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

SECTION 5.5 | Other Standardized Distributions Based on z-Scores 147
29
43 57 71 85
X
,− Original scores (m 5 57 and s 5 14)
22
21 0 11 12
z
,− z-Scores (m 5 0 and s 5 1)
30
40
50
60
70
X
,− Standardized scores (m 5 50 and s 5 10)
Joe
F I G U R E 5.9
The distribution of exam scores from Example 5.9. The original distribution was standardized to produce a distribution
with μ = 50 and σ = 10. Note that each individual is identified by an original score, a z-score, and a new, standardized
score. For example, Joe has an original score of 43, a z-score of −1.00, and a standardized score of 40.
Figure 5.9 provides another demonstration of the concept that standardizing a distribution
does not change the individual positions within the distribution. The figure shows the
original exam scores from Example 5.9, with a mean of μ = 57 and a standard deviation of
σ = 14. In the original distribution, Joe is located at a score of X = 43. In addition to the
original scores, we have included a second scale showing the z-score value for each location
in the distribution. In terms of z-scores, Joe is located at a value of z = −1.00. Finally,
we have added a third scale showing the standardized scores where the mean is μ = 50
and the standard deviation is σ = 10. For the standardized scores, Joe is located at X = 40.
Note that Joe is always in the same place in the distribution. The only thing that changes is
the number that is assigned to Joe: For the original scores, Joe is at 43; for the z-scores, Joe
is at −1.00; and for the standardized scores, Joe is at 40.
LEARNING CHECK
1. A distribution with μ = 47 and σ = 6 is being standardized so that the new mean
and standard deviation will be μ = 100 and σ = 20. What is the standardized score
for a person with X = 56 in the original distribution?
a. 110
b. 115
c. 120
d. 130