56 MEASURES OF LOCATION AND VARIABILITY
Table 5.1 Sampling Properties of Means and Variances22
2,lO2,8
10,2
10,lO
10,4
10,8
42
4,lO
4,4
43
8,2
8,lO
8,4
8,8
c2,42 -4
6 0
3 -3
5 -1
6 0
10 4
7 1
9 3
3 -3
7 1
4 -2
6 0
5 -1
9 3
6 0
8 2
96 016 0
0 32
9 2
1 18
0 32
16 0
1 18
9 2
9 2
1 18
4 0
0 8
1 18
9 2
0 8
4 0
80 160The variance of the sample means is computed fromor by summing the next-to-last column and dividing by N = 16.
The mean of the population is p = 6 and the mean of the sample means is px = 6.
It is no coincidence that py = p; this example illustrates a general principle. If all
possible samples of a certain size are drawn from any population and their sample
means computed, the mean of the population consisting of all the sample means is
equal to the mean of the original population.
A second general principle is that a+ = a2/n; here a2/n = 1012 = 5 and
as = 5. The variance of a population of sample means is equal to the variance of
the population of observations divided by the sample size.
It should be noted that 4 rather than 3 was used in the denominator in the calculation
of 02, Similarly, 16 was used instead of 15 in calculating 0%. This is done when the
variance is computed from the entire population.
The formula 05 = a2/n is perfectly general if we take all possible samples of a
fixed size and if we sample with replacement. In practice, we usually sample without
replacement and the formula must be modified somewhat to be exactly correct (see
Kalton [1983]). However, for samples that are a small fraction of the population, as
is generally the case, the modification is slight and we use the formula 0% = a2/n
just as if we were sampling with replacement. In terms of the standard deviation, the
formula becomes ax = o/fi. This is called the standard error of the mean.