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3)
2
by itsel
f (
−
and equals 9. When we sum these squared deviations
Y
)
2
(
Y
−
we cre
ate
a value known as the
sum of squares
.Inthepresent
Y
)
2
case,
SS
=
(
Y
−
=
34. By dividing the sum of squares by the sample
size minus one or
n
1, we determine the variance, which is 8.5 in the
present example. In Chapter 3, we will describe this denominator or
n
−
1
value as the
degrees of freedom
and discuss it in more detail. For now,
think of the degrees of freedom as a way of adjusting the sum of squares.
Thus, in practice the variance is equal to the sum of squares divided by
the degrees of freedom, or symbolically,
s
2
−
=
SS
/
df
.
2.7.3 COMPUTATIONAL FORMULA FOR THE VARIANCE
The computational formula for the variance takes the following form:
Y
)
2
n
(
Y
2
−
s
2
=
.
(2.3)
−
n
1
This formula is based upon calculating two separate sums:
Y
2
(read “summation
Y
squared”), the sum of the squared scores.
Y
)
2
(read “summation
Y
quantity squared”), the square of the total
Y
-score sum
(
.
The calculations for this computational formula can be more readily
understood by examining the numerical example of Table 2.4, which uses
the same raw scores used with the defining formula example (shown in
Table 2.3).
Ta b l e 2 . 4 .
Computational formula calculation steps for the variance
Y
i
Y
i
(or Score
2
)
Score
Y
1
8
64
Y
2
9
81
Y
3
10
100
Y
4
13
169
Y
5
15
225
n
=
5
Y
=
55
Y
2
=
639
(
Y
)
2
=
3,025
(
Y
)
2
n
3,025
5
=
Y
2
−
=
−
=
−
=
SS
639
639
605
34
Y
n
=
55
5
=
11
Y
=
(
Y
)
2
n
3,025
5
Y
2
−
639
−
639
−
605
4
34
4
=
s
2
=
=
=
=
.
8
5
n
−
1
5
−
1
