What is Kendall's Tau?

Kendall's Tau (τ) is a non-parametric rank correlation coefficient that measures the strength and direction of the monotonic association between two ordinal or continuous variables. It is based on counting concordant pairs (where both variables move in the same direction) and discordant pairs (where they move in opposite directions). Tau-b is the version that corrects for ties. Values range from -1 (perfect negative association) to +1 (perfect positive association).

When should I use Kendall's Tau instead of Spearman's rho?

Kendall's Tau is preferred over Spearman's rho when: (1) sample sizes are small (n < 30), because Tau has better asymptotic efficiency and its exact distribution is easier to compute; (2) there are many tied ranks, because Tau-b has a more principled correction; (3) you want a more interpretable coefficient — Tau can be directly interpreted as a probability difference (P(concordant) - P(discordant)); (4) you want robust results less influenced by outliers in the rank space.

What is the difference between Kendall's Tau-a and Tau-b?

Tau-a = (C - D) / (n(n-1)/2) does not account for tied pairs and always ranges from -1 to +1 only when there are no ties. Tau-b corrects for ties in both variables: Tau-b = (C - D) / sqrt((n0 - n1)(n0 - n2)), where n0 is total pairs, n1 is pairs tied in X, and n2 is pairs tied in Y. Tau-b always ranges between -1 and +1 regardless of ties, making it the standard choice for data with ties.

What sample size is needed for Kendall's Tau to be reliable?

Kendall's Tau can be computed for any n ≥ 3, but interpretation requires adequate power. For exact p-values, the minimum sample size to detect significance at α = 0.05 (two-tailed) is n = 5 (requiring perfect concordance). For practical detection of medium effects (τ ≈ 0.30) at 80% power, approximately n = 60 is needed. For large effects (τ ≈ 0.50), n ≈ 22 suffices. The asymptotic normal approximation is generally accurate for n ≥ 10.

How do you interpret Kendall's Tau effect size?

Based on Cohen (1988) adapted for rank correlations: Negligible |τ| < 0.10, Small 0.10 ≤ |τ| < 0.30, Medium 0.30 ≤ |τ| < 0.50, Large |τ| ≥ 0.50. Kendall's Tau can also be interpreted directly as the probability difference: τ = P(concordant pair) - P(discordant pair). For example, τ = 0.40 means there is a 40% higher probability of drawing a concordant pair than a discordant pair.

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Q: What is the difference between Kendall's Tau-a and Tau-b?

Tau-a = (C - D) / (n(n-1)/2) does not account for tied pairs and always ranges from -1 to +1 only when there are no ties. Tau-b corrects for ties in both variables: Tau-b = (C - D) / sqrt((n0 - n1)(n0 - n2)), where n0 is total pairs, n1 is pairs tied in X, and n2 is pairs tied in Y. Tau-b always ranges between -1 and +1 regardless of ties, making it the standard choice for data with ties.

Q: What sample size is needed for Kendall's Tau to be reliable?

Kendall's Tau can be computed for any n ≥ 3, but interpretation requires adequate power. For exact p-values, the minimum sample size to detect significance at α = 0.05 (two-tailed) is n = 5 (requiring perfect concordance). For practical detection of medium effects (τ ≈ 0.30) at 80% power, approximately n = 60 is needed. For large effects (τ ≈ 0.50), n ≈ 22 suffices. The asymptotic normal approximation is generally accurate for n ≥ 10.

Q: How do you interpret Kendall's Tau effect size?

Based on Cohen (1988) adapted for rank correlations: Negligible |τ| < 0.10, Small 0.10 ≤ |τ| < 0.30, Medium 0.30 ≤ |τ| < 0.50, Large |τ| ≥ 0.50. Kendall's Tau can also be interpreted directly as the probability difference: τ = P(concordant pair) - P(discordant pair). For example, τ = 0.40 means there is a 40% higher probability of drawing a concordant pair than a discordant pair.

Kendall's Tau: Statistical Background

Kendall's Tau (τ), originally proposed by Maurice Kendall (1938), is a non-parametric rank correlation coefficient that quantifies the strength and direction of the monotonic association between two variables. The coefficient makes no distributional assumptions, does not presuppose linearity, and is computed entirely from the ordinal ranking of paired observations rather than from their raw magnitudes.

The statistic is grounded in a pairwise comparison principle. For every distinct pair of observations \((X_i, Y_i)\) and \((X_j, Y_j)\), one determines whether the two variables co-vary in the same direction (a concordant pair) or in opposite directions (a discordant pair). Kendall's Tau is the normalised difference between these two counts across all \(\binom{n}{2}\) possible pairs.

The Probabilistic Interpretation of Kendall's Tau-b Kendall's Tau-b admits a direct probabilistic interpretation that no other standard correlation coefficient provides: \(\tau_b = P(\text{concordant pair}) - P(\text{discordant pair})\). A coefficient of \(\tau_b = 0.40\), for instance, indicates that the probability of randomly drawing a concordant pair from the population exceeds the probability of drawing a discordant pair by 40 percentage points. This interpretability makes Tau particularly valuable for communicating findings to non-statistical audiences.

Concordant Pairs, Discordant Pairs, and Ties

For any two distinct observation pairs \((X_i, Y_i)\) and \((X_j, Y_j)\) with \(i \neq j\), the classification proceeds as follows:

Concordant Pair (C)

Both variables are ordered consistently: \((X_i - X_j)(Y_i - Y_j) > 0\). The observation with the larger X value also has the larger Y value. Each concordant pair contributes +1 to the concordance sum S.

Discordant Pair (D)

The orderings of X and Y are reversed: \((X_i - X_j)(Y_i - Y_j) < 0\). The observation with the larger X value has the smaller Y value. Each discordant pair contributes −1 to S.

Tied Pair

At least one variable takes equal values across the pair: \(X_i = X_j\) or \(Y_i = Y_j\). Tied pairs are neither concordant nor discordant. Tau-b corrects the denominator to account for their presence; Tau-a does not.

The concordance sum is \(S = C - D\), and the total number of pairs is \(n_0 = \binom{n}{2} = \frac{n(n-1)}{2}\).

The Formulas

Tau-a (no tie correction)

\( \tau_a = \dfrac{C - D}{\dfrac{n(n-1)}{2}} = \dfrac{S}{n_0} \)

Tau-a does not account for tied pairs and can range from −1 to +1 only in the absence of ties. It underestimates the association when ties are present.

Tau-b: The Tie-Corrected Standard

\( \tau_b = \dfrac{S}{\sqrt{(n_0 - n_1)(n_0 - n_2)}} \)

Here \(n_0 = \frac{n(n-1)}{2}\) denotes the total number of observation pairs; \(n_1 = \sum_k \frac{t_k(t_k-1)}{2}\) is the number of pairs tied on X, summed over all tie groups \(t_k\) in X; and \(n_2 = \sum_k \frac{u_k(u_k-1)}{2}\) is the number of pairs tied on Y. The denominator \(\sqrt{(n_0-n_1)(n_0-n_2)}\) represents the geometric mean of the untied pair counts in each variable, and serves as the maximum value S could take given the observed tie structure. As a consequence, Tau-b is bounded strictly within \([-1, +1]\) regardless of the degree of tying.

Hypothesis Testing

Exact p-Value for Small Samples

Under H₀: \(\tau = 0\), all \(n!\) permutations of the ranks of one variable relative to the other are equally probable. The exact null distribution of the concordance sum S is therefore enumerable in principle, and in practice is computed efficiently via a dynamic programming recurrence. The exact p-value is the proportion of permutations yielding a value of |S| at least as large as the observed |S₀|. This approach is employed when n ≤ 50 and no ties are present in X, conditions under which the exact distribution is well-defined and computationally tractable.

Asymptotic Test for Large Samples or Tied Data

When the sample is large or ties are present in either variable, the exact permutation distribution is replaced by its normal approximation. Under H₀, the concordance sum S is asymptotically normally distributed with mean zero and variance given by the Kendall (1970) formula:

\( \text{Var}(S) = \dfrac{v_0 - v_t - v_u}{18} + \dfrac{v_1}{2n(n-1)} + \dfrac{v_2}{9n(n-1)(n-2)} \)

In this expression, \(v_0 = n(n-1)(2n+5)\) is the untied term; \(v_t = \sum_k t_k(t_k-1)(2t_k+5)\) and \(v_u = \sum_k u_k(u_k-1)(2u_k+5)\) are correction terms for tie groups \(t_k\) in X and \(u_k\) in Y respectively; \(v_1 = \sum_k t_k(t_k-1) \cdot \sum_k u_k(u_k-1)\) and \(v_2 = \sum_k t_k(t_k-1)(t_k-2) \cdot \sum_k u_k(u_k-1)(u_k-2)\) account for joint tie structure. The standardised test statistic is \(z = S / \sqrt{\text{Var}(S)}\), referred to the standard normal distribution. The approximation is satisfactory for n ≥ 10 in most practical applications.

Variants of the Kendall Coefficient: Tau-a, Tau-b, and Tau-c

Tau-a

The denominator is the total number of pairs \(n(n-1)/2\) with no adjustment for ties. The coefficient attains the bounds ±1 only in the complete absence of ties and systematically underestimates the degree of association when tied observations are present. It is rarely reported in contemporary research.

Tau-b (Recommended Standard)

The denominator is corrected for ties in both X and Y, guaranteeing that the coefficient lies in \([-1,+1]\) for any data configuration. Tau-b is the default reported by SPSS, SAS, R, and Python, and is appropriate for continuous, discrete, and ordinal data alike.

Tau-c (Stuart's Tau)

Developed specifically for rectangular contingency tables in which the numbers of row and column categories differ. The denominator incorporates an additional factor \(m(m-1)n^2/2\) where \(m = \min(\text{rows}, \text{cols})\). This variant is not appropriate for continuous or ranked data.

Selection Among Rank Correlation Coefficients

Grounds for Selecting Kendall's Tau

Kendall's Tau is the preferred rank correlation measure in the following circumstances. When samples are small (n < 30), the exact permutation distribution of Tau is computable and well-calibrated, whereas Spearman's rho relies on approximations that perform poorly at small n. When tied ranks are prevalent, Tau-b employs a denominator correction that is theoretically better motivated than the Spearman formula modified for ties. When interpretability is paramount, Tau-b's probability statement (P(concordant) − P(discordant)) carries an intuitive meaning absent from rho. Tau also possesses superior asymptotic efficiency relative to Spearman under a broad class of alternatives, and its sampling distribution is less sensitive to extreme rank differences.

When Alternative Measures Are More Appropriate

Spearman's rho is justified when the researcher requires direct comparability with a substantial existing literature that has uniformly employed rho, or when sample sizes are moderate to large and ties are infrequent. Pearson's product-moment correlation is appropriate only when both variables can be assumed continuous, the joint distribution is approximately bivariate normal, and the association is linear in form. Pearson's r is sensitive to departures from normality and to outliers in ways that rank-based measures are not.

Effect Size Classification

The following thresholds, adapted from Cohen (1988) for rank-based correlation coefficients, provide conventional benchmarks for interpreting the practical magnitude of Kendall's Tau-b. These benchmarks are descriptive guidelines and should be contextualised against domain-specific expectations and prior literature.

Negligible

|τ_b| < 0.10. The monotonic association is so small as to be of no practical consequence in most research contexts. The probability excess of concordant over discordant pairs is below 10 percentage points.

Small

0.10 ≤ |τ_b| < 0.30. A real but weak association that is typically imperceptible without systematic measurement. Detection requires adequately powered samples.

Medium

0.30 ≤ |τ_b| < 0.50. A moderate association that is substantively meaningful across most applied disciplines. Corresponds to a 30–49% probability excess of concordance.

Large

|τ_b| ≥ 0.50. A strong association with at least a 50-percentage-point excess of concordant over discordant pairs. Such a relationship is generally apparent through graphical inspection of the data.

Comparability with Spearman's Rho

Kendall's Tau-b values are systematically smaller than Spearman's rho computed on the same dataset, with the ratio approximating |τ| ≈ (2/3)|ρ| under many distributional conditions. The two coefficients should not be compared numerically across studies.

Assumptions Underlying the Test

Conditions Required for Valid Inference

Independence of observation pairs. Each bivariate observation \((X_i, Y_i)\) must be statistically independent of all other pairs. Note that independence within a pair is not required; X and Y may be correlated, as that is precisely the quantity under investigation. Repeated-measures designs and clustered sampling schemes violate inter-pair independence and require specialised methods.
Ordinal measurement scale. Both variables must be measured on at least an ordinal scale, meaning a consistent rank ordering of values must be meaningful. Kendall's Tau is applicable to continuous, discrete, and ordered categorical data, but it is not defined for purely nominal categories lacking a natural ordering.
Random sampling. For inferences to generalise to the target population, observations must constitute a probability sample from that population. Results based on convenience samples may not support population-level conclusions regardless of statistical significance.

Distributional Assumptions Kendall's Tau places no parametric distributional requirement on X or Y. The test remains valid for non-normal, skewed, bounded, and ordinally scaled data. This distributional robustness is the principal advantage of Tau over Pearson's r for non-continuous or non-Gaussian data. The asymptotic normal approximation to the null distribution of S, used when n is large or ties are present, is well-supported for n ≥ 10 across a wide variety of population distributions.

Selected Methodological Questions

Bounds of the Kendall's Tau Coefficient

Tau-b is bounded within the closed interval \([-1, +1]\) by construction. Tau-a reaches these bounds only when the data contain no ties. The value +1 is obtained when all pairs are concordant, which occurs when the rankings of X and Y are in perfect agreement. The value −1 arises when all pairs are discordant, corresponding to a perfect reversal of rankings.

Interpreting a Non-Significant Result

A non-significant Kendall's Tau indicates that the observed data do not provide sufficient evidence to reject the null hypothesis of no monotonic association at the chosen significance level. This does not constitute evidence that the two variables are independent. Insufficient statistical power, arising from a small sample size or a small true effect, may produce a non-significant result even when a real association exists. For this reason, the point estimate and confidence interval for Tau-b should always be reported alongside the significance decision.

Computation of the Confidence Interval for Tau-b

The confidence interval is constructed using the asymptotic standard error \(\text{SE}(\tau_b) = \sqrt{\text{Var}(S)} / \sqrt{(n_0 - n_1)(n_0 - n_2)}\), yielding the approximate Wald interval \(\tau_b \pm z_{\alpha/2} \cdot \text{SE}(\tau_b)\), constrained to \([-1, +1]\). For small samples (n < 25) or when Tau-b falls near the boundary of its range, bootstrap-based confidence intervals are preferable to the Wald approximation.

Application to Ordinal Likert-Scale Data

Kendall's Tau-b is well suited to Likert-scale responses. Likert items produce ordered categorical data (e.g., 1 = strongly disagree through 5 = strongly agree) that naturally induce ties, and Tau-b's denominator correction handles this tie structure in a theoretically principled manner. Pearson's r, by contrast, treats Likert responses as interval-level measurements with equal spacing, an assumption that is generally unjustified and that inflates or deflates the apparent magnitude of the association depending on the score distribution.

Kendall's Tau (τ) Rank Correlation: Reference & Calculator