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

494 CHAPTER 15 | Correlation
either the original scores or the new scores. The current figure shows the original scores,
but if the values on the X-axis (0, 1, 2, 3, and so on) were doubled (0, 2, 4, 6, and so on),
then the same figure would show the pattern formed by the new scores. Multiplying either
the X or the Y values by a negative number, however, does not change the numerical value
of the correlation but it does change the sign. For example, if each X value in Figure 15.4
were multiplied by –1, then the current data points would be moved to the left-hand side of
the Y-axis, forming a mirror image of the current pattern. Instead of the positive correlation
in the current figure, the new pattern would produce a negative correlation with exactly the
same numerical value. In summary, adding a constant to (or subtracting a constant from)
each X and/or Y value does not change the pattern of data points and does not change the
correlation. Also, multiplying (or dividing) each X or each Y value by a positive constant
does not change the pattern and does not change the value of the correlation. Multiplying
by a negative constant, however, produces a mirror image of the pattern and, therefore,
changes the sign of the correlation.
■ The Pearson Correlation and z-Scores
The Pearson correlation measures the relationship between an individual’s location in the
X distribution and his or her location in the Y distribution. For example, a positive correlation
means that individuals who score high on X also tend to score high on Y. Similarly, a
negative correlation indicates that individuals with high X scores tend to have low Y scores.
Recall from Chapter 5 that z-scores identify the exact location of each individual score
within a distribution. With this in mind, each X value can be transformed into a z-score,
z X
, using the mean and standard deviation for the set of Xs. Similarly, each Y score can
be transformed into z Y
. If the X and Y values are viewed as a sample, the transformation
is completed using the sample formula for z (Equation 5.3). If the X and Y values form a
complete population, the z-scores are computed using Equation 5.1. After the transformation,
the formula for the Pearson correlation can be expressed entirely in terms of z-scores.
For a sample, r 5 oz X z Y
sn 2 1d
For a population, ρ 5 oz X z Y
N
Note that the population value is identified with a Greek letter rho.
(15.4)
(15.5)
LEARNING CHECK
1. What is the sum of products (SP) for the following data?
a. 6 X Y
b. –5
2 4
c. 43
5 2
d. None of the other 3 choices is correct. 3 5
2 5
2. A set of n = 5 pairs of X and Y values has SS X
= 5, SS Y
= 20 and SP = 8. For these
data, the Pearson correlation is _____.
a. r = 8
100 = 0.08
b. r = 8
10 = 0.80
c. r = 8
25 = 0.32
d. r = 8
20 = 0.40