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considered individually when we
focus on the total sum of squares.
That is, when considering the total
sumofsquares,weignorethe
fact that there are multiple groups
involved in the study and simply
pool all of the data together.
As you may recall from Chap-
ter 2 (Section 2.7.2), the numer-
ator of the variance computation
is the sum of squares, which calls
for summing the squared devia-
tions of scores from a mean. In the
case of the total variance, the scores
in the sum of squares computa-
tion are those shown in Figure 3.1.
The mean from which these scores deviate is the average of all these scores.
Such a mean, based on all of the individual scores, is known as the
grand
mean.
It is this grand mean that becomes the reference point for the deviations
we speak of concerning the total sum of squares. For the total sum of
squares we deal with differences (variability) of each individual score
from the grand mean. Thus,
17
11
18
21
17
14
12
13
16
19
20
15
22
16
Figure 3.1
Total variance of the depen-
dent variable.
=
(individual score
grand mean)
2
−
.
total sum of squares
Our notation system calls for using uppercase
Ys
to represent the scores
on the dependent variable. The formula for computing the total sum of
squares is therefore written as
=
(
Y
i
−
Y
T
)
2
,
SS
To t a l
(3.1)
whe
re
SS
To t a l
is the total sum of squares,
Y
i
is the score for a given case,
and
Y
T
is the grand mean.
3.4.2 BETWEEN-GROUPS SUM OF SQUARES
The focus of the between-groups sum of squares is on the group means,
and it is therefore the between-groups source of variance that represents
the effect of the independent variable. As we saw from Table 3.1, the scores
for the Red Room condition are generally lower than those for the Blue
Room condition. These scores generate respective group means that are
quite different from each other - a mean of 14 for the Red Room group
and a mean of 19 for the Blue Room group. These means summarize the
scores by providing a single value that captures (denotes, represents) the
center of their respective distributions and serve as proxies for the sets of
scores that gave rise to them.
