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In-Depth Information
Ta b l e 2 . 3 .
Defining or deviational formula calculation steps for the variance
Y
i
Score
Deviation from mean
Squared deviation
Y
1
8
−
3
9
Y
2
9
−
2
4
Y
3
10
−
1
1
Y
4
13
2
4
Y
5
15
4
16
SS
=
(
Y
−
Y
)
2
n
=
5
Y
=
55
(
Y
−
Y
)
=
0
=
34
Y
n
=
55
5
=
(
Y
−
Y
)
2
n
−
1
34
4
=
=
s
2
=
=
.
Y
11
8
5
2.7.2 DEFINING OR DEVIATIONAL FORMULA
The defining formula for the variance is as follows:
Y
)
2
=
(
Y
i
−
s
2
variance
=
.
(2.2)
n
−
1
This formula indicates that the variance is a function of the sum of the
squared deviations from the mean divided by the sample size minus one.
This formula and definition are easier to understand within the context
of some actual data, as in Table 2.3.
Note that in Table 2.3 we begin with a distribution of five scores whose
sum (
Y
) is equal to 55. We divide this sum by the number of scores to
obtain the treatment mean of 11. For example,
=
Y
n
=
55
5
=
Y
11
.
Recall that the goal or purpose of measures of variability such as the
variance is to describe how the scores in a distribution deviate or vary on
or about the mean. One way to approach this goa
l i
s to take each
Y
i
score
and subtract the treatment mean from it (
Y
i
−
Y
). We have made this
calculation in the middle of Table 2.3 under the heading “Deviation from
mean.” For example, the first score
Y
1
has a value of 8 and we then subtract
the mean of 1
1 to
obtain a value of
−
3. Symbolically and computationally,
we have (
Y
1
−
3. This is done for e
ac
h of the five scores.
An interesting feature of these deviations (
Y
)
=
(8
−
13)
=−
Y
) is that their sum
equals zero. This will always be true regardless of how much or how little
a set of scores is dispersed about the mean. This is happening because the
negative values are balancing out the positive values.
The fact that the sum of the deviations from the mean always equals
zero poses a minor quandary for statisticians. However, we can sidestep
this dilemma by eliminating the negative valences on our deviation scores.
This can be achieved by squaring all the deviation scores (i.e., multiplying
each deviation by itself). These calculations are provided on the far right
side of Table 2.3. For example, the first deviation score,
Y
−
−
3, is multiplied
