Distribution of Errors
51
Standard Deviation
The standard deviation is the square root of the variance (s
2 ). The variance is
almost the same as the average of the squares of the deviations of the measurements from the average (x); it is defined as
variance = ,» =
(
*' " ^
+ <*• ~ ^
+ (x * ~ *>'
+ ' ' '
+ (x » ~ &
n - 1
n - 1
(5-2)
For reasons that we need not discuss here, n - 1 is used as the denominator
instead of n. Of course, for very large values of n (say, 1000), there is no
appreciable difference between n and n - I. Thus, for very large numbers of
measurements, you really can say that the variance is the average of the
squares of the deviations. Because the standard deviation is the square root of
the variance, we have
standard deviation = s = \; ———-r—
(5-3)
V
n - I
The standard deviation will have the same units as the original measurements,
and the same units (but not the same value) as the average deviation.
PROBLEM:
Calculate the standard deviation (s) of the measurements made in the preceding
problem.
SOLUTION:
As in the last problem, first find the average of the measurements, and then
subtract it from each of the individual measurements to get the deviation. The sum
of the squares of these deviations divided by n - 1 is the variance.
Measurement
x t
10.325 m
10.320
10.315
10.313
10.327
= 51.600m
= 10.320m
Deviation
(x t - x)
0.005 m
0.000
-0.005
-0.007
0.007
(Deviation)'
2
(x t - x)'
2
0.000025 m
2
0.000000
0.000025
0.000049
0.000049
0.000148 m
2 =
Standard deviation = s = \
:
— =
V n - 1