Research Methodology - Dr. Krishan K. Pandey

140 Research Methodology
where X i = ith value of X variable
X = mean of X
Y i = ith value of Y variable
Y = Mean of Y
n = number of pairs of observations of X and Y
σ X = Standard deviation of X
σ Y = Standard deviation of Y
In case we use assumed means (A x and A y for variables X and Y respectively) in place of true means,
then Karl Person’s formula is reduced to:
where ∑ dx = ∑ X − A
∑ ⋅
−F HG
∑
− ∑ F
dx dy
n
dx dx I
HG KJ n n
∑
∑ ⋅
−F HG
∑
− ∑ F
dx dy
n
dx dx I
HG KJ n n
∑
b
d
b
d
b
g
i
2 g2
i
gd i
i i x
∑ dy = ∑ Y − A
i i y
2
i
2
i x
i i y
∑ dx = ∑ X − A
∑ dy = ∑ Y − A
∑dx ⋅ dy = ∑ X − A Y − A
i i i x i y
∑dx ⋅ ∑dy
n
i i i i
2 2 2 2
i i i i
dy
n
∑dx ⋅∑dy
n
i i i i
dy
n
I
KJ F I
HG KJ I
KJ F I
HG KJ − ∑dy
n
2 2 2 2
i i i i
− ∑dy
n
n = number of pairs of observations of X and Y.
This is the short cut approach for finding ‘r’ in case of ungrouped data. If the data happen to be
grouped data (i.e., the case of bivariate frequency distribution), we shall have to write Karl Pearson’s
coefficient of correlation as under:
∑ f ⋅ dx ⋅ dy
∑ fdx
n
n
F
HG
− ∑ fdx
n
F
HG
I
KJ ∑
− ∑ fdx
n
⋅ ∑ f dy
n
ij i j i i j j
2 2 2
fdy
i i i i i j j j
n
F
HG
I
KJ
− ∑ f dy
n
where f ij is the frequency of a particular cell in the correlation table and all other values are defined
as earlier.
I
KJ