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

SECTION 15.5 | Alternatives to the Pearson Correlation 517
TABLE 15.4
The same data are organized
in two different formats.
On the left-hand side, the
data appear as two separate
samples appropriate for
an independent-measures
t hypothesis test. On the
right-hand side, the same
data are shown as a single
sample, with two scores
for each individual: the
number of puzzles solved
and a dichotomous score
(Y) that identifies the group
in which the participant is
located (Well-lit = 0 and
Dimly lit = 1). The data on
the right are appropriate for
a point-biserial correlation.
Number of Solved Puzzles
Well-Lit Room Dimly Lit Room Participant
Data for the Point-Biserial Correlation.
Two scores X and Y for each of the
n = 16 participants
Puzzles
Solved X
Group Y
11 6 7 9 A 11 0
9 7 13 11 B 9 0
4 12 14 15 C 4 0
5 10 16 11 D 5 0
n = 8 n = 8 E 6 0
M = 8 M = 12 F 7 0
SS = 60 SS = 66 G 12 0
H 10 0
I 7 1
J 13 1
K 14 1
L 16 1
M 9 1
N 11 1
O 15 1
P 11 1
When the data in Table 15.4 were originally presented in Chapter 10, we conducted an
independent-measures t hypothesis test and obtained t = –2.67 with df = 14. We measured
the size of the treatment effect by calculating r 2 , the percentage of variance accounted for,
and obtained r 2 = 0.337.
Calculating the point-biserial correlation for these data also produces a value for r.
Specifically, the X scores produce SS = 190; the Y values produce SS = 4.00, and the
sum of the products of the X and Y deviations produces SP = 16. The point-biserial
correlation is
r 5
SP 16
5
ÏSS X
SS Y
Ïs190ds4d 5 16
27.57 5 0.58
Notice that squaring the value of the point-biserial correlation produces r 2 = (0.58) 2 =
0.336, which, within rounding error, is the same as the value of r 2 we obtained measuring
effect size.
In some respects, the point-biserial correlation and the independent-measures hypothesis
test are evaluating the same thing. Specifically, both are examining the relationship
between room lighting and cheating behavior.
1. The correlation is measuring the strength of the relationship between the two
variables. A large correlation (near 1.00 or –1.00) would indicate that there is a
consistent, predictable relationship between cheating and the amount of light in the
room. In particular, the value of r 2 measures how much of the variability in cheating
can be predicted by knowing whether the participants were tested in a will-lit
or a dimly lit room.
2. The t test evaluates the significance of the relationship. The hypothesis test determines
whether the mean difference in grades between the two groups is greater
than can be reasonably explained by chance alone.
As we noted in Chapter 10 (pp. 316–317), the outcome of the hypothesis test and the
value of r 2 are often reported together. The t value measures statistical significance and r 2