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

SECTION 9.3 | Measuring Effect Size for the t Statistic 283
TABLE 9.2
Calculation of SS, the sum
of squared deviations, for
the data in Figure 9.6. The
first three columns show
the calculations for the
original scores, including
the treatment effect. The
last three columns show
the calculations for the
adjusted scores after the
treatment effect has been
removed.
Calculation of SS including the treatment
effect
Score
Deviation from
μ = 10
Squared
Deviation
Calculation of SS after the treatment
effect is removed
Adjusted
Score
Deviation
from μ = 10
Squared
Deviation
8 –2 4 8 – 3 = 5 –5 25
10 0 0 10 – 3 = 7 –3 9
12 2 4 12 – 3 = 9 –1 1
12 2 4 12 – 3 = 9 –1 1
13 3 9 13 – 3 = 10 0 0
13 3 9 13 – 3 = 10 0 0
15 5 25 15 – 3 = 12 2 4
17 7 49 17 – 3 = 14 4 16
17 7 49 17 – 3 = 14 4 16
SS = 153 SS = 72
Thus, removing the treatment effect reduces the variability by 52.94%. This value is called
the percentage of variance accounted for by the treatment and is identified as r 2 .
Rather than computing r 2 directly by comparing two different calculations for SS, the
value can be found from a single equation based on the outcome of the t test.
t2
r 2 5
t 2 1 df
(9.5)
The letter r is the traditional symbol used for a correlation, and the concept of r 2 is discussed
again when we consider correlations in Chapter 15. Also, in the context of
t statistics, the percentage of variance that we are calling r 2 is often identified by the Greek
letter omega squared (ω 2 ).
For the hypothesis test in Example 9.2, we obtained t = 3.00 with df = 8. These values
produce
r 2 5 32
3 2 1 8 5 9 5 0.5294 s52.94%d
17
Note that this is exactly the same value we obtained with the direct calculation of the percentage
of variability accounted for by the treatment.
Interpreting r 2 In addition to developing the Cohen’s d measure of effect size, Cohen
(1988) also proposed criteria for evaluating the size of a treatment effect that is measured
by r 2 . The criteria were actually suggested for evaluating the size of a correlation, r, but
are easily extended to apply to r 2 . Cohen’s standards for interpreting r 2 are shown in
Table 9.3.
According to these standards, the data we constructed for Examples 9.1 and 9.2 show a
very large effect size with r 2 = .5294.
TABLE 9.3
Criteria for interpreting
the value of r 2 as proposed
by Cohen (1988).
Percentage of Variance Explained, r 2
r 2 = 0.01
Small effect
r 2 = 0.09
Medium effect
r 2 = 0.25
Large effect