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

152 CHAPTER 5 | z-Scores: Location of Scores and Standardized Distributions
F I G U R E 5.11
The distribution of
weights for the population
of adult rats.
Note that individuals
with z-scores near 0
are typical or representative.
However, individuals
with z-scores
beyond +2.00
or −2.00 are extreme
and noticeably different
from most
of the others in the
distribution.
Extreme
individuals
(z beyond 22.00)
360
380
Representative
individuals
(z near 0)
Population
of
nontreated rats
m 5 400 420 440
Extreme
individuals
(z beyond 12.00)
X
22.00
21.00
0
1.00 2.00
z
X 5 418
X 5 450
hormone injection. Specifically, our injected rat would be located near the center of the
distribution for regular rats with a z-score of
z 5 X 2m
s
5
418 2 400
20
5 18
20 5 0.90
Because the injected rat still looks the same as a regular, nontreated rat, the conclusion
is that the hormone does not appear to have an effect.
Now, assume that our injected rat weighs X = 450 g. In the distribution of regular rats
(see Figure 5.11), this animal would have a z-score of
z 5 X 2m
s
5
450 2 400
20
5 50
20 5 2.50
In this case, the hormone-injected rat is substantially bigger than most ordinary rats, and
it would be reasonable to conclude that the hormone does have an effect on weight. ■
In the preceding example, we used z-scores to help interpret the results obtained from a
sample. Specifically, if the individuals who receive the treatment in a research study have
extreme z-scores compared to those who do not receive the treatment, we can conclude that
the treatment does appear to have an effect. The example, however, used an arbitrary definition
to determine which z-score values are noticeably different. Although it is reasonable
to describe individuals with z-scores near 0 as “highly representative” of the population,
and individuals with z-scores beyond ±2.00 as “extreme,” you should realize that these
z-score boundaries were not determined by any mathematical rule. In the following chapter
we introduce probability, which gives us a rationale for deciding exactly where to set the
boundaries.