Q.Ten students were ranked on the basis of their proficiency in sports and general knowledge. The rank correlation coefficient obtained from the data was found to be 0.2 later on, it was noticed that the difference in the ranks of the two attributes for one of the students was taken as 3 instead of 2. Find the correct value of rank correlation coefficient.

When the ranks match exactly for every observation, $\Sigma d^2 = 0$ and Spearman's rank correlation coefficient becomes $r = 1$ — option (c).

Spearman's rank correlation coefficient (GSEB Class-12 Statistics, Linear Correlation) is:

$r = 1 - \frac{6\,\Sigma d^2}{n(n^2 - 1)}, \qquad d = R_x - R_y$

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They are equal when the data are ranks $1,2,\dots,n$ with no ties, since Pearson's $r$ on such ranks reduces exactly to Spearman's formula.

Spearman's rank correlation coefficient

$r_R = 1 - \frac{6\sum d^2}{n(n^2 - 1)}$

is actually Karl Pearson's coefficient of correlation computed on the ranks of the two variables, under the assumption that the ranks are the integers $1, 2, \dots, n$ each occurring once (no ties).

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