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D. Mean Values

This section discusses the significant figures you can report when you compute the mean (average) of several measurements.


Suppose we want to measure the length of a certain object with a meter stick, with smallest graduations equal to 1 mm, and find the length to be between


15.3 cm and 15.4 cm.


Suppose we are able to estimate the value of the length to within .01 cm, and assume that 3 estimations give us


15.39 cm , 15.37 cm , and 15.37 cm .


Then the mean value is 15.3766666 . . . cm.


Now how many figures must we retain? Clearly taking 4 digits is insufficient because it does not illustrate the fact that we have taken several readings and obtained the mean. The question is how many more than 4 figures can we justify?


To understand the answer, suppose we change one measurement by 0.01 cm so that our readings are:


15.38 cm , 15.37 cm , and 15.37 cm .


Now the mean value is 15.373333 . . . cm.


This change alters the fifth digit counting from the left. Therefore it is meaningless to keep any figure beyond the fifth. Our answer should then have five digits and be


15.377 cm.


It is generally true that when a mean value is obtained, we keep one more figure than in the original observations. The exception would be averaging widely spread values. The average deviation discussed in the next section can guide us in this case.

  • The data table from Section B is reproduced below. Find the average length of the rod that should be reported by each student group. Display your answer in the text box showing the correct number of significant figures. Then click on the bar in the last column for the correct answer.

Student Group
Student 1
Student 2
Student 3
Student 4
Student 5
Group A
10.1
10.4
9.6
9.9
10.8
Group B
10.135
10.227
10.201
10.011
10.155
Group C
12.14
12.17
12.15
12.14
12.18
Group D
10.05
10.82
8.01
11.5
10.77
Group E
10
11
10
10
10

1. Group A: Length =
2. Group B: Length =
3. Group C: Length =
4. Group D: Length =
5. Group E: Length =

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