Information Technology Reference
In-Depth Information
additional variables enabling us to explore complex interrelationships
among potentially important variables.
The within-groups sum of squares represents error variability - vari-
ability within each group that cannot be attributed to the level of the
independent variable those participants experienced. In the context of
sum of squares, the scores within each group are the scores from which a
mean is subtracted. The mean that is subtracted is the mean of the group
containing the score. Thus,
= (score
group mean) 2
within-groups sum of squares
.
In our notation system the formula for computing the within-groups sum
of squares is therefore written as
SS S / A = ( Y i
Y j ) 2 ,
(3.3)
where SS S / A is t he within-groups sum of squares, Y i is the score in a
given group, and Y j is the mean of the particular group. The heart of this
computation is the residual values when we subtract the group mean from
each of the scores. Because some of these residuals would be positive (the
score would be higher than the mean) and others would be negative, such
that their sum would be zero (the sum of deviations around the mean
must add to zero), it is necessary to square these residuals or deviations to
obtain values that, when added, will not necessarily produce a zero value.
3.4.4 SUMS OF SQUARES ARE ADDITIVE
One feature of the sum of squares worth noting at this point is its additive
nature: The between-groups sum of squares added to the within-groups
sum of squares is equal to the total sum of squares. This allows us to take
the proportion of the total sum of squares accounted for by the between-
groups sum of squares. We will use such a proportion to quantify the
“strength of effect” of the independent variable indexed by eta squared in
Chapter 4.
3.5 DEGREES OF FREEDOM ( df )
The degrees of freedom, commonly abbreviated as df, that are associated
with each source of variance are contained in the third column of the
summary table in Table 3.2. These are, very roughly, computed by sub-
tracting the value of 1 from the number of scores that are being processed
in a computation.
3.5.1 A BRIEF EXPLANATION OF DEGREES OF FREEDOM
Consider a set of scores with these two constraints:
There are a total of three scores (negative values are allowed).
Their sum is 11.
 
Search WWH ::




Custom Search