Page 182 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 8: Correlation Analysis

The coefficient of correlation obtained on the basis of ranks is called ‘Spearman’s Rank Correlation’ Notes
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  1
The means of X and Y, i.e., . Also,
2 2
1 n ( 1) 2 1 n n 1)(2 n 1) n ( 1) 2 n 2 1
(
2 2 2 2 2
X Y [1 2  n ]
n 4 n 6 4 12
Let d be the difference in ranks of the i th individual, i.e.,
i
d = X - Y X i X Y i Y  X Y
i i i
Squaring both sides and taking sum over all the observations, we get
2 2
d i X i X Y i Y
2 2
X i X Y i Y 2 X i X Y i Y
Dividing both sides by n, we get
1 2 1 2 1 2 2
d i X i X Y i Y X i X Y i Y
n n n n
2 2 2Cov X ,Y 2 2 2Cov X ,Y  2 2
X Y X X Y

Cov X,Y
2
2
2
2
2 2 2 2 2 1 
X X Y X X X
X Y
2
1 d i
From this, we can write 1
n 2 2 X
2 2 2
1 d i 1 d i 12 6 d i
or 1 n 2 2 X 1 n 2 n 2 1 1 n n 2 1
Note: This formula is not applicable in case of a bivariate frequency distribution.

8.2.1 Case of Tied Ranks

In case of a tie, i.e., when two or more individuals have the same rank, each individual is
assigned a rank equal to the mean of the ranks that would have been assigned to them in the
event of there being slight differences in their values. To understand this, let us consider the
series 20, 21, 21, 24, 25, 25, 25, 26, 27, 28. Here the value 21 is repeated two times and the value 25
is repeated three times. When we rank these values, rank 1 is given to 20. The values 21 and
21 could have been assigned ranks 2 and 3 if these were slightly different from each other. Thus,
each value will be assigned a rank equal to mean of 2 and 3, i.e., 2.5. Further, the value 24 will be
assigned a rank equal to 4 and each of the values 25 will be assigned a rank equal to 6, the mean
of 5, 6 and 7 and so on.

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