# Measurements and Signiﬁcant Figures Units of Measurement All ```Measurements and Significant Figures
Units of Measurement
All measurements done in lab must be expressed using appropriate units. In any measurement, it
is best to keep the data in original units and convert them to SI units or other appropriate units
later in calculations. This practice saves you a lot of trouble. If you make a unit conversion
error in calculation, you can always check the data and unit conversion. For example, if
measurements are made with a meter stick, it is best to record the readings in centimeters, rather
than in meters. The reason for this is that if you make a mistake in conversion and the reading
was, for example, 25.62 cm, you may record the data incorrectly, as 0.02562 m, instead of
correct, 0.2562 m. Once you make an error in recording your data, it is very difficult to find the
error in the calculations. The most common mistake students make in calculations in the physics
lab 1s, in fact, the conversion of units.
Personal (Human) Error
Personal errors, often called human errors, arise from carelessness in reading an instrument,
performing an experiment, recording data, or calculations including unit conversion. Personal
errors are not legitimate errors in experiments and can always be avoided with caution. You
should always be careful not to make personal errors during an experiment.
Uncertainty in Measurement
Every measured number has some degree of uncertainty. What is the thickness of your lab
table? Is it 4.7 centimeters, 4.73 centimeters, 4.730 centimeters or 4.7301 centimeters? You
cannot state its exact measurement with absolute certainty.
The uncertainty shows the margin of error in a measurement. The uncertainty in a
measurement depends on the precision of the measuring device and the skill of the person who
uses it. Errors due to uncertainty in laboratory measurements have a different meaning from
“human error.” Uncertainties related to the precision of your measuring instruments cannot be
avoided.
Measurements convey two kinds of information: (1) the magnitude of the measurement and; (2)
the precision of the measurement. The precision of the measurement depends on the measuring
device and the method of measurement.
Significant Figures
Significant figures are the digits in any measurement that are known with certainty plus one digit
that is estimated, and hence, is uncertain. The right-most digit is always an estimated digit.
Only one estimated digit is ever recorded as part of a measurement. On the other hand, counting
numbers are different from measured quantities. Counting numbers are exact. In other words, a
counting number has an infinite number of significant figures.
Rule 1: In numbers that do not contain zeros, all the digits are significant.
e example 1:
3.5781 five significant figures
19 4 three significant figures
1621 four significant figures
Rule 2: All zeros between significant digits are significant.
e example 2:
4.028 four significant figures
4028 four significant figures
105 three significant figures
3002 four significant figures
Rule 3: Zeros to the left of the first non-zero digit serve only to fix the position of the decimal
point, therefore, these zeros are not significant.
e example 3:
0.21 two significant figures
0.0567 three significant figures
0.0000031 two significant figure
Rule 4: In a number with digits to the right of the decimal point, zeros to the right of the last
non-zero digit are significant.
sexamples 4:
75.0 three significant figures
75.00 four significant figures
0.00200 three significant figures
0.40050 five significant figures
1.0020 five significant figures
Rule 5: In a number that has no decimal point and that ends in one or more zeros (such as 1800),
the zeros that end the number may or may not by significant.
The number is ambiguous in terms of significant figures. The confusion is avoided when
numbers are expressed in scientific notations. All digits are taken to be significant when
expressed using powers of ten.
e example 5:
1.8x 10° two significant figures
1.80x 10° three significant figures
1.800x 10° four significant figures
2x 107 one significant figure
2.0% 10” two significant figures
Rounding Off
A calculator displays eight or more digits. How do you round off such a display of digits to, say,
three significant figures? Three simple rules govern the process of deleting unwanted (not
significant) digits from a calculator number.
Rule 1: Tf the first digit to the right of the last significant figure is less than 5, that digit and all
the digits that follow are simply dropped.
o example 6: 43.549 rounded off to three significant figure becomes 43.5
Rule 2: If the first digit to be dropped is a digit equal to or greater that 5, the excess digits are
dropped and the last retained digit 1s increased in value by one unit.
e example 7: 54.36, 54.359, and 54.3598 rounded off to three significant figures all become
54.4.
Significant Figures of Calculated Quantities
Two separate rules apply for multiplication and division, and for addition and subtraction.
Rule for Multiplication and Division
For multiplication and division, an answer should have the number of significant figures found in
the number with the fewest significant figures.
e example 8: 1.356 x 0.452 = 0.612912 (calculator answer)
The input with the fewest significant figures is 0.452, which has three significant figures.
Therefore, the calculator answer 0.612912 must be rounded off to 0.613.
The next example is a rare case when a calculator gives a less number of significant figures than
expected.
e example9: 17.612 — 1.258 = 14 (calculator answer)
The inputs contain five and four significant figures. Therefore, the correct answer must contain
four significant figures and must be written as 14.00. >
e example 10: calculation involving a counting number. Suppose you measured the time for
a pendulum to make 10 complete oscillations and want to find the period (time for one
oscillation) of the pendulum:
15.6s-10=156s
The time 15.6 s has three significant figures and 10 is a counting number (exact number), so the
answer should have three significant figures, same as that of the measured value.
e example 11: calculation involving a constant.
When a constant is used in calculation, you should use one extra number of significant figures
than that of a measured value so that you do not have a round-off error in the result. All other
rules mentioned earlier apply in determining the number of significant figures in the result.
Area of circle of radius 5.25 cm: A = m“ =3.142 x (5.25 ст)? = 86.6 ст”
For addition and subtraction, the answer should not have digits beyond the last digit position
common to all the numbers being added or subtracted.
e example 12: 7.26
+28.1
+33 2
68.56—68.6
The last digit position common to all numbers is the first decimal place. The answer must be
rounded off to 68.6.
e example 13: 345.293
- 215.15
130.143—130.14
The last digit position common to both numbers is the second decimal place. Therefore,
the answer should be rounded off to 130.14.
e example 14: percent error. À percent error 1s defined as follows:
| measured value - accepted value | р
percent error = x 100%
accepted value
If the measured value of the acceleration of gravity is found to be 9.73 m/s”, and the accepted
value is 9.80 m/s”, then the percent error is
| 9.73 т/ \$ - 9.80 т/ 5”
| ZZ > sr x 100% = EU = 0714% = 0.7%
980 m/ s 9.80
percent error =
Note that since the absolute value of the difference of the measured value and the accepted value
has only one significant figure in this case, the result should be rounded off to 0.7%.
6
* example 15: If the number of significant figures of an experimental result is different from
that of an accepted value, one of them should be rounded off so that both have the same number
of significant figures in calculating percent errors.
267g / em” —2.7g/ em: 2.78 /em* —2.7g ! em”
%error = Е x 100% = - -x 100% = 0%
2.78 / cm 2.781 em
This is a percent error calculation for the density of aluminum. In this case, the accepted value
has only two significant figures, so the experimental value of 2.67 g/cm’ was rounded off to 2.7
g/cm’ and the percent error is 0%.
® example 16: Percent difference. In order to compare two experimental values, percent
difference 1s used in stead of percent error.
Percent difference = —————— x100%
(E,+E,)I2
IE, - E,
where E, and F, are two experimental values.
The percent difference between two measured values of 9.71 m/s? and 9.9 m/s? is
9.71-9.9| 0.2
percent difference =—————x 100% = —x 100% = 2%
(9.71+9.9)/2 9.8
Note that in this case, the two values have different number of significant figures, so the sum and
the difference of the two values are rounded off to the first decimal place to b= 0.2 and 9.8,
respectively. Since the numerator 0.2 has only one significant figure, the result has also one
significant figure, 2%.
=
Standard Deviation G
The precision of the measurement can be determined using a quantity called the standard
deviation, o,_,, which is defined by
1 y AT
Li | 2 я (1)
where N is the number of repeated measurements, x, is the ith value of the measured quantity,
and X is the mean (average) of the measurements. Using the standard deviation, it is possible to
determine the probability that the measurements fall within any specific range about the mean
(average) of the data. For example, approximately 68.3% of all repeated measurements fall
within a range of one c from the mean. Furthermore, 99.73% of all measurements should fall
within 3c of the mean. This indicates that if one of the measurements is more than 36 farther
from the mean, it is likely the result of a personal or systematic error.
The precision of the repeated measurements can be expressed by using a quantity called the
standard error, & , which is defined by
Per 2
de 2)
If several groups of N measurements are made, 68.3% of the means produced from each group
should fall in the range x +a. Or, one can say that there is a 68.3% probability that the true
value lines in this range.
=
Example
Assume five measurements are made of some quantity x, and the measured values are 12.6, 13.4,
11.9, 12.0, and 13.1. The mean (average) of the values x is given by
x =(12.6+13.4+11.9+12.0+13.1)/5 =12.6
The standard deviation for these values 1s
1 2 2 2 42 +
Tn = E (12.6-12.6)° + (13.4 -12.6)* + (11.9-12.6)? + (12.0-12.6)* + (13.1-12.6)* ]=0.7
_ 0.66 _
JN AS
We can write the mean of the data with the standard error a as
0.3
x=126+1L3
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