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

142 CHAPTER 5 | z-Scores: Location of Scores and Standardized Distributions
3. The Standard Deviation The distribution of z-scores will always have a standard
deviation of 1. In Figure 5.6, the original distribution of X values has μ = 100
and σ = 10. In this distribution, a value of X = 110 is above the mean by exactly
10 points or 1 standard deviation. When X = 110 is transformed, it becomes
z = +1.00, which is above the mean by exactly 1 point in the z-score distribution.
Thus, the standard deviation corresponds to a 10-point distance in the X distribution
and is transformed into a 1-point distance in the z-score distribution. The advantage
of having a standard deviation of 1 is that the numerical value of a z-score is
exactly the same as the number of standard deviations from the mean. For example,
a z-score of z = 1.50 is exactly 1.50 standard deviations from the mean.
In Figure 5.6, we showed the z-score transformation as a process that changed a distribution
of X values into a new distribution of z-scores. In fact, there is no need to create a
whole new distribution. Instead, you can think of the z-score transformation as simply relabeling
the values along the X-axis. That is, after a z-score transformation, you still have the
same distribution, but now each individual is labeled with a z-score instead of an X value.
Figure 5.7 demonstrates this concept with a single distribution that has two sets of labels:
the X values along one line and the corresponding z-scores along another line. Notice that
the mean for the distribution of z-scores is zero and the standard deviation is 1.
When any distribution (with any mean or standard deviation) is transformed into
z-scores, the resulting distribution will always have a mean of μ = 0 and a standard deviation
of σ = 1. Because all z-score distributions have the same mean and the same standard
deviation, the z-score distribution is called a standardized distribution.
DEFINITION
A standardized distribution is composed of scores that have been transformed
to create predetermined values for μ and σ. Standardized distributions are used
to make dissimilar distributions comparable.
A z-score distribution is an example of a standardized distribution with μ = 0 and
σ = 1. That is, when any distribution (with any mean or standard deviation) is transformed
into z-scores, the transformed distribution will always have μ = 0 and σ = 1.
F I G U R E 5.7
Following a z-score transformation, the X-axis
is relabeled in z-score units. The distance that is
equivalent to 1 standard deviation on the X-axis
(σ = 10 points in this example) corresponds to
1 point on the z-score scale.
80 90 100 110 120
m
s
22 21 0 11 12
m
X
z