Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 4.5 | Sample Variance as an Unbiased Statistic 117
4.5 Sample Variance as an Unbiased Statistic
LEARNING OBJECTIVES
11. Define biased and unbiased statistics.
12. Explain why the sample mean and the sample variance (dividing by n 2 1) are
unbiased statistics.
■ Biased and Unbiased Statistics
Earlier we noted that sample variability tends to underestimate the variability in the corresponding
population. To correct for this problem we adjusted the formula for sample
variance by dividing by n 2 1 instead of dividing by n. The result of the adjustment is that
sample variance provides a much more accurate representation of the population variance.
Specifically, dividing by n 2 1 produces a sample variance that provides an unbiased estimate
of the corresponding population variance. This does not mean that each individual
sample variance will be exactly equal to its population variance. In fact, some sample
variances will overestimate the population value and some will underestimate it. However,
the average of all the sample variances will produce an accurate estimate of the population
variance. This is the idea behind the concept of an unbiased statistic.
DEFINITIONS
A sample statistic is unbiased if the average value of the statistic is equal to the
population parameter. (The average value of the statistic is obtained from all the
possible samples for a specific sample size, n.)
A sample statistic is biased if the average value of the statistic either underestimates
or overestimates the corresponding population parameter.
The following example demonstrates the concept of biased and unbiased statistics.
EXAMPLE 4.9
We begin with a population that consists of exactly N 5 6 scores: 0, 0, 3, 3, 9, 9. With a
few calculations you should be able to verify that this population has a mean of m 5 4 and
a variance of s 2 5 14.
Next, we select samples of n 5 2 scores from this population. In fact, we obtain every
single possible sample with n 5 2. The complete set of samples is listed in Table 4.1. Notice
TABLE 4.1
The set of all the possible
samples for n 5 2 selected
from the population
described in Example 4.9.
The mean is computed
for each sample, and the
variance is computed two
different ways: (1) dividing
by n, which is incorrect
and produces a biased
statistic; and (2) dividing
by n 2 1, which is correct
and produces an unbiased
statistic.
Sample
First
Score
Second
Score
Mean
M
Biased
Variance
(Using n)
Sample Statistics
Unbiased
Variance
(Using n 2 1)
1 0 0 0.00 0.00 0.00
2 0 3 1.50 2.25 4.50
3 0 9 4.50 20.25 40.50
4 3 0 1.50 2.25 4.50
5 3 3 3.00 0.00 0.00
6 3 9 6.00 9.00 18.00
7 9 0 4.50 20.25 40.50
8 9 3 6.00 9.00 18.00
9 9 9 9.00 0.00 0.00
Totals 36.00 63.00 126.00