In this section the average deviation is discussed. Then the average deviation is used to report the uncertainty for the mean, and finally to determine how many significant figures should be stated with the result.
An observer can improve the accuracy of a reading by making several trials and computing the mean. Suppose she or he measures the length of the object referred to in the previous section 8 times instead of 3 and obtains:
15.39
15.37
15.37
15.37
15.37
15.38
15.39
15.38
123.02
Deviation from mean
0.012
0.008
0.008
0.012
0.002
0.008
0.008
0.002
0.060
123.02 / 8 = 15.378 (mean)
0.060 / 8 = .008 (A.D.)
The deviations from the mean, neglecting sign, are given in the second column, and the mean of these deviations is obtained and labeled A.D. (Average Deviation). This last quantity shows how much any given measurement can be expected to differ from the mean of a number of observations. Since these deviations are as likely to be on one side of the mean as the other we can write the results of our measurements as 15.378 ± 0.008 cm.
Note that the average deviation can tell us the number of significant digits to keep when reporting the mean. When the mean was originally computed the result was: mean = 15.3775. When the average deviation was originally computed the result was: A.D. = 0.0075. The average deviation was then rounded to one significant figure and the mean was rounded to contain a matching number of decimal places. If in this example the A.D. had been 0.01 we would report the result as 15.38 ± 0.01 cm.
Note the general rule: One significant figure in the uncertainty:
It is generally meaningless to report an uncertainty or A.D. to more than one significant figure. There are very few exceptions to this rule.
If the individual measurements are spread widely about the mean then the average deviation will be relatively large. Even then it will still tell us the number of digits we can report.
Last Updated November 19, 2014
