STATISTICAL PROCESS CONTROL

```STATISTICAL
PROCESS
CONTROL
Statistical Process Control
Statistical Process Control
Table of Contents
I.
STATISTICAL PROCESS CONTROL....................................................................1
A.
INTRODUCTION.........................................................................................1
1.
Concepts...........................................................................................1
2.
What is SPC?....................................................................................1
3.
The Terms.........................................................................................2
4.
Control Methods................................................................................3
Figure 1-1 The Classic Control Cycle...............................................3
Figure 1-2 The SPC Control Cycle...................................................4
5.
SPC Benefits.....................................................................................4
6.
For Success......................................................................................5
7.
Deming and the SPC Story...............................................................6
8.
Obligations of Management..............................................................8
9.
Deming's 14 Points...........................................................................8
10. Calculator..........................................................................................9
B.
GLOSSARY.............................................................................................. 10
1.
SPC Terms..................................................................................... 10
2.
Basic Symbols................................................................................ 20
C.
VARIABILITY............................................................................................ 22
1.
Variability Defined.......................................................................... 22
2.
Distributions................................................................................... 23
3.
Causes of Variability....................................................................... 23
Inherent Causes............................................................................. 24
Assignable Causes........................................................................ 24
4.
The Role of SPC............................................................................ 25
5.
Variables and Attributes................................................................. 26
6.
Summary........................................................................................ 27
D.
DATA DISPLAY AND DISTRIBUTION...................................................... 27
1.
Tabulated Data............................................................................... 28
Figure 1-3 Sample Data Sheet...................................................... 28
2.
Frequency Tally.............................................................................. 29
Figure 1-4 Frequency Tally............................................................ 29
3.
Histograms..................................................................................... 30
Figure 1-5 Histogram..................................................................... 30
4.
Normal Distribution......................................................................... 32
Figure 1-6 Normal Distribution....................................................... 32
Figure 1-7 Normal Distribution - Few Classes............................... 33
5.
Constructing a Histogram.............................................................. 34
k-Graph.......................................................................................... 34
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E.
F.
Figure 1-8 The k-Graph................................................................. 35
Example Data................................................................................ 37
Figure 1-9 Conventional Histogram or Frequency Tally................. 38
Figure 1-10 Completed Histogram................................................. 41
6.
Interpretation.................................................................................. 42
Figure 1-11 Variation Example (Ideal Situation)............................. 42
Figure 1-12 Variation Example (Off Center)................................... 43
Figure 1-13 Variation Example (Well Centered)............................ 43
Figure 1-14 Variable Example (Out-Of-Limit Pieces)..................... 44
Figure 1-15 Variable Example (Parts Outside Both Limits)............ 44
Figure 1-16 Variable Example (Double Distribution)...................... 45
Figure 1-17 Variable Example (Total Spread Greater)................... 45
Figure 1-18 Variable Example (Off-Center)................................... 46
Figure 1-19 Variable Example (100% Inspection Ineffective)........ 46
Figure 1-20 Variable Example (Salvage Limit, Incorrect Gage
Set-Up, Operator Difficulties.......................................................... 47
Figure 1-21 Variable Example (Well Centered Principle
Distribution).................................................................................... 47
Figure 1-22 Variable Example (Favorite Readings)....................... 48
7.
Histogram Exercise........................................................................ 49
Figure 1-23 Sample Data Sheet 1................................................. 49
DESCRIPTIVE STATISTICS..................................................................... 51
1.
Measures of Central Tendency...................................................... 51
Mean.............................................................................................. 51
Median........................................................................................... 52
Mode.............................................................................................. 52
Example......................................................................................... 53
2.
Measures of Dispersion................................................................. 54
Range............................................................................................ 54
Standard Deviation........................................................................ 54
3.
Descriptive Statistics Exercise....................................................... 56
NORMAL DISTRIBUTION CURVE........................................................... 57
1.
General Shape............................................................................... 58
Figure 1-24 Normal Distribution Curve.......................................... 58
2.
Symmetry....................................................................................... 58
Figure 1-25 Symmetrical Distributions........................................... 59
3.
Probability...................................................................................... 59
4.
Area Under the Curve.................................................................... 60
5.
Capability....................................................................................... 60
Figure 1-26 Standardized Normal Distributions............................. 61
6.
Using the Normal Distribution........................................................ 62
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G.
H.
I.
7.
Exercises........................................................................................ 62
Figure 1-27 Normal Distribution Exercises.................................... 63
SAMPLE VERSUS POPULATIONS......................................................... 64
Figure 1-28 Plotted Graph........................................................................ 64
Figure 1-29 Completed Histogram............................................................ 65
1.
Central Limit Theorem.................................................................... 66
Figure 1-30 Control Limits.............................................................. 67
Figure 1-31 Factors for Control Charts.......................................... 68
Figure 1-32 Sampling Distribution of Averages.............................. 69
Figure 1-33 Sampling Distribution of Averages.............................. 70
2.
Sample Versus Population Exercises............................................ 71
CONTROL CHARTS................................................................................. 71
Figure 1-34 Sampling Data Sheet 2......................................................... 72
1.
Preparation.................................................................................... 74
2.
Process Control Charts.................................................................. 74
3.
Control Chart Functions................................................................. 75
4.
Variables Control Charts................................................................ 76
Average-Range Charts.................................................................. 76
Construction Steps for X-R Charts................................................. 77
Figure 1-35 Factors For Control Charts......................................... 80
Exercises........................................................................................ 82
Figure 1-36 Sampling Data Sheet 2............................................... 83
Figure 1-37 X-R Chart Exercise 1.................................................. 84
Figure 1-38 X-R Chart Exercise 2.................................................. 85
Median-range Charts..................................................................... 86
Summary of Median Range Charts Steps..................................... 87
Median-range Charts Exercises..................................................... 88
Figure 1-39 Sampling Data Sheet 2............................................... 89
Figure 1-40 X-R Chart Exercise..................................................... 90
Average-Standard Deviation Charts.............................................. 91
Average-Standard Deviation Charts Summary.............................. 91
5.
Attribute Control Charts................................................................. 92
p Charts.......................................................................................... 93
Construction Steps For Constructing p Charts.............................. 93
Figure 1-41 p Chart Conversion Chart........................................... 93
p Chart Exercise............................................................................. 97
Figure 1-42 Attribute Data Sheet 1................................................ 97
Figure 1-43 Attribute Control Chart................................................ 98
INTERPRETATIONS................................................................................. 99
1.
Nonrandom Patterns...................................................................... 99
Figure 1-44 Random Pattern........................................................100
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2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Freaks...........................................................................................101
Figure 1-45 Freaks........................................................................101
Sudden Shift In Level....................................................................102
Figure 1-46 Sudden Shift In Level................................................102
Trends...........................................................................................104
Figure 1-47 A Trend......................................................................104
Cycles...........................................................................................105
Figure 1-48 Cycles........................................................................105
Grouping.......................................................................................106
Figure 1-49 Grouping....................................................................106
Instability.......................................................................................107
Figure 1-50 Instability....................................................................107
Mixtures.........................................................................................108
Figure 1-51 Mixture.......................................................................109
Stratification..................................................................................110
Figure 1-52 Stratification...............................................................110
Process Capability........................................................................111
z Scores........................................................................................113
Figure 1-53 z Scores.....................................................................113
Figure 1-54 Capability Index.........................................................114
Capability Options.........................................................................115
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Statistical Process Control
I.
STATISTICAL PROCESS CONTROL
A. Introduction
Through the use of Statistical Process Control (SPC) industry can improve
productivity, quality, human relations and profit. These results take time,
patience, and commitment on the part of everyone in an organization - from
management to manufacturing personnel.
SPC is not a quick fix for problems. It is a method of quality management
that operates on facts rather than guesswork. It is a tool for the more efficient
management of business. The goal of SPC is to reduce process and product
variability to increase the quality level of goods as they are produced.
1.
Concepts
The basic philosophies of SPC are:
•
Improved quality leads to improved productivity.
•
Improved productivity leads to lower costs and lower prices.
•
Improved quality and lower prices lead to improved market
share.
•
2.
Improved market share leads to more jobs.
What Is SPC?
Statistical Process Control is more than the use of statistics to solve
business problems. It is a way of thinking about how to manage
operations by continuing to improve both processes and people.
SPC is a fast feedback system. It evaluates people, materials, methods,
machines, and processes by stressing prevention rather than detection.
SPC is a method of managing a process by gathering information
about it and using that information to adjust the process to prevent the
same problem from happening again.
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3.
The Terms
The meaning of the words which make up SPC’s name — statistics,
process, and control — should be understood.
•
Statistics is a scientific method of collecting, classifying,
presenting, and interpreting numerical information. Statistics
is a science that aids in making reasonable decisions in
an uncertain world. It is a body of techniques for gathering
accurate knowledge from incomplete information.
•
A process is any set of conditions, or set of causes, which
work together to produce a given result. In manufacturing,
it refers to the combination of machines, equipment, people,
raw materials, methods and environment that produces a
given product or a specific property of a product. A process
can be a single machine, a group of many machines, a single
person, a group of many people, a piece of test equipment, a
method of measurement, a method of assembly, or a method of
processing.
•
Control is measuring the actual performance of the process,
comparing the results to the standard, and acting on the
difference. Control is how a process is made to behave the way
it should.
Remember: SPC is the use of statistical techniques to analyze a
process or its output so that appropriate actions can be taken to achieve
and maintain a state of control.
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4.
Control Methods
Traditionally, American quality control has been inspection; a sorting
method in which the good is sorted from the bad after production is
completed. This method has led to high costs, high scrap, and lower
quality products. It is referred to as the “Classic” Control Cycle (Figure
1-1).
FIGURE 1-1
The Classic Control Cycle
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The SPC Control Cycle (Figure 1-2) is different. In SPC, the process
is monitored during the production and adjustments are made to the
process before it produces out-of-specification parts or products.
This reduces variability, scrap, and inspection costs while improving
quality.
FIGURE 1-2
The SPC Control Cycle
5.
SPC Benefits
The SPC method of quality control, rather than the inspection-sorting
method, is good because it:
•
Increases customer satisfaction by producing a more troublefree product.
•
Decreases scrap, rework, and inspection costs by controlling
the process.
•
Decreases operating costs by increasing the frequency of
process adjustments and changes.
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•
Improves productivity by identifying and eliminating the
causes of out-of-control conditions.
•
Sets a predictable and consistent level of quality.
•
Reduces the need for receiving inspection by the purchaser.
•
Provides management with an effective and impersonal basis
for making decisions.
•
Increases the effectiveness of experimental studies.
•
Helps in selecting equipment and processes.
•
Helps people to work together to solve problems.
6.For Success
Six areas should be evaluated in any manufacturing process if a
successful quality control program is to exist. They are:
•
Control of the quality of the materials coming into the process
(this is beyond the scope of this training).
•
The accuracy, stability, and variation of the measuring system.
•
The capability of the process measured over a short period of
time.
•
The ability and method to control the process over a long period
of time.
•
An audit of the process to ensure that the control techniques are
operating properly.
•
Ways to ensure continuous improvement.
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Statistical Process Control is not an overnight cure for production
problems. To work well, SPC must be used as an ongoing program
involving all levels of personnel with a joint goal of improving quality
efficiently and continuously.
We know how the process acts normally, and we know how each cause
affects the process; then we can make educated corrections when the
process strays from the norm.
Remember: If the effect each cause has on the process and what can
be expected from that process are known, then corrective action can
be taken when the results are not those desired. This is a key concept
in SPC.
7.Deming and the SPC Story
Dr. W. Edwards Deming has been called the “Father of SPC” because
his work in developing and promoting its basic concept played an
important role in SPC’s growing use.
In the mid-1920s Deming left a teaching post at the University of
Wyoming to join the Federal Bureau of Statistics as a mathematical
advisor. In Washington, he was also responsible for teaching courses
in mathematics and statistics for the Department of Agriculture from
1933-1946. His interest in process control, however, was apparently
sparked by his meeting Dr. W. W. Shewhart in 1928. Dr. Shewhart,
developer of the SPC control charts still in use today, was then a
member of the technical staff at Bell Telephone Laboratories.
Deming recognized the impact Shewhart’s methods could have on
American industry and he soon had a chance to put those methods
to work. After the beginning of World War II, Stanford University
wanted to aid the war effort. Deming suggested that Stanford teach
the simple yet powerful techniques of statistics to engineers and
others. He believed this would bring about better precision and higher
productivity to the nation’s plants. Stanford accepted his offer to teach
the first few courses, and after the first class in July of 1942, Deming
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taught 23 similar courses at various universities. The courses taught
by Deming and others were attended by more than 10,000 people from
800 organizations.
Despite his success, Deming’s course had little effect on the quality
control functions of most organizations because it failed to involve
and educate top management in the use of these techniques. As a
result, control charts had appeared in many organizations, and were
very effective — to a point. Management did not want to hear the bad
news the charts often brought, and gradually the charts disappeared
from use.
In 1945, the Japanese government asked Deming for help in its studies
of nutrition, housing, agriculture, and fishing. He was invited back
for the same reasons in 1948. Then, in 1949, the Japanese Union of
Scientists and Engineering asked him to teach statistical methods to
industry. Deming had his doubts, and feared that SPC would be used
for a short time before “burning itself out.”
Deming’s fears were never realized. After 45 top-level executives
were brought together to hear Deming speak, SPC gained the foothold
it needed to begin changing Japanese industry. The change did not
occur overnight, and Deming made many trips to Japan after 1950.
But the Japanese eagerly accepted Deming’s advice to view quality
improvement as part of a total system.
Today, the term “Made in Japan” no longer means “cheap, poorlymade products.” Japanese products have risen to high levels of quality,
while many U.S. products are now considered inferior or over-priced.
It may be that the quality of U.S. products has not gotten worse,
but that the quality of foreign products has improved. The Japanese
industrial success story is now known worldwide, and Japan has
recognized Deming’s contribution by naming its highest award for
industrial excellence, “The Deming Prize.”
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8.
Obligations of Management
According to Deming, 85 percent of a company’s problems can be
solved solely by management, and only 15 percent can be solved by
the workers. If a company is to see significant gains in productivity,
Deming said, it must change its management methods and styles.
Deming developed his “Fourteen Obligations of Management” to ensure
success in a company’s attempts to improve quality, productivity, and
its competitive position. The 14 points can apply anywhere - to small
organizations and large ones, to service industries, and manufacturing.
They can even apply to a division within a company.
9.Deming's 14 Points
Here is a summary of Deming’s 14 management points:
1. Work toward improving products and services, in order to be
competitive, to stay in business, and to provide jobs.
2. Adopt a new philosophy. With foreign competition, American’s
can no longer live with old styles of management, with
commonly accepted levels of delays, mistakes, and defective
products.
3. Build quality into the product first. Don’t depend on inspection
to achieve quality.
4. Keep the total cost as low as possible. Don’t award business on
the basis of price tag.
5. Improve the production and service system to improve quality
and productivity, and so constantly lower costs.
6. Train on the job.
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7. Improve supervision. Supervision should be able to help people
and machines do a better job. Provide training for management
and production workers.
8. Drive out fear so that everyone can work well for the
company.
9. Break down barriers between departments. All departments
must work as a team to find and solve problems that may be
found in the product or service.
10.
Use slogans and targets that are realistic for the work force.
Unrealistic targets, such as "zero defects" or production levels
that are too high, only create bad feelings between management
and workers. Most of the causes of low quality and low
productivity belong to the system, and cannot be corrected by
the work force alone.
11. Use aids and helpful supervision to meet production
requirements. Don’t use work standards that set numerical
quotas for the day.
12. Encourage “pride of workmanship,” both for hourly workers
and management. Stress quality instead of sheer numbers.
13. Start a strong program of education and training.
14. Put everyone in the company to work to accomplish the
transformation. The transformation is everyone’s job.
10. The Calculator
The electronic calculator is a tool used to solve mathematical problems
more quickly and easily than relying on pencil and paper. Since the
calculator is able to do only what the user tells it to do, its features and
functions must be understood.
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Calculator functions will vary from manufacturer to manufacturer.
Some calculators feature only the most basic functions, such as addition,
subtraction, multiplication and division. More advanced scientific
calculators feature keys for more complex algebraic, mathematical and
statistical work. For calculating some of the formulae used in SPC, the
calculator should carry these more advanced statistical functions.
Because calculators differ, this manual will not attempt to detail steps
for computing the formulae used in SPC. The calculator’s functions as
described in its individual user’s manual or handbook, or as outlined
by the instructor, should be understood.
B. GLOSSARY
1.
SPC Terms
Accuracy - Freedom from mistake or error.
Attribute Data - Qualitative data that typically shows only the number
of articles conforming and the number of articles failing to conform to
a specified criterion. Sometimes referred to as Countable Data.
Average - The sum of the numerical values in a sample divided by the
number of values.
Average Line - The horizontal line in the middle of a control chart
that shows the average value of the items being plotted. Also called
the centerline.
Axis - One of the reference lines of a coordinate system.
Bar Chart - A chart that uses bars to represent data. This type of chart
is usually used to show comparisons of data from different sources.
Bimodel Distribution - A distribution with two modes that may
indicate mixed data.
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Binomial Distribution - A distribution resulting from measured
data from independent evaluation, where each measurement results
in either success or failure and where the true probability of success
remains constant from sample to sample.
Capability - The competency, power, or fitness of a process for an
indicated use or development.
Cells - The bars on a histogram with each cell representing a subgroup
of data.
Central Tendency - A broad term for numerous characteristics of
the distribution of a set of values or measurements around a value
or values at or near the middle of the set. The standard measures of
central tendency are the Mean (average), Median, and Mode.
Class Interval - Interval for dividing variable's values: any of
the variables into which adjacent discrete values of variables are
divided.
Common Cause - A factor or event that produces normal variation
that is expected in a given process.
Control Chart - A chart that shows plotted values, a central line, and
one or two control limits and is used to monitor a process over time.
The types of control charts are:
a. X-bar chart - A control chart where the average of a subgroup
is the measure that is being calculated and plotted.
b. R chart – The range of a subgroup is the measure that is being
calculated and plotted.
c. Median chart – The middle value (median) of a subgroup is
the statistical measure that is being plotted.
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d. p chart – Used for data that consists of the ratio of the number
of occurrences of a defect as compared to total occurrences.
Generally used to report the percent non-conforming.
e. np chart – Similar to a p chart but tracks the number of occurrences of a defect or event.
f. c chart - Used for data that counts the number of units that
contain one or more occurrences of a characteristic.
g. u chart - Similar to a c chart but is used to track the average
number of defects per unit in a sample of n units (constant
sample size).
Control Limits - A line or lines on a control chart used as a basis
for judging the significance of variation from subgroup to subgroup.
Variation beyond a control limit shows that special causes may be
affecting the process. Control limits are usually based on the three
standard deviations around an average or centerline.
Coordinate - Any set of numbers used in specifying the location of a
point on a line, surface, or in space.
Countable Data - The type of data obtained by counting. Attribute
data.
Curve - a.
A graphic representation of a variable affected
by conditions.
b.
A graphic indication of development or
progress.
Data - Facts, usually expressed in numbers, used in making decisions.
Data are gathered by either counting or measurement.
Data Collection - The process of gathering information upon which
decisions to improve the process can be based.
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Detection - A form of product control, (not process control), that is
based on inspection that attempts to sort good and bad output. This is
an ineffective and costly method.
Dimension - Physical form or proportions.
Distribution - A group of data that is described by a certain
mathematical formula. A common distribution observed in industry is
the Normal Distribution. A graphical representation of the variability.
Environment - The complex system of factors including climate,
humidity, temperature, light, etc. which surrounds any process.
Fluctuation - Uncertain, unstable shifts in a sequence of values or
events.
Frequency Distribution - A visual means of showing the variation
that occurs in a given group of data. When enough data have been
collected, a pattern can usually be observed. It exhibits how often
each variable occurs.
Graph - a.
b.
A diagram, which represents the variation of a
variable in comparison with that of one or more
other variables.
The collection of all points whose coordinates
satisfy a given functional relation. Histogram - A bar chart that represents data in cells of equal width.
The height of each cell is determined by the number of observations
that occur in each cell.
Horizontal Axis - The line across the bottom of a chart.
k - The symbol that represents the number of subgroups of data. For
example, the number of cells in a given histogram.
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k-graph - A graph representing the optimum number of class intervals
for the number of measurements available.
Lower Control Limit - The line below the centerline on a control
chart.
Mean - The average value of a set of measurements; see Average.
Median - The middle value (or average of the two middle values) of
a set of observations when the figures have been arranged according
to size.
Mode - The most frequent value in a distribution. The mode is the
peak of a distribution.
Measurable Data - The type of data obtained by measurement. This
is also referred to as Variables data. An example would be diameter
measured in millimeters.
n - The symbol that represents the number of items in a group or
sample.
np - [np-bar] — The symbol that represents the number of
nonconforming items in samples of a constant size.
np - The symbol that represents the centerline on an np chart.
Nonconformities - Something that does not conform to a drawing or
specification; an error or reason for rejection.
Non-random - Having a definite plan, purpose, or pattern. Relating
to a set of elements that do not have a definite probability of occurring
with a specific frequency.
Normality - Occurring naturally.
Numerical - Denoted by a number.
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Out-of-Control - The condition describing a process from which all
special causes of variation have not been eliminated. This condition is
evident on a control chart by the presence of points outside the control
limits or by nonrandom patterns within the control limits.
p - The symbol on a p chart that represents the proportion of
nonconforming units in a sample.
p [p bar] - The symbol on a p chart that represents the average
proportion of nonconforming units in a series of samples.
Pareto Charts - A bar chart that arranges data in order of importance.
For example, the bar representing the item that occurs or costs the
most is placed on the left-hand side to the horizontal axis. The
remaining items are placed on the axis in descending (most to least)
order. Typically, a few causes account for most of the output; hence
the phrase “vital few and trivial many.”
Points Beyond Control Limits - The occurrence of points above or
below the control limits on a control chart. This may be an indication
that a special cause of variation is present.
Poisson Distribution - An approximation to the Binomial distribution.
This distribution is used for np charts.
Population - All members, or elements, of a group of items. For
example, the population of parts produced by a machine includes all
of the parts the machine has made. Typically, in SPC we use samples
that are representative of the population.
Prevention - A process control strategy that improves quality by
directing analysis and action towards process management that is
consistent with the philosophy of continuous quality improvement.
Probability - A mathematical basis for prediction that for an exhaustive
set of outcomes is the ratio of the outcomes that would produce a
given event to the total number of possible outcomes.
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Process - Any set of conditions or causes working together to produce
an outcome. For example, how a product is made.
Process Capability - The common cause variation of a process; the
short-term variation under controlled conditions. This variation will
always be present in a process and the capability measured is the best
the process will ever produce unless changed. This is sometimes called
the short-term capability.
Process Control - Using data gathered about a process to control the
output. This may include the use of controls including SPC techniques
and the establishment of a feedback loop to prevent the manufacture
of nonconforming products.
Process Flow Chart or Diagram - A chart that presents a picture of
the steps followed in making a product.
Process Performance - The statistical measure of the two types
of variation exhibited by a process, within subgroup and between
subgroup. Performance is determined from a process study, which is
conducted over an extended period of time under normal operating
conditions.
Product - What is produced; the outcome of the process.
Proportion - A comparison of the number of nonconformities to the
total number of items checked.
Quality - Conformance to requirements or specifications; i.e., how
well a product is made.
Quantitative - Able to be expressed in terms of quantity or amount.
Random - Lacking a definite plan, purpose, or pattern.
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Random Sampling - A data collection method used to ensure that
each member of a population has an equal chance of being part of the
sample. This method leads to a sample that is representative of the
entire population.
Range - The difference between the highest and lowest values in a
subgroup.
Run Chart - A line chart that plots data from a process to indicate
how it is operating.
Regression Analysis - A mathematical method of modeling the
relationships among three or more variables. It is used to predict the
value of one variable given the values of the others. For example, a
model might estimate sales based on age and gender. A regression
analysis yields an equation that expresses the relationship.
Sample - A small portion of a population.
Sampling - A data collection method in which only a portion of
everything produced is checked on the basis of the sample being
representative of the entire population. Scale - The way in which an axis is divided to show measurements.
Scales are shown on horizontal and vertical axis.
Scatter Diagram - A diagram that shows if a relationship exists
between two variables.
Skewed Distribution - A distribution that tapers off in one direction.
It indicates that something other than normal, random factors are
affecting the process.
Special Cause - Intermittent source of variation that is unpredictable,
or unstable; sometimes called an assignable cause. It is signaled by
a point beyond the control limits or a run or other nonrandom pattern
or points within the control limits. The goal of SPC is to control the
special cause variation in a process.
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Specification - The established limits of acceptable variation for a
product.
Spread - The extent by which values in a distribution differ from one
another; the amount of variation in the data.
Standard Deviation ( σ ) - The measure of dispersion that indicates
how data spreads out from the mean. It gives information about the
variation in a process. Also called sigma.
Statistical Control - The condition describing a process from which
all special causes of variation have been eliminated and only common
causes remain, evidenced by the absence of points beyond the control
limits and by the absence of non-random patterns or trends within the
control limit.
Statistical Methods - The means of collecting, analyzing, interpreting,
and presenting data to improve the work process.
Statistical Process Control (SPC) - The use of statistical methods
and techniques (such as control charts) to analyze a process or its
output so as to take appropriate actions to achieve and maintain a state
of statistical control.
Statistics - A branch of mathematics that involves collecting,
analyzing, interpreting, and presenting masses of numerical control.
Subgroup - A group of consecutively produced units or parts from a
given process.
Successive - Following each other without interruption.
Symmetrical - Capable or being divided by a longitudinal plane into
similar halves.
Tabular - Set up in rows and columns.
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Tally or Frequency Tally - A display of the number of items of a
certain measured value. A frequency tally is the beginning of data
display and is similar to a histogram.
Tolerance - The allowable deviation from standard; i.e., the permitted
range of variation about a nominal value. Tolerance is derived from
the specification and is not to be confused with a control limit.
Trend - A pattern that changes consistently over time.
u - The symbol used to represent the number of nonconformities per
unit in a sample which may contain more than one unit.
Upper Control Limit - The line above the central line on a control
chart.
Variables - A part of a process that can be counted or measured, for
example, speed, length, diameter, time, temperature and pressure.
Variable Data Measurable Data.
Data that can be obtained by measuring. See
Variation - Measurements of the differences in product or process.
A change in the value of a measured characteristic. The two types of
variation are within subgroup and between subgroup. The sources of
variation can be grouped into two major classes: common causes and
special causes.
Vertical Axis - The line that runs up and down on the left side of a
chart or graph.
z Score - The number of sigma units between the process average and
the specification limits.
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Statistical Process Control
2.Basic Symbols
20
A2
Multiplier of R for calculating X chart control limits. X
refers to sample averages.
A2
Multiplier of A for calculating X-chart control limits. X
refers to sample medians.
A3
Multiplier of s for calculating X-chart control limits. This
is when for calculating X-chart control limits. This is when
sample standard deviation is plotted instead of sample
ranges.
B3
Multiplier of s to determine s-chart LCL.
B4
Multiplier of s to determine s-chart UCL.
c
Number of nonconformities or defects in a specified
inspection unit [sample size].
d2
Factor for estimating σ from R.
D3
Multiplier of R to determine A-chart LCL.
D4
Multiplier of R to determine A-chart UCL.
k
Number of subgroups or samples such as the number of
cells in a histogram.
LCL
Lower Control Limit
LSL
Lower Specification Limit
µ
True population average: "mu".
n
Sample size: number of items in sample.
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np
Number of nonconforming items or defectives in a sample
of size n.
np
The central line on a np chart - average number
nonconforming.
p
The central line on a p chart - average number
nonconforming.
R
Range: X highest - X lowest.
R
Average of sample ranges.
s
Sample standard deviation
s
Average of sample standard deviations.
σ
Standard deviation.
σx
Standard deviation of a frequency distribution of individual
measurements [X's].
σx
Standard error of the mean.
σ
True population standard deviation "sigma prime".
X
A random variable: an individual measurement upon which
other subgroup statistics are based.
X
Sample average: X1 + X2 + ... + Xn/n.
X
Average of the averages: "grand average".
X
True population average: "X-bar prime" = µ.
X
Sample median.
X
The average of the medians.
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u
Nonconformities or defects per inspection unit.
UCL
Upper Control Limit.
LCL
Lower Control Limit.
C. VARIABILITY
No two snowflakes examined closely under a magnifying glass have exactly
the same structure or dimensions. They will melt at different rates when
exposed to heat. It cannot be predicted, based on observing two snowflakes,
what the next one will look and act like if it is observed for the characteristics
mentioned above.
In manufacturing a product, it is also impossible to build each part exactly
like the one before it. Each part or product, though it appears identical, will
not be perfectly identical to the one produced before or after it.
No two things are exactly the same, neither in nature nor in a manufacturing
process, due to the law of variability. Understanding how variability works is
vital to producing products that meet some standard of acceptance.
1.
Variability Defined
Variability is defined as the net result of the many (sometimes
immeasurable) factors, which are constantly affecting the process. In
the case of snowflakes, both fall through the same obvious environment,
and therefore some force other than wind, temperature, humidity, etc.,
has acted on the snowflakes to make them slightly different from each
other.
In the case of a manmade product, again many factors are acting at
the same time to affect the finished dimension. Bearing wear, tool
wear, material hardness, dye concentration, pigments, paint viscosity,
temperature and constancy of the power supply are just a few of the
factors that could ultimately affect the finished product.
Remember: By observing natural and manmade products, it must be
concluded that:
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•
Variability is always present.
•
No two objects are exactly alike.
•
Variability in manufacturing is inevitable.
From a manufacturing viewpoint, the total variation must be traced
back to its source and an attempt made to control that variation if
quality products are to be produced every time.
2.Distributions
In any manufacturing process, pieces vary from each other. If only one
measurement is taken, very little about the variability of the process
can be learned. By continuing to take measurements, however, and
plotting the individual measurements on a chart or graph, a form
of distribution occurs that resembles a “bell-shaped” curve. This
distribution could be displayed as a “point-to-point” distribution, a
histogram (which will be discussed later) or a normal bell-shaped
curve.
Distribution (or a graphic representation of the variability) may differ
in location (a situation where the central value has shifted either to
left or the right), in size (where the central value has been reduced and
the spread of the distribution has increased), or in shape (where most
of the measurements are clustered at a point that is not the central
value).
Because distributions are subject to all of the above, some method is
needed to measure and control the variability that always exists.
Even though individual things are unpredictable, groups of things
sampled together from the same system of causes form a predictable
distribution and so are predictable when analyzed as a group.
3.
Causes of Variability
The causes of variability can be categorized into those causes that are
inherent and those that are assignable.
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Inherent Causes
Inherent causes are those, which randomly affect the system. They
are always present and built into the process itself. Inherent variation
represents random changes in the manufacturing process, equipment,
environment, etc. Inherent variation is also called common or chance
variation.
Inherent variation generally cannot be identified to a particular cause
because of lack of knowledge or because identification would be too
costly. These are usually many small, sometimes immeasurable causes
which when acting together add to the total variability. In general,
they cannot be reduced or eliminated without major changes in the
process itself.
Assignable Causes
Variation due to assignable causes represents nonrandom variations in
the process, which can be identified to a particular cause.
Because assignable variation can be identified, it is usually worth the
cost to discover the reason for the variation and correct or eliminate it.
Normally, only a few assignable causes are acting on a system and are
usually things that come and go over time. Assignable variation might
be caused by any one, or more, of the following:
24
•
People – setup (speeds, feeding), accuracy, training, experience,
motivation, etc.
•
Machines – accuracy, calibration, wear, sensitivity, etc.
•
Materials – different compounds, mixing accuracy, calibration,
etc.
•
Measurements – repeatability, precision, accuracy, calibration,
etc.
•
Environment – temperature, humidity, etc.
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4.
The Role of SPC
Any process can vary due to inherent causes and assignable causes.
SPC’s job is to help determine when the variation is only due to the
small, random causes that are inherent, or common, in any system, or
to signal the operator when assignable causes are at work adding to
the overall variation.
Assignable variation might occur suddenly in a process, or over a
considerable period of time. A sudden change in the performance of a
process can generally be detected immediately (for example, operator
change, material change, etc.). A gradual change (trend), or cyclical
change in the performance of a process cannot be detected immediately
(for example, bearing wear, slow loss of calibration, a gradual change
in ambient conditions, arterial changes between lots, etc.).
Process control can be achieved only when the assignable causes in
the system can be identified and controlled.
Remember the following differences between inherent and assignable causes:
Inherent Causes
Assignable Causes
1.
A large number are in
1.
effect at any time.
Very few are in effect at
any time.
2.
Each has an individual
effect that is too small to
mention.
The effect is measurable.
3.
Only a change in the system
3.
will reduce that part of the
variability.
2.
They can be found and
eliminated.
4.
Only management has the 4.
ability to make changes.
The machine operator is
best able to discover and
make changes.
5.
Occur infrequently in
unpredictable fashion.
Chance causes remain
5.
constant over time.
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If a process is said to be in control, it means only inherent causes of
variation are present.
When a process is said to be out-of-control, it means that assignable
causes of variation are present.
If a process is to be controlled, and quality parts produced, it must be
determined which category of variability is acting on the process at any
time. The variability must be categorized, because the responsibility
for improvement action may lie with different levels of management.
In the case of inherent causes of variability, this can usually
be considered a system fault. Management (design engineers,
manufacturing engineers, and/ or industrial engineers) must spearhead
the effort to reduce the variability.
In the case of assignable causes of variation, the fault is usually the
responsibility of the operator or the first-line supervisor.
In SPC, statistics provide a method of identifying when assignable
causes are present in a process. SPC also helps in separating assignable
causes from the inherent causes in a manufacturing process.
The primary objective of SPC is to identify and correct the assignable
causes within the process at the time they occur, rather than find that a
large number of bad or unacceptable products must either be scrapped
or reworked.
5.
Variables and Attributes
One of the important distinctions in the technical language of statistics
is the distinction made between variables and attributes.
A variable is generally known as a measurable characteristic. This
can be inches, meters, thousandths of an inch, temperature, viscosity
or any other measurable characteristic.
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An attribute is generally referred to as countable data. For example, a
record showing only the number of articles conforming to specifications
and the number failing to conform would represent attribute data.
These two concepts will be discussed in more detail in the sections
that deal with the actual construction of control charts. The two major
divisions in control charts occur between Variables Control Charts
and Attribute Control Charts.
6.
Summary
Remember:
•
All manufacturing processes have variability.
•
The control of quality is largely the control of variability.
•
Causes of variability are either inherent or assignable.
•
Assignable causes may be found and eliminated.
•
The future can be predicted in terms of past behavior.
•
The only economical way to improve a process that is "in
control" is to change the system.
D.DATA DISPLAY AND DISTRIBUTION
Very little can be learned about a process if only one measurement, or
sample, is taken from it. If a series of measurements are taken, however, the
differences, or variability, between them can be discovered and steps taken
to eliminate that variability if it is due to assignable causes.
When collecting data, or a series of measurements, it is necessary to display
it in a form that is easily understood. There are several different ways of
displaying data. Each is important, but none by themselves can provide all
the information that might be needed about a process.
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1.
Tabulated Data
It is a common practice to first tabulate, or list, a series of measurements
or readings on a sample data sheet as shown in Figure 1-3.
FIGURE 1-3
Sample Data Sheet
Data displayed as tabulated successive readings may be of limited use
because it does not accurately portray the nature of the distribution
from which the sample was drawn. Tabulated data is only useful when
the order of the samples is important.
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2.Frequency Tally
A type of presentation similar to tabulated data is the frequency tally.
However, the frequency tally provides more information than the
simple tabulation. A frequency tally of the data provided in Figure 1-3
is shown as Figure 1-4.
FIGURE 1-4
Frequency Tally
A frequency tally is simply a tally, or number, of times a particular
measurement or reading occurs in the data. This tally provides the
analyst with the general shape of the distribution he or she is working
with. More than that, specification tolerances could be added to the
distribution which would show if any measurements were outside the
specification tolerances.
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3.
Histograms
Another way of presenting data is a histogram, or bar chart. Histograms
are special types of frequency distributions.
Histograms are the recommended method of displaying data because
they aid in analyzing the distribution of data for centering, spread, and
shape. Histograms can be used to determine whether the process is
operating the way it is desired. Histograms can also be used to identify
the factors that cause the process to vary from what is wanted.
A histogram created from the data in Figure 1-3 is shown as Figure
1-5.
FIGURE 1-5
Histogram
The histogram in Figure 1-5 is constructed with the measured
dimensions (collected data) shown on a horizontal line, and the
frequency of the readings shown on a vertical line.
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One of the problems with using histograms is that the data may be
distorted if the sample is large with small frequencies for some values,
or if the sample is small with a large spread of data values.
It is important that a method be found for grouping the data to provide
a more compact variation pattern. The best method of grouping data
for histogram construction will be covered in detail later. To show as
much information as possible about the distribution, the number of
class intervals in histogram construction must be chosen carefully.
A class interval is an interval for dividing variable's values: any of
the variables into which adjacent discrete values of variables are
divided. If the number of class intervals is too small or too large, the
population’s estimated true shape may not be easily seen.
The bases of the histogram rectangles are always equal, and one class
interval in width. All measurements within any class are characterized
by the midpoint of the interval. Each rectangle height is proportional
to the class frequency in such a way that the histogram total area is
proportional to the total frequency.
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4.
Normal Distribution
If there is enough data to form a large number of classes, in most cases
the histogram takes on the shape of what is referred to as a normal
distribution. Figure 1-6 shows a histogram with a very large number
of classes, representing a normal distribution. Even in a histogram
with fewer classes (Figure 1-7), the shape of the normal distribution
can be seen.
FIGURE 1-6
Normal Distribution
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FIGURE 1-7
Normal Distribution - Few Classes
A histogram will generally be tall in the center and shallow toward
the ends if it follows a normal distribution. If a smooth curve is traced
over the peaks of the histogram bars, the familiar bell-shaped curve
can be seen.
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5.
Constructing a Histogram
Before constructing a histogram, the optimum number of class intervals
for the number of measurements available must be determined. A kgraph is provided for this purpose.
k-Graph
A k-graph is shown in Figure 1-8. The horizontal axis on the k-graph
is scaled for the number of measurements available. The number of
classes (“k”) is shown on the vertical axis of the graph.
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FIGURE 1-8
The K-Graph
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To determine the number of classes to use for a histogram:
1.
Locate on the horizontal axis the number of measurements
taken.
2.
Project upward from that location to the intersection of the
curve.
3.
At the intersection of the curve, read directly across the graph
to the intersection of the vertical axis.
4.
At the intersection of the vertical axis, read the optimum
number of classes to be used for the measurements available.
The number of classes will seldom be located at a whole number;
therefore, the closest whole number of classes should be selected
when developing a histogram.
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Example Data
The following measurements are for the width of 30 bolts of cloth:
Width (Inches)
61.5
61.4
61.3
61.2
61.1
61.0
60.9
60.8
60.7
60.6
60.5
60.4
60.3
60.2
60.1
60.0
59.9
59.8
59.7
59.6
59.5
59.4
59.3
59.2
59.1
No. of Bolts of Fabric
1
1
3
3
3
7
5
3
3
1
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If the data is plotted on a histogram that has on the horizontal axis all
the data points that might be encountered, the histogram would be like
the plot in Figure 1-9.
FIGURE 1-9
Conventional Histogram or Frequency Tally
Although the data appears to be taking on the shape of the normal,
bell-shaped distribution, a much better histogram could be created by
determining the class size using the k-graph.
The following are step-by-step instructions for constructing a
histogram, using the data for the 30 bolts of cloth in the previous
example:
1.
Determine the range of the data. The range is found by
subtracting the lowest value from the highest value:
61.5"
- 59.1"
2.4"
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2.
Determine the number of classes and class size.
•
From k-graph for 30 samples, the number of classes is
6.
•
Class size is range divided by the number of classes: 2.4 '' ÷ 6 = 0.4 ''
3.
Express class width as class size rounded up to the next half
number:
Use 0.5"
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Statistical Process Control
4.
Establish class midpoints and class limits.
a.
Select either the highest or the lowest reading for the
following midpoint:
Midpoint
61.5
61.0
0.5''
0.5''
60.5
60.0
59.5
59.0
b.
0.5"
0.5"
0.5"
Determine the class limits by dividing the class width
by 2 and adding and subtracting the result from the midpoint.
Class limits = Midpoint ± (Class width ÷ 2)
Upper Class Limit = Midpoint + (.5'' ÷ 2)
UCL = Midpoint + .25
Lower Class Limit = Midpoint - (.5'' ÷ 2)
LCL = Midpoint - .25
Midpoint
UCL
61.5
61.0
60.5
60.0
59.5
59.0
61.75
61.25
60.75
60.25
59.75
59.25
LCL
61.25
60.75
60.25
59.75
59.25
58.75
Frequency
1
7
10
8
3
1
(Note: The frequency of the data is found by determining
within what class each of the data points lie.)
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5.
Construct the histogram (Figure 1-10).
a.
The frequency scale of the vertical axis should slightly exceed
the largest class.
b.
The measurement scale on the horizontal axis should be at
regular intervals independent of class width.
FIGURE 1-10
Completed Histogram
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Statistical Process Control
6.
Interpretation
The simplest histogram is helpful in making an analysis, but its use is
limited because it:
•
Requires many measurements.
•
Does not take time into consideration.
•
Does not separate the two kinds of variation - assignable and
inherent.
•
Does not show trends.
A histogram is a picture of the process at one particular time. It portrays
a situation that has already occurred. Since the histogram does not
consider the time factor, it may provide a false picture if a change in
the process over a specific time frame is being determined.
Following are variation examples using histograms (Figures 1-11
through 1-22). An explanation is provided with each example.
FIGURE 1-11
Variation Example (Ideal Situation)
42
An ideal situation where the process
spread is substantially within the specified
limits and is well-centered.
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Statistical Process Control
FIGURE 1-12
Variation Example (Off Center)
The process has drifted off center and is
producing pieces outside the limits.
FIGURE 1-13
Variation Example (Well Centered)
A process with a spread approximately
the same as the specification limits and is
well-centered.
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Statistical Process Control
FIGURE 1-14
Variable Example (Out-Of-Limit Pieces)
A process with a spread approximately the
same as the specification limits which has
drifted off-center and is producing out-oflimit pieces.
FIGURE 1-15
Variable Example (Parts Outside Both Limits)
A process with a spread greater than
the specification limits producing parts
outside both limits.
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FIGURE 1-16
Variable Example (Double Distribution)
A double distribution suggesting that two
different machines or two different setups are involved.
FIGURE 1-17
Variable Example (Total Spread Greater)
A double distribution with total spread
greater than Figure 1-16 resulting in
increased rework and/ or scrap.
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Statistical Process Control
FIGURE 1-18
Variable Example (Off-Center)
A process operating off-center where
pieces have been 100% inspected and the
defective ones removed. This might also
indicate a histogram of run-out where
only plus readings from zero are being
measured.
FIGURE 1-19
Variable Example (100% Inspection Ineffective)
A process resembling Figure 1-18 in
which 100% inspection has not been
entirely effective.
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FIGURE 1-20
Variable Example (Salvage Limit, Incorrect Gage Set-Up,
Operator Difficulties)
A process similar to Figure 1-18 and 119, but indicating the possibility of the
use of a salvage limit, the gage being
set up incorrectly, or an operator having
difficulty deciding borderline cases.
FIGURE 1-21
Variable Example (Well Centered Principle Distribution)
A well-centered principle distribution with
another small distribution that may be the
result of set-up pieces being included in
the lot.
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Statistical Process Control
FIGURE 1-22
Variable Example (Favorite Readings)
A distribution where the operator has
favorite readings because the gaging was
inadequate or difficult to interpret.
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7.
Histogram Exercise
Using the data provided in Sample Data Sheet 1 (see Figure 1-23),
construct a histogram of the data following all steps. Blank graph
paper on which to draw the histogram is also provided on back of this
sheet.
FIGURE 1-23
Sample Data Sheet 1
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E.DESCRIPTIVE STATISTICS
Basic statistics are used to define how like items compare with each other
- that is, how they tend to be the same as a group and, at the same time, how
they differ from each other individually. For example, one group of people
will have an “average” age, yet each person will have a unique age which is
probably different from all the others.
To quantitatively describe large sets of data, two general categories of
statistical measure must be used: the measure of central tendency and the
measure of dispersion.
These concepts will be important in the next sections dealing with normal
distributions and control charts.
1.
Measures Of Central Tendency
A frequency distribution shows approximately where the data is
clustered, but usually a closer estimate (one number) is needed. This
closer estimate can be found by calculating the measure of central
tendency, which indicates where the center of the distribution is
located.
The three measures of central tendency are the mean, median and
mode. Of the three, the most frequently used is the mean, also called
the average.
Mean
The mean is calculated by adding all the observations and dividing
the total by the number of observations. The advantages of using the
mean as a measure of central tendency are that it:
•
Is the most commonly-used measure of central tendency.
•
Is easy to compute.
•
Is easily understood.
•
Lends itself to algebraic manipulation.
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The disadvantage of using the mean is that it is strongly affected by
extreme values and so may not be representative of the distribution.
The mean is commonly calculated with the formula:
∑
X = nX
X = The sum of X divided by n.
X = an individual value or score.
n = the number of individual values or observations in the subgoup or
sample.
∑ = the sum of the observations or values.
Median
If the sample values (the number of the sample values is n) are
arranged in ascending or descending order, the median is the middle
value. However, if there are an even number of values, the median is
the average of the two middle values.
Mode
The sample mode is defined as the value that has the largest frequency.
In most cases this value can be read directly from the frequency
distribution.
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Example
The following measurements of a quality characteristic, X, were made
during a day shift:
60
62 59 60 63 58
58 61
62
60
60
57 63 62 60 60
61
61
60 61 59
58
59
59
61
A frequency distribution could be an easy way of summarizing and
displaying the data. Expressed in a tabular fashion, the frequency
distribution is:
57
58
59
60
61 62
63 1
3
4
7
5
3
2
The sample mean can be found by adding all the Xs and dividing by
the number of measurements (in this case 25).
X = (60 + 62 +... 61 ÷ 25
X = 60.16
The sample median can be found by finding the value, n/2+.5, in the
data listed above. The sample median is the 13th value. Counting
down from the tabular frequency distribution, the 13th value is 60.
Since the sample mode is the value that has the highest frequency,
it can be read directly from the frequency distribution. The sample
mode is 60.
Remember: When the data is a true normal distribution, the value of
the mean, median, and mode will be identical.
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2.
Measures Of Dispersion
Not only is it important to know the central tendency of a distribution,
but also to know the amount of scatter around the central point. It is
possible that the data may be closely grouped near the central point. It
may be uniform or there may be relatively large numbers of extreme
values. It is obvious that some description of spread is needed. This can
be done by calculating a measure of dispersion, which is an indication
of the amount of scatter, or spread, around the central point.
The most important measurements of dispersion are the range and the
standard deviation.
Range
The range is calculated by subtracting the smallest observation from
the largest.
The advantages of using the range are:
•
It is easily understood.
•
It is easily calculated.
The disadvantages are that the range is affected by extreme values and
is inefficient because it ignores some information.
Range, defined in statistical terms, is:
R = X (max) - X (min)
Standard Deviation
Standard deviation, also known as sigma , ( σ ) is a more efficient
estimator of dispersion. Unfortunately, it is somewhat more difficult
to calculate.
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The sample standard deviation can be found by the following
formula:
σ
(X2 - X )2
n-1
Because statistical calculators are available, calculators will generally
be used to obtain the standard deviation. The standard deviation of the
previously discussed data is 1.57.
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Statistical Process Control
3.Descriptive Statistics Exercise
Find the median, mean, mode, range and standard deviation of the
following sample data:
16.7
17.0 16.8 16.9 17.1 17.1
17.0 16.9 16.8 16.9 16.9
17.0
16.8
16.7 16.6
Mean _________________
Median _________________
56
Mode
_________________
σ
_________________
R
_________________
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Statistical Process Control
F.
NORMAL DISTRIBUTION CURVE
The primary distribution utilized in SPC is the normal distribution.
Many things in nature, such as the heights of all males in the United States,
follow what has become known as the normal distributions. In the late 1800s,
a group of scientists in Great Britain who were studying the human anatomy
discovered that the data they collected followed a certain pattern. As the
researchers recorded other data, such as the length of the thigh bones, they
again found that the data displayed similar patterns. Many other researchers
studying different subjects discovered the same types of patterns in their
data.
All data sets, however, do not follow the normal distribution. For example,
the distribution of all the heights of the people in the United States, both
male and female, would have two separate groupings. One grouping would
be for females, who are typically smaller than males, and another grouping
for males.
In industry, a machine might run to the “high side” and produce a distribution
that would have a larger number of higher values. The data collected from
this machine would not follow a normal curve.
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Statistical Process Control
1.
General Shape
A normal distribution curve is shown in Figure 1-24. Because the
mean, median, and mode are exactly equal, the curve has the bell
shape, which is characteristic of a normal distribution.
FIGURE 1-24
Normal Distribution Curve
2.
Symmetry
The curve in Figure 1-24 is also exactly symmetrical. If the curve
were cut in half, each side would be a mirror image of the other.
Although the normal distribution is symmetrical, all symmetrical
distributions are not normal distributions. Characteristics other than
symmetry must be examined before the normality of the data can be
determined.
58
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FIGURE 1-25
Symmetrical Distributions
In Figure 1-25, all four of the distributions are symmetrical. Only
one (distribution D) is normal. This illustrates that all symmetrical
distributions are not normal, but a normal distribution is
symmetrical.
3.
Probability
Under a normal curve, the total area of the distribution is 1. This
means that if the probabilities of all possible outcomes in a set of data
are considered, the total of those probabilities must equal 1.
When a single die is tossed, the probability of getting a one is 1/6; the
probability of getting a two is 1/6; the probability of getting a three is
1/6 and so on. In fact, the probability of getting any number between
one and six is 1/6, and if all of the probabilities are added together, the
total would be 1. This characteristic of the normal distribution can be
applied to a variety of situations.
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Statistical Process Control
4.
Area Under The Curve
In a normal distribution, the area under the curve can be determined
because the curve is completely described by a mathematical formula.
Combining two characteristics of the normal distribution (the fact that
the total area is equal to 1 and the fact that this area can be determined)
allows the areas of the normal curve to be converted to probabilities.
If the mean, or average, and the standard deviation are known (each
is described in the previous section), the normal distribution can be
fully described.
5.
Capability
The normal distribution has a number of other important characteristics,
as follows:
•
The areas on either side of the mean are equal.
•
About 68.25 percent of the total area is included within a
distance of ± 1 standard deviation from the mean.
•
About 95.45 percent of the total area is included within a
distance of ± 2 standard deviations from the mean.
•
About 99.73 percent (or nearly all) of the area is included
within a distance of ± 3 standard deviations from the mean.
The curves in Figure 1-26 are the percentages of area under the curve
for ± 1, 2 and 3 standard deviations for a distribution with a mean of
0 and a standard deviation of 1.
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FIGURE 1-26
Standardized Normal Distributions
Since almost all of the area under the curve is included within ± 3
standard deviations from the mean, American industry has defined
capability as ± 3 standard deviations, or 6 standard deviations. This
6-sigma rule of capability will be discussed in greater detail in the
section covering Control Chart Interpretation.
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6.Using The Normal Distribution
The normal distribution is important in quality control for two
reasons:
7.
•
Many distributions of quality characteristics of a product are
reasonably similar to the normal distribution. This makes it
possible to use the normal distribution or estimating percentages
of product that are likely to fall within certain limits — that is,
a process’s capability.
•
Even when the distribution of product is quite far from normal,
many distributions of statistical quantities, such as averages,
tend to distribute themselves in accordance with the normal
distribution. For this reason, the normal distribution has
important uses in statistical theory, including some of the
theory that underlines control charts.
Exercises
1.
2.
62
Using the fours sets of data provided in the Normal
Distribution Exercises (see Figure 1-27), use the concepts of
normal distribution to calculate the largest and smallest parts
that could be expected. Do this by calculating:
a.
The mean and standard deviation of each data set.
b.
The value of ± 3 standard deviations for each data set.
Compare this rough estimation of capability obtained in each
of the Normal Distribution Exercises with the specifications
listed on the data sheet. Can these processes be improved?
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figure 1-27
Normal Distribution Exercises
80 PCS
45 PCS
1.249
1.250
1.251
1.252
1.253
1.254
1.255
1.256
1.250
1.255
( x ) Mean
( ) STD. DEV
x+3
x -3
75.25 mm
75.35 mm
75.25
75.26
75.27
75.28
75.29
75.30
75.31
75.32
75.33
75.34
75.35
75.36
75.37
75.38
75.39
75.40
( x ) Mean
( ) STD. DEV
x+3
x -3
.869
.875
.8 mm
.4 mm
20 PCS
80 PCS
.867
.868
.869
.870
.871
.872
.873
.874
.875
.876
.877
.6
.8
.7
.5
.6
.4
.8
.9
.6
.4
.5
.6
.7
.5
.6
.7
.8
.5
.7
.6
( x ) Mean
( x ) Mean
( ) STD. DEV
( ) STD. DEV
x+3
x+3
x -3
x -3
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G. SAMPLE VERSUS POPULATIONS
Often, data is presented in a chart that looks like this:
356 358
361 357
350 349
356
355 358
357 359 359
365
355
354
360
361
361
359
358
The chart above really does not provide much information about the data.
Few conclusions can be drawn from it. The data is simply a group of numbers
with little meaning.
The same data can also be plotted on a graph. A graph of the above data is
shown in Figure 1-28.
FIGURE 1-28
Plotted Graph
A plotted graph provides a little more information about the data, but
still not enough for many conclusions to be drawn. How often each
number occurs is, however, more readily seen.
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A histogram will provide even more information about the data. A
histogram of the same data is shown in Figure 1-29.
FIGURE 1-29
Completed Histogram
The histogram provides considerably more information about the
data. It provides a clearly understood picture how often each number
or measurement occurs. Using the histogram, the data can be analyzed
for spread, centering and shape.
If the same average (mean) and the standard deviation (sigma) are
calculated, there is even more information which can be used to
analyze the data. For the data discussed here, the average is 357.4 and
the standard deviation is 3.73.
After the data has been seen in several different forms, and the average
and standard deviation have been calculated, conclusions about both
the sample itself and the population from which it can be drawn. In
order to draw conclusions about an entire population based on sample
data, it is important to understand the applications of the Central
Limit Theorem.
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1.
Central Limit Theorem
A population, also called a universe, parent distribution, or distribution
of individuals, can be thought of as the potentially unlimited output of
a manufacturing process, or as a particular lot of manufactured articles.
Because it is obviously impractical to measure every item produced
by a process, samples must be relied on to provide information about
the process.
Samples and populations from which they are taken are related by a
mathematical law called the Central Limit Theorem. Because of this
relationship, the population actually determines the center and spread
(mean and standard deviation) of the sample, and to a certain extent,
the sample distribution’s shapes.
Some of these relationships are beyond the scope of this SPC manual,
but it is important to understand the concepts behind the sample
distribution of averages. The most common control chart used in SPC,
the average-range chart (X-R chart) is based on sample averages, the
average of sample averages, and the standard deviation of sample
averages (standard error of the mean, σ or x ).
According to the Central Limit Theorem:
66
•
The average (center) of the sample averages will be the same as
the population average.
•
Samples selected from a normally distributed population will
also be normally distributed.
•
The averages ( Xs ) of sample selected from an abnormally
distributed population will be normally distributed for sample
sizes of 30 or more.
•
The standard deviation of the distribution of sample averages
(or standard error of the mean) will equal the standard deviation
of the population divided by the square root of the number of
samples per subgroup. Control limits on X - R charts (Figure 1AIDT - Statistical Process Control - October 5, 2006
Statistical Process Control
30) are actually the 3 sigma limits for the sampling distribution
of averages, or ± 3 σ x . This factor has already been calculated
for different subgroup sizes and included in the Table of
Control Chart Factors in Figure 1-31. This table will be used in calculating control limits on control charts.
FIGURE 1-30
Control Limits
Several examples of samples taken from different distributions are
shown on the following pages as Figures 1-32 and 1-33 to further
demonstrate the Central Limit Theorem. In each of the four different
cases, the averages remain the same.
The Central Limit Theorem holds true in almost all cases when
samples of 30 or more are selected from a population, regardless of
the population’s distribution. As the sample size increases, the bellshaped pattern or the normal distribution becomes more evident when
the samples are plotted onto the graph.
Remember: The Central Limit Theorem provides critical information
needed to understand and analyze how any process is operating based
on random samples of 30 or more selected from the population. Using
this theorem, it is unnecessary to inspect or measure a large portion of
the population.
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FIGURE 1-31
Factors For Control Charts
68
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FIGURE 1-32
Sampling Distribution of Averages
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69
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FIGURE 1-33
Sampling Distribution of Averages
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2.
Sample Versus Population Exercises
For the 125 data points on Sample Data Sheet 2 (Figure 1-34),
calculate:
1. The average and standard deviation of the 125 values.
2. The sample average and range of the 25 subgroups of 5 values
each.
3. The average of the averages from Step 2 and the standard
deviation of the subgroup average. Compare them to the values
calculated in Step 1.
4. Construct histograms of the individual values and of the sample
averages using the blank graph paper provided with Figure 134). Note the similarities and differences between the two.
(Note: A histogram of the individual values was done in Section D on
the Display/Distribution of Data.)
H. CONTROL CHARTS
The Control Chart is one of the most important tools of SPC. Control charts
are simple, yet powerful tools for checking the stability of a process over
time, as well as verifying the results of any improvement actions taken.
Remember: The measured quality of any manufactured product is subject
to a certain amount of variation as the result of chance. A stable “system
of chance causes” is inherent in any scheme of production and inspection.
This variation is unavoidable as long as the production and inspection system
remain unchanged. However, causes of variation outside this stable pattern
can be discovered and corrected.
The power of the control chart is in its ability to separate these assignable
causes of quality variation from inherent, unavoidable causes.
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FIGURE 1-34
Sampling Data Sheet 2
72
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73
Statistical Process Control
In this regard, the purpose of control chart analysis is to identify:
•
Evidence that the inherent process variability and the process average
are no longer operating on stable, controlled levels.
•
Evidence that one or both are out of control (unstable).
•
The need for corrective action.
While the term “control chart” is widely accepted and used, it must be
remembered that the control chart does not actually control anything. It
simply provides a basis for taking action. It is effective only if those who are
responsible act on the information the chart reveals.
1.
Preparation
Before a control chart can be used, several steps must be taken:
2.
1. Management must prepare a responsive environment.
2. The process that is to be studied must be understood.
3. The characteristics to be controlled must be determined.
4. The measurement system must be defined.
5. Unnecessary variation must be minimized.
Process Control Charts
A control chart is a simple, three-line graph — a graphic display of
how data occurs over time. Special, assignable causes of variability
(process instabilities suggesting a process which is out-of-control)
occur or evolve in unusual ways, which will be reflected by the data.
Determining the presence of such a source of variability (assignable
causes) in the presence of the stable variability (inherent causes) is
possible by identifying unusual patterns and unexpected data points
on the control chart.
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In this three-line graph, the centerline represents the average
performance of the process for a particular statistic (mean, range,
percent, defective, etc.). The two outer lines are called control limits.
These control limits are usually set up symmetrically above and
below the centerline so that there is a 97.7 percent chance that a point
will fall between the two limits as long as the average performance
of the process has not changed. A point outside the control limits or
nonrandom (unusual) patterns in the data within the two control limits
indicate that a change in the average performance of the process has
occurred.
If the control chart shows no points outside the control limits, and no
unusual patterns within the control limits, then the process is under
control and there are no assignable variations present.
3.
Control Chart Functions
Awareness of the reference distribution underlying a particular control
chart is of primary importance.
The control chart provides clear documentation of process variation
in an easily understood form. As variability is reduced, there is also
less masking of the smaller effects of any corrective actions made in
a process.
Some basic functions of control charts are to:
•
Monitor process performance over time.
•
Describe what control there is.
•
Help attain control by detecting change in the performance of
the process.
•
Estimate the capability of the process.
•
Signal when corrective action is needed.
•
Verify the results of any corrective action.
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Statistical Process Control
The remainder of this section will discuss the two categories of control
charts: Variables Control Charts and Attribute Control Charts.
4.
Variables Control Charts
Variables Control Charts are used to record and monitor process
performance with respect to the selected variable.
Remember: Variables are those parameters which can be measured.
The three types of variables charts are:
•
Average and Range Charts ( X - R charts).
•
Median and Range Charts ( X - R charts).
•
Average and Standard Deviation Charts ( X - S charts).
Average-Range Charts
Average and Range charts are developed from measurements of
particular characteristics of the process output.
The X - R is one of the most powerful and sensitive SPC tools.
An X chart and an R chart, as a pair, represent a single product
characteristic. The data is reported in small subgroups, usually including
from 2 to 5 consecutive pieces, with subgroups taken periodically (for
example, once every 15 minutes, twice per shift, etc.).
In order for the charts to be effective:
76
•
Each item must be accurately measured for the characteristic
being observed.
•
The subgroups must be chosen so that the variation among the
units represents the variability that cannot be controlled in the
short-run. However, variation between subgroups can ref﻿lect
changes in the process that can be controlled.
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Statistical Process Control
Construction Steps for X-R Charts
In preparing to plot and construct X - R charts, f﻿irst the inspection
data must be gathered, recorded and plotted on the chart according
to a def﻿inite plan. For an initial study of a process, the subgroup
should consist of 4 to 5 consecutively produced pieces that represent
production from a single tool, machine, head, die cavity, etc. This will
ensure that the pieces within a subgroup are all produced under very
similar conditions during a short time interval.
Following are step- by-step procedures to be used in constructing
X - R charts:
1.
2.
Label the chart.
a. Enter the name of the part.
b. Enter the part number.
c. Enter the specif﻿ications.
d. Enter the plant identif﻿ication.
e. Identify department.
f. Enter the machine number.
g. Enter operation number or other information needed to
identify the process.
Enter basic information for each sample, including:
a.
Date and time the sample was taken.
b. The initials of the checker (operator).
c. The shift on which the samples were taken.
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Statistical Process Control
3.
Enter measured data.
Raw data for at least 25 subgroups (representing 125 or more
individual readings) should be collected before accurate control
limits can be established.
4.
Calculate total, average and range for each subgroup.
Place the result in the space provided on the control chart
form.
5.
Establish scales.
The establishment of the vertical scales is generally determined
by the person doing the plotting. Scales should be developed
which make it easy to plot the data. As a rule of thumb, the
range of values of the scale should at least include the larger
of:
•
The product specification or
•
Two times the difference between the highest and lowest
subgroup averages.
For the R chart, values should extend from a lower value
of zero to an upper value of 1.5 to 2 times the largest range
encountered during the initial period.
6.
Plot data.
On the chart form, there are lines drawn from the center of the
piece number data blocks to the bottom of the chart. These lines
are used to plot the average and range data calculated from the
samples taken previously. Plot the data points, both average
and range, on their respective charts. Connect the points with
lines so that patterns and trends can be seen.
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7.
Determine centerline.
Calculate X (the average of the averages). X will be the
centerline for the X chart. Calculate R (the average of the
ranges.) R will be the centerline for the R chart.
a. To find X, add all of the subgroup averages together and
divide by the number of subgroups.
b. To find R, add all of the sample ranges together and
divide by the number of subgroups.
Locate the corresponding points on the vertical axis on the
scales established on the respective charts. Draw the centerlines
on the chart form as solid lines.
8.
Calculate control limits.
Without control limits, there is no way to determine if a process
is operating in control. The control limits represent the mean
range and the process average plus or minus an allowance for
the inherent variation that can be expected.
Control limits are based on the subgroup sample size and the
amount of variability reflected in the range.
The upper and lower control limits are based on moving out 3
standard deviations from the average. Since a subgroup sample
that exceeds the upper or lower control limits is a signal to
look for assignable causes in the process, control limits must
be wide enough so that time will not be spent searching for
assignable causes when the signal is false. If the limits are too
widely spread, there is a risk that a timely or significant change
in the process will not be found.
Remember: Most American industries accept the 3-sigma
limit, which limits the false signals to 0.27 percent in normal
populations.
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Statistical Process Control
Tables have been developed to assist in calculating the control
limits (see Figure 1-35). Using relationships between samples and
populations derived from the Central Limit Theorem, the following
formulae indicate how to calculate the 3-sigma control limits.
FIGURE 1-35
Factors For Control Charts
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Statistical Process Control
UCL for X = X + A2R
LCL for X = X - A2R
UCL for R = D4R
LCL for R = D3R
The values for X and R have already been calculated. The other values
in the formulae are found in Figure 1-35 and depend on the size of the
sample subgroup.
9.
Plot the control limits.
The control limits are drawn as dashed horizontal lines starting at the
corresponding points on the chart scale. These lines should be labeled
UCL an LCL respectively.
10.
Interpret the chart.
If the process short-term variability and the process average remain
constant at their present levels, the individual subgroup ranges and
averages would vary by chance alone, but they would seldom go
beyond the control limits.
The object of control chart analysis is to identify that the process
variability or the process average are no longer operating at previously
established levels of acceptance, and to signal the need for appropriate
corrective action.
Summary of Steps
Here is a summary of the steps used in preparing an X - R chart:
1.
Properly label the chart.
2.
Collect and record data.
3.
Select scales.
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81
Statistical Process Control
4.
Plot data.
5.
Develop R chart first.
a.
Establish centerline (R).
b.
Calculate control limits.
UCL for R = D4R
LCL for R = D3R
6.
Develop X chart:
a.
Establish centerline (X).
b.
UCL X = X + A2R
LCL X = X - A2R
7.
Draw lines on control chart.
8.
Interpret chart.
Exercises
1.
Using the 125 data points on Sample Data Sheet 2, Figure
1-36, construct an X - R chart. A blank Control Chart Form
(Figure 1-37, X - R Chart Exercise 1) is provided.
2.
Using the data recorded on the Control Chart Form (Figure 138) labeled X - R Chart Exercise 2, do the following:
a.
b.
82
Construct a histogram.
Complete X - R chart.
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Statistical Process Control
FIGURE 1-36
Sampling Data Sheet 2
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83
Statistical Process Control
FIGURE 1-37
X - R Chart Exercise 1
84
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FIGURE 1-38
X - R Chart Exercise 2
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85
Statistical Process Control
Median-range Charts
Median and Range control charts X - R charts is very similar to those
for X - R charts, but fewer calculations are needed. The steps for
constructing X - R charts are as follows:
1.
Label chart.
2.
Enter basic information for each sample.
3.
Enter measured data.
Usually a sample size of 3 or 5 will be used on X - R charts.
If a sample size of 3 is used, 40-50 subgroups are needed
in order to establish control limits. The median and range
must be calculated for each subgroup. Important: Medians
are calculated by arranging the observations in ascending
or descending order and selecting the middle observation,
rather than calculating the actual middle value as previously
discussed.
4. Establish scales.
5. Plot the data.
For X - R charts, every individual piece of data is plotted. The
middle data points (medians) should be connected by lines.
6. Establish the centerlines.
For the X chart, the centerline is the average of the medians
(X). For the R chart, the centerline is the median of the ranges
(R). Draw the centerlines on each chart as solid lines.
7. Calculate control limits.
The control limits for X - R charts are based on the 3-sigma
limits, just as in X - R charts. The formulas used to calculate
these control limits are as follows:
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UCL R = D4R
LCL R = D3R
UCL X + X + A2R
LCL X = X - A2R
The values for A, D, and D depend on the size of the subgroup.
Typically, subgroup sizes of 3 or 5 are used and the values of
these constants are given in the table below.
A 2
D 3
D 4
8. n = 3
1.19
0.00
2.574
n=5
0.69
0.00
2.114
Plot the control limits.
The control limits are drawn as dashed lines on the chart and
labeled UCL and LCL respectively.
9. Interpret the chart.
The same rules for interpreting X - R charts apply to X - R
charts.
Summary of Median Range Charts Steps
Here is a summary of the steps used to develop X - R charts:
1. Label chart.
2.
Collect data, establish scales and plot all data.
3.
Develop R chart first.
a.
Centerline = R
b.
UCL = D4R; LCL = D3R
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87
Statistical Process Control
4.
Develop X chart.
a.
Centerline = R
b.
UCL = X + A2 R; LCL = X - A2 R
5.
Draw centerlines and control limits.
6.
Interpret chart.
Median-Range Charts Exercise
Using Sampling Data 2, Figure 1-39, construct an X - R chart. Use
the Blank Control Chart Form labeled X - R Chart Exercise 1 (Figure
1-40).
88
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FIGURE 1-39
Sampling Data Sheet 2
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89
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FIGURE 1-40
X - R Chart Exercise
90
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Statistical Process Control
Average-Standard Deviation Charts
The same approach is taken in developing Average and Standard
Deviation Charts ( X - S ) as for X - R charts. The difference in these
two types of charts is that the sample standard deviation is plotted
instead of the sample range.
All of the steps used in constructing X - R charts apply to these charts,
but the standard deviation ( σ ) must be calculated instead of R.
X - S charts are used for sample sizes greater than 10, and are seldom
used by operators in production facilities. The X - S chart is used
primarily in laboratories and in research and development work.
Average-Standard Deviation Charts Summary
Because the X - S chart is seldom used in production operations, the
steps for developing them are simply summarized below:
1.
Label chart.
2.
Collect data. Calculate X and S for each sample. Establish
scales and plot data.
3.
Calculate X and S to develop the centerlines. Draw the
centerlines on the chart as solid lines.
4.
Calculate the control limits. The values for the constants in the
formulae below are found in special tables for X - S charts.
Those tables are not included in this manual.
UCL S = B4S
LCL S = B3S
UCL X = X + A2S
LCL X = X - A2S
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91
Statistical Process Control
5.
5.
Draw the control limits on the chart as dashed lines.
6.
Interpret the chart.
Attribute Control Charts
Although variables charts have their advantages for use in a
production operation, their use is limited to only a fraction of the
quality characteristics specified for manufactured products. They are
charts for variables, or quality characteristics that can be measured
and expressed in numbers.
Many quality characteristics can be observed only as attributes, which
cannot be listed and plotted on a numerical chart. Attributes generally
fall into two classes: either good or bad, go or no-go, conforming or
nonconforming.
In some cases, characteristics, which could be measured and plotted
as variables data, are controlled by attribute data strictly due to the
number of dimensions that would have to be controlled if the data
were handled as variable data.
There are several different types of attribute control charts, which may
be used in these cases:
•
p charts, for the fraction rejected as nonconforming to
specification.
• np charts, for plotting the number of nonconforming items.
• c charts, for the number of nonconformities.
• u charts, for the number of nonconformities per unit.
This manual will outline the steps for constructing p charts, because
they are the most widely used attribute charts.
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p Charts
Remember: The p chart is a control chart for the fraction rejected as
nonconforming to specifications, often referred to as defects.
Construction Steps For Constructing p Charts
The following are steps for constructing p charts. A graphical display
of the steps are shown in Figure 1-41.
FIGURE 1-41
p Chart Conversion Chart
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1.
Gather data.
For p charts to be used effectively, the sample size should be
at least 50. If possible, the sample size should be the same for
each sample, but because the p chart is often used to monitor
lots, the sample size may vary. Generally, if the sample size
does not vary more than 25 percent, control limits based on the
average sample are acceptable. If the sample size varies more
than 25 percent, control limits for that individual sample must
be calculated based on that sample size.
The frequency of the sample should be often enough to detect
variation in the process being charted.
The sample size should be great enough to include a number
of the nonconforming units per sample. Another rule of thumb
is that the sample size should be large enough to include 4
or 5 nonconforming unit spc sample, to ensure that process
variability can be detected.
2.
Calculate p.
After the number of nonconforming units per sample have been
recorded, the fractions of nonconformities (p) are calculated.
As an example, if a sample of 500 is found to have 12
nonconforming units, the fraction of nonconformities would
be calculated as follows:
p = the number of nonconformities divided by the total
number of items in the sample.
In the example above:
p = 12 ÷ 500
p = .024
This calculation must be done for each sample.
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3.
Plot the data.
The p chart form is different from a variables control chart
form. Because only one value, the fraction nonconforming, is
plotted, it has only one section.
Like the X - R chart, an appropriate scale must be established.
The scale should start at 0 and should include at least 1.5 times
the highest sample point.
Once the scale has been established, the fractions of
nonconformities per sample (or p values) are plotted.
4.
Calculate control limits.
First, the average proportion of nonconformities per sample
( p ) is calculated. The average proportion is determined by
summing all the nonconforming items and dividing the total by
the total number of items inspected.
As an example, if 405 nonconformities were found in a total of
12,500 inspected items:
p = ∑np ÷ ∑n
p = 405 ÷ 12,500
p = .0324
p becomes the centerline of the p chart.
Next, the control limits for the process are calculated. UCL is
the upper control limit for the fraction of nonconforming items.
LCL is the lower control limit for the fraction of nonconforming
items. These are the 3 sigma limits previously discussed. UCL
and LCL are calculated using the following formulae:
UCL = p + 3
p(1-p)
n
LCL = p + 3
p(1-p)
n
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p = the average proportion of nonconforming parts.
n = the number of items per sample.
In a case where the sample size varies, if the sample size does
not vary more than 25 percent, then the average sample size
could be used for n.
Figure 1-41 contains the completed calculations using these
formulae:
1.
Plot the upper and lower control limits on the chart using
dashed lines.
2.
In some cases where p and n are small, the LCL may
become a negative number when it is calculated. In this
event, 0 should be used for the LCL.
Note: There is no set rule for calculating control limits when
sample subgroup sizes vary. They may be calculated as detailed
above, or for each individual subgroup. Other techniques
include calculating a single set of control limits based on the
average subgroup size, or calculating using separate sets of
control limits based on each subgroup size. Another widely used
method is to calculate three sets of control limits - one based
on the average subgroup size, a second based on the smallest
subgroup size, and a third based on the largest subgroup size.
When using these methods, either color code or clearly label
the control limits on the chart to avoid confusion over which
control limits apply for the data points.
Each of the preceding steps is illustrated in Figure 1-41.
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p Chart Exercise
Construct a p chart from the data contained in Attribute Data Sheet 1
( Figure 1-42). Because the sample size is not constant, a control limit
for each data point should be calculated. The formulae for nonconstant
sample size is the same as the calculation for constant sample size.
Note: Since the formulae “3 times the square root of p times (1 - p)
will be a constant, it is only necessary to divide the sample size by the
square root of n in each case to find 3-sigma. When the 3-sigma value
is added or subtracted from p , the upper and lower control limits for
each sample are found.
A Blank Attribute Control Chart is shown in Figure 1-43.
FIGURE 1-42
Attribute Data Sheet 1
A value assembly is inspected 100%. Data on 3 weeks production is given below.
Day
Production
Nonconforming
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
80
80
80
95
95
95
95
60
60
60
60
60
115
115
115
3
2
2
7
2
2
1
3
0
2
3
2
4
0
3
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FIGURE 1-43
Attribute Control Chart
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I.
Interpretations
1.
Nonrandom Patterns
A control chart is used to determine if a process is in control or out of
control. When a process is in control, there are no assignable variations
working in that process. When it is out of control, there are assignable
variations influencing the process.
The following basic criteria should be used for determining an out-of
control process:
•
A point outside the control limits.
This indicates that an external influence, or influences, exists
or that an assignable cause is present.
•
A run.
A change in the process can occur even when no points fall
outside the control limits. The change can be observed when
successive points are on one side of the centerline but still
within the control limits, a rule of thumb for detecting a run is
7 or more points on the same side of the centerline.
•
A trend.
Sometimes there is a steady, progressive change in the
performance of the process. This is called a trend, and may be
caused by wear or deterioration. A rule of thumb for detecting
a trend is 6 or more points moving upward or downward.
If the points on a chart fluctuate near the centerline without a distinct
pattern, then the process can be considered a randomly operating
system. However, joints in a nonrandom system will form a distinct
pattern. The ability to interpret a control chart depends on the ability
to distinguish between random and nonrandom patterns.
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If the pattern is random (Figure 1-44), the process is under control.
FIGURE 1-44
Random Pattern
When a process is in control, no point will fall outside the control limits
and the points within the control limits will not exhibit any unusual
patterns. A process that is in control will also show the following
characteristics on a control chart:
•
Most of the points will be near the centerline.
•
A few of the points will be near the control limits.
•
None of the points will exceed the control limits.
All three of these characteristics must occur simultaneously to consider
the process under control.
If the pattern is nonrandom, the process is not under control.
Nonrandom patterns always involve the absence of one of the three
characteristics of a random pattern. Nonrandom patterns either
fluctuate widely about the centerline, do not fluctuate widely enough,
or group themselves on one side of the centerline.
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The following patterns are considered nonrandom:
•
Freaks
•
Grouping
•
Sudden Shifts •
Instability
•
Trends
•
Mixture
•
Cycles
•
Stratification
The patterns named above are the most commonly encountered
nonrandom patterns, but there are others that might be encountered.
2.Freaks
Freaks (Figure 1-45) are the result of a single unit or a single measurement
being greatly different from the other units or measurements. Freaks
are generally due to outside causes. On rare occasions, measurements
that appear to be freaks are in reality a normal part of the process.
FIGURE 1-45
Freaks
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Another common cause of freaks is a mistake in calculation. Failure
to divide by the proper number in calculating the average, or not
subtracting correctly to obtain the range point will sometimes have
this effect. A freak may also be caused by a plotting error, as when a
person plotting the point misreads the scale on the charts. Accidental
damage or mishandling may also result in freaks.
Freaks are the easiest of the nonrandom patterns to recognize, and in
most cases their causes are the easiest to identify.
Typical causes of freaks on the X or p chart may include: a wrong
setting, corrected immediately; an error in measurement, an error in
plotting, an incomplete operation, an omitted operation, an equipment
breakdown or accidental inclusion of experimental items.
Typical causes of freaks on the R charts may include: accidental
damage of items, an incomplete operation, an omitted operation,
equipment breakdown, inclusion of experimental items, inclusion of
setup items, an error in subtraction, a measurement error or a plotting
error.
3.
Sudden Shift In Level
A sudden shift in level (Figure 1-46) will be indicated by a positive
change in direction, either up or down, on the chart. Sudden shifts
may appear on any of the control charts.
FIGURE 1-46
Sudden Shift In Level
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On an X or p chart, this indicates the sudden appearance of a new
element or cause into the process which moves the center of distribution
and then ceases to act on it further. The process shifts up or down
rapidly and is then established at the new level.
On an R chart, a sudden increase in level generally indicates the
introduction of a new population. A sudden drop in level might indicate
that one or more populations have been removed.
Typical causes of sudden shifts on X or p charts may include: a change
to a different type of material, a new operator, a new inspector, the
use of new test equipment, use of a new or modified machine, a new
machine setting or a change in setup or production method.
Typical causes of sudden shifts on R charts may include: a change in
motivation on the part of the operator, a new operator, use of new or
modified equipment, use of different material or supplies.
On R charts, the following causes will make the pattern rise:
•
Greater carelessness on the operator’s part.
•
Inadequate maintenance.
•
Positioning or holding devices in need of repair.
•
Use of a less uniform material.
The following causes will make the pattern drop on R charts:
•
Improved workmanship.
•
Improved materials.
•
Improved machines or equipment.
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4.
Trends
A trend (Figure 1-47) is a continuous movement, up or down, indicated
by a long series of points without a change in direction. Trends can
result from any causes which work on the process gradually rather
than suddenly.
FIGURE 1-47
A Trend
If a trend appears on an X or p chart, the cause is one which moves
the center of the distribution steadily from low to high or from high
to low.
If it appears on the R chart, the cause is gradually increasing or
decreasing the spread.
Typical causes of trends on X and p charts include: tool wear (wear of
bearings, threads, holding devices, gauges, etc.), aging, deterioration
of solutions, inadequate maintenance, seasonal affects (including
temperature and humidity), operator fatigue, changes in production or
poor housekeeping.
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On an R chart, typical causes of an increasing trend include: the dulling
of tools, something loosening or wearing gradually or the existence of
two or more populations (machines, shifts, etc.)
Decreasing trends on R charts are frequently caused by: a gradual
improvement in an operator’s performance, an improved maintenance
program, implemented process controls or the production of a more
uniform product.
5.
Cycles
Cycles (Figure 1-48) are short trends in data which occur in repeated
patterns. Any tendency of the pattern to repeat by showing a series of
high points interspersed by a series of low points is called a cycle.
FIGURE 1-48
Cycles
The causes of cycles are generally processing variables that come
and go on a more or less regular basis. They may be associated with
fatigue patterns, schedules, shifts, etc. They may also be associated
with seasonal effects, which come and go more slowly.
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Typical causes of cycles on the X or p chart include: seasonal effects
(temperature, humidity, etc.), worn positions or treads on locking
devices, concentric dies, rollers, etc.; operator fatigue, rotation
of people, differences between gauges used by operators, voltage
fluctuations or shift changes.
On the R chart, typical causes of cycles include: maintenance
schedules, operator fatigue, the rotation of fixtures, gauges, etc.; shift
change, wear of tool or die or a tool in need of sharpening.
6.
Grouping
On a control chart, grouping (Figure 1-49) is represented by the
clustering or bunching of measurements in a nonrandom manner.
FIGURE 1-49
Grouping
Grouping is an indication that assignable causes are present. When
measurements cluster in a nonrandom fashion, it indicates that a
different system of causes has been introduced to the process.
Grouping generally occurs on R charts or on individual charts, but
sometimes occurs on an X chart.
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Typical causes for grouping on an X chart include: measurement
difficulties, changes in calibration of the test equipment, a different
person taking the measurements, the existence of two or more
populations (machines, shifts, etc.) or sampling mistakes.
On R charts, typical causes of grouping include freaks in the data or
the existence of two or more populations (machines, shifts, etc.).
7.
Instability
Instability (Figure 1-50) on a control chart will be indicated by
unusually large fluctuations between data points. This pattern is
characterized by scattered ups and downs, resulting in points on both
sides of the control chart. The fluctuations between points appear to
be too great for the control limits.
FIGURE 1-50
Instability
Instability patterns may arise in either of two ways:
•
A single cause capable of affecting the average or spread of the
distribution.
•
A group of causes, each capable of shifting the average, the
spread, or both.
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For both X and p charts, the common causes can be broken into simple
and complex categories.
Common causes of instability on X or p charts include:
•
Simple causes
Over-adjustment of machine, fixtures not holding work in place
properly, carelessness of operator in setting temperature or time
device, different lots of material mixed in storage, different
codes, difference in test equipment, deliberately running on
high or low side of specification, erratic behavior of automatic
controls.
•
Complex causes
Effect of many process variables on the characteristic, effect of
screening and sorting at different stations, effect of experimental
or development work being done.
Common causes of instability on R charts include:
8.
•
On the high side
Instability on high side, untrained operator, too much play in
holding fixture, mixture of material, machine in need of repair,
unstable test equipment, operator carelessness, assemblies offcenter, equipment worn or not fitting together properly.
•
On the low side
Instability on low side, better operator, more uniform material,
better work habits, possible effect due to implementing control
charts.
Mixtures
Mixtures (Figure 1-51) are indicated when the points on a control
chart tend to fall near the upper and lower limits with an absence of
fluctuations near the centerline. This pattern can be recognized by the
unusual length of the lines joining the points, which create an obvious
seesaw appearance.
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FIGURE 1-51
Mixture
Mixture patterns may appear on X or p charts when samples are taken
separately from different sources of product.
The mixture pattern may appear on an R chart when random samples
are taken from different sources (machines, shifts, departments, etc.).
Mixture patterns are closely related to instability, grouping and freaks.
Generally, the detection and elimination of mixtures will make other
nonrandom patterns easier to interpret.
Common causes of mixture patterns on X and p charts include:
consistent differences in material, operators, machines, shifts, etc.;
different lots of raw material, differences in codes, differences in test
equipment, improper sampling, lack of machine alignment, overadjustment by operators.
On R charts, common causes for mixture include: different lots of raw
materials, a frequent drift or jump in automatic controls, difference in
test equipment or unreliable holding devices or fixtures.
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9.
Stratification
Stratification (Figure 1-52) is a form of mixture. If differs, however,
in that instead of fluctuating near the control limits, a stratification
pattern appears to hug the centerline with a few points at any distance
from the centerline. Stratification, then, is indicated by unnaturally
small fluctuations or an absence of points near the control limits.
FIGURE 1-52
Stratification
Stratification may be caused by any element in the process, which is
consistently being spread across the samples. For example, the cause
will probably be the machine, if one item is taken from each machine.
It will probably be the spindle, if items are taken from each spindle. It
will be the boxes of product, if one item for the sample is taken from
each box.
Common causes of stratification on X or p charts include: differences
between materials, machines, operators, etc.; differences in lots
of raw materials, differences in test equipment, improper sampling
technique, misplacing a decimal point during the calculation or an
incorrect calculation of control limits.
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On R charts, common causes of stratification include: different lots
of raw materials, differences in test equipment, frequent changes in
operating conditions and a mixing of product lines.
10.
Process Capability
After it is determined from a control chart that a process is in control,
the next step is to determine if the process is capable. This is done
by comparing the average and range of the process’ output with the
specifications.
But because the control chart contains only the average and range of
samples taken from the process, the average and range of the total
process output (the process population) is not known. To estimate the
possible range of values for the total process output, the standard
deviation (sigma) must first be calculated. The standard deviation was
discussed in detail in the sections on Descriptive Statistics, Normal
Distributions and Samples Versus Populations.
Remember: Standard deviation is a measure of how closely values
are grouped around the average.
When the standard deviation is known for a sample taken from a
process, which has a normal distribution, the possible range of values
for the total process output can be computed.
Remember: When a process has a normal distribution, 68 percent
of the values will occur under the center of the curve within an area
which is 2-sigmas wide; 95 percent of the values will occur within
an area which is 4-sigmas wide; 99.7 percent of the values will
occur within an area which is 6-sigmas wide. Since a 6-sigma range
includes 99.7 percent of the population, a 6-sigma process is generally
considered capable. That 6-sigma range, however, must remain within
the specification limits for the process to be considered capable.
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As an example, a completed control chart has a centerline or average
( X ) of .2. Based on the sample data appearing on the chart, sigma is
calculated to equal 1.7.
σ = 1.7
X = .2 By adding 1-sigma to the average, and subtracting 1-sigma from the
average, it can be predicted that 68 percent of the values in the total
population from which the sample was taken will fall between - 1.5
and 1.9.
.2
.2
-1.7
-1.5
€
+ 1.7
1.9
€
There is a risk of not knowing where 32 percent of the values will
fall.
By adding and subtracting 2-sigmas from the average, it can be
predicted that 95 percent of the total values in the population from
which the sample was taken will fall between -3.2 and 3.6.
.2
-3.4
-3.2
.2
+3.4
3.6
€
Still, there is a risk of not knowing where 5 percent of the values will
fall.
By adding and subtracting 3-sigmas to the average, it can be predicted
that 99.7 percent of the total values in the total population from which
the sample was taken will fall between -3.2 and 3.6
.2
-5.1
-4.9
.2
+5.1
5.3
€
There is risk of not knowing where only .3 percent of the values will
fall.
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Each time another sigma is added to both sides of the curve, the risk of
finding values outside the range is reduced. Most American industries
are satisfied with a 6-sigma range (3-sigmas on either side of the
average). With 99.7 percent of the values falling within this range,
a manufacturer may produce 1,000 pieces using a process with this
capability, and only three could potentially fall beyond the 6-sigma
range.
11.
z Scores
Process capability is often described in terms of the number of sigma
units between the process average and the specification limits. These
specification limits are not the responsibility of the machine or
process operator to set, but are established by the process engineers
or management. The operator, however, can use control charts and
standard deviation to determine the number of sigma units between
the average and the specification limits, or the z score.
Generally, to be capable, a process must have at least 3-sigmas between
the process average and each specification limit. This is illustrated in
Figure 1-53.
FIGURE 1-53
z Scores
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The following formulae are applied to determine z scores:
(USL - X ) ÷ σ = Upper z
(5 - 1) ÷ 1.33 = Upper z
4 ÷ 1.33 = 3.007
(LSL - X ) ÷ σ = Lower z
[(-5) - 1)] ÷ 1.33 = Lower z
-6 ÷ 1.33 = 4.511
Generally, a process must have at least 3-sigmas between the process
average and each specification limit. The process in Figure 1-53 is
capable because the smallest z score (also called z minimum) is greater
than 3-sigma. To determine a process’ capability index (Cpk), divide z
minimum by 3. In the example above, z minimum divided by 3 equals
a capability index greater than 1.
z min. ÷ 3 = Cpk
3.007 ÷ 3 = 1.002
1 < 1.002
Study the process illustrated in Figure 1-54.
FIGURE 1-54
Capability Index
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The z score formulae would be used to determine the capability index
of the process in Figure 1-54, as follows:
(LSL - X ) ÷ σ = Lower z
[(- 3) - (-.22) ÷ 1.74 = Lower z
-2.78 ÷ 1.74 = 1.59 σ
1.59 ÷ 3 = Cpk
1.59 ÷ 3 = .53
(USL - X ) ÷ σ = Upper z
[3-(-.22)] ÷ 1.74 = Upper z
3.22 ÷ 1.74 = 1.85 σ
1.85 ÷ 3 = Cpk
1.85 ÷ 3 = .61
1 > .53
1 > .61
The process in Figure 1-54 is not capable, because neither the lower
nor the upper z score is greater than 3-sigma. Either z score divided by
3 gives a capability index of less than 1.
12. Capability Options
If a process is capable, it should be allowed to continue running.
If the process is only marginally capable, a choice must be made
to either stop the process and make the necessary adjustments or
continue to run it, sorting a certain segment of process output for
defects (nonconformities).
If the process is not capable, the process should be stopped and
the necessary corrections made. When a process is not capable,
management should be alerted immediately to determine why the
process is not performing as desired.
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