Question

In a singing competition, two judges rank the seven contestants as follows:

$\begin{array}{cccccccc}\text{Judge 1}& 5& 4& 7& 3& 1& 2& 6\\ \text{Judge 2}& 6& 5& 2& 1& 3& 4& 7\end{array}$

Calculate the coefficient of rank correlation.

Answer 1

$\begin{array}{cccc}\text{Rank by judge 1}& \text{Rank by judge 2}& D=R_1-R_2& D^2\\ \left(R_1\right)& \left(R_2\right)& & \\ 5& 6& -1& 1\\ 4& 5& -1& 1\\ 7& 2& 5& 25\\ 3& 1& 2& 4\\ 1& 3& -2& 4\\ 2& 4& -2& 4\\ 6& 7& -1& 1\\ N=7& & & \sum D^2=40\end{array}$

$r_k=1-\frac{6\sum D^2}{N^3-N}=1-\frac{6\left(40\right)}{7^3-7}=1-\frac{240}{336}=0.286$

There is a low degree of positive correlation.