Spearman's Rank Correlation Coefficient

Sometimes the data are not exact measurements but ranks — the order in which judges place competitors, or the positions of students in two subjects. Sometimes the characteristic (beauty, honesty, efficiency) cannot be measured numerically at all, only ranked. For such data the British psychologist Charles Spearman gave a coefficient based only on the ranks, called the rank correlation coefficient, denoted $R$ (or $r_s$).

When there are no repeated ranks (no ties):

where $d_i = R_{1i} - R_{2i}$ is the difference between the two ranks of the $i$-th individual and $n$ is the number of individuals. Like Pearson's $r$, $R$ lies between $-1$ and $+1$.

Steps: (i) rank each variable separately (say, largest value gets rank 1 — be consistent for both variables); (ii) find $d = R_1 - R_2$ for each pair; (iii) square and total to get $\sum d^2$; (iv) substitute in the formula. As a check, $\sum d$ must always equal $0$.

When ranks are repeated (tied ranks): each set of tied values is given the average of the ranks they would occupy, and a correction term is added for every tie. If a value is repeated $m$ times, the correction is $\dfrac{m(m^2 - 1)}{12}$, and the formula becomes

the inner sum running over every tied group in both variables.