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

376 CHAPTER 12 | Introduction to Analysis of Variance
Because ANOVA formulas
require ΣX for each
treatment and ΣX for
the entire set of scores,
we have introduced new
notation (T and G) to
help identify which ΣX
is being used. Remember:
T stands for treatment
total and G stands
for grand total.
5. The sum of all the scores in the research study (the grand total) is identified by G.
You can compute G by adding up all N scores or by adding up the treatment totals:
G = ΣT.
6. Although there is no new notation involved, we also have computed SS and M for
each sample, and we have calculated ΣX 2 for the entire set of N = 15 scores in the
study. These values are given in Table 12.2 and are important in the formulas and
calculations for ANOVA.
Finally, we should note that there is no universally accepted notation for ANOVA.
Although we are using Gs and Ts, for example, you may find that other sources use other
symbols.
TABLE 12.2
The same data that
appeared in Table 12.1
with summary values and
notation appropriate for
an ANOVA.
Treatment 1
No Phone
(Sample 1)
Telephone Conditions
Treatment 2
Hand-Held
(Sample 2)
Treatment 3
Hands-Free
(Sample 3)
4 0 1 ΣX 2 = 106
3 1 2 G = 30
6 3 2 N = 15
3 1 0 k = 3
4 0 0
T 1
= 20 T 2
= 5 T 3
= 5
SS 1
= 6 SS 2
= 6 SS 3
= 4
n 1
= 5 n 2
= 5 n 3
= 5
M 1
= 4 M 2
= 1 M 3
= 1
■ ANOVA Formulas
Because ANOVA requires extensive calculations and many formulas, one common problem
for students is simply keeping track of the different formulas and numbers. Therefore,
we will examine the general structure of the procedure and look at the organization of the
calculations before we introduce the individual formulas.
1. The final calculation for ANOVA is the F-ratio, which is composed of two
variances:
F 5
variance between treatments
variance within treatments
2. Each of the two variances in the F-ratio is calculated using the basic formula for
sample variance.
sample variance 5 s 2 5 SS
df
Therefore, we need to compute an SS and a df for the variance between treatments
(numerator of F), and we need another SS and df for the variance within treatments
(denominator of F). To obtain these SS and df values, we must go through
two separate analyses: First, compute SS for the total study, and analyze it into two