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

SECTION 5.1 | Introduction to z-Scores 133
5.1 Introduction to z-Scores
LEARNING OBJECTIVE
1. Describe the two general purposes for transforming X values into z-scores.
In the previous two chapters, we introduced the concepts of the mean and standard deviation
as methods for describing an entire distribution of scores. Now we shift attention to the
individual scores within a distribution. In this chapter, we introduce a statistical technique
that uses the mean and the standard deviation to transform each score (X value) into a
z-score or a standard score. The purpose of z-scores, or standard scores, is to identify and
describe the exact location of each score in a distribution.
The following example demonstrates why z-scores are useful and introduces the general
concept of transforming X values into z-scores.
EXAMPLE 5.1
Suppose you received a score of X = 76 on a statistics exam. How did you do? It should be
clear that you need more information to predict your grade. Your score of X = 76 could be
one of the best scores in the class, or it might be the lowest score in the distribution. To find
the location of your score, you must have information about the other scores in the distribution.
It would be useful, for example, to know the mean for the class. If the mean were
μ = 70, you would be in a much better position than if the mean were μ = 85. Obviously,
your position relative to the rest of the class depends on the mean. However, the mean by
itself is not sufficient to tell you the exact location of your score. Suppose you know that
the mean for the statistics exam is μ = 70 and your score is X = 76. At this point, you
know that your score is 6 points above the mean, but you still do not know exactly where it
is located. Six points may be a relatively big distance and you may have one of the highest
scores in the class, or 6 points may be a relatively small distance and you are only slightly
above the average. Figure 5.2 shows two possible distributions. Both distributions have a
mean of μ = 70, but for one distribution, the standard deviation is σ = 3, and for the other,
σ = 12. The location of X = 76 is highlighted in each of the two distributions. When the
standard deviation is σ = 3, your score of X = 76 is in the extreme right-hand tail, one of
the highest scores in the distribution. However, in the other distribution, where σ = 12,
your score is only slightly above average. Thus, the relative location of your score within
the distribution depends on the standard deviation as well as the mean.
■
(a)
(b)
s53
s512
70
73
X
70
82
X
X 576
X 576
FIGURE 5.2
Two distributions of exam scores. For both distributions, μ = 70, but for one distribution, σ = 3, and for the other,
σ = 12. The relative position of X = 76 is very different for the two distributions.