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

398 CHAPTER 12 | Introduction to Analysis of Variance
If you are having trouble predicting the outcome of the ANOVA, read the following hints,
and then go back and look at the data.
Hint 1:
Hint 2:
Remember: SS between
and MS between
provide a measure of how much difference
there is between treatment conditions.
Find the mean or total (T) for each treatment, and determine how much difference
there is between the two treatments.
You should realize by now that the data have been constructed so that there is zero difference
between treatments. The two sample means (and totals) are identical, so SS between
= 0,
MS between
= 0, and the F-ratio is zero.
■
Conceptually, the numerator of the F-ratio always measures how much difference exists
between treatments. In Example 12.7, we constructed an extreme set of scores with zero
difference. However, you should be able to look at any set of data and quickly compare
the means (or totals) to determine whether there are big differences between treatments or
small differences between treatments.
Being able to estimate the magnitude of between-treatment differences is a good
first step in understanding ANOVA and should help you to predict the outcome of an
ANOVA. However, the between-treatment differences are only one part of the analysis.
You must also understand the within-treatment differences that form the denominator of
the F-ratio. The following example is intended to demonstrate the concepts underlying
SS within
and MS within
. In addition, the example should give you a better understanding of
how the between-treatment differences and the within-treatment differences act together
within the ANOVA.
EXAMPLE 12.8
The purpose of this example is to present a visual image for the concepts of betweentreatments
variability and within-treatments variability. In this example, we compare two
hypothetical outcomes for the same experiment. In each case, the experiment uses two
separate samples to evaluate the mean difference between two treatments. The following
data represent the two outcomes, which we call experiment A and experiment B.
Experiment A
Treatment
Experiment B
Treatment
I II I II
8 12 4 12
8 13 11 9
7 12 2 20
9 11 17 6
8 13 0 16
9 12 8 18
7 11 14 3
M = 8 M = 12 M = 8 M = 12
s = 0.82 s = 0.82 s = 6.35 s = 6.35
The data from experiment A are displayed in a frequency distribution graph in Figure 12.9(a).
Notice that there is a 4-point difference between the treatment means (M 1
= 8 and M 2
= 12).
This is the between-treatments difference that contributes to the numerator of the F-ratio.
Also notice that the scores in each treatment are clustered close around the mean, indicating
that the variance inside each treatment is relatively small. This is the within-treatments