Page 212 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Unit 10: Correlation

Properties of Correlation: Notes

(i) The value of correlation coefficient  varies between [–1, +1]. This indicates that the value
of does not exceed unity.
(ii) Sign of  depends on sign of the covariance.

(iii) If = –1, the variables are perfectly negatively correlated.
(iv) If = +1, the variables are perfectly positively correlated.
(v) If  = 0, the variables are not correlated in a linear fashion. There may be nonlinear
relationship between variables.
(vi) Correlation coefficient is independent of change of scale and shifting of origin. In other
words, shifting the origin and change the scale do not have any effect on the value of
correlation.
Let us see the following example to understand the concept, ‘if  = 0, the variables are not
correlated in a linear fashion. There may be nonlinear relationship between variables’.

Example 4: If X and Y are given as below, we calculate the correlation coefficient.
i i
X Y X 2 Y X Y
2
i i i i i i
-3 9 9 81 -27
-2 4 4 16 -8
-1 1 1 1 -1
0 0 0 0 0
1 1 1 1 1
2 4 4 16 8
3 9 9 81 27
2
2
2
2
 X =0  Y =28  X =28  Y =196  X Y =0
i
i
i
i
i
i


n X Y   X i  Y i
i
i
 =
2
 n X    2   n Y    2 
2
 X
 Y


 i i   i i 
7 x0  0x28 
 =
   
   0  
 7 x28  2 7 x196  28 2
   
0
 =
196x588
0
 = 196x588 = 0
Since  = 0 it does not mean that the variables X and Y are uncorrelated. It can only be said that
i i
the variables are linearly uncorrelated. In fact if we closely look at the data of X and Y , it can be
i i
observed that Y = X is the relationship existing between X and Y . This is a nonlinear relationship
2
i i i i
between the variables. Karl Pearson’s coefficient of correlation can not measure nonlinear
relationship between the variables.
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