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 195
However, the z-scores and probabilities that we have considered so far are limited to situations
in which the sample consists of a single score. Most research studies involve much
larger samples such as n = 25 preschool children or n = 100 American Idol contestants.
In these situations, the sample mean, rather than a single score, is used to answer questions
about the population. In this chapter we extend the concepts of z-scores and probability
to cover situations with larger samples. In particular, we introduce a procedure for transforming
a sample mean into a z-score. Thus, a researcher is able to compute a z-score that
describes an entire sample. As always, a z-score value near zero indicates a central, representative
sample; a z-value beyond +2.00 or –2.00 indicates an extreme sample. Thus, it
is possible to describe how any specific sample is related to all the other possible samples.
In addition, we can use the z-score values to look up probabilities for obtaining certain
samples, no matter how many scores the sample contains.
In general, the difficulty of working with samples is that a sample provides an incomplete
picture of the population. Suppose, for example, a researcher randomly selects a sample of
n = 25 students from the state college. Although the sample should be representative of the
entire student population, there are almost certainly some segments of the population that are
not included in the sample. In addition, any statistics that are computed for the sample will
not be identical to the corresponding parameters for the entire population. For example, the
average IQ for the sample of 25 students will not be the same as the overall mean IQ for the
entire population. This difference, or error between sample statistics and the corresponding
population parameters, is called sampling error and was illustrated in Figure 1.2 (page 7).
DEFINITION
Sampling error is the natural discrepancy, or amount of error, between a sample
statistic and its corresponding population parameter.
Furthermore, samples are variable; they are not all the same. If you take two separate
samples from the same population, the samples will be different. They will contain different
individuals, they will have different scores, and they will have different sample means.
How can you tell which sample gives the best description of the population? Can you even
predict how well a sample will describe its population? What is the probability of selecting
a sample with specific characteristics? These questions can be answered once we establish
the rules that relate samples and populations.
■ The Distribution of Sample Means
As noted, two separate samples probably will be different even though they are taken from
the same population. The samples will have different individuals, different scores, different
means, and so on. In most cases, it is possible to obtain thousands of different samples
from one population. With all these different samples coming from the same population,
it may seem hopeless to try to establish some simple rules for the relationships between
samples and populations. Fortunately, however, the huge set of possible samples forms a
relatively simple and orderly pattern that makes it possible to predict the characteristics of
a sample with some accuracy. The ability to predict sample characteristics is based on the
distribution of sample means.
DEFINITION
The distribution of sample means is the collection of sample means for all the possible
random samples of a particular size (n) that can be obtained from a population.
Notice that the distribution of sample means contains all the possible samples. It is necessary
to have all the possible values to compute probabilities. For example, if the entire