Page 210 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Unit 10: Correlation

Karl Pearson’s Method – Without Deviations (Short-cut Method) Notes

When the arithmetic means of both sets of numerical items are not whole numbers and involve
decimals, calculating the coefficient of correlation by direct method becomes tedious. To
overcome this difficulty the following modified short-cut method formula is used:

 X Y
Cov(X , Y ) = i i  X Y
i i n

 X 2 i  Y i 2
2
V(X ) =  X ; V(Y ) =  Y 2
i i
n n
Cov(X ,Y )
i
i
 =
{V(X )V(Y )}
i
i

n X Y   X i  Y
 = i i i
2
2
 n X    2   n Y    2 


 X
 Y
 i i   i i 
Example 2: Calculate the Karl Pearson’s coefficient of correlation for the following data
between sales and advertising expenditure.
Let sales represents Xi variable and advertise expenditure represents Yi variable to calculate the
correlation coefficient using the following formula:
n X Y   X i  Y

 = i i i
2
2
 n X    2   n Y    2 
 X
 Y


 i i   i i 
i X i Y i X i Y 2 i X Y
2
i
1 3 1 9 3
2 15 4 225 30
3 6 9 36 18
4 20 16 400 80
5 9 25 81 45
6 25 36 625 150
 X =21  Y =78  X =91  Y =1376  X Y = 326
2
2
i
i
i
i
i
i

6x326  21x78 
 =
 6x91  21 2 6x1376  78 2
   
    
   
318
 = (10.247 x 46.605)
 = 0.667
This suggests that a fairly high degree of correlation between X and Y series i.e. between sales
and advertising expenditure.
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