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 213
A sample of n = 100 students should produce a sample mean that represents the population
much more accurately than a sample of n = 4 or n = 1. As shown in Figure 7.8(c), there is
very little error between M and µ when n = 100. Specifically, you would expect on average
only a 2-point difference between the population mean and the sample mean.
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In summary, this example illustrates that with the smallest possible sample (n = 1), the
standard error and the population standard deviation are the same. As sample size increases,
the standard error gets smaller, and the sample means tend to approximate μ more closely.
Thus, standard error defines the relationship between sample size and the accuracy with
which M represents μ.
IN THE LITERATURE
Reporting Standard Error
As we will see in future chapters, the standard error plays a very important role in
inferential statistics. Because of its crucial role, the standard error for a sample mean,
rather than the sample standard deviation, is often reported in scientific papers. Scientific
journals vary in how they refer to the standard error, but frequently the symbols
SE and SEM (for standard error of the mean) are used. The standard error is reported
in two ways. Much like the standard deviation, it may be reported in a table along
with the sample means (Table 7.3). Alternatively, the standard error may be reported
in graphs.
Figure 7.9 illustrates the use of a bar graph to display information about the sample
mean and the standard error. In this experiment, two samples (groups A and B) are given
different treatments, and then the subjects’ scores on a dependent variable are recorded.
The mean for group A is M = 15, and for group B, it is M = 30. For both samples, the
standard error of M is σ M
= 4. Note that the mean is represented by the height of the
bar, and the standard error is depicted by brackets at the top of each bar. Each bracket
extends 1 standard error above and 1 standard error below the sample mean. Thus, the
graph illustrates the mean for each group plus or minus 1 standard error (M ± SE).
When you glance at Figure 7.9, not only do you get a “picture” of the sample means, but
also you get an idea of how much error you should expect for those means.
TABLE 7.3
The mean self-consciousness scores for
participants who were working in front of
a video camera and those who were not
(controls)
n Mean SE
Control 17 32.23 2.31
Camera 15 45.17 2.78
Figure 7.10 shows how sample means and standard error are displayed in a line
graph. In this study, two samples representing different age groups are tested on a task
for four trials. The number of errors committed on each trial is recorded for all participants.
The graph shows the mean (M) number of errors committed for each group on
each trial. The brackets show the size of the standard error for each sample mean. Again,
the brackets extend 1 standard error above and below the value of the mean.