Page 314 - DMTH404_STATISTICS

Statistics

Notes 22.8 Spearman's Rank Correlation

This is a crude method of computing correlation between two characteristics. In this method,
various items are assigned ranks according to the two characteristics and a correlation is computed
between these ranks. This method is often used in the following circumstances:
(i) When the quantitative measurements of the characteristics are not possible, e.g., the results
of a beauty contest where various individuals can only be ranked.
(ii) Even when the characteristics is measurable, it is desirable to avoid such measurements
due to shortage of time, money, complexities of calculations due to large data, etc.

(iii) When the given data consist of some extreme observations, the value of Karl Pearson's
coefficient is likely to be unduly affected. In such a situation the computation of the rank
correlation is preferred because it will give less importance to the extreme observations.
(iv) It is used as a measure of the degree of association in situations where the nature of
population, from which data are collected, is not known.

The coefficient of correlation obtained on the basis of ranks is called 'Spearman's Rank Correlation'
or simply the 'Rank Correlation'. This correlation is denoted by  (rho).
Let X be the rank of i th individual according to the characteristics X and Y be its rank according
i i
to the characteristics Y. If there are n individuals, there would be n pairs of ranks (X , Y ), i = 1, 2,
i i
...... n. We assume here that there are no ties, i.e., no two or more individuals are tied to a
s
s
particular rank. Thus, X ' and Y ' are simply integers from 1 to n, appearing in any order.
i i
1 2 n  n n 1  n 1


The means of X and Y, i.e., X  Y    . Also,
n 2n 2
2
(
1 n (  1) 2 1  n n 1)(2 n 1)  n (  1) 2 n  1


2 2 2 2 2

 X    [1  2   n ]   
Y
n 4 n   6   4 12
Let d be the difference in ranks of the i th individual, i.e.,
i
Y
d = X - Y  X  X  Y  Y  d  X  i
  i
i i i i
Squaring both sides and taking sum over all the observations, we get
2
2  
 d     X  X  Y  Y  
i
  i
i
2 2
Y 
   X  X     i  Y  2  X  X Y   Y
i
i
 i
Dividing both sides by n, we get
1 2 1 2 1 2 2
Y 
 d    X  X     i Y     X  X Y  Y 
i
i
 i
i
n n n n
2
2
2
 2    2 Cov X Y,   2  2 Cov X Y,   d 2   i
X Y X X Y
 Cov X,Y  
2
2
2
 2  2   2  2  2 2 1     
X X Y X X X  
   Y 
X
306 LOVELY PROFESSIONAL UNIVERSITY