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

118 CHAPTER 4 | Variability
We have structured
this example to mimic
“sampling with replacement,”
which is covered
in Chapter 6.
that the samples are listed systematically to ensure that every possible sample is included.
We begin by listing all the samples that have X 5 0 as the first score, then all the samples
with X 5 3 as the first score, and so on. Notice that the table shows a total of 9 samples.
Finally, we have computed the mean and the variance for each sample. Note that the
sample variance has been computed two different ways. First, we make no correction for
bias and compute each sample variance as the average of the squared deviations by simply
dividing SS by n. Second, we compute the correct sample variances for which SS is divided
by n 2 1 to produce an unbiased measure of variance. You should verify our calculations
by computing one or two of the values for yourself. The complete set of sample means and
sample variances is presented in Table 4.1.
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First, consider the column of biased sample variances, which were calculated dividing
by n. These 9 sample variances add up to a total of 63, which produces an average value
of 63
9 5 7. The original population variance, however, is s2 5 14. Note that the average
of the sample variances is not equal to the population variance. If the sample variance is
computed by dividing by n, the resulting values will not produce an accurate estimate of
the population variance. On average, these sample variances underestimate the population
variance and, therefore, are biased statistics.
Next, consider the column of sample variances that are computed using n 2 1. Although
the population has a variance of s 2 5 14, you should notice that none of the samples has
a variance exactly equal to 14. However, if you consider the complete set of sample variances,
you will find that the 9 values add up to a total of 126, which produces an average
value of 126
9 5 14.00. Thus, the average of the sample variances is exactly equal to the original
population variance. On average, the sample variance (computed using n 2 1) produces
an accurate, unbiased estimate of the population variance.
Finally, direct your attention to the column of sample means. For this example, the
original population has a mean of m 5 4. Although none of the samples has a mean exactly
equal to 4, if you consider the complete set of sample means, you will find that the 9 sample
means add up to a total of 36, so the average of the sample means is 36
9 5 4. Note that the
average of the sample means is exactly equal to the population mean. Again, this is what
is meant by the concept of an unbiased statistic. On average, the sample values provide
an accurate representation of the population. In this example, the average of the 9 sample
means is exactly equal to the population mean.
In summary, both the sample mean and the sample variance (using n 2 1) are examples
of unbiased statistics. This fact makes the sample mean and sample variance extremely valuable
for use as inferential statistics. Although no individual sample is likely to have a mean
and variance exactly equal to the population values, both the sample mean and the sample
variance, on average, do provide accurate estimates of the corresponding population values.
LEARNING CHECK
1. What is meant by a biased statistic.
a. The average value for the statistic overestimates the corresponding population
parameter.
b. The average value for the statistic underestimates the corresponding population
parameter.
c. The average value for the statistic either overestimates or underestimates the corresponding
population parameter.
d. The average value for the statistic is exactly equal to the corresponding population
parameter.