Kendall rank correlation coefficient (${\tau }_a=(C-D)/\left(\genfrac{}{}{0ex}{}{n}{2}\right)$)

A family of ordinal association measures based on concordant pairs and discordant pairs. Without a tie adjustment, the coefficient is ${\tau }_a=(C-D)/\left(\genfrac{}{}{0ex}{}{n}{2}\right)$: the arithmetic mean, over unordered pairs, of the product of the two coordinate comparison signs. Reversing one ordering reverses the sign, and strictly increasing relabellings preserve it. Kendall's tau-b makes an explicit adjustment for tied pairs. The name commemorates Maurice Kendall.

Let $C,D$ count concordant pairs and discordant pairs, and let $T_X,T_Y$ count pairs tied only on the indicated variable. Thenwhen both factors in the denominator are positive. Unlike Goodman and Kruskal's gamma, this coefficient accounts for ties. Its absolute value is at most one: form vectors of the comparison signs for all unordered pairs. Their inner product is $C-D$ and their squared norms are the two denominator factors, in reverse variable order. The Cauchy-Schwarz inequality supplies the bound. A variable constant throughout gives a zero denominator, not a meaningful zero association.^[ref]