448 Chapter 10:Analysis of Variance
When computing by hand, the quantitySSbdefined by
SSb=n
∑m
i= 1
(Xi.−X..)^2
should be computed first. OnceSSbhas been computed,SSWcan be determined from the
sum of squares identity. That is,
∑m
i= 1
∑n
j= 1 X
2
ij andX
2
..should also be computed and
thenSSWdetermined from
SSW=
∑m
i= 1
∑n
j= 1
Xij^2 −nmX..^2 −SSb
EXAMPLE 10.3b Let us do the computations of Example 10.3a by hand. The first thing to
note is that subtracting a constant from each data value will not affect the value of the test
statistic. So we subtract 220 from each data value to get the following information.
Gas Mileage
∑
jXij
∑
jX
2
ij
1 0 31 6 26 40 103 3,273
2 24 15 12 22 5 78 1,454
3 32 52 30 18 36 168 6,248
Nowm=3 andn=5 and
X1.=103/5=20.6
X2.=78/5=15.6
X3.=168/5=33.6
X..=(103+ 78 +168)/15=23.2667, X..^2 =541.3393
Thus,
SSb= 5 [(20.6−23.2667)^2 +(15.6−23.2667)^2 +(33.6−23.2667)^2 ]=863.3335
Also,
∑∑
Xij^2 =3,273+1,454+6,248=10,975
and, from the sum of squares identity,
SSW=10,975−15(541.3393)−863.3335=1991.5785