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

SECTION 12.3 | ANOVA Notation and Formulas 381
For the data in Table 12.2, there are three different treatment conditions (three
T values or three sample means), so the between-treatments degrees of freedom are
computed as follows:
df between
= 3 – 1
= 2
Notice that the two parts we obtained from this analysis of degrees of freedom
add up to equal the total degrees of freedom:
df total
= df within
+ df between
14 = 12 + 2
The complete analysis of degrees of freedom is shown in Figure 12.6.
As you are computing the SS and df values for ANOVA, keep in mind that the labels that
are used for each value can help you understand the formulas. Specifically,
1. The term total refers to the entire set of scores. We compute SS for the whole set of
N scores, and the df value is simply N – 1.
2. The term within treatments refers to differences that exist inside the individual
treatment conditions. Thus, we compute SS and df inside each of the separate
treatments.
3. The term between treatments refers to differences from one treatment to another.
With three treatments, for example, we are comparing three different means (or
totals) and have df = 3 – 1 = 2.
■ Calculation of Variances (MS) and the F-Ratio
The next step in the ANOVA procedure is to compute the variance between treatments and
the variance within treatments, which are used to calculate the F-ratio (see Figure 12.4).
In ANOVA, it is customary to use the term mean square, or simply MS, in place of the
term variance. Recall (from Chapter 4) that variance is defined as the mean of the squared
deviations. In the same way that we use SS to stand for the sum of the squared deviations,
df total
N21
FIGURE 12.6
Partitioning the degrees of freedom
(df) for the independentmeasures
ANOVA.
df between treatments
df within treatments
k 21
S(n 21) 5 N2k