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

134 CHAPTER 5 | z-Scores: Location of Scores and Standardized Distributions
The purpose of the preceding example is to demonstrate that a score by itself does
not necessarily provide much information about its position within a distribution. These
original, unchanged scores that are the direct result of measurement are called raw scores.
To make raw scores more meaningful, they are often transformed into new values that
contain more information. This transformation is one purpose for z-scores. In particular,
we transform X values into z-scores so that the resulting z-scores tell exactly where the
original scores are located.
A second purpose for z-scores is to standardize an entire distribution. A common
example of a standardized distribution is the distribution of IQ scores. Although there
are several different tests for measuring IQ, the tests usually are standardized so that they
have a mean of 100 and a standard deviation of 15. Because most of the different tests
are standardized, it is possible to understand and compare IQ scores even though they
come from different tests. For example, we all understand that an IQ score of 95 is a little
below average, no matter which IQ test was used. Similarly, an IQ of 145 is extremely
high, no matter which IQ test was used. In general terms, the process of standardizing
takes different distributions and makes them equivalent. The advantage of this process
is that it is possible to compare distributions even though they may have been quite
different before standardization.
In summary, the process of transforming X values into z-scores serves two useful purposes:
1. Each z-score tells the exact location of the original X value within the distribution.
2. The z-scores form a standardized distribution that can be directly compared to
other distributions that also have been transformed into z-scores.
Each of these purposes is discussed in the following sections.
LEARNING CHECK
1. If your exam score is X = 60, which set of parameters would give you the
best grade?
a. μ = 65 and σ = 5
b. μ = 65 and σ = 2
c. μ = 70 and σ = 5
d. μ = 70 and σ = 2
2. For a distribution of exam scores with μ = 70, which value for the standard
deviation would give the highest grade to a score of X = 75?
a. σ = 1
b. σ = 2
c. σ = 5
d. σ = 10
3. Last week Sarah had a score of X = 43 on a Spanish exam and a score of X = 75
on an English exam. For which exam should Sarah expect the better grade?
a. Spanish
b. English
c. The two grades should be identical.
d. Impossible to determine without more information
ANSWERS
1. A, 2. A, 3. D