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 377
The final goal for the
ANOVA is an F-ratio
F 5
Variance between treatments
Variance within treatments
Each variance in
the F-ratio is
computed as SS/df
Variance
SS between
between 5
treatments df between
Variance
SS within
within 5
treatments df within
FIGURE 12.4
The structure and
sequence of calculations
for the ANOVA.
To obtain each of
the SS and df values,
the total variability
is analyzed into the
two components
SS between
SS total
SS within
df between
df total
df within
components (between and within). Then compute df for the total study, and analyze
it into two components (between and within).
Thus, the entire process of ANOVA requires nine calculations: three values for SS, three
values for df, two variances (between and within), and a final F-ratio. However, these nine
calculations are all logically related and are all directed toward finding the final F-ratio.
Figure 12.4 shows the logical structure of ANOVA calculations.
■ Analysis of Sum of Squares (SS)
The ANOVA requires that we first compute a total sum of squares and then partition this
value into two components: between treatments and within treatments. This analysis is
outlined in Figure 12.5. We will examine each of the three components separately.
1. Total Sum of Squares, SS total
. As the name implies, SS total
is the sum of squares for
the entire set of N scores. As described in Chapter 4 (pp. 110–112), this SS value
can be computed using either a definitional or a computational formula. However,
SS Total
G 2
S X 2 2
N
FIGURE 12.5
Partitioning the sum of squares
(SS) for the independentmeasures
ANOVA.
SS between treatments
n (SS for the treatment means)
or
T 2 G 2
S 2
n N
SS within treatments
SSS inside each treatment