Question

Two judges in a beauty competition rank the 12 entries as follows:

X	1	2	3	4	5	6	7	8	9	10	11	12
Y	12	9	6	10	3	5	4	7	8	2	11	1

Calculate the Spearman's rank coefficient of correlation.

Answer 1

X	Y	R1	R2	D=R1−R2	D2
1 2 3 4 5 6 7 8 9 10 11 12	12 9 6 10 3 5 4 7 8 2 11 1	1 2 3 4 5 6 7 8 9 10 11 12	12 9 6 10 3 5 4 7 8 2 11 1	−11 −7 −3 −6 2 1 3 1 1 8 0 11	121 49 9 36 4 1 9 1 1 64 0 121
N = 12					$\Sigma D^2$=416

${\mathrm{r}}_{\mathrm{k}}=1-\frac{6\left[{\mathrm{\Sigma D}}^2-{\displaystyle \frac{1}{12}}\left({\mathrm{m}}^3-\mathrm{m}\right)+{\displaystyle \frac{1}{12}}\left({\mathrm{m}}^3-\mathrm{m}\right)+...\right]}{{\mathrm{N}}^3-\mathrm{N}}$
Here, m = 0
$$\begin{gathered} r_{\mathit{k}}=1-\frac{6\Sigma D^2}{N^{\mathit{3}}\mathit{-}N} \\ \mathrm{or},r_k=1-\frac{6\times 416}{1728-12} \\ \mathrm{or},r_k=1-\frac{2496}{1716} \\ \Rightarrow r_k=-0.454 \end{gathered}$$

Hence, spearmen's rank coefficient of correlation = -0.454