Kendall Tau Calculator
Correlation Built From Pairwise Agreement
Kendall's tau takes a different route to measuring association than Pearson or Spearman: instead of working with variances or ranks directly, it counts pairs of observations. For every pair, it asks whether the two variables agree on which one is larger (concordant) or disagree (discordant). The balance between these two counts, adjusted for ties, becomes tau-b — a measure that many statisticians consider more directly interpretable than rho, since it's literally a difference in pair-agreement probabilities.
The Formula
where C = concordant pairs, D = discordant pairs, n0 = n(n−1)/2, n1 and n2 are tie-adjustment terms for x and y
Significance is tested with: z = 3 × tau-b × √(n(n−1)) / √(2(2n+5))
Worked Example
x = 1, 2, 3, 4, 5. y = 2, 1, 4, 3, 5.
| Quantity | Value |
|---|---|
| Concordant pairs (C) | 8 |
| Discordant pairs (D) | 2 |
| tau-b | 0.6000 |
| z | 1.4697 |
| Two-tailed p-value | 0.1416 |
Computed directly from the calculator's concordant/discordant pair-counting formula.
Out of the 10 possible pairs among 5 points, 8 agree on ordering and only 2 don't — giving tau-b = (8−2)/10 = 0.6, a moderately strong positive association.
Interpreting Tau-b
| |tau-b| range | Strength |
|---|---|
| ≥ 0.70 | Strong |
| 0.50 – 0.69 | Moderate |
| 0.30 – 0.49 | Weak |
| < 0.30 | Very weak / negligible |
Bands taken directly from the calculator's classification logic. Note tau-b values are typically smaller in magnitude than Spearman's rho for the same data.
Where This Calculation Matters
- Small samples with many ties — tau-b's explicit tie correction makes it a reliable choice when data has repeated values, more so than Spearman's rho.
- Ranking agreement — comparing two independent rankings of the same items (e.g., two judges' rankings of contestants) for consistency.
- Robustness to outliers — because it only cares about pairwise ordering, not magnitudes, tau-b is highly resistant to extreme values.
How to Use This Calculator
- Enter the X values as a comma-separated list.
- Enter the Y values as a comma-separated list, in the same order (at least 3 paired values).
- Select Calculate to get tau-b, the z-score, and the two-tailed p-value.
Related Calculations
For a rank-correlation measure based on rank differences rather than pair concordance, see the Spearman Rank Correlation Calculator. For a linear relationship on raw values, use the Pearson Correlation Calculator.