Q.In which situation, the values of Karl Pearson's correlation coefficient and Spearman's rank correlation coefficient are equal?

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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