Question

In a music contest, 3 judges X, Y, and Z give scores out of 10. The scores of 4 contestants are as follows. Find which judges disagree with each other the most?
$\begin{array}{cccc}Contestants& X& Y& Z\\ C_1& 10& 7& 8\\ C_2& 7& 9& 4\\ C_3& 6& 8& 5\\ C_4& 8& 10& 9\end{array}$

• A. Judges X and Y
• B. Judges Y and Z
• C. Judges X and Z
• D. Cannot be concluded

Answer 1

The correct option is C

Judges X and Z

i) Convert the ratings of X and Y to corresponding ranks and calculate deviations.

$\begin{array}{ccccc}Contestants& X& Y& D_i=X-Y& D_i^2\\ C_1& 1& 4& -3& 9\\ C_2& 3& 2& 1& 1\\ C_3& 4& 3& 1& 1\\ C_4& 2& 1& 1& 1\\ & & & & \sum =12\end{array}$

$r_k=1-\frac{6\sum D_i^2}{n^3-n}=1-\frac{6\times 12}{64-4}=1-\frac{72}{60}=-0.2$

ii) Convert the ratings of Y and Z to corresponding ranks and calculate deviations.

$\begin{array}{ccccc}Contestants& Y& Z& D_i=Y-Z& D_i^2\\ C_1& 4& 3& 1& 1\\ C_2& 2& 1& 1& 1\\ C_3& 3& 2& 1& 1\\ C_4& 1& 4& -3& 9\\ & & & & \sum =12\end{array}$

$r_k=1-\frac{6\sum D_i^2}{n^3-n}=1-\frac{6\times 12}{64-4}=1-\frac{72}{60}=-0.2$

iii) Convert the ratings of X and Z to corresponding ranks and calculate deviations.

$\begin{array}{ccccc}Contestants& Y& Z& D_i=Y-Z& D_i^2\\ C_1& 1& 3& -2& 4\\ C_2& 3& 1& 2& 4\\ C_3& 4& 2& 2& 4\\ C_4& 2& 4& -2& 4\\ & & & & \sum =16\end{array}$

$r_k=1-\frac{6\sum D_i^2}{n^3-n}=1-\frac{6\times 16}{64-4}=1-\frac{96}{60}=-0.6$

The rank coefficient of -0.6 indicates that judges X and Z disagree with each other the most.