Spearman's Rank Correlation Coefficient

When the data available are already in the form of ranks — or consist of a qualitative attribute that can only be ordered, not measured (e.g., ranking contestants by beauty, honesty, or leadership quality) — Karl Pearson's method cannot be applied directly, since it needs actual measured values. Spearman's Rank Correlation Coefficient (denoted $\rho$, rho) measures the degree of correlation using ranks instead.

Without repeated (tied) ranks: let $d$ be the difference between the ranks given to the same item by the two rankings ($d = R_x - R_y$), and $n$ the number of items ranked:

With repeated (tied) ranks: when two or more items receive the same rank, each is given the average of the ranks they would jointly have occupied (e.g., if two items tie for 2nd and 3rd place, both are given rank $\frac{2+3}{2}=2.5$). A correction term is then added to $\sum d^2$ for every group of $m$ tied items (in either ranking):

(Every tied group anywhere in either of the two rankings contributes its own correction term to the same sum.)

Merits: simple to compute; the only method usable when data is genuinely ordinal (ranks/qualities) rather than measured; also convenient as a quicker check for a small number of items even when actual values ARE available. …