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

SECTION 7.4 | More about Standard Error 211
sample mean. Also, notice that the sample means tend to pile up around the population
mean (μ), forming a normal-shaped distribution as predicted by the central limit theorem.
The distribution shown in Figure 7.7 provides a concrete example for reviewing the
general concepts of sampling error and standard error. Although the following points may
seem obvious, they are intended to provide you with a better understanding of these two
statistical concepts.
1. Sampling Error The general concept of sampling error is that a sample typically
will not provide a perfectly accurate representation of its population. More
specifically, there typically is some discrepancy (or error) between a statistic
computed for a sample and the corresponding parameter for the population. As you
look at Figure 7.7, notice that the individual sample means are not exactly equal
to the population mean. In fact, 50% of the samples have means that are smaller
than μ (the entire left-hand side of the distribution). Similarly, 50% of the samples
produce means that overestimate the true population mean. In general, there will be
some discrepancy, or sampling error, between the mean for a sample and the mean
for the population from which the sample was obtained.
2. Standard Error Again, looking at Figure 7.7, notice that most of the sample
means are relatively close to the population mean (those in the center of the distribution).
These samples provide a fairly accurate representation of the population.
On the other hand, some samples produce means that are out in the tails of the
distribution, relatively far from the population mean. These extreme sample means
do not accurately represent the population. For each individual sample, you can
measure the error (or distance) between the sample mean and the population mean.
For some samples, the error will be relatively small, but for other samples, the error
will be relatively large. The standard error provides a way to measure the “average”,
or standard, distance between a sample mean and the population mean.
Thus, the standard error provides a method for defining and measuring sampling error.
Knowing the standard error gives researchers a good indication of how accurately their
sample data represent the populations they are studying. In most research situations, for
example, the population mean is unknown, and the researcher selects a sample to help
obtain information about the unknown population. Specifically, the sample mean provides
information about the value of the unknown population mean. The sample mean is not
expected to give a perfectly accurate representation of the population mean; there will be
some error, and the standard error tells exactly how much error, on average, should exist
between the sample mean and the unknown population mean. The following example demonstrates
the use of standard error and provides additional information about the relationship
between standard error and standard deviation.
EXAMPLE 7.6
A recent survey of students at a local college included the following question: How many
minutes do you spend each day watching electronic video (online, TV, cell phone, tablet,
etc.). The average response was μ = 80 minutes, and the distribution of viewing times
was approximately normal with a standard deviation of σ = 20 minutes. Next, we take a
sample from this population and examine how accurately the sample mean represents the
population mean. More specifically, we will examine how sample size affects accuracy by
considering three different samples: one with n = 1 student, one with n = 4 students, and
one with n = 100 students.
Figure 7.8 shows the distributions of sample means based on samples of n = 1, n = 4,
and n = 100. Each distribution shows the collection of all possible sample means that could
be obtained for that particular sample size. Notice that all three sampling distributions are