0.6
8
2
0
10
6
0.8
0
0.6
0.6
8
2
0
10
6
0.8
0
0.6
Kendall's Tau (τ) is a non-parametric rank correlation coefficient that measures the ordinal association between two ranked variables. Unlike Pearson's correlation, which assumes linearity and normality, Kendall's Tau makes no distributional assumptions and is robust to outliers. It is based on comparing every pair of observations and counting whether they are concordant (both variables agree in direction) or discordant.
This calculator accepts 5 paired data points and computes Kendall's Tau-a by examining all 10 possible pairs, counting concordant and discordant pairs, and reporting the correlation coefficient with its interpretation. Tau-a is the simplest form, defined as the net concordance divided by the total number of pairs.
For n observations, there are n(n−1)/2 distinct pairs. For each pair (i, j), we check the sign of (Xᵢ − Xⱼ)(Yᵢ − Yⱼ):
Kendall's Tau-a is defined as:
$$\tau_a = \frac{C - D}{n(n-1)/2}$$
Where C is the count of concordant pairs and D is the count of discordant pairs. For n = 5 data points, there are 5(4)/2 = 10 pairs to evaluate.
The value ranges from −1 to +1. A value of +1 means all pairs are concordant (perfect positive association), −1 means all pairs are discordant (perfect negative association), and 0 means equal numbers of concordant and discordant pairs.
When ties are present, Kendall's Tau-b adjusts the denominator:
$$\tau_b = \frac{C - D}{\sqrt{(n_0 - n_1)(n_0 - n_2)}}$$
where n₀ = n(n−1)/2, n₁ = tied pairs on X, n₂ = tied pairs on Y. Tau-b is bounded by [−1, +1] even with ties, while Tau-a may not reach the extremes when ties exist.
Kendall's Tau is interpreted similarly to other correlation coefficients:
Note that Kendall's Tau values tend to be smaller in magnitude than Spearman's Rho for the same data. A rough conversion is τ ≈ (2/3)ρ, though this is only approximate. Tau has a cleaner probabilistic interpretation: it represents the difference between the probability of concordance and discordance.
Inputs
Results
Five students' study hours (X) and exam scores (Y): mostly concordant pairs with 8 out of 10 concordant. τₐ = 0.60, indicating strong positive correlation.
Inputs
Results
Perfect agreement in rankings: all 10 pairs are concordant, yielding τₐ = 1.0 (perfect positive ordinal correlation).
Tau-a divides the net concordance (C − D) by the total number of pairs n(n−1)/2, regardless of ties. Tau-b adjusts the denominator to account for ties on each variable separately, ensuring that the coefficient can still reach ±1 when ties are present. Tau-b is generally preferred when ties exist in the data.
Both are rank-based correlation measures, but they use different approaches. Spearman's Rho is the Pearson correlation of ranks, while Tau is based on concordant/discordant pair counts. Tau has a clearer probabilistic interpretation and better statistical properties for small samples, but tends to give smaller absolute values. Both test the same null hypothesis of independence.
This calculator is designed for 5 data points (10 pairs) to keep the AST computation tractable. For larger datasets, the formula is identical — you simply enumerate all n(n−1)/2 pairs. Statistical software can handle hundreds or thousands of data points efficiently.
Yes. Because Tau is based on the signs of differences (ordinal comparisons) rather than the magnitudes, it is highly robust to outliers. An extreme value in one observation affects only the pairs involving that observation, and the effect is limited to changing a pair from concordant to discordant or vice versa — the magnitude of the outlier is irrelevant.
Use Kendall's Tau when: (1) data are ordinal rather than continuous, (2) the relationship may be monotonic but not linear, (3) data contain outliers, (4) normality assumptions are violated, or (5) sample size is small. Pearson's r is more powerful when its assumptions are met, but Tau is more robust when they are not.
Tau = 0 means the number of concordant pairs exactly equals the number of discordant pairs. There is no net ordinal tendency. However, this does not necessarily mean the variables are independent — it only means there is no monotonic association. A non-monotonic relationship (e.g., U-shaped) would produce a Tau near zero despite a strong relationship.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Random Number Generator
Advanced & Specialized Statistical Tools
Central Limit Theorem Calculator
Advanced & Specialized Statistical Tools
Empirical Rule Calculator
Advanced & Specialized Statistical Tools
Chebyshev's Theorem Calculator
Advanced & Specialized Statistical Tools
Monte Carlo Estimation Calculator
Advanced & Specialized Statistical Tools
Power Analysis Calculator
Advanced & Specialized Statistical Tools
