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

SECTION 17.4 | Effect Size and Assumptions for the Chi-Square Tests 583
does not necessarily mean a large effect, it is generally recommended that the outcome of a
hypothesis test be accompanied by a measure of the effect size. This general recommendation
also applies to the chi-square tests presented in this chapter.
Cohen (1992) introduced a statistic called w that provides a measure of effect size for
either of the chi-square tests. The formula for Cohen’s w is very similar to the chi-square
formula but uses proportions instead of frequencies.
w 5ÎS (P o 2 P e )2
P e
(17.6)
In the formula, the P o
values are the observed proportions in the data and are obtained by
dividing each observed frequency by the total number of participants.
observed proportion 5 P o
5 f o
n
Similarly, the P e
values are the expected proportions that are specified in the null hypothesis.
The formula instructs you to
1. Compute the difference between the observed proportion and the expected
proportion for each cell (category).
2. For each cell, square the difference and divide by the expected proportion.
3. Add the values from step 2 and take the square root of the sum.
The following example demonstrates this process.
EXAMPLE 17.5
A researcher would like to determine whether students have any preferences among four
pizza shops in town. A sample of n = 40 students is obtained and fresh pizza is ordered
from each of the four shops. Each student tastes all four pizzas and then selects a favorite.
The observed frequencies are as follows:
Shop A Shop B Shop C Shop D
6 12 8 14 40
The null hypothesis says that there are no preferences among the four shops so the
expected proportion is P = 0.25 for each. The observed proportions are 6/40 = 0.15 for
shop A, 12/40 = 0.30 for shop B, 8/40 = 0.20 for shop C, and 14/40 = 0.35 for shop D.
The calculations for w are summarized in the table.
P o
P e
(P o
– P e
) (P o
– P e
) 2 (P o
– P e
) 2 /P e
Shop A 0.15 0.25 0.10 0.01 0.04
Shop B 0.30 0.25 –0.05 0.0025 0.01
Shop C 0.20 0.25 0.05 0.0025 0.01
Shop D 0.35 0.25 –0.10 0.01 0.04
0.10
S (p o 2 p e )2
p e
= 0.10 and w = Ï0.10 = 0.316
■
Cohen (1992) also suggested guidelines for interpreting the magnitude of w, with values
near 0.10 indicating a small effect, 0.30 a medium effect, and 0.50 a large effect. By these
standards, the value obtained in Example 17.5 is a medium effect.