Question

Calculate the coefficient of rank correlation from the following data.

X	:	48	33	40	9	16	16	65	25	15	57
Y	:	13	13	24	6	15	4	20	9	6	19

Answer 1

X	R1	Y	R2	(R1 − R2) D	D2
48	8	13	5.5	2.5	6.25
33	6	13	5.5	0.5	0.25
40	7	24	10	−3	9
9	1	6	2.5	−1.5	2.25
16	3.5	15	7	−3.5	12.25
16	3.5	4	1	2.5	6.25
65	10	20	9	1	1
25	5	9	4	1	1
15	2	6	2.5	−0.5	0.25
57	9	19	8	1	1
					$\sum D^2=39.5$

Given:
N = 10
When, ranks are repeated more than once, the following formula is used to calculate coefficient of rank correlation.
$$\begin{gathered} r_k=1-\frac{6\left[\mathrm{\Sigma }{D}^2+{\displaystyle \frac{1}{12}}\left(m^3-m\right)+{\displaystyle \frac{1}{12}}\left(m^3-m\right)+{\displaystyle \frac{1}{12}}\left(m^3-m\right)\right]}{N^3-N} \\ =1-\frac{6\left[39.5+{\displaystyle \frac{1}{12}}\left(2^3-2\right)+{\displaystyle \frac{1}{12}}\left(2^3-2\right)+{\displaystyle \frac{1}{12}}\left(2^3-2\right)\right]}{1000-10} \\ =1-\frac{6\times 41}{990}=1-\frac{246}{990}=1-0.248 \\ =0.751 \end{gathered}$$
Thus, the coefficient of rank correlation is 0.751.