Page 195 - DMGT404 RESEARCH_METHODOLOGY

Unit 9: Correlation and Regression

å X Y - nXY = å X Y - ) Notes
Also, i i (X - )( i Y
i
å
2
2
and å X - nX = (X - X ) 2
i
i
å (X - X Y - Y )
)( i
 b = i 2 .... (6)
å (X - X )
i
å x y i
i
or b = å x i 2 .... (7)
where x and y are deviations of values from their arithmetic mean.
i i
Dividing numerator and denominator of equation (6) by n we have
1 å (X - X Y - Y )
b = n i )( i  Cov ( ,X Y ) .... (8)
1 å (X - X ) 2 s X 2
n i
The expression for b, which is convenient for use in computational work, can be written from
equation (5) is given below:

å X Y - n å X i å Y i å X Y - (å X i )(å Y i )
×
i
i
i
i
b = æ n å X ö n  (å n ) 2
2
å X - n ç i ÷ å X - X i
2
2
i
è n ø i n
Multiplying numerator and denominator by n, we have
nå X Y - (å X )(å Y )
b = i i i 2 i .... (9)
nå X - (å X )
2
i i
To write the shortcut formula for b, we shall show that it is independent of change of origin but
not of change of scale.
As in case of coefficient of correlation we define
X - A
u = i
i h
Y - B
and v = i
i k
or X = A + hu
i i
and Y = B + kv
i i
 X = A hu+
and Y = B kv+

also (X - X ) = ( h u - ) u
i
i
k v -
and Y - Y = ( i ) v
i

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