Basic Statistics

11 8 VARIANCES: ESTIMATION AND TESTS

brief discussion is given of the use of nonparametric tests that do not require a normal
distribution.

9.1 POINT ESTIMATES FOR VARIANCES AND STANDARD

DEVIATIONS

First, a reminder as to how to estimate a population variance or standard deviation.

From a single sample, the $ample variance s2 is calculated and used as the best point

estimate of the population variance c2, and the sample standard deviation s is used

as the point estimate for o.

Sometimes, however, more than one sample is available for estimating a variance.

In Section 7.5.3, for example, when group comparisons were made in the study of

gains in weight under two diets, it was assumed that the variability of gains in weight

was the same under the two diets. In other words, the population of gains in weight

under the supplemented diet and the population of gains in weight under the standard

diet had the same variance, 02. The two sample variances were pooled, and sg was

used to estimate 02, with

(nl - 1)s: + (n2 - 1)s;

121 + n2 - 2

s= P

When there are several samples rather than just two for estimating a single variance,
the sample variances are pooled in a similar way. In general, for k samples, the pooled
estimate of the variance is

(n1 - 1)s: + (122 - 1)s; +... + (nk - 1)st

s2 = -

P nl + 122 +. .. + nk - k

where the variances of the Ic samples are s:, sz,.... st and the sample sizes are

n1, n2.... , nk.

For example, if s: = 20. sz = 30, si = 28. n1 = 12.722 = 16, and n3 = 5, then

• = 26.07
  ll(20) + l5(30) + 4(28) - 782
  30

s= -

P 12 + 16+5 - 3

and sp = 5.1.

9.2 TESTING WHETHER TWO VARIANCES ARE EQUAL: FTEST

Frequently, with two samples we wish to make a test to determine whether the popu-

lations from which they come have equal variances. For example, with two methods

for determining blood counts, we may wish to establish that one is less variable than

the other. Another situation arises when we need to pool the two sample variances

and use si as an estimate for both population variances. In making the usual t test

to test the equality of two means, we assume that the two populations have equal