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Unit 10: Correlation: Scatter Diagram Method, Karl Pearson's Coefficient of Correlation

Notes
( x )∑ ( d d y ) ∑
∑dd –
xy
r = N 2
∑ x – ( 2 N X )∑d 2 ∑d d y 2 – ( N y ) ∑d

where d refers to deviations of X series from an assumed mean, i.e., (X – A); d refers to deviations of
x y
Y series from an assumed mean, i.e., (Y – A); ∑dd = sum of the product of the deviations of X and
x y
Y series from their assumed means; ∑d x 2 = sum of the squares of the deviations of X series from an

assumed means; ∑d y 2 = sum of the squares of the deviations of Y series from an assumed mean; ∑d x
= sum of the deviations of X series from an assumed mean; ∑ d = sum of the deviations of Y series
y
from an assumed mean.
It may be pointed out that there are many variations of the above formula. For example, the above
formula may be written as:

N ∑ xy ( { ∑dd x ) d ( d y ) ∑ – }
r =
N ∑ 2 – ( x d x )∑d 2 N ∑ 2 – ( y d y ) ∑d 2

But the form given above is the easiest to apply.
Note: While applying assumed mean method, any value can be taken as the assumed mean and
the answer will be the same. However, the nearer the assumed mean to the actual mean,
the lesser will be the calculations.
Steps

(i) Take the deviations of X series from an assumed mean, denote these deviations by d and
x
obtain the total, i.e., ∑d .
x
d
(ii) Take the deviations of Y series from an assumed mean, denote these deviations by y and
obtain the total, i.e., ∑d .
y

(iii) Square d and obtain the total ∑d x 2 .
x
d
(iv) Square y and obtain the total ∑d y 2 .
d
d
(v) Multiply x and y and obtain the total ∑dd .
x y
(vi) Substitute the values of ∑dd , ∑d , ∑ d , ∑d x 2 and ∑d y 2 in the formula given above.
y
y
x
x
The following examples shall illustrate the procedure:

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