Recall that when the variables XX and
YY are independent, the variance of the
sum or difference between XX and
YY can be written as follows:
σ
X
±
Y
2=
σ
X
2+
σ
Y
2
σ
X
±
Y
2
σ
X
2
σ
Y
2
which is read "The variance of XX
plus or minus YY is equal the variance
of XX plus the variance of
YY.
When XX and YY
are correlated, the following formula should be used:
σ
X
±
Y
2=
σ
X
2+
σ
Y
2±2ρ
σ
X
σ
Y
σ
X
±
Y
2
±
σ
X
2
σ
Y
2
2
ρ
σ
X
σ
Y
where ρρ is the correlation between
XX and YY in
the population. For example, if the variance of verbal SAT were
1000010000, the variance of quantitative SAT were
1100011000 and the correlaton between these two tests
were 0.500.50, then the variance of total SAT (verbal
+ quantitative) would be:
σ
verbal
+
quant
2=10000+11000+2×0.501000011000
σ
verbal
+
quant
2
10000
11000
2
0.50
10000
11000
which is equal to 3148831488. The variance of the
difference is:
σ
verbal
-
quant
2=10000+11000-2×0.501000011000
σ
verbal
-
quant
2
10000
11000
2
0.50
10000
11000
which is equal to 1051210512.
If the variances and the correlation are computed in a sample,
then the following notation is used to express the variance sum law:
s
X
±
Y
2=
s
X
2+
s
Y
2±2r
s
X
s
Y
s
X
±
Y
2
±
s
X
2
s
Y
2
2
r
s
X
s
Y
