Business Mathematics and Statistics · Ch 9 — Correlation and Regression Analysis

Spearman's Rank Correlation Coefficient

Pearson's $r$ needs actual numerical values to work with. Sometimes the available data is only an order — two judges ranking the same set of contestants, two examiners ranking the same essays — with no underlying numerical score at all, or the ranking itself (rather than the raw score) is what genuinely matters for the comparison. Spearman's rank correlation coefficient measures the association between two such rankings of the same $n$ items.

Note

Spearman's Rank Correlation Coefficient

For $n$ items each given two ranks, let $d_i$ be the difference between an item's two ranks. Then:

$r_s = 1 - \dfrac{6\sum d_i^2}{n(n^2-1)}$

Like Pearson's $r$ , Spearman's $r_s$ always lies between $-1$ and $+1$ : $r_s$ near $+1$ means the two rankings largely agree; $r_s$ near $-1$ means they are largely reversed; $r_s$ near $0$ means the two rankings show little relationship. If the two rankings are identical, every $d_i=0$ , giving $r_s=1$ exactly. …

Definition 4Spearman's Rank Correlation Coefficient (r_s)

A measure of association between two rankings of the same $n$ items, $r_s = 1-6\sum d^2/(n(n^2-1))$ , where $d$ is the difference between an item's two ranks; used for ranked (ordinal …

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