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

166 CHAPTER 6 | Probability
FIGURE 6.3
The normal distribution. The exact shape of the normal
distribution is specified by an equation relating each X value
(score) with each Y value (frequency). The equation is
Y 5 1
Ï2s 2 e2sX 2md2 y2s 2
s
(p and e are mathematical constants). In simpler
terms, the normal distribution is symmetrical
with a single mode in the middle.
The frequency tapers off as you
move farther from the middle
in either direction.
m
X
that fall in each of these sections. For example, the section between the mean (z = 0) and
the point that is 1 standard deviation above the mean (z = 1) contains 34.13% of the scores.
Similarly, 13.59% of the scores are located in the section between 1 and 2 standard deviations
above the mean. In this way it is possible to define a normal distribution in terms of
its proportions; that is, a distribution is normal if and only if it has all the right proportions.
There are two additional points to be made about the distribution shown in Figure 6.4.
First, you should realize that the sections on the left side of the distribution have exactly
the same areas as the corresponding sections on the right side because the normal distribution
is symmetrical. Second, because the locations in the distribution are identified by
z-scores, the percentages shown in the figure apply to any normal distribution regardless
of the values for the mean and the standard deviation. Remember: When any distribution is
transformed into z-scores, the mean becomes zero and the standard deviation becomes one.
Because the normal distribution is a good model for many naturally occurring distributions
and because this shape is guaranteed in some circumstances (as you will see in Chapter 7),
we devote considerable attention to this particular distribution. The process of answering
probability questions about a normal distribution is introduced in the following example.
34.13%
13.59%
2.28%
FIGURE 6.4
The normal distribution following
a z-score transformation.
22 21 0
m
11 12
z