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

148 CHAPTER 5 | z-Scores: Location of Scores and Standardized Distributions
2. A distribution with μ = 35 and σ = 8 is being standardized so that the new mean
and standard deviation will be μ = 50 and σ = 10. In the new, standardized distribution
your score is X = 60. What was your score in the original distribution?
a. X = 45
b. X = 43
c. X = 1.00
d. impossible to determine without more information
3. Using z-scores, a population with μ = 37 and σ = 6 is standardized so that the new
mean is μ = 50 and σ = 10. How does an individual’s z-score in the new distribution
compare with his/her z-score in the original population?
a. new z = old z + 13
b. new z = (10/6)(old z)
c. new z = old z
d. cannot be determined with the information given
ANSWERS
1. D, 2. B, 3. C
5.6 Computing z-Scores for Samples
LEARNING OBJECTIVES
7. Transform X values into z-scores and transform z-scores into X values for a sample.
8. Describe the effects of transforming an entire sample into z-scores and explain the
advantages of this transformation.
Although z-scores are most commonly used in the context of a population, the same
principles can be used to identify individual locations within a sample. The definition of a
z-score is the same for a sample as for a population, provided that you use the sample mean
and the sample standard deviation to specify each z-score location. Thus, for a sample, each
X value is transformed into a z-score so that
1. The sign of the z-score indicates whether the X value is above (+) or below (−) the
sample mean, and
2. The numerical value of the z-score identifies the distance from the sample mean by
measuring the number of sample standard deviations between the score (X) and the
sample mean (M).
Expressed as a formula, each X value in a sample can be transformed into a z-score as
follows:
z 5 X 2 M
(5.3)
s
Similarly, each z-score can be transformed back into an X value, as follows:
X = M + zs (5.4)
You should recognize that these two equations are identical to the population equations
(5.1 and 5.2) on pages 136 and 137, except that we are now using sample statistics, M and