign trade, banking and currency, railways, post and telegraphs, and
population are entirely under the Central Government. The items like agriculture and
education are to be looked after by the State Government. There is, however, a common
category of subjects. For example, industry, where both the Central and State Governments
collect statistics to meet their respective requirements. Further, everywhere the States have
primary authority for the collection of statistics. The Central Government acts as a
coordinating agency for the maintenance and publication of data on the all-India basis. The
493
subjects allocated to the Centre and States are further divided among the different Central
Ministries and the State Government departments respectively.
17.2. Statistical organisations at the Centre
Most of the central ministries either have a full fledged statistical department, a division, or a
section depending upon their needs and the stage of development of statistics being collected.
Some of these statistical departments located within the administrative departments are
engaged in the processing of data being collected as a part of the administrative process.
Some of these agencies are the office of Income Tax Department, Central Board of Revenue,
Railways, Post and Telegraphs and the Directorate General of Supplies and Disposal. The
Textile commissioner’s Office and Iron and Steel Controller’s Office also collect the data
required for the control of production and distribution of the product.
Some of the organisations specifically established by the Government for collecting statistics
are briefly presented below:
Office of the Registrar, Government of India
This office was set up on a permanent basis in 1949 for organizing and conducting population
census for every 10 years in the country. Since 1960, the work of collection, compilation and
publication of the statistics relating to the population is being done by this office.
Department of Commercial Intelligence and Statistics
This organisation was set up in 1895 and was responsible for the collection, consolidation
and publication of important statistical data till the Second World War. Subsequently with the
formation of statistical units in different ministries, several of the functions of this office were
transferred to them and this office was left with only commercial intelligence and trade
statistics.
Labour Bureau, Ministry of Labour, Employment and Rehabilitation
This bureau was set up in 1946 with the task of collection, compilation, publication and
dissemination of labour statistics. This bureau also prepares the consumer price index number
for working class.
Directorate of Economics and Statistics, Ministry of Food and Agriculture
The directorate was set up in 1947 and given the responsibility of compiling and publishing
of agricultural statistics on an all-India basis. The data collected and published by the
directorate covers area, population, yield of major crops, fisheries, livestock and forests.
These data are mostly collected by the State Governments.
Army Statistical Organisation, Ministry of Defence
This organisation has the responsibility to render technical advice on statistics to the army.
This organisation collects data relating to personnel, vehicles, animals, etc. and it has a
research unit working on the application of sampling techniques to various areas of interest to
army.
National Sample Survey (NSS)
Initially this organisation was set up under the ministry of finance in 1950 for collecting data
on sample basis relating to all aspects of national economy on a continuing basis. The
organization was subsequently transferred to the cabinet secretariat in 1957. It has been
regularly conducting surveys and so far published several hundred reports relating to various
activities of the economy.
494
Central Statistical Organisation (CSO)
CSO is responsible for computation and publishing the annual survey of industries. It also
computes and publishes national income estimates. The CSO is also engaged in improving
the statistical standard in the country, particularly with regard to concepts, definition,
classification and statistical methodology. CSO also publishes annual statistical abstracts and
monthly abstracts of statistics, giving data regarding different aspects of the economy.
In addition to the organisations described above, various ministries and departments have
statistical cells attached to them. Periodically these cells publish valuable data relating to their
area of concern. For example, department of agriculture and cooperation, Ministry of
Agriculture, Government of India, bring out a Hand Book on Fisheries Statistics every year.
17.3. Important publications
A few important publications containing official statistics for general use are given below:
Statistical abstract of India: This annual publication is brought out by the
CSO and contains the statistics of various sectors of the Indian
economy at least for the last five years. This also gives state-wise
statistics at least for the last year.
India- Pocketbook economic information: This publication is brought out
annually by the Ministry of Finance. It deals with the various aspects
of economy particularly financial, foreign aid and international
economic comparisons.
Basic statistics relating to Indian economy: This is brought out by the
statistics and surveys division of planning commission on an annual
basis. It contains basic indicators on various aspects of economy for a
number of back years on time series basis.
India- a reference manual: This is published annually by the Ministry of
Information and Broadcasting and it contains information on various
aspect of Indian economy.
Estimation of National Product, savings and capital formation- This is
generally known as a white paper on national income and is published
annually by the CSO. It contains estimates of national income,
savings, capital formation and consumption, expenditure together
with national and public sector accounts.
495
Agriculture and Cooperation (Directorate of Economics and Statistics)
on a monthly basis. This publication contains all available current
statistics together with notes and articles on the assessment of
current agriculture situation in the country. Available district-wise
data is also reported from time to time.
Monthly statistics on production of selected industries in India: This
publication of CSO contains data on output, capacity and index
numbers and several other variables relating to more than 90
industries.
Reserve Bank of India Bulletin: This is published by the RBI on a monthly
basis and it contains elaborated data on various aspects of Indian
economy including a detailed data on currency and finance situation.
Economic survey: This is published annually by the Department of
Economics Affair and Ministry of Finance on the eve of the
presentation of budget. The document contains an elaborated review
of all aspects of Indian economy.
Banking statistics- Basic Statistical returns: This is published twice in a
year by the Reserve Bank of India. This contains general information
on banking data on distribution of deposits and advances district-wise
according to type and bank group-wise. The regional distribution of
advance published, is according to district of utilization.
Directorate of Economics and Statistics, Ministry of Agriculture. This
publication brings out data and notes on the integrated picture of
area, production, yield, market arrivals, imports, exports, distribution,
prices, etc. of commercial crops.
Economics and Statistics, Ministry of Agriculture and Cooperation
496
annually.
It
contains
data
on
production,
market
arrivals,
procurement, imports, distribution, prices, availability, etc. of food
grains in India together with information relating to various aspects of
subsidiary foods.
17.4. Statistical Organisations in the State
The focal point of the collection of statistics at the state level is the statistical bureaux. These
bureaux have the following as their main functions:
15.1. Coordination of statistics collected by different departments.
16.1. Publication of statistical abstracts by assembling all essential
statistics.
17.1. Organizing all enquiries and surveys.
18.1. Liaison between statistical organization of the centre and other states,
and
19.1. Statistical work relating to planning.
However there are some differences in the functioning of these state statistical bureaux.
While in some states the collection of statistics is almost centralized, in case of some others
the collection of agriculture, labour and vital statistics are done by the other departments.
Most of the state statistical bureaux participate in NSS programmes of socio-economic
surveys. These bureaux also have the responsibility of computing state income.
The state statistical bureaux function through district statistical offices for a speedy collection
of data. Statistical units attached to various state departments for collection, compilation and
publication of statistics are responsible to those department. These units also collect data at
the instance of their counterparts at the Centre.
17.5. Non-Governmental Agencies
Reserve Bank of India (RBI): A major research department was set up in RBI in 1945 for
processing statistics of banking activities. This department also maintains balance of payment
statistics. This department reports its assessment of country’s external and internal economic
situation and publishes the RBI bulletin on a monthly basis. Reserve Bank also publishes the
annual report of the bank and the report on currency and finance. The report on currency and
finance gives the data on financial year (April-March) basis, except where otherwise
specified. Since 1970-71 forms the base year for most of the official series of index numbers,
the data are presented commencing with this year. The statement on (i) industrial production,
(ii) capital raised by non-Government and Government companies, (iii) estimated
employment in public and private sector, (iv) employment in public and private sector in
major industries and services, (v) number of applicants on Live Register of Employment
Exchange and (vi) domestic production and imports of crude oil and petroleum products are
presented on calendar year basis, as the statistics are compiled by the source agencies is on
that basis. Further, data relating to (i) agricultural production, (ii) financial assistance
sanctioned by the Industrial Finance Corporation of India, Industrial Development Bank of
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India, Agricultural Refinance and Development Corporation, National Bank of Agriculture
and Rural Development and Unit Trust of India and (iii) consumer price index numbers for
agricultural labourers.
Industrial Development Bank of India (IDBI): The IDBI is an apex term lending
institution in the country. The institution brings out annual report on development of banking
of India, since 1977-78. In earlier years this publication was called operational statistics. The
purpose of the publication is to disseminate basic information on the operations of the
developmental finance institution during the year.
Indian Statistical Institute (ISI): The Institute was established substantially in developing
the Indian statistical system. In the initial years of NSS, this institution was incharge of all
technical works relating to the sample surveys conducted by the NSS. It is now actively
engaged in research and professional training and publication of a journal called ‘The
Sankhya’.
Institute of Agricultural Research Statistics: This institution has been working in the field
of agricultural statistics since 1931. It has contributed to the application of the method of
random sampling to the official procedure for the estimation of yield of crops and evolved
suitable designs for experimentation in cultivators’ field. This institution has also conducted
research and training in agriculture and animal husbandry statistics.
17.6. Labour statistics
With the industrialisation, the need for comprehensive statistical information on labour has
become more intense. The process of collection of labour statistics has also received
substantial attention because of the lead given by the international Labour Organizations
(ILO). The labour statistics include employment and non-employment data, wages and
earning, statistics of different categories of gainfully employed persons, statistics relating to
trade unions, industrial injuries, industrial disputes, absenteeism and social security.
In India, labour statistics are largely the by-product of the administration of various labour
laws like the Factories Act, the Payment of Wages Act, Minimum Wages Act, and others.
The Labour Bureau, Ministry of Labour, Government of India, is responsible for the
20.1. Collection, complication and publication of labour statistics.
21.1. Maintenance of consumer price index numbers for urban and rural
areas.
22.1. Construction of consumer price index numbers for agricultural
workers.
23.1. Keeping factual data relating to working conditions.
24.1. Conducting research into specific problems of individual workers.
25.1. Publication of labour statistics.
In addition to the Labour Bureau, employees and employers also collect data for making their
own studies. The research centres of the universities and other research organisations also
collect and compile labour statistics.
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Employment: The decennial population census conducted in 1951, 1961, 1971, 1981, 1991
and 2001 presents an overall employment data.
Factories: The State Chief Inspector of Factories collects the statistics regarding daily
employment in registered factories and the number of factories on a half yearly and yearly
basis under the statutory provision of the Factories Act 1948. The statistics are compiled and
processed by the Labour Bureau and Published in its annual publication Indian Labour.
Statistics: The employment statistics are available for different industries and for different
states. These statistics also cover employment of women for major industries based on returns
submitted by the factory. The annual return to be filled by the factories have to give the
information relating to (a) average number of workers employed, (b) days worked, (c)
number of hours worked and intervals.
In addition to the above, the annual survey of industries conducted by the industrial statistics
wing of CSO also publishes the data on average daily employment separately for men,
women and children for more than 200 industries.
Mining: The information regarding the number of persons employed in and around mines
and quarries covering wage earners, salaried employees, foreman and apprentices is collected
by the Director General of Mines Safety. These statistics are published in the Statistics of
Mines in India which is published in two volumes: volume 1 deals with the coal mines and
volume 2 covers the mines other than coal. These statistics are also included in the
publication of Indian Labour Statistics and Labour Year Book brought out by the Labour
Bureaus.
Plantations: The Directorate of Economic and Statistics compiles the statistics of the average
daily employment in coffee, rubber and tea plantations. These statistics are compiled on the
basis of returns from the individual plantations which are collected by the State Government.
The figures of average daily employment are obtained by dividing the total attendance during
the year by 300. These statistics are also combined and included in the Indian Labour
Statistics.
Transport and communications: The Directorate General of Posts and Telegraphs collects
the statistics on employment in the Post and Telegraphs department and these figures are also
published in the Indian Labour Statistics. In addition, the figures of employment in Railways
are given in the Annual Report of the Railway Board. The Indian Labour Statistics also
covers data on workers employed by port authorities, seamen registered with the shipping
offices and seamen’s employment offices.
Employment in the Public Sector: The Directorate General of Employment and Training
collects the statistics on employment in the Central Government, State Governments, quasi
Governments and local bodies on quarterly basis. These figures are published in the
Employment Review, brought by the Directorate General of Employment and Training.
Motor Transport Undertakings: The Motor Transport workers Act 1961 provides for
collection of data from all transport companies which employ five or more workers. The
state-wise data was published for first time in the Indian Labour Statistics in 1971.
Other data: The Indian Labour Statistics also includes the data on employment on shop and
commercial establishment of selected urban centres of some of the states. The data given in
the employment review includes (i) total employment, zone-wise and state-wise, (ii) public
sector employment classified by different branches, (iii) employment in selected industries
for the public and the private sectors separately and (iv) work seekers by age and by
education level.
All these information give details about women employees and women work seekers.
Biennial survey conducted by the Director General of Employment and Training presents the
industry-wise and state-wise employment figures in smaller establishments. This directorate
also publishes data on occupational pattern in India separately for public and private sectors.
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Wages and earnings: Since 1948, the data regarding agricultural wages is collected on a
monthly basis for all the districts in each state. At least one representative village is selected
from each district for this purpose. The statistics is compiled by the Directorate of Economics
and Statistics in the publication called Agricultural Wages in India. The data is also reported
on monthly basis in Agriculture situation in India.
Wages in manufacturing industry: The task of compiling, consolidating and
publishing the wage data is undertaken by the Labour Bureau and it is reported in
the Indian Labour Statistics. These statistics include the per capita annual income
of workers classified according to states, according to industries and according to
the components of wages. In computing the annual average earnings, the data from
seasonal industries like those making food products, tobacco, cotton ginning and
pressing is excluded for reasons of comparability with others. The annual survey of
industries also publishes data of the industry-wise wages and salaries separately for
men, women and children in different states.
Other data on wages: The Indian Labour Statistics and the Statistics of Mines in India
presents the statistics of earnings of employees in different mines. The Indian labour statistics
also publishes similar data for plantation and dock labourers. Annual report of the railway
board includes the statistics of average annual earnings under selected categories in railways.
17.7. Trade union and other miscellaneous statistics
submit annual return giving data on membership and finance to the labour bureau. This data
is compiled and published in the Trade Unions in India. This data is also published in Indian
Labour Statistics. No systematic information is available on unregistered trade unions.
Industrial disputes: The industrial dispute statistics collected on a voluntary basis by the
State Labour Department and Regional Labour Commissioner include the following: (i)
Number of disputes, (ii) Number of workers involved, (iii) Number of man-days lost (the
total number of man days available). All these statistics are published in Indian Labour
Statistics.
Absenteeism: For the purpose of collection of statistics, the absenteeism is measured by
taking the percentage of man-shifts lost industries and Industry Association compile the
statistic related to absenteeism. Such statistics for selected industries at important centres are
published in the Indian Labour Statistics.
Industrial injuries: The statistics on industrial injuries are collected by the Labour Bureau
from the following: (i) factories, (ii) mines, (iii) railways and dock workers. The injuries are
classified into two categories, fatal and non-fatal. The industry-wise statistics are published
by the Bureau in the Industrial Statistics.
Social security: The Labour Bureau collects and publishes, the statistics relating to social
security benefits. The data for this is derived from the administration of the following Acts:
(i) Workmen’s Compensation Act 1923, (ii) Employees State Insurance Act, (iii) Maternity
Benefits Act, and (iv) Employees Provident Fund Act 1952.
Employment and Underemployment: The Indian Labour Statistics publishes the data from
employment exchanges regarding the number of persons seeking work at the end of each
month classified according to different occupation group, number of applicants placed in the
employment, number of vacancies notified and number of persons registered.
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17.8. Industrial Statistics
Manufacturing industries in India are classified into two sectors as organized industries
(factory establishments) and unorganised small industries.
All manufacturing establishments employing ten or more workers using power and 20 or
more workers if not using power are covered by the organized sector. The organized sector is
further subdivided into large scale industrial establishments and small scale factories. Among
the organized sector establishments, those employing 50 or more workers if using power and
10 or more workers if not using power constitute large scale establishments and remaining
ones are classified as small scale organized factories.
Factory Establishment
The important sources of Industrial Statistics for organized sector in India are (i) Census of
Indian Manufacturers (CMI) (ii) Sample Survey of Manufacturing Industries (SSMI), (iii)
Annual Survey of Industries (ASI) and (iv) Monthly Statistics of Production of Selected
Industries of India (MSPSI).
(a)
Census of Indian Manufacturers: These censuses were carried out in India between
1944 and 1958. Subsequently this census was replaced by the Annual Survey of
Industries (ASI) in 1959. The census carried out between this period divided
manufacturing industries into 63 groups and the data were actually collected for only
first 29 industries and the census covered only the organized sector excluding those
factories which were under the control of defence ministry. The census data were
published in the census of Indian manufactures. Usually the time lag observed
between the collection and publication of data was about three years. The census data
published for each industry included the quantity and value of different products and
by-products in the industry, different kinds of fuels and materials consumed in value
and quantity terms, details of employment along with wages and salaries, componentwise fixed and working capital employed, depreciation, value added, etc. These data
also included summary for each industry by size of employment and type of
ownership.
(b)
Sample survey of manufacturing industry: This was started in 1950 and it was
conducted by the Directorate of Industrial Statistics. This survey was conducted on
calendar year basis covering the whole of India except Jammu and Kashmir and
Andaman, Nicobar Islands. The Jammu and Kashmir was subsequently added in 1953
onwards. The items of information were uniform in all the reports and they included
the following items among the value of inputs: (i) fuel, lubricants and electricity
consumed; (ii) raw materials and (iii) work done by other concerns. The report also
included the data on value and output, capital employment, total value of inputs, value
added, employment, emoluments drawn and few such broad items of information for
the sector as a whole. The survey was discontinued and replaced by the annual survey
of industries in 1959.
(c)
Annual survey of industries: This survey replaced both the CMI as well as the SSMI
and it is carried out on the Collection of Statistics Rule 1959. The ASI covers the
entire factory sector, factories being those registered under the Factories Act (1948).
The required fieldwork for the survey is conducted by NSS. The census part of the
survey is compiled and tabulated by the CSO and the sample part of the data is
compiled by the ISI. The basis for conducting the census is the reference list of
factories (classified according to industry) provided by the chief inspector of factories.
The reference period of all industries is the calendar year except for sugar where the
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year ending 30th June is used as base. A standard set of forms have been prescribed
and used for the purpose of the census and sample surveys since 1960. The items of
information, coverage, concepts and definitions used are the same as in the case of
CMI.
In case of sampling part also the concepts used are the same as those for the census
part. The sampling frame is based on the list of all factories, excluding those covered
by the census part. The sample size is allocated to different industries, i.e., proportion
to the total all India employment.
Since the sampling based reports of ASI are not detailed enough for planning and
development needs of small scale industries, the CSO collects separate data on
priority group of industries on census basis, and these include (i) metal products
including machineries, (ii) chemical and related products, (iii) textile, and (iv) other
industries. The data collected for small scale sector is also published under annual
survey of industries (small industries sector).
The information presented in these ASI reports include data on 15 characteristics: (i)
fixed capital, (ii) productive capital, (iii) invested capital, (iv) workers, (v) employees,
(vi) wages, (vii) total emoluments, (viii) fuels consumed, (ix) materials consumed, (x)
total inputs, (xi) products and by-products, (xii) total output, (xiii) depreciation, (xiv)
value added, and (xv) outstanding loans.
(d)
Monthly statistics of production of selected industries: The census data referred
above takes a considerable time in publication process. Therefore, it is difficult to use the
data for assessing the short term behaviour of production. This gap is being bridged by the
publication of monthly data in MSPSI, published by the industrial statistics wing of the CSO.
The statistics included in the publication relates to production, installed capacity and stocks
(in physical quantities). The data is collected and compiled from the returns received from
owners of factories. In case of coal, sugar, vegetable oil products, salt, cotton textiles and
most other industries submit the returns on voluntary basis. In some cases where the units do
not furnish the data in a particular month, the production figures are estimated and revised in
the next month. The installed capacity data is based on the estimates of the agencies
responsible for collection of the data or some other appropriate agencies which are
specialized for that purpose. In case of each type of industry this capacity is estimated based
on the technology-continuous process or batch process, and number of days of operations of
the units. For example, in case of sugar, the production potential of individual factories
depend on (i) daily cane crushing capacity (ii) number of actual working days and (iii)
average percentage recovery of sugar from cane.
Trade can be divided into external and internal. External trade can be goods imported or
exported by air, through land route or by sea. The internal trade can be coastal or by road, rail
or through the rivers.
Foreign trade- Foreign trade statistics are mainly collected for (i) analysing the balance of
payment position and determination of size and changes in the foreign exchange holdings, (ii)
preparing and administering the barter and other trade agreements between the countries, (iii)
identifying markets and planning for export promotion and (iv) for estimating the national
income of the country.
The figures of exports and imports are classified commodity-wise and country-wise. The
classification has now been made uniform by following the list prepared by the United
Nations Organisation (UNO). The classification by country is arrived at by deciding the stage
at which the transaction is to be taken into account. For example, for exports the following
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stages are possible: (i) country of final destination, (ii) country of sale, and (iii) country of
immediate destination.
Similarly for imports the stage could be: (i) country of purchase and (ii) country of immediate
destination.
Source of information: The major source of the statistics relating to foreign trade is the
Monthly Statistics of Foreign Trade in India which is compiled by the Department of
Commercial Intelligence and Statistics. This publication follows an international
classification recommended by the Economic and Social Council of United Nations.
The data presented in the publication relates to the figures of foreign trade registered by
customs authorities at Indian Seaports, Airports and Land customs stations. In estimating
these figures a general system of statistical recording is followed. The imports comprise
goods brought across the customs frontier whether they are intended for home consumption
or they are meant for re-exports. ‘Exports’ mean exports of Indian merchandise. ‘Re-exports’
mean exports of foreign merchandise previously imported in India.
The figures relate to quantity and value of commodities traded with foreign countries. The
figures in terms of quantity are based on the declaration made by importers on Bills of Entry,
as subsequently checked by the custom authorities. These figures represent the net weights
exclusive of packing. The value of these goods is assessed by customs authority for their
purposes and it is based on wholesale market prices and represents the wholesale cash prices
for which the like kind and quantity are sold or are capable of being sold at the time of
importation or exportation (as the case may be) without any deduction, except the duties
payable on importation (in the case of goods imported).
The imports are classified as received from countries of consignment. The countries of
consignment may not in all the cases be the countries where it has been produced. Similarly the
exports are credited to the countries of final destination. The publication provides information
relating to the following: (i) value of foreign trade, (ii) overall balance of trade, (iii) foreign
in groups of commodities with each currency area, (vi) index numbers, (vii) value of principal
article of export and import, (viii) foreign trade in treasure, and (ix) foreign trade with selected
countries.
Compilation of export and import statistics by Reserve Bank of India (RBI): The RBI
also compiles the statistics regarding the exports and imports based on the exchange control
data for the balance of payment purposes. In case of export receipts, the main document is
GR which is submitted to customs authorities in triplicate. The first copy of the GR is to be
sent to RBI after adjusting the valuation changes, if any. The exporter gives the other two
copies to the authorised dealers in foreign exchange. The one copy sent to the RBI is the main
source of statistics collected by RBI for exports. In case of imports, the main document of
payment information are the ‘A’ and ‘S’ forms submitted by the importers. ‘S’ form is
submitted by the importers when they deposit rupees with the Government account in
payment for aid financed imports under the direct settlement procedure. The exchange
control data on imports are compiled on the basis of when the exchange is sold or transferred
and not on the basis of the receipt of goods actually imported. The RBI data on imports and
exports is published on quarterly and annual basis in the monthly Bulletin of the RBI. There
are some differences between the RBI data and the data compiled by Department of
Commercial Intelligence and Statistics because of the coverage, difference in valuation,
difference in timings and difference in the imports and exports for repairs and improvement.
Imports and exports licensing statistics: In addition to the above data, the Directorate of
Research and Statistics Office of Chief Controller of Import and Export prepare the detailed
imports licensing statistics. Similarly the value of controlled export trade (which is very small
as compared to total export trade) is also compiled by the Directorate. The data is reported by
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the Directorate in the Weekly Bulletin of Industrial Licenses, Import Licences and Export
Licences.
Coastal Trade: These statistics relate to the movement of merchandise and treasures
between various seaports of the country. These data are published in the Statistics of the
Coastal Trade of India. For the purpose of these statistics, all the seaports have been grouped
into 12 maritime block: (i) West Bengal, (ii) Orissa, (iii) Andhra Pradesh, (iv) Tamil Nadu,
(v) Kerala, (vi) Karnataka, (vii) Maharashtra, (viii) Gujarat, (ix) Pondicherry, (x) Goa, (xi)
Andaman & Nicobar Islands and (xii) Laccadive, Ninicoy and Amindive Islands. The
statistics are derived from daily import returns compiled by customs authorities from the
relevant bills of entry.
Inland Trade (Rail and River Borne): The statistics are reported in the accounts relating to
the inland (rail and river borne) trade of India. The source of material for compiling this
statistics are the invoices relating to the consignment of the related commodities reached at
each railway and steamer station from trade block other than the area in which it is situated.
The figures relate to mainly quantities traded. The quantities are, however, presented in net
weights excluding packing.
Balance of payments: The Reserve Bank of India data presents the balance of payment
statistics on the current account and the capital account. Then current account include (i)
mercandize (import and export of goods), (ii) travel account, (iii) transportation services (iv)
investment accounts, (v) insurance, (vi) government services not included elsewhere (vii)
miscellaneous (for example agency services, film rental and maintenance of state, (viii)
transfer payment and (ix) errors and omission. The capital accounts cover the following
items:
26.1. Private: Private long term capital and private short term capital
movements.
27.1. Banking: The changes in the assets and liabilities of the banking
sector are shown separately under banking.
28.1. Official: (a) Loan: loans raised abroad by the government of India
including drawing on the IMF constitute receipts under loans while
loans extended to foreign governments constitute the payments.
(b) Amortisation: foreign payments under this account are the repayments of loans
secured by the Government of India, while the receipts are the repayments of
loans by foreign governments to India.
(c) Miscellaneous: all other residual capital transactions come under this head, e.g.
changes in PL-480 rupee balances held by the U.S. Government etc.
(d) Reserves: changes in the foreign exchange reserves of this country as a result of
the rest of the transactions.
17.10. Index numbers
In India index numbers are constructed for a wide range of economic subjects and their use is
constantly increasing. The construction of index numbers started in India as early as the last
quarter of 19th century. The main use of index numbers is to facilitate the assessment of
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average changes over the years with regard to wide range of economic activities. Many
official and non-official agencies compile and publish index numbers of various kind.
Index number of industrial production
The Index Number of Industrial production was first compiled by the Office of the Economic
Adviser to the Government of India with base 1937 = 100. The index has been revised from time
to time with respect to the base year, the basis of weighing and the coverage of items. The current
index number is with base 1970 = 100 and the items covered are divided into four groups, viz., (i)
basic industries, (ii) capital goods industries, (iii) intermediate goods industries and (iv) consumer
goods industries.
Index numbers of commodity prices
Index Number of commodity prices may be broadly classified as index number of wholesale
prices and index number of retail prices.
(a)
Index number of wholesale prices- The wholesale price index number are of two
types: (i) The general purpose index is constructed with a view to reflect the changes
taking place in the general price level; hence it includes a large number of
commodities. (ii) A sensitive index on the other hand serves as an indicator of the
movements of the general price levels and it includes only few important commodities
which generally react quickly to the market trends. The current index number of
wholesale prices has 1970-71 as the base year.
(b)
Index number of retail prices- The chief retail price index numbers compiled in
India are (i) Labour bureau index number of retail prices for urban centres and (ii)
Labour bureau index number of retail prices for rural centres.
The Labour Bureau, Ministry of Labour, Government of India, compiles and publishes
the index number of retail prices for 18 selected urban centres and 11 selected
rural centres in various parts of the country on monthly basis. The index
number initially had 1944 as the base. Now the construction of this index
number is discontinued and it is replaced by simple price relatives of certain
selected articles of consumption with the calendar year 1960 as base year.
(c)
Consumer price index numbers- The consumer price index number compiled and
published by the Labour Bureau are important indicators of the changing economic
situation in the country. At all India level three different series of consumer price
index numbers are compiled. They are (i) consumer price index of industrial workers,
(ii) consumer price index for non-manual employees and (iii) consumer price index
for agricultural labourers.
The index numbers of foreign trade of India are compiled by the office of Director General of
Department of Commercial Intelligence and Statistics. These series relate to: unit value
indices of imports, volume indices for imports, unit-value indices of exports, volume indices
for exports, and index of terms of trade, (i.e., ratio of export price index to import price
index). These index numbers are compiled on a monthly basis and published in the
supplement to monthly statistics of Foreign Trade of India and in the monthly Bulletin of
Reserve Bank of India. The annual index numbers are also computed from these figures.
Twenty-six items are included in the index numbers and these are divided into nine groups.
They are: (i) food, (ii) beverages and tobacco, (iii) crude materials, (iv) mineral fuels and
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lubricants, (v) animal and vegetable oils and facts, (vi) chemicals, (vii) manufactured goods
classified chiefly as materials, (viii) machinery and transport equipments, and (ix)
miscellaneous manufactured articles.
Index number of security prices
Index number of security prices are compiled and published by government and commercial
enterprises. The official index number of security prices were compiled and published by the
Economic Adviser with base year 1927-28 up to the year 1949. Thereafter this task was
transferred to Reserve Bank of India. This series of index number of security price is called
Since January 1946 the RBI started a weekly series of security price index number with the
year 1938 as base. This series was revised and the RBI started a new series from July 1957
with the year 1952-53 as base. This new series was revised from time to time and at present
the index number of security price with base 1970-71 = 100 is compiled. In this series the
quotation of scrips are obtained from the published list of Ahemdabad, Bombay, Madras,
Calcutta and Delhi Stock Exchanges.
17.11. Self-test questions
1.
Describe the current statistical system in India.
2.
What is the role of Central Statistical Organisations? What is the role
of the statistical organisations in the states?
3.
Describe the role of various non-government agencies in the statistical
system of our country.
4.
What is the need of labour statistics? Who is responsible for the
collection and compilation of labour statistics in India? Give some
sources of labour statistics in our country.