StatReason

Kendall Tau Calculator

Measure how consistently two variables rank observations in the same order, using Kendall's tau-b, a concordance-based alternative to Pearson and Spearman correlation that handles tied values explicitly.

What this answers

This calculator answers "how often do these two variables agree on which of any two observations ranks higher?" rather than measuring linear or even strictly monotonic association the way Pearson and Spearman do. It is often preferred with small samples or data containing many tied values.

How it is calculated

Every possible pair of observations is compared: a pair is concordant if both variables agree on which observation is larger, discordant if they disagree, and tied if either variable has equal values in that pair. Tau-b is the difference between the concordant and discordant pair counts, divided by a denominator adjusted downward for the number of tied pairs in each variable separately, which is what distinguishes tau-b from the simpler tau-a that ignores ties. The reported p-value uses a standard normal approximation to tau-b's sampling distribution; this calculator's version is unadjusted for ties in the variance term, a documented simplification worth knowing about with heavily tied data.

Worked example

For x and y both equal to 1, 2, 3, 4 in the same order, every one of the 6 possible pairs is concordant and none is discordant, giving a tau-b of exactly 1, the maximum possible value, reflecting perfect agreement in ranking between the two variables.

Assumption audit

Calculated from your data: the concordant and discordant pair counts, the number of tied pairs in each variable, and the resulting tau-b and approximate p-value.
Evidence to review: how many tied pairs your data contains; a large proportion of ties in either variable makes the normal-approximation p-value above less reliable, since it does not fully adjust for ties in its variance term.
You must verify: that your pairs are genuinely independent observations and that an ordinal or ranking relationship, not necessarily a linear one, is the question you actually want answered.

What this result does not mean

A high tau-b means the two variables rank observations similarly, not that one causes the other, and not that the relationship between them is linear; a strongly nonlinear but monotonic relationship can still produce a tau-b near 1, the same way it can for Spearman's rank correlation.

Limitations

Kendall's tau and Spearman's rank correlation both measure ordinal association and often move together, but they are not interchangeable numbers; tau-b is typically smaller in magnitude than Spearman's rho on the same data, a normal property of the different counting methods, not evidence that one measure is wrong.