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Unformatted text preview: arson, a great biometrician and statistician,
suggested a mathematical method for measuring the magnitude of
linear relationship between the two variables. It is most widely
used method in practice and it is known as pearsonian coefficient of
correlation. It is denoted by ‘ r’ . The formula for calculating ‘ r’ is
C ov( x, y )
(i) r =
where σ x , σ y are S.D of x and y
σ x .σ y
respectively.
∑ xy
(ii) r =
n σx σ y
(iii) r = Σ XY
∑ X2 .∑ Y2 , X = x− x , Y = y− y when the deviations are taken from the actual mean we can apply
any one of these methods. Simple formula is the third one.
The third formula is easy to calculate, and it is not
necessary to calculate the standard deviations of x and y series
respectively.
Steps:
1. Find the mean of the two series x and y.
2. Take deviations of the two series from x and y.
X = x− x , Y = y− y
3. Square the deviations and get the total, of the respective
squares of deviations of x and y and denote by X2 ,
Y2 respectively.
4. Multiply the deviations of x and y and get the total and
Divide by n. This is covariance.
5. Substitute the values in the formula.
r= cov( x, y )
=
σx.σy ∑ ( x − x) ( y  y ) / n ∑( x − x) 2 ∑( y − y ) 2
.
n
n
196 The above formula is simplified as follows
Σ XY
r=
, X = x− x , Y = y− y
∑ X2 .∑ Y2
Example 1:
Find Karl Pearson’ s coefficient of correlation from the following
data between height of father (x) and son (y).
X
64
65
66
67
68
69
70
Y
66
67
65
68
70
68
72
Comment on the result.
Solution:
x
Y
XY
X2 Y = y − y Y2
X = x− x
X = x – 67
Y = y  68
64
66
3
9
2
4
6
65
67
2
4
1
1
2
66
65
1
1
3
9
3
67
68
0
0
0
0
0
68
70
1
1
2
4
2
69
68
2
4
0
0
0
70
72
3
9
4
16
12
469 476
0
28
0
34
25
469
476
x=
= 67 ; y =
= 68
7
7
25
25
25
Y
ΣX
r=
=
=
=
= 0.81
2
2
30.85
28 × 34
952
∑ X. ∑ Y
Since r = + 0.81, the variables are highly positively correlated. (ie)
Tall fathers have tall sons.
Working rule (i)
We can also find r with the following formula
C ov( x, y )
We have r =
σ x .σ y
Cov(...
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 Winter '08
 Moshiri
 Business

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