Two judges rank six paintings in a school art show, with 1 for the best…

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Two judges rank six paintings in a school art show, with $1$ for the best. The first judge's ranks of the six paintings are $1, 2, 3, 4, 5, 6$ . The second judge's ranks for the same paintings are $2, 1, 4, 3, 6, 5$ . Find the differences between the two ranks, the sum of their squares and Spearman's rank correlation coefficient.

Need a reminder? The key ideaShow

$r = \dfrac{S_{xy}}{\sqrt{S_{xx}\,S_{yy}}}$ and $-1 \le r \le 1$ . From totals, the numerator is $n\sum xy - \sum x \sum y$ . The denominator has the factors $n\sum x^2 - \left(\sum x\right)^2$ and $n\sum y^2 - \left(\sum y\right)^2$ under one root. $r$ is unchanged by a change of units, and for ranks $r_s = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}$ .

The worked solution

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