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

SECTION 7.1 | Samples, Populations, and the Distribution of Sample Means 197
As a result, most of the sample means should be relatively close to the population
mean.
2. The pile of sample means should tend to form a normal-shaped distribution. Logically,
most of the samples should have means close to μ, and it should be relatively
rare to find sample means that are substantially different from μ. As a result, the
sample means should pile up in the center of the distribution (around μ) and the
frequencies should taper off as the distance between M and μ increases. This
describes a normal-shaped distribution.
3. In general, the larger the sample size, the closer the sample means should be to the
population mean, μ. Logically, a large sample should be a better representative
than a small sample. Thus, the sample means obtained with a large sample size
should cluster relatively close to the population mean; the means obtained from
small samples should be more widely scattered.
As you will see, each of these three commonsense characteristics is an accurate description
of the distribution of sample means. The following example demonstrates the process
of constructing the distribution of sample means by repeatedly selecting samples from a
population.
EXAMPLE 7.1
Remember that random
sampling requires sampling
with replacement.
Consider a population that consists of only 4 scores: 2, 4, 6, 8. This population is pictured
in the frequency distribution histogram in Figure 7.1.
■
We are going to use this population as the basis for constructing the distribution of
sample means for n = 2. Remember: This distribution is the collection of sample means
from all the possible random samples of n = 2 from this population. We begin by looking
at all the possible samples. For this example, there are 16 different samples, and they are
all listed in Table 7.1. Notice that the samples are listed systematically. First, we list all the
possible samples with X = 2 as the first score, then all the possible samples with X = 4 as
the first score, and so on. In this way, we are sure that we have all of the possible random
samples.
Next, we compute the mean, M, for each of the 16 samples (see the last column of
Table 7.1). The 16 means are then placed in a frequency distribution histogram in Figure 7.2.
This is the distribution of sample means. Note that the distribution in Figure 7.2 demonstrates
two of the characteristics that we predicted for the distribution of sample means.
1. The sample means pile up around the population mean. For this example, the population
mean is μ = 5, and the sample means are clustered around a value of 5. It
should not surprise you that the sample means tend to approximate the population
mean. After all, samples are supposed to be representative of the population.
FIGURE 7.1
Frequency distribution
histogram for a population
of 4 scores: 2, 4, 6, 8.
Frequency
2
1
0
0 1 2 3 4 5 6 7 8 9
Scores