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APPENDIX D
Deviational Formula for Sums of Squares
From our discussion in Chapter 3, we noted that the total variability in
an experiment or research study consists of between-groups variability
(i.e., the effects of the independent variable plus error) and within-groups
variability (i.e., the effects of error alone). This variability is a function of
how the scores on the dependent variable deviate from the grand mean,
or the individual treatment group means, and also how each group mean
deviates from the grand mean.
More specifically, we can note that the total variability (total deviation)
in a study is composed of how each dependent variable score deviates from
the grand mean. It can be expressed as
total deviation
=
Y
ij
−
Y
T
.
(D.1)
As noted above, this total deviation has two component parts, the
between-groups and within-groups deviations. The between-groups devi-
ation is a function of each treatment group mean from the grand mean
and can be expressed as
between-groups deviation
=
Y
j
−
Y
T
.
(D.2)
The within-group deviation is based on the deviation of each score
from its respective treatment group mean. It can be expressed as
within-group deviation
=
Y
ij
−
Y
j
.
(D.3)
Thus, we can summarize these relationships in the following manner:
total deviation
=
between-groups deviation
+
within-groups deviation;
Y
ij
−
Y
T
=
Y
j
−
Y
T
+
Y
ij
−
Y
j
.
(D.4)
These symbolic relationships can be more readily understood within
the context of a numerical example. Consider the data summarized
in Table D1. These data represent the hypothetical study investigating
the effects of study time on SAT performance we first encountered in
Chapter 6.
The far left column of Table D1 depicts the scores on the depen-
dent variable (SAT scores) broken by the five levels of the independent
variable (study time in months). The three columns on the right-hand
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