Spearman's Rank Correlation Coefficient

Pearson's correlation coefficient needs actual numerical values of $x$ and $y$ to work with. Sometimes, though, the data available is only an order — two judges ranking the same contestants, or two examiners ranking the same essays — with no underlying numerical score at all, or the numerical scores are themselves converted to ranks because the ranking itself (not the raw score) is what is being compared. Spearman's rank correlation coefficient measures the strength of association between two such sets of ranks.

Like Pearson's $r$, Spearman's $r_s$ always lies between $-1$ and $+1$, and is read the same way: $r_s$ near $+1$ means the two rankings largely agree (an item ranked high on one criterion tends to be ranked high on the other too); $r_s$ near $-1$ means the rankings are largely reversed; $r_s$ near $0$ means the two rankings show little relationship. If the two rankings are identical, every $d_i = 0$ and $r_s = 1$ exactly.

⚠️ Scope note: the formula above is the standard version, used when no two items share the same rank on either characteristic (no tied ranks). This is the formulation this syllabus's own topic line specifies; a further correction term exists in the wider subject for the case of tied ranks, but is not part of this chapter's stated scope, so every worked example and exercise here uses data with distinct ranks throughout. …