τ-a is 0.714286 and τ-c is 0.937500 — a spread of 0.2232 on the same data.
τ-a
0.714286
ignores ties entirely
τ-b
0.872872
the usual default
τ-c
0.937500
for a non-square table
Spearman ρ
0.930175
a different scale
Concordant pairs
20
ordered the same way
Discordant pairs
0
ordered oppositely
Tied pairs
8
5 in x, 1 in y, 2 in both
Two-tailed p
0.013348
normal approximation
τ-a is (20 − 0) ÷ 28 = 0.714286, and that is a probability difference. Pick two observations at random. 71.43% of the time they are ordered the same way in both variables; 0.00% of the time oppositely. τ-a is the gap between those two percentages. No other correlation coefficient has a reading that direct — Spearman’s ρ and Pearson’s r are both ratios of sums of squares with no comparably plain statement attached.
Three definitions, three answers: 0.7143, 0.8729 and 0.9375. τ-a divides by every pair including the tied ones, so ties drag it toward zero and it cannot reach 1 on tied data at all. τ-b divides by the geometric mean of the untied pairs in each variable, which restores the full range and is what R’s cor(method="kendall") returns. τ-c rescales by the smaller number of distinct levels, which matters when the underlying table is not square — five categories against three, say. A paper reporting “τ = 0.73” has not said which of these it means.
Spearman gives 0.930175 and τ-b gives 0.872872 — a ratio of 1.066. They are not on the same scale. For moderately associated data ρ runs about 1.5 times τ, so a ρ of 0.6 and a τ of 0.4 describe the same relationship and neither is “stronger”. Comparing a Kendall from one paper against a Spearman from another is a unit error, and it is a common one. The Spearman page covers the other half.
The p-value uses a normal approximation, which needs about ten observations. Under independence, C − D has mean zero and variance n(n − 1)(2n + 5)/18, and that ratio is approximately standard normal. Below n = 10 the exact distribution is enumerable and noticeably different; with heavy ties the variance formula above is the untied one and understates the true spread, so the p-value comes out optimistic. A permutation test on your own data avoids both problems.
What this tool shows
“Kendall’s tau” names three statistics. On the tied preset here they are 0.7143, 0.8729 and 0.9375 — a spread of 0.223 from the same eight pairs. Software defaults differ and papers rarely say which. All three are computed, with the pair counts they are built from.
τ-a, τ-b and τ-c side by side, with the spread between them
Concordant, discordant and tied pair counts — the raw material of the statistic
τ read as a probability difference, which no other correlation coefficient offers
Spearman’s ρ alongside, since the two are on different scales
A normal-approximation p-value, with its small-sample and tie limits stated
Which definition each piece of software returns by default
Three definitions Pair counts shown Against Spearman Probability reading
0.7143, 0.8729 or 0.9375 — all correctly called τ.
Updated 12 September 2026 · Works in any browser, no installation
Kendall’s tau counts pairs. For every pair of observations it asks whether they are ordered the same way in both variables or oppositely, and tau is the difference between those two proportions. That gives it a plain-language reading neither Spearman’s rho nor Pearson’s r has.
At a glance
Formula shown
With C concordant and D discordant pairs out of n(n − 1)/2: τ-a = (C − D)/n₀ divides by every pair, so ties pull it toward zero. τ-b = (C − D)/√((n₀ − n₁)(n₀ − n₂)) divides by the geometric mean of the untied pairs in each variable. τ-c = 2m(C − D)/(n²(m − 1)) rescales by the smaller number of distinct levels, for a non-square table.
Scenario support
Ordinal categories, agreement between two rankings, small samples where an exact distribution matters, and any relationship where the natural question is “how often do these two agree about the ordering” rather than “how close is this to a line”.
Educational estimate
Planning support from the values you enter — not professional advice.
Three taus, and which one your software returns
The three differ only in what they divide by, and the divisor is a decision about how to treat tied pairs.
τ-a divides by every pair. Tied pairs are counted in the denominator and in neither the concordant nor the discordant count, so they drag it toward zero — and on tied data τ-a cannot reach 1 even when every untied pair agrees.
τ-b divides by the geometric mean of the untied pairs in each variable, which removes the tied ones from the denominator and restores the full −1 to 1 range. It is the usual default: R’s cor(method="kendall") returns it, and so does most other software.
τ-c rescales by the smaller number of distinct levels. It exists for tables that are not square — five ordinal categories against three — where even perfect association cannot fill the table’s diagonal and τ-b is capped below 1.
On the tied preset they are 0.7143, 0.8729 and 0.9375. A spread of 0.223, which is the difference between reporting a moderate association and a very strong one.
Without ties all three coincide exactly. Which is why the distinction can be taught as a footnote and then forgotten — and why it resurfaces on exactly the ordinal data Kendall is usually chosen for.
So name the one you used. “τ = 0.73” is not a complete report, and the three are far enough apart to change how a result reads.
The probability reading nothing else has
This is the reason to prefer Kendall when you have to explain a number to somebody who does not do statistics.
τ-a is P(concordant) − P(discordant). Pick two observations at random. τ-a is the probability they are ordered the same way in both variables, minus the probability they are ordered oppositely.
So τ = 0.5 has a plain meaning. Concordant pairs outnumber discordant ones by 50 percentage points — roughly 75% against 25% when nothing ties.
ρ and r have no equivalent statement. Both are ratios of sums of squares, and “r = 0.5” means the covariance is half the product of the standard deviations, which explains nothing to anybody.
The tool prints both percentages. Not just the difference — the concordant and discordant shares separately, because the difference alone hides how many pairs were tied and therefore counted in neither.
It also makes the interpretation robust to the scale. Since only orderings enter, nothing about the reading changes if you transform either variable monotonically. A τ computed on raw incomes and on log incomes is the same number.
Against Spearman: the ratio is about 1.5
Both are rank correlations, both run −1 to 1, and they are compared as though a 0.6 in one meant a 0.6 in the other.
ρ runs roughly 1.5 times τ for moderate associations. So a ρ of 0.6 and a τ of 0.4 describe the same relationship, and calling one stronger is a unit error.
The relation is exact in a special case. For bivariate normal data both are functions of the underlying correlation: τ = (2/π)·arcsin(r) and ρ = (6/π)·arcsin(r/2). The ratio approaches exactly 1.5 as r approaches zero.
τ is more stable on small samples. Its sampling distribution approaches normal faster, and its exact null distribution is enumerable for small n where Spearman’s is not as tractable.
ρ is more familiar and slightly more powerful under normality. It is what most readers expect and what most software reports first, which is a real argument when the audience matters.
τ handles ties more transparently. Its three variants make the tie treatment an explicit choice; Spearman’s tie handling is a correction buried inside a formula, and the Spearman page shows what happens when it is skipped.
Neither is a substitute for the other in a meta-analysis. Converting between them needs the arcsine relations above and an assumption of bivariate normality, which is usually the thing a rank method was chosen to avoid assuming.
Testing tau, and where the approximation fails
The p-value comes from a normal approximation to the distribution of C − D, and it has two specific limits.
Under independence, C − D has mean zero and variance n(n − 1)(2n + 5)/18. That ratio is approximately standard normal, and the approximation is good from about n = 10.
Below n = 10 the exact distribution is enumerable and different. With n = 6 there are 720 possible orderings, so the p-value is coarse and a continuous approximation to it is optimistic. Exact tables exist for small n and are worth using.
The variance formula assumes no ties. With ties the true variance is smaller and the formula above overstates it, which makes the p-value conservative for τ-a and unreliable for τ-b, whose null distribution depends on the tie pattern.
A permutation test avoids both. Shuffle one variable, recompute τ, repeat. The null distribution is then the one your actual tie structure produces, with no approximation anywhere.
And significance is not strength. With n = 500 a τ of 0.08 is significant and means that 54% of pairs agree against 46% disagreeing — which is barely distinguishable from a coin flip, and is exactly the kind of result the probability reading makes hard to overstate.
When to reach for Kendall
Spearman is the more common default. There are four situations where Kendall is the better choice, and one where it is clearly not.
When the number has to be explained. The probability reading survives a conversation with a non-statistician in a way that no other correlation coefficient does.
When the sample is small. Its distribution is enumerable, its normal approximation kicks in earlier, and its estimates are less variable.
When there are many ties. Ordinal scales with few levels produce tables full of ties, and τ-b and τ-c handle them with a stated rule rather than a hidden correction.
When outliers are present. Both rank measures resist them, but τ depends only on orderings of pairs and is affected even less than ρ by a single extreme value.
Not when you need speed on a large dataset. The naive pair count is O(n²), so a million observations means half a trillion comparisons. Efficient O(n log n) algorithms exist; this tool uses the direct count, which is exact and fine for the sizes a browser will see.
And not when the spacing between values carries information. Ranking discards the magnitudes. If the difference between 10 and 20 genuinely means twice as much as the difference between 10 and 15, Pearson uses information Kendall throws away.
Where tau turns up outside correlation
The concordant-minus-discordant construction is more general than the correlation coefficient it is usually taught as, and it appears under other names.
Goodman and Kruskal’s gamma is the same count with a different denominator. (C − D)/(C + D), which ignores tied pairs entirely rather than adjusting for them — so it is systematically larger than τ-b and is the most optimistic of the family.
Somers’ d is the asymmetric version. It adjusts for ties in one variable only, which is what you want when one variable is an outcome and the other a predictor rather than the two being interchangeable.
The Mann-Kendall trend test is τ against time. It is the standard non-parametric test for monotone trend in environmental and hydrological series, and it is exactly this statistic with x set to the time index.
Concordance is also what the C-statistic measures. The area under an ROC curve is the probability that a randomly chosen positive scores above a randomly chosen negative — the same pairwise-ordering idea, which is why AUC and Mann-Whitney U are algebraically related.
And it underlies rank aggregation. Kendall’s tau distance — the number of discordant pairs between two rankings — is the metric used to combine search rankings and voting orders, where the question is how far apart two orderings are rather than how correlated.
Method. Concordant, discordant and tied pairs are counted directly over all n(n − 1)/2 pairs rather than through a rank-based shortcut, so the tie breakdown is exact and separable into ties in x, ties in y, and ties in both. All three taus are computed from that same count. The p-value uses the exact null variance of C − D for untied data, with the tie and small-sample limits stated rather than hidden. The suite checks τ-a against an independently written brute-force enumeration across 300 datasets, asserts that all three definitions coincide exactly without ties and diverge with them, that every pair is accounted for exactly once, and that a perfect ordering gives τ-b = 1. That engine is verified on every change against 67 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Spearman CorrelationComputes rho correctly as Pearson on the midranks, and beside it the 6Σd²/(n(n²−1)) shortcut every textbook teaches — which is exact only when no two values tie, and overstates the correlation when they do.
Correlation CoefficientReports Pearson, Spearman and Kendall together with the scatter plot, and ships Anscombe's quartet built in — four datasets with an identical r of 0.816 that Spearman tells apart.
Mann-Whitney UMann-Whitney U with the tie correction and both medians shown, because it is not a test of medians — two samples with identical medians of 4 are separated at p = 0.01.
Cohen's KappaKappa with the two figures that explain it: the maximum the marginals permit, and PABAK. Two built-in tables with identical 85% agreement give kappas of 0.6995 and 0.3219, and a third with 94.4% agreement gives −0.0234.
CovarianceSample and population covariance with a unit multiplier that shows why the magnitude is unreadable — rescaling x multiplies the covariance while leaving the correlation untouched.
Coefficient of DeterminationR-squared across five models at once with adjusted R-squared beside it, because adding a term can never lower it and the value most tools report for a curve fit was computed in log space.
An educational tool. The normal approximation for testing tau assumes no ties and about ten or more observations; with heavy ties or a small sample, a permutation test or an exact table is the appropriate alternative.
Published a Kendall calculator that prints THREE taus, because the name covers three statistics and they disagree. On the built-in tied preset tau-a is 0.7143, tau-b is 0.8729 and tau-c is 0.9375 — a spread of 0.223 from the same eight pairs. Software defaults differ and a paper reporting 'tau = 0.73' has not said which.
Reports the concordant, discordant and tied pair counts the statistic is built from, because tau's reading is entirely about pairs: tau-a is the probability that a random pair is ordered the same way in both variables minus the probability that it is ordered oppositely. No other correlation coefficient has a statement that direct, and the tool prints both percentages rather than only their difference.
Shows Spearman's rho alongside with the ratio between them, since rho runs about 1.5 times tau for moderate associations — a ratio that is exactly 1.5 in the limit for bivariate normal data. Comparing a Kendall from one paper against a Spearman from another is a unit error.
States that the normal-approximation p-value assumes no ties and about ten observations, and that with heavy ties the variance formula is the untied one and the p-value comes out optimistic.
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