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

SECTION 12.2 | The Logic of Analysis of Variance 373
Analyzing the total variability into these two components is the heart of ANOVA. We
will now examine each of the components in more detail.
■ Between-Treatments Variance
Remember that calculating variance is simply a method for measuring how big the differences
are for a set of numbers. When you see the term variance, you can automatically
translate it into the term differences. Thus, the between-treatments variance simply measures
how much difference exists between the treatment conditions. There are two possible
explanations for these between-treatment differences:
1. The differences between treatments are not caused by any treatment effect but are simply
the naturally occurring, random and unsystematic differences that exist between
one sample and another. That is, the differences are the result of sampling error.
2. The differences between treatments have been caused by the treatment effects. For
example, if using a telephone really does interfere with driving performance, then
scores in the telephone conditions should be systematically lower than scores in the
no-phone condition.
Thus, when we compute the between-treatments variance, we are measuring differences
that could be caused by a systematic treatment effect or could simply be random and unsystematic
mean differences caused by sampling error. To demonstrate that there really is a
treatment effect, we must establish that the differences between treatments are bigger than
would be expected by sampling error alone. To accomplish this goal, we determine how
big the differences are when there is no systematic treatment effect; that is, we measure
how much difference (or variance) can be explained by random and unsystematic factors.
To measure these differences, we compute the variance within treatments.
■ Within-Treatments Variance
Inside each treatment condition, we have a set of individuals who all receive exactly the
same treatment; that is, the researcher does not do anything that would cause these individuals
to have different scores. In Table 12.1, for example, the data show that five individuals
were tested while talking on a hand-held phone (sample 2). Although these five individuals
all received exactly the same treatment, their scores are different. Why are the scores different?
The answer is that there is no specific cause for the differences. Instead, the differences
that exist within a treatment represent random and unsystematic differences that occur
when there are no treatment effects causing the scores to be different. Thus, the withintreatments
variance provides a measure of how big the differences are when H 0
is true.
Figure 12.3 shows the overall ANOVA and identifies the sources of variability that are
measured by each of the two basic components.
■ The F-Ratio: The Test Statistic for ANOVA
Once we have analyzed the total variability into two basic components (between treatments
and within treatments), we simply compare them. The comparison is made by computing
an F-ratio. For the independent-measures ANOVA, the F-ratio has the following structure:
F 5
variance between treatments
variance within treatments
5
differences including any treatment effects
differences with no treatment effects
(12.1)
When we express each component of variability in terms of its sources (see Figure 12.3),
the structure of the F-ratio is
systematic treatment effects 1 random, unsystematic differences
F 5
random, unsystematic differences
(12.2)