Use this Kendall Tau correlation calculator to measure monotonic association by comparing concordant and discordant pairs. Enter raw data, upload a spreadsheet, or paste ordinal ratings to instantly calculate Kendall τ, pair counts, z-statistic, p-value, confidence intervals, and a full step-by-step pair comparison that is especially useful for small samples and tie-heavy datasets.
Best for continuous variables with a linear relationship.
Pearson r = 0.9991 based on 8 valid pairs, p = 0.0000.
Average the X values and the Y values before measuring joint movement.
Subtract the mean from every X and Y value to get centered deviations.
Multiply each pair of deviations and add them up.
Compute the denominator from the independent spread of X and Y.
The covariance term is normalized by both standard-deviation components.
Enter paired observations manually, paste two columns from Excel, or upload a CSV or Excel file.
Choose Kendall Tau when your variables are ordinal, your sample is small, or tied values are common.
Read τ, τ², z, p-value, confidence intervals, and the dedicated pair analysis card with concordant and discordant counts.
Expand the step-by-step section to inspect every pair comparison and the exact τ-a versus τ-b logic used.
Kendall tau, also called the Kendall rank correlation, is a non-parametric measure of monotonic association based on pair ordering rather than raw numerical distance. Maurice Kendall introduced the statistic in 1938 to answer a practical question: when you compare two observations at a time, how often do the two variables agree about which observation ranks higher? The coefficient is written as , and it ranges from -1 to +1. Positive values indicate that concordant pairs outnumber discordant pairs, negative values indicate the reverse, and values near zero indicate little systematic agreement in ordering.
The unique strength of Kendall tau is that it stays close to the real ordering question analysts often care about. A pair is concordant when both variables move in the same direction across two observations. A pair is discordant when the order flips. Because the method is built on pair comparisons, Kendall rank correlation is especially appealing for small samples, ordinal ratings, and datasets with many ties. It does not require normality and is less influenced by outliers than Pearson correlation.
Kendall tau also has an unusually intuitive interpretation. If , then . In plain language, a randomly chosen pair of observations has an 80% chance of being in the same order across both variables. That direct probability interpretation is one of the reasons Kendall Tau is preferred in fields such as survey analysis, preference ranking, medicine, and any domain where tied or small-sample ordinal data appears frequently.
| Symbol | Meaning |
|---|---|
| Number of concordant pairs | |
| Number of discordant pairs | |
| Number of X-tied pairs | |
| Number of Y-tied pairs | |
| Number of valid observations | |
| The smaller dimension in a rectangular contingency table for τ-c |
is the original no-ties formula. It simply compares the difference between concordant and discordant pairs against the total number of possible pairs. That works well in toy examples, but real data often contains tied values.
is therefore the default on this page. It corrects the denominator for ties in X and Y separately, which is why it is the standard version used by major statistical packages and research workflows. is included for completeness because it is useful in rectangular contingency tables, but it is less common for ordinary two-variable ranking data. The calculator computes the large-sample statistic shown above to estimate the two-tailed p-value.
List all possible observation pairs. With n observations, Kendall compares n(n - 1)/2 pairs.
For each pair, decide whether the order is concordant, discordant, tied in X, or tied in Y.
Count C, D, T_x, and T_y from those pair classifications.
Apply τ-b when ties are present. If there are no ties, τ-a and τ-b are identical.
Convert τ to a z-statistic, estimate the p-value, and review the pair analysis card before drawing a conclusion.
The worked panel above follows the exact same workflow with your own dataset. It lists the pair comparisons, shows the counts of concordant and discordant pairs, reveals when ties force a switch from to , and then computes the final significance test.
Kendall Tau is built on pair classification. Take two observations and . They are concordant when both variables agree on the order: either and , or and . They are discordant when one variable says observation i is larger while the other says observation j is larger. If or , the pair is tied rather than concordant or discordant.
This pair logic is why Kendall Tau feels so intuitive. Instead of summarizing raw distances, it asks a ranking question repeatedly and counts the outcomes. With observations there are distinct pairs to compare. The coefficient rises when concordant pairs dominate and falls when discordant pairs dominate. That makes Kendall Tau a natural fit for small ordered datasets such as judge rankings, Likert scales, and preference lists where pair order matters more than numeric spacing.
Start with the sign and the magnitude, just as you would with Pearson or Spearman. Then use Kendall Tau's biggest advantage: the probability interpretation. Because , you can convert the coefficient into an intuitive estimate of how often two randomly selected observations keep the same order across both variables. Pair counts, ties, and the p-value then add the remaining context.
Ties are a direct reason to prefer Kendall Tau over simpler rank measures. In the no-ties version, uses the full pair count in the denominator. Once ties appear, that denominator becomes too large relative to the number of comparable pairs, which pushes the coefficient downward and underestimates the true ordered agreement.
fixes that by correcting the denominator with tie counts in X and Y. If a value appears times in one variable, it contributes tie pairs. This calculator detects those ties automatically, reports and in the pair analysis card, and switches from to whenever the correction is needed.
This is why τ-b is the default result returned by the calculator whenever ties are found.
Kendall Tau is the most pair-focused and the most tie-aware of the common correlation measures. Compared with Spearman correlation, it is often preferred in small samples and with many tied ranks. Compared with Pearson correlation, it makes no normality assumption and does not require a straight-line relationship. The tradeoff is that Kendall Tau is computationally heavier because it compares observation pairs directly.
| Kendall τ-b | Spearman ρ | Pearson r | |
|---|---|---|---|
| Data type | Ordinal or continuous | Ordinal or continuous | Continuous |
| Distribution requirement | No distribution assumption | No distribution assumption | Approximately normal for formal inference |
| Small samples (n < 20) | Best choice | Usable | Often not preferred |
| Tie handling | τ-b tie correction | Average tied ranks | Not applicable |
| Probability interpretation | Direct and intuitive | Indirect | Usually via r² |
| Outlier robustness | Strongest | Strong | Weakest |
| Computation | O(n²) | O(n log n) | O(n) |
| Common defaults | SPSS, R, Python support τ-b | Widely available | Most common overall |
| Link | Current page | /spearman-correlation/ | /pearson-correlation/ |
A compact ranking dataset intended for Kendall Tau interpretation.
This dataset is useful when rank order matters more than distance and ties are likely to appear in the observations.
Two 1 to 5 survey scales with many ties, ideal for Kendall Tau-b.
This dataset is useful when rank order matters more than distance and ties are likely to appear in the observations.
Kendall Tau (τ) is a non-parametric measure of rank correlation that quantifies the ordinal association between two variables. It counts concordant pairs, where both variables move in the same order, minus discordant pairs, where the order reverses, and then normalizes that difference. Developed by Maurice Kendall in 1938, it is especially reliable for small samples and datasets with many tied ranks.
Kendall τ-a uses the simple formula (C - D) divided by n(n - 1)/2 and assumes there are no ties. Kendall τ-b modifies the denominator to account for ties in X and Y separately, which makes it the standard choice for real-world data. τ-c is designed for rectangular contingency tables. This calculator defaults to τ-b whenever ties are present.
Kendall Tau ranges from -1 to +1. One of its main advantages is the probability interpretation: P(concordant) = (1 + τ) / 2. So if τ = 0.6, then a randomly chosen pair of observations has an 80% chance of being concordant. As a rough guide, absolute values below 0.3 are weak, 0.3 to 0.7 are moderate, and above 0.7 are strong.
Prefer Kendall Tau when your sample is small, when your data has many tied values, when you want a direct probability interpretation, or when you need the most robust measure of monotonic association. Spearman is more commonly reported and faster to compute, but Kendall Tau is often statistically more efficient for small samples.
First list all n(n - 1)/2 possible pairs of observations. Next classify each pair as concordant, discordant, tied in X, or tied in Y. Then count C, D, T_x, and T_y. Finally apply τ-b = (C - D) / √((C + D + T_x)(C + D + T_y)). This calculator shows every pair comparison in the step-by-step section.
A concordant pair occurs when two observations keep the same relative order in both variables. If x_i is larger than x_j and y_i is also larger than y_j, or if both are smaller, the pair is concordant. In intuitive terms, both variables agree on which observation ranks higher.
Kendall Tau is one of the best choices for small samples. It can still be informative with about 8 observations, and the normal-approximation z test is often used from around 10 observations upward. For very tiny samples, exact permutation p-values are better than the large-sample approximation.
No. Kendall Tau measures association between two variables, while Kendall's W measures agreement among three or more raters or rankings. They come from the same family of rank-based statistics, but they answer different questions.