Question

Find out rank difference correlation of X and Y.

$\begin{array}{cccccccc}X& 80& 78& 75& 58& 67& 60& 59\\ Y& 12& 13& 14& 14& 16& 15& 17\end{array}$

Answer 1

Calculation of Rank Correlation

$\begin{array}{cccccc}X& R_1& Y& R_2& R_1-R_2=D& D^2\\ 80& 1& 12& 8& -7& 49\\ 78& 2& 13& 7& -5& 25\\ 75& 3.5& 14& 5& -1.5& 2.25\\ 58& 3.5& 14& 5& -1.5& 2.25\\ 67& 8& 16& 2& +3& 9\\ 60& 5& 15& 3& +3& 9\\ 59& 6& 17& 1& +3& 9\\ & 7& & & 6& 36\\ & & & & & \sum D^2=141.50\end{array}$
Here, N=8, $\sum D^2=141.50,m_1=2and{m}_2=3$
$rk=1-\frac{6[\sum D^2+\frac{1}{12}(m_1^3-m_1)+\frac{1}{12}(m_2^3-m_2\left)\right]}{N^3-N}$
$=1-\frac{6[141.50+\frac{1}{12}(2^3-2)+\frac{1}{12}(3^3-3\left)\right]}{8^3-8}=1-\frac{6[141.50+\frac{6}{12}+\frac{24}{12}]}{512-8}$
$=1-\frac{6(141.50+0.50+2)}{504}$
$=1-\frac{6\times 144}{504}=1-\frac{864}{504}=1-1.712=-0.712$
It indicates that there is moderate degree of negative correlation.