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

372 CHAPTER 12 | Introduction to Analysis of Variance
12.2 The Logic of Analysis of Variance
LEARNING OBJECTIVE
4. Identify the sources that contribute to the variance between-treatments and the
variance within-treatments, and describe how these two variances are compared in
the F-ratio to evaluate the null hypothesis.
The formulas and calculations required in ANOVA are somewhat complicated, but the
logic that underlies the whole procedure is fairly straightforward. Therefore, this section
gives a general picture of ANOVA before we start looking at the details. We will introduce
the logic of ANOVA with the help of the hypothetical data in Table 12.1. These data represent
the results of an independent-measures experiment using three separate samples,
each with n = 5 participants, to compare performance in a driving simulator under three
telephone conditions.
TABLE 12.1
Hypothetical data from
an experiment examining
driving performance
under three telephone
conditions.
Treatment 1 No
Phone
(Sample 1)
Treatment 2
Hand-Held
(Sample 2)
Treatment 3
Hands-Free
(Sample 3)
4 0 1
3 1 2
6 3 2
3 1 0
4 0 0
M = 4 M = 1 M = 1
One obvious characteristic of the data in Table 12.1 is that the scores are not all the
same. In everyday language, the scores are different; in statistical terms, the scores are
variable. Our goal is to measure the amount of variability (the size of the differences) and
to explain why the scores are different.
The first step is to determine the total variability for the entire set of data. To compute
the total variability, we combine all the scores from all the separate samples to obtain one
general measure of variability for the complete experiment. Once we have measured the
total variability, we can begin to break it apart into separate components. The word analysis
means dividing into smaller parts. Because we are going to analyze variability, the process
is called analysis of variance. This analysis process divides the total variability into two
basic components.
1. Between-Treatments Variance. Looking at the data in Table 12.1, we clearly see
that much of the variability in the scores results from general differences between
treatment conditions. For example, the scores in the no-phone condition tend to be
much higher (M = 4) than the scores in the hand-held condition (M = 1). We will
calculate the variance between treatments to provide a measure of the overall differences
between treatment conditions. Notice that the variance between treatments
is really measuring the differences between sample means.
2. Within-Treatment Variance. In addition to the general differences between treatment
conditions, there is variability within each sample. Looking again at Table 12.1,
we see that the scores in the no-phone condition are not all the same; they are variable.
The within-treatments variance provides a measure of the variability inside
each treatment condition.