Kendall's Tau Rank Correlation
Also known as: Kendall tau, Kendall rank correlation, tau-b, tau-c
Kendall's tau is a nonparametric measure of the ordinal association between two variables. It quantifies how consistently the relative ordering of one variable matches the ordering of another across all observation pairs, making it robust to outliers and suitable for ordinal or non-normally distributed data.
Key highlights
- Distribution-free: requires no normality assumption for either variable.
- Robust to outliers because only the rank ordering — not the raw values — matters.
- Handles ordinal data naturally, including Likert-scale responses.
- Has a clear probabilistic interpretation: the difference in probabilities of concordance and discordance.
- Better small-sample properties than Spearman's rho due to a more accurate normal approximation.
- Tau-b and tau-c variants accommodate tied observations in a principled way.
Intuition
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How it works
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When to use it
Use Kendall's tau when you want to measure the monotone association between two ordinal variables, or two continuous variables that violate normality or contain outliers. It is preferred over Spearman's rho in small samples because its sampling distribution is better approximated. Avoid it when you need a Pearson-style linear correlation or when both variables are measured at the interval/ratio level and normally distributed — Pearson's r is more powerful in that case. Also note that tau values are numerically smaller than the equivalent Spearman rho and the two are not directly interchangeable.
Strengths & limitations
- Distribution-free: requires no normality assumption for either variable.
- Robust to outliers because only the rank ordering — not the raw values — matters.
- Handles ordinal data naturally, including Likert-scale responses.
- Has a clear probabilistic interpretation: the difference in probabilities of concordance and discordance.
- Better small-sample properties than Spearman's rho due to a more accurate normal approximation.
- Tau-b and tau-c variants accommodate tied observations in a principled way.
- Tau values are systematically lower in magnitude than Pearson's r or Spearman's rho for the same dataset, making cross-method comparisons misleading.
- Less statistically powerful than Pearson's r when data truly are bivariate normal.
- Computationally intensive for very large datasets because all n(n-1)/2 pairs must be evaluated.
- Does not capture non-monotone (e.g. U-shaped) relationships.
Common pitfalls
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Applications
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Frequently asked
What is the difference between tau-a, tau-b, and tau-c?
Tau-a does not correct for ties and is rarely used in practice. Tau-b adjusts for ties on both variables and is appropriate for square or non-rectangular contingency tables; it is the default in most software. Tau-c (also called Stuart's tau-c) further corrects for the number of rows and columns and is preferred when the two variables have different numbers of possible values.
When should I use Kendall's tau instead of Spearman's rho?
Both are nonparametric rank correlations. Kendall's tau is preferable in small samples because its normal approximation is more accurate, and its interpretation as a probability difference is more intuitive. Spearman's rho tends to be slightly more powerful in medium to large samples. The two are not numerically interchangeable — Spearman rho values are typically larger in absolute value than the corresponding Kendall tau.
Can Kendall's tau be used as an effect size?
Yes. Tau-b is a bounded coefficient between -1 and +1 and is directly interpretable as an effect size. Common benchmarks are |tau| = 0.10 (small), 0.30 (medium), 0.50 (large), though these are approximate and context-dependent.
What is the Mann-Kendall trend test?
The Mann-Kendall test applies the logic of Kendall's tau to a single time series. It treats time as one ranking and the measured variable as the other, testing whether there is a monotone upward or downward trend. It is widely used in environmental and climate science.
Does Kendall's tau require independent observations?
Yes. Like all standard correlation tests, it assumes that each pair of observations is independent of other pairs. It should not be applied to repeated-measures or clustered data without appropriate modifications.
Sources
- 1.Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1/2), 81–93.
- 2.Kendall, M. G. (1948). Rank Correlation Methods. Charles Griffin & Company.
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Cite this page
ScholarGate. (2026, June 3). Kendall's tau. ScholarGate. https://scholargate.app/statistics/kendalls-tau