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

204 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
TABLE 7.2
Calculations for the points
shown in Figure 7.3.
Again, notice that the
size of the standard error
decreases as the size of
the sample increases.
Sample Size (n)
Standard Error
1 σ M
= 10
Ï1
4 σ M
= 10
Ï4
9 σ M
= 10
Ï9
16
10
σ M
=
Ï16
25
10
σ M
=
Ï25
49
10
σ M
=
Ï49
64
10
σ M
=
Ï64
100
10
σ M
=
Ï100
= 10.00
= 5.00
= 3.33
= 2.50
= 2.00
= 1.43
= 1.25
= 1.00
EXAMPLE 7.2
If samples are selected from a population with µ = 50 and σ = 12, then what is the standard
error of the distribution of sample means for n = 4 and for n = 16? You should obtain
answers of σ M
= 6 for n = 4 and σ M
= 3 for n = 16.
■
■ Three Different Distributions
Before we move forward with our discussion of the distribution of sample means, we will
pause for a moment to emphasize the idea that we are now dealing with three different but
interrelated distributions.
1. First, we have the original population of scores. This population contains the
scores for thousands or millions of individual people, and it has its own shape,
mean, and standard deviation. For example, the population of IQ scores consists
of millions of individual IQ scores that form a normal distribution with a mean of
μ = 100 and a standard deviation of σ = 15. An example of a population is shown
in Figure 7.4(a).
2. Next, we have a sample that is selected from the population. The sample consists
of a small set of scores for a few people who have been selected to represent the
entire population. For example, we could select a sample of n = 25 people and
measure each individual’s IQ score. The 25 scores could be organized in a frequency
distribution and we could calculate the sample mean and the sample standard
deviation. Note that the sample also has its own shape, mean, and standard
deviation. An example of a sample is shown in Figure 7.4(b).
3. The third distribution is the distribution of sample means. This is a theoretical
distribution consisting of the sample means obtained from all the possible random
samples of a specific size. For example, the distribution of sample means for
samples of n = 25 IQ scores would be normal with a mean (expected value) of