Introduction to Probability and Statistics for Engineers and Scientists

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