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In-Depth Information
6.5.3 SUM OF SQUARES
Throughout this text, we will be providing you with what are called “com-
putational formulas” for computing the various sums of squares by hand.
Computational formulas provide a faster and more efficient means of
conducting the sometimes intricate calculations involved in producing
sums of squares. However, this mathematical efficiency or elegance comes
at a cost; computational formulas are not intuitive, nor foster conceptual
understanding. That is, it is sometimes difficult to see how the variance
is actually partitioned when using these computational procedures. A
more intuitive, but more mathematically cumbersome, approach to cal-
culating sums of squares is to use
deviational formulas
. Such formulas
depict very concretely how the total variability within the study is bro-
ken into between-groups and within-groups variability, all on the basis
of how the original raw scores on the dependent variable deviate from
the grand mean (total variability), how these same scores deviate from
their respective group mean (within-groups variability), and how each
group mean deviates from the grand mean (between-groups variability).
We will illustrate these deviational formulas with the current numerical
example in Appendix E, and keep our discussion focused on computa-
tional procedures within each chapter. Remember, either the deviational
or computational formula will produce the exact same result.
The sum of squares values (shown as
SS
in Table 6.2) are shown in the
second column of Table 6.2. The preliminary calculations and summary
statistics shown in Table 6.1 become the ingredients for computing the
various sums of squares. In a one-way between-subjects ANOVA, there
are three sums of squares that must be calculated:
Sum of squares between groups or treatment,
SS
A
,inwhichthe
subscript
A
represents the single independent variable or Factor
A
.
Sum of squares within groups or error,
SS
S
/
A
,inwhichthesubscript
S
A
indicates subjects within each level of the
A
treatment.
Sum of squares total,
SS
T
.
/
6.5.3.1 Sum of Squares for the Treatment Effect
(Between-Groups Variance):
SS
A
The formulas for these three sums of squares and the summary of their
calculations based on the data from Table 6.1 are as follows:
A
2
n
T
2
(
a
)(
n
)
SS
A
=
−
(2,890)
2
(3,320)
2
(3,870)
2
(4,300)
2
(4,360)
2
+
+
+
+
=
7
(
18,740
)
2
35
−
=
230,496.57
.
(6.1)
