A First Course in Quality Engineering, 2nd Edition

482 A First Course in Quality Engineering

The variability not explained by the regression line is

SS residual

( )

= −

( )

=

∑

y y

i i

i

n

2

1

ˆ

.

It can be shown (Hogg and Ledolter 1987) that

y y y y y y

i i i

i

n

i

i

n

i

n

−

( )

= −

( )

+ −

( )

= ==

∑ ∑∑

2 2

1

2

11

ˆ ˆ

.

That is, SS(total) = SS(residual) + SS(regression)�

The term R

2

is dened as the ratio of the variability explained by regres-

sion to the total variability� Therefore,

R

2

=

( )

( )

=

SS regression

SS total

SS(total) - SS(residual

SS total

)

(( )

= −

( )

( )

1

SS residual

SS total

If the value of R

2

 for a model is large, close to 1�0, the regression line is able

to explain a major portion of the variability in the values of y, ...