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Mr. Haley
Introduction to Measurements Handout
When making quantitative measurements that involve continuous variables, the level of
uncertainty must be reported. Better instruments and laboratory procedures will yield
results closer to the actual result. It is important to note that obtaining the exact answer is
not important, but learning how to report a measured value along with the level of
uncertainty is very important. In other words, you must be honest when reporting values.
Definitions
1. Measurement error – The difference between the measured value of a quantity and the
accepted value of the quantity.
Error = Measured value – Accepted value
Example 1 – The accepted value of pi to seven decimal places is 3.1415926. If a
circumference experiment yields a value of pi to be 3.16, then what is the error in the
measurement?
Solution – First, we need to round off the actual value to the same decimal place as the
measured value since it would be pointless to keep more digits.
Error = 3.16 – 3.14 = 0.02
2. Relative error – The error of a quantity divided by the accepted value of the quantity.
If x is the actual/accepted value and x1 is the measured value, then
x  x1
Relative error =
x
3. Percent error – 100% times the relative error. This is used when comparing a
measured value to an accepted value. The percent error provides a measure of the
accuracy of the measurement.
Example 2 – What is the percent error in the measurement of pi = 3.16?
Solution – We found the error to be 0.02 in the previous example. The relative error is
then 0.02/3.14 = 0.00637 to a few decimal places. The percent error is then
100% x 0.00637 = 0.64%
4. Percent difference =
x1  x 2
x1  x2  2
 100%
When comparing two measured values, the percent difference provides a measure of the
precision of the experiment. Notice the denominator is the average of the values.
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5. Random error – Usually small errors of observation. Random errors occur when the
instrument is used at the limit of its accuracy. Random errors are equally likely to occur
positively or negatively and thus repeating the measurement is likely to reduce the error
in the measurement. Random errors can be analyzed using statistics.
6. Systematic error – Errors resulting from incorrectly calibrated equipment, poor
laboratory habits, and/or incorrect zero point positioning. Repeating the measurement
will not reduce the error. Systematic errors cannot be analyzed using statistics.
7. Uncertainty – The estimated amount by which a measured value may differ from the
actual value. The uncertainty identifies the reliability of the measured value.
8. Relative Uncertainty – The uncertainty of a value divided by the value.
9. Accuracy - The ability of a measurement to match the actual value of the quantity
being measured. Systematic errors will affect the accuracy of the measurements.
Example 3 – If the actual value of gravity is accepted to be 9.8 m/s2, then which
measured value is more accurate, 9.7 m/s2 or 9.5 m/s2?
Solution – 9.7 m/s2 is the correct answer since it is closer to the accepted value.
10. Precision – The ability of a measurement to be consistently reproduced. The number
of significant figures in the reported value indicates the level of precision of the
measuring instrument. Small random errors lead to higher precision.
Example 4 – Which group of measured values has a greater precision, (25 m, 26 m, and
24 m) or (22 m, 28 m, and 32 m)?
Solution – 25 m, 26 m, and 24 m is a more precise grouping since the repeated
measurement is closer in each case.
11. Significant figures – All the digits in a measurement that are certain plus one that is
estimated. When measurements are added or subtracted, the answer can contain no more
decimal places than the measurement with the least number of decimal places. When
measurements are multiplied or divided, the answer can contain no more significant
figures than the least precise measurement.
Rules for counting significant figures:
a. The most significant digit is the leftmost nonzero digit. In other words, zeros at the left
are never significant.
b. If there is no decimal point explicitly given, the rightmost nonzero digit is the least
significant digit.
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c. If a decimal point is explicitly given, the rightmost digit is the least significant digit,
regardless of whether it is zero or nonzero.
d. The number of significant digits is found by counting the places from the most
significant to the least significant digit.
Note that zeros can cause some confusion when counting significant figures. To clear
this confusion, write potentially ambiguous values in scientific notation.
Example 5 – How many significant figures does 8000 have?
Solution – By the above method 8000 should have one significant figure.
Example 6 – How can you report the value 8000 to have two significant figures?
Solution – Rewrite 8000 as 8.0 x 10 3.
Example 7 – 9.001 cm + 2.1 cm = 11.101 cm, but is reported as 11.1 cm.
Example 8 – 9.001 cm x 2.1 cm = 18.9021 cm2, but is reported as 19 cm2.
12. Precision of the measuring tool – The smallest subdivision that can be read directly.
If a single value is measured to be 25.0 cm with a tool of precision 1 mm = 0.1 cm, then
the value should be reported as (25.0 ± 0.1) cm.
Reporting Values and Dealing With Random Errors
Mean and Standard Deviation – Random errors have an equal likelihood to be low or
high compared to the true value. So, taking the mean x of many measurements
x1, x2, …, xn is a natural way to reduce the effect of random errors. The mean is defined
as
x
x1  ...  xn
n
and is the best value obtained from all the measurements. A general rule is to assign one
more significant figure to the mean value.
Statistical analysis will show that the sample standard deviation
x 
( x1  x ) 2  ...  ( x n  x ) 2
n 1
is a good measure of the precision of the measurements.
Standard error,  
x
n
, measures the precision of the mean.
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Reporting the uncertainty – The standard deviation will substitute as the uncertainty for
the mean of many measurements. It is necessary to report it correctly to the reader. Use
the following format
x   x or x   x
It is important to note that it is necessary to keep no more than two significant figures in
the standard deviation and the standard error. Be sure to keep the same decimal place in
the mean as in the standard deviation and standard error.
Example 9 – Given the following measurements find the mean and standard deviation
and report it in the correct format. 2.45 m, 2.47 m, 2.43 m, 2.51 m, 2.44 m.
Solution – The mean is
(2.45 m  2.47 m  2.43 m  2.51 m  2.44 m)
= 2.460 m.
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The standard deviation works out to be 0.0316 m. The correct form for reporting is
x=
2.460 ± 0.032 m or 2.46 ± 0.03 m
Sometimes the standard deviation will be calculated to be too small and will seem to be
zero. In this case, we must use the precision of the measuring tool and the measurer’s
technique to estimate the uncertainty.
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Propagation of Error
It is not entirely trivial how to include the uncertainties in calculations involving more
than one quantity with an uncertainty. It is important to use the method of propagation of
error. There are two forms of equations that will be discussed here.
A) Z = a xb yc wd
B) Z = ax + by + cw
Assume that Z, x, y, and w are variables with uncertainties σZ, σx, σy, σw respectively and
a, b, and c are constants without uncertainties.
For equation (A) Z  ax b y c w d the propagation of error will be found using the following
equation
(A1)
Z2
Z2
b
2
 x2
x2
c
2
 y2
y2
d2
 w2
w2
For equation (B) Z  ax  by  cw the propagation will be found using the following
equation
(B1)  Z  a 2 x  b 2 y  c 2 w
2
2
2
2
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Exercises
For questions (1) – (8) use the following information. You have measured the length of a
table to be 205.0 cm, 205.8 cm, 205.4 cm, 204.6 cm, and 204.9 cm five independent
times. You measured the width of the same table to be 60.1 cm, 60.4 cm, 60.2 cm, 60.0
cm, and 60.5 cm five independent times.
1) Calculate the mean length L of the table.
2) Calculate the standard deviation of the mean length σL of the table.
3) Calculate the mean width W of the table.
4) Calculate the standard deviation of the mean width σW of the table.
5) Calculate the area A = L x W of the table.
6) Using the correct equation for propagation of error, calculate the uncertainty of
the area σA of the table.
7) Calculate the perimeter P = 2L + 2W of the table.
8) Using the correct equation for propagation of error, calculate the uncertainty of
the perimeter σP of the table.
9) Report the mean length, mean width, area, and perimeter including their
uncertainties in the correct format discussed above.
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