Formula for Spearman's Rank Correlation Coefficient

Theorem

Let $S$ be a set of entities.

Let $\sequence {x_i}$ and $\sequence {y_i}$ be rankings of $S$ based on an arbitrary property of the elements of $S$.

Let there be no tied ranks in either $\sequence {x_i}$ or $\sequence {y_i}$.

The Spearman's rank correlation coefficient may be calculated using the formula:

$r_S = 1 - \dfrac {6 T} {n \paren {n^2 - 1} }$

where $T$ denotes the sum of squares of the differences between the ranks for each of the pairs $\tuple {x_1, y_1}$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): correlation coefficient: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): correlation coefficient: 2.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): rank correlation coefficient