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

388 CHAPTER 12 | Introduction to Analysis of Variance
Calculation of Mean Squares. Because we already found the df values (Step 2), we
now can compute the variance or MS value for each of the two components.
MS between
5 SS between
df between
5 84 2 5 42
MS within
5 SS within
df within
5 88
15 5 5.87
Calculation of F. We compute the F-ratio:
F 5 MS between
MS within
5 42
5.87 5 7.16
STEP 4
Make a decision. The F value we obtained, F = 7.16, is in the critical region (see Figure
12.8). It is very unlikely (p < .05) that we would obtain a value this large if H 0
is true.
Therefore, we reject H 0
and conclude that there is a significant treatment effect. ■
Example 12.2 demonstrated the complete, step-by-step application of the ANOVA procedure.
There are two additional points that can be made using this example.
First, you should look carefully at the statistical decision. We have rejected H 0
and
concluded that not all the treatments are the same. But we have not determined which ones
are different. Is answering prepared questions different from making up and answering
your own questions? Is answering prepared questions different from simply rereading?
Unfortunately, these questions remain unanswered. We do know that at least one difference
exists (we rejected H 0
), but additional analysis is necessary to find out exactly where this
difference is. We address this problem in Section 12.5.
Second, as noted earlier, all of the components of the analysis (the SS, df, MS, and F)
can be presented together in one summary table. The summary table for the analysis in
Example 12.2 is as follows:
Source SS df MS
Between treatments 84 2 42.00 F = 7.16
Within treatments 88 15 5.87
Total 172 17
Although these tables are very useful for organizing the components of an ANOVA, they
are not commonly used in published reports. The current method for reporting the results
from an ANOVA is presented on p. 389.
■ Measuring Effect Size for ANOVA
As we noted previously, a significant mean difference simply indicates that the difference
observed in the sample data is very unlikely to have occurred just by chance. Thus, the
term significant does not necessarily mean large, it simply means larger than expected by
chance. To provide an indication of how large the effect actually is, it is recommended that
researchers report a measure of effect size in addition to the measure of significance.
For ANOVA, the simplest and most direct way to measure effect size is to compute the
percentage of variance accounted for by the treatment conditions. Like the r 2 value used to
measure effect size for the t tests in Chapters 9, 10, and 11, this percentage measures how
much of the variability in the scores is accounted for by the differences between treatments.
For ANOVA, the calculation and the concept of the percentage of variance is extremely