Introduction to Probability and Statistics for Engineers and Scientists

456 Chapter 10:Analysis of Variance

where

∑m

i= 1

αi=

∑n

j= 1

βj= 0

The valueμis called thegrand mean,αiis thedeviation from the grand mean due to row i,
andβjis thedeviation from the grand mean due to column j.
Let us now determine estimators of the parametersμ,αi,βj,i=1,...,m,j=1,...,n.
To do so, continuing our use of “dot” notation, we let

Xi.=

∑n

j= 1

Xij/n=average of the values in rowi

X.j=

∑m

i= 1

Xij/m=average of the values in columnj

X..=

∑m

i= 1

∑n

j= 1

Xij/nm=average of all data values

Now,

E[Xi.]=

∑n

j= 1

E[Xij]/n

=μ+

∑n

j= 1

αi/n+

∑n

j= 1

βj/n

=μ+αi since

∑n

j= 1

βj= 0

Similarly, it follows that

E[X.j]=μ+βj

E[X..]=μ

Because the preceding is equivalent to

E[X..]=μ

E[Xi.−X..]=αi

E[X.j−X..]=βj