Question

Calculate the coefficient of correlation of the following data by Spearman's rank correlation method:

$\begin{array}{ccccccc}\text{X}& 19& 24& 12& 23& 19& 16\\ \text{Y}& 9& 22& 20& 14& 22& 18\end{array}$

Answer 1

$\begin{array}{cccccc}\text{X}& \text{Y}& R_1& R_2& (R_1-R_2)& D^2\\ & & & & D& \\ 19& 9& 3.5& 1& +2.5& 6.25\\ 24& 22& 6& 5.5& +0.5& 0.25\\ 12& 20& 1& 4& -3& 9\\ 23& 14& 5& 2& +3& 9\\ 19& 22& 3.5& 5.5& -2& 4\\ 16& 18& 2& 3& -1& 1\\ N=16& & & & & \sum D^2=29.50\end{array}$

Here number 19 is repeated twice in series X and number 22 is repeated twice in series Y. Therefore, in X, m = 2 and in Y, m = 2.

Rank Coefficient of Correlation $\left(r_k\right)=$

$1-\frac{6[\sum D^2+\frac{1}{12}(m_1^3-m_1)+\frac{1}{12}(m_2^3-m_2)+.....]}{N^3-N}$

$\sum D^2=29.50$;

$=1-\frac{6[29.50+\frac{1}{12}(2^3-2)+\frac{1}{12}(2^3-2\left)\right]}{6^3-6}$

$=1-\frac{6[29.50+0.5+0.5]}{201}$

$=1-\frac{6\times 30.5}{210}$

$=1-\frac{183}{210}=1-0.871$

= 0.129

Rank Coefficient of Correlation = 0.129. There is a very low degree of positive correlation.