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Experiments, samples and populations

Example 1.4. Find the sample variance and sample standard deviation of the data
given in table 2.1. ,

Solution. We have already found that the sample mean is 185.0 to one decimal place. However,
when the mean is to be used in the subsequent calculation of the sample variance it is better
to use the most accurate value available. In this case the exact value is 184.975, and so using
(1.6),

1 1608.36
2
2
2
s = [(188.7 − 184.975) + · · · + (200.0 − 184.975) ] = = 201.0,
8 8
where once again we have quoted the result to one decimal place. The sample standard deviation
√
is then given by s = 201.0 = 14.2. As it happens, in this case the difference between the true
mean and the rounded value is very small compared to the variation of the individual readings
about the mean and using the rounded value makes negligible difference; however, this would
not be so if the difference were comparable to the sample standard deviation.

Using the definition (1.7), it is clear that in order to compute the standard deviation of a sample
we must first compute the sample mean. This requirement can be avoided, however, by using an
2
alternative form for s . From (1.6), we see that
N N N N
1 ∑ 1 ∑ 1 ∑ 1 ∑
2
2
2
2
s = (x i − ¯x) = x − 2x i ¯x + ¯ x =
i
N N N N
i=1 i=1 i=1 i=1
2
2
2
2
= ¯x − 2¯x + ¯x = x − ¯x .
¯ 2
2
We may therefore write the sample variance s as
( ) 2
N N
1 ∑ 1 ∑
2
2
2
¯ 2
s = x − ¯x = x − x i , (1.8)
i
N N
i=1 i=1
from which the sample standard deviation is found by taking the positive square root. Thus, by
∑ N ∑ n 2
evaluating the quantities x i and x for our sample, we can compute the sample mean
i=1 i=1 i
and sample standard deviation at the same time.
∑ N ∑ N 2
Example 1.5. Compute x i and x for the data given in table 2.1 and hence
i=1 i=1 i
find the mean and standard deviation of the sample. ,
From table 2.1, we obtain
N
∑
x i = 188.7 + 204.7 + · · · + 200.0 = 1479.8,
i=1
N
∑
2
2
2
2
x = (188.7) + (204.7) + · · · + (200.0) = 275334.36.
i
i=1
Since N = 8, we find as before (quoting the final results to one decimal place)
√
( ) 2
1479.8 275334.36 1479.8
¯ x = = 185.0, s = − = 14.2.
8 8 8
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