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

212 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
(a)
Distribution of M
for n 5 1
s M
5 s 5 20
(b)
Distribution of M
for n 5 4
s M
5 10
(c)
Distribution of M
for n 5 100
s M
5 2
20
10
2
80
FIGURE 7.8
The distribution of sample means for (a) n = 1, (b) n = 4, and (c) n = 100 obtained from a normal population with
μ = 80 and σ = 20. Notice that the size of the standard error decreases as the sample size increases.
80
80
normal (because the original population is normal), and all three have the same mean,
μ = 80, which is the expected value of M. However, the three distributions differ greatly
with respect to variability. We will consider each one separately.
The smallest sample size is n = 1. When a sample consists of a single student, the mean
for the sample equals the score for the student, M = X. Thus, when n = 1, the distribution
of sample means is identical to the original population of scores. In this case, the standard
error for the distribution of sample means is equal to the standard deviation for the original
population. Equation 7.1 confirms this observation.
s M
5 s Ïn 5 20
Ï1 5 20
When the sample consists of a single student, you expect, on average, a 20-point difference
between the sample mean and the mean for the population. As we noted earlier,
the population standard deviation is the “starting point” for the standard error. With the
smallest possible sample, n = 1, the standard error is equal to the standard deviation (see
Figure 7.8(a)).
As the sample size increases, however, the standard error gets smaller. For a sample of
n = 4 students, the standard error is
s M
5 s Ïn 5 20
Ï4 5 20 2 5 10
That is, the typical (or standard) distance between M and μ is 10 points. Figure 7.8(b) illustrates
this distribution. Notice that the sample means in this distribution approximate the
population mean more closely than in the previous distribution where n = 1.
With a sample of n = 100, the standard error is still smaller.
s M
5 s Ïn 5 20
Ï100 5 20
10 5 2