Kendall's Tau Calculator

How to Use the Kendall's Tau Calculator

1. 1. Enter the pairs. Type two lists in matching order, or one "x, y" pair per line. Raw measurements, ratings and ranks all work — only the order of the values matters.
2. 2. Choose the hypothesis. The default is a two-sided test of τ = 0. Pick "positive" or "negative" only if you committed to a direction before seeing the data.
3. 3. Read τ-b and the p-value. τ-b is the usual choice when either variable has ties. The widget says whether the p-value is exact or a normal approximation.
4. 4. Check the pair counts. The table shows how many pairs are concordant, discordant or tied. For up to ten observations every pair is listed, so you can verify the count by hand.

What Kendall's Tau Measures

Kendall's τ asks a simple question of every pair of observations: do the two variables put this pair in the same order? If the person who scored higher on x also scored higher on y, the pair is concordant. If the order flips, it is discordant. A pair with equal values on x, on y, or on both is tied. τ is built from the balance of concordant against discordant pairs, so it is the difference between the probability of seeing a concordant pair and the probability of seeing a discordant one.

τ-a divides by every pair, so ties pull it towards zero and it can never reach ±1 when ties exist. τ-b shrinks the denominator to the pairs that are not tied on each variable, which restores the full −1 to +1 range; it is the version SciPy, R and SPSS report. With no ties the two are identical.

Worked Example

Two judges rank ten entries in a poster competition. Judge A's ranking is 1 to 10; Judge B ranked the same entries 2, 1, 4, 3, 7, 8, 5, 10, 6 and 9. Of the 45 pairs of entries, 37 are concordant and 8 are discordant, so S = C − D = 29 and τ = 29 ÷ 45 = 0.6444. There are no ties, and with n = 10 the calculator uses the exact test: p = 0.0091. The judges agree substantially, and the chance of this much agreement between unrelated rankings is under 1%. Spearman's ρ for the same data is 0.8182 — larger, because the two coefficients are on different scales, not because they disagree.

Now take the practice-hours data from the Spearman page: hours 1, 2, 2, 3, 4, 5, 6, 8, 10, 12 against scores 35, 46, 40, 52, 58, 57, 64, 66, 69, 68. Two students share 2 hours, so one pair is tied on x. Out of 45 pairs, 42 are concordant, 2 discordant and 1 tied. τ-a = 40 ÷ 45 = 0.8889, but τ-b = 40 ÷ √(44 × 45) = 0.8989. Because of the tie the normal approximation is used: z = 3.5921, p = 0.0003.

Exact and Approximate P-Values

When every value is distinct and n is below 50, the calculator enumerates the null distribution: for unrelated variables, all n! orderings are equally likely, and the number of concordant pairs follows a known distribution. The p-value is read straight off it, with no approximation. With ties, or for larger samples, it falls back on the normal approximation to S = C − D, using Kendall's variance formula with the correction for ties. These are the same rules as R's cor.test(x, y, method = "kendall"), and the results agree to the digits shown. A continuity correction is optional and off by default, as in R.

Reading the Size of τ

τ is smaller in magnitude than Spearman's ρ or Pearson's r for the same data — for bivariate normal data, τ is roughly two-thirds of ρ. So the usual bands for r overstate how weak a τ is. The widget calls |τ| ≥ 0.5 strong, 0.3–0.5 moderate and 0.1–0.3 weak, but these are conventions that depend on your field. τ has a direct reading that r lacks: in the first example, 37 of 45 pairs were concordant, so picking two entries at random, the judges agree on their order about 82% of the time.

Frequently Asked Questions

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

Both divide the excess of concordant over discordant pairs by a count of pairs. τ-a uses all n(n − 1) ÷ 2 pairs, so ties shrink it. τ-b removes pairs tied on each variable from the denominator, so it can still reach −1 or +1 with ties. If your data have no ties they are the same number; with ties, report τ-b.

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

Kendall's τ is often preferred for small samples, for data with many tied ranks, and when you want a coefficient with a probability interpretation (the chance that a random pair is concordant minus the chance it is discordant). Spearman's ρ is easier to compute by hand and is more familiar. They almost always agree on direction and significance.

How is the p-value for Kendall's tau calculated?

With no ties and fewer than 50 observations, it is exact, computed from the distribution of concordant pairs across all possible orderings. Otherwise it is a normal approximation: z = S ÷ √Var(S), where the variance is corrected for ties. This matches R's cor.test with method = "kendall".

What counts as a strong Kendall's tau?

There is no universal cut-off. Because τ runs smaller than r or ρ, a τ of 0.5 is already a strong association in many fields. As a rough guide, below 0.1 is negligible, 0.1–0.3 weak, 0.3–0.5 moderate, and above 0.5 strong. Judge by what is typical for your type of data.

How do I report Kendall's tau in APA style?

Give the coefficient without a leading zero, the p-value, and n, for example: Kendall’s τb = .64, p = .009, n = 10. Say which version you used (τ-a or τ-b) and whether the p-value is exact or approximate. The widget builds this line for you to copy.

Can Kendall's tau handle ordinal or Likert data?

Yes — that is one of its main uses. Only the order of values matters, so ratings on a 1–5 scale work, and the many ties they produce are handled by τ-b and the tie-corrected variance.