Is Kendall's tau the same as Kendall's W?

No. Tau compares paired ordering; W summarizes concordance across several rankings.

Yes. Negative values indicate more discordant than concordant ordering under the chosen normalization.

Does tau require normally distributed measurements?

No. Its calculation relies on ordering, though valid inference still depends on appropriate sampling assumptions.

AI Glossary / Statistics and probability / Intermediate

Kendall's tau

Definition

Kendall's tau is a rank-based measure of association between two variables, calculated by comparing the ordering of pairs of observations. A pair is concordant when the two variables order it in the same direction and discordant when they order it in opposite directions. The coefficient summarizes the balance between those pair types. Positive values indicate aligned ordering, negative values indicate opposing ordering, and values near zero indicate little net concordance. Several versions exist, particularly to handle ties, so the reported variant should be identified.

What it means

Unlike Pearson correlation, tau does not measure how closely points follow a straight line in their original units. It asks whether a larger value of one variable tends to accompany a larger or smaller value of the other. A monotonic transformation that preserves ordering leaves the relevant rankings unchanged. This can be useful for ordinal data or nonlinear monotonic relationships. However, a tau near zero does not prove independence or rule out a strong nonmonotonic pattern. The statistic also does not establish that either variable causes changes in the other.

How it works

For n paired observations without ties, count concordant pairs C and discordant pairs D among the n(n − 1)/2 unordered pairs. Tau-a is (C − D)/[n(n − 1)/2]. When ties occur, tau-b commonly uses (C − D)/√[(C + D + T)(C + D + U)], where T counts pairs tied only in the first variable and U counts pairs tied only in the second. Pairs tied in both are excluded from those two tie counts. Tau-c uses another normalization. Inference can use exact or approximate procedures where appropriate; the method must account for ties and the sampling design.

Three practical examples

Ranking comparison: compare two methods' ordering of the same candidates or products.
Ordinal research: examine whether higher ordered satisfaction categories tend to accompany higher ordered service ratings.
Model evaluation: investigate whether numerical predictions preserve the ordering of observed outcomes, alongside measures that assess error in the original units. Good ordering alone does not establish accurate numerical predictions.

Think of it this way

Think of choosing every possible pair of objects and asking the same question: do both measurements put these two objects in the same order? Tau balances the votes for agreement against the votes for disagreement. Tied pairs require explicit treatment because one or both measurements may not provide an ordering. This pairwise viewpoint explains why tau is not simply the correlation of rank numbers and why different tie conventions can change the answer.

Worked example

Take x = (1, 2, 3, 4, 5) and y = (1, 3, 2, 5, 4). There are ten pairs. Eight are concordant and two are discordant, so tau-a = (8 − 2)/10 = 0.6. With no ties, tau-b has the same value. Now consider x = (1, 1, 2, 3) and y = (1, 2, 2, 3). Among six pairs, four are concordant, none are discordant, one is tied only in x and one only in y. Tau-b is 4/√[(4 + 1)(4 + 1)] = 0.8. Tau-a would instead be 4/6, approximately 0.6667. These different values reflect different normalizations rather than arithmetic disagreement. In neither example is the coefficient a p-value or the proportion of exactly matching observed scores.

Illustrative case study

Fictional case study: Hazel Reviews compares an automated ordering of support tickets with an expert urgency ordering. Many expert assessments share the same ordinal category, so the analyst reports tau-b and documents tie handling. The team separately checks whether the highest-priority cases are missed, because an overall ordering statistic can hide consequential errors in a small subset. It also evaluates numerical calibration if the automated system supplies probabilities. Concordance is one useful dimension of evaluation, not a complete assessment of the system's practical performance.

Common misconceptions and mistakes

Common mistakes include treating tau as identical to Spearman correlation, using a no-ties formula on tied data without explanation, and interpreting zero as proof of no relationship of any kind. Pairwise comparisons are dependent because each observation appears in multiple pairs; ordinary calculations must not treat all pairs as independent observations for uncertainty estimation. Constant variables can make tau-b undefined. Exact-test availability depends on the method and tie pattern. Repeated, clustered or time-ordered observations may require inference that reflects their dependence rather than the default independent-sample calculation.

Business use

Businesses can use tau to compare rankings and assess monotonic associations when order matters more than numerical spacing. Report sample size, the chosen variant, tie handling and uncertainty or test method where relevant. Inspect plots and task-specific errors, especially when the top or bottom of a ranking has disproportionate consequences. Use original-unit measures if prediction magnitude matters. A favorable association can support further investigation, but it should not be presented as proof of interchangeability between scoring systems or as causal evidence.

Check your understanding

Why is tau-b 0.8 rather than 1 in the tied example?

Answer: Although no pair is discordant, each variable ties a different pair. The tie-adjusted denominator reflects that the two variables do not supply fully matching order information.

Related terms

Concordant pairs
Discordant pairs
Monotonic relationship
concordant pair
discordant pair
Spearman correlation
ordinal data
Kendall's W
tie correction

Learn next

enumerate pairs in a small tied dataset and compare tau-a with tau-b. Then examine an example with a strong U-shaped relationship to see why a low rank-association coefficient does not establish independence.

Sources

Sources: D. C. Howell, statistical glossary, https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html. SciPy, kendalltau documentation, https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kendalltau.html (variants, tie counts and inference). Both worked datasets and the fictional case are original illustrations.

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