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

SECTION 5.4 | Using z-Scores to Standardize a Distribution 143
■ Demonstration of a z-Score Transformation
Although the basic characteristics of a z-score distribution have been explained logically,
the following example provides a concrete demonstration that a z-score transformation
creates a new distribution with a mean of zero, a standard deviation of 1, and the same
shape as the original population.
EXAMPLE 5.8
We begin with a population of N = 6 scores consisting of the following values: 0, 6, 5, 2, 3, 2.
This population has a mean of μ = 18
6 = 3 and a standard deviation of σ = 2 (check the
calculations for yourself).
Each of the X values in the original population is then transformed into a z-score as
summarized in the following table.
X = 0
X = 6
X = 5
X = 2
X = 3
X = 2
Below the mean by 1 1 2 standard deviations z = −1.50
Above the mean by 1 1 2 standard deviations z = +1.50
Above the mean by 1 standard deviation z = +1.00
Below the mean by 1 2 standard deviation z = −0.50
Exactly equal to the mean—zero deviation z = 0
Below the mean by 1 2 standard deviation z = −0.50
The frequency distribution for the original population of X values is shown in Figure 5.8(a)
and the corresponding distribution for the z-scores is shown in Figure 5.8(b). A simple comparison
of the two distributions demonstrates the results of a z-score transformation.
1. The two distributions have exactly the same shape. Each individual has exactly the
same relative position in the X distribution and in the z-score distribution.
2. After the transformation to z-scores, the mean of the distribution becomes μ = 0.
For these z-scores values, N = 6 and Σz = −1.50 + 1.50 + 1.00 + −0.50 + 0 +
−0.50 = 0. Thus, the mean for the z-scores is μ = Σz/N = 0/6 = 0.
(a)
Frequency
2
1
s
0
1 2 3
m
4
5 6
X
(b)
F I G U R E 5.8
Transforming a
distribution of raw
scores (a) into
z-scores (b) does not
change the shape of
the distribution.
Frequency
2
1
–1.5
–1.0 –0.5 0
m
s
+0.5
+1.0 +1.5
z