Kendall's Tau Critical Value Calculator
What Is the Kendall's Tau Critical Value Table?
Kendall's Tau (τ, also written tau) is a nonparametric statistic that measures the strength and direction of the monotonic relationship between two ranked variables. Proposed by Maurice Kendall in 1938, it is calculated from the number of concordant and discordant pairs in the data rather than from means or variances, which means it requires no distributional assumptions.
A computed τ is not automatically evidence of a real population relationship — chance sampling could produce a non-zero τ even when no monotonic relationship exists. The critical value τcrit is the threshold that separates values plausibly due to chance from values too extreme to attribute to chance at the chosen α level.
Core decision rule: τ is what you calculate from your data; τcrit is what you look up. If |τ| ≥ τcrit, reject H₀ and conclude a statistically significant monotonic relationship exists. This rule applies regardless of α level or test direction.
Kendall's Tau Critical Value Table (Complete Reference)
All values below are τcrit — the minimum |τ| for statistical significance at the chosen α and sample size n. Select the tab matching your test direction and significance level. Click any cell to load it into the calculator above.
Reject H₀ if |τ| ≥ τcrit. n = sample size (number of paired observations). Values for n ≤ 40 based on exact permutation distributions; n > 40 use normal approximation z = 3τ√[n(n−1)] / √[2(2n+5)]. Sources: Kendall, M.G. (1948). Rank Correlation Methods. Charles Griffin & Company. Conover, W.J. (1999). Practical Nonparametric Statistics, 3rd ed. Wiley.
How Kendall's Tau Hypothesis Testing Works
Testing the significance of Kendall's Tau follows a structured six-step procedure. Each step corresponds to a specific statistical decision, and each can be performed by hand for small samples or verified in R, Python, or SPSS for any sample size.
Step 1 — State Hypotheses and Choose α
The null hypothesis is always H₀: τ = 0 (no monotonic association in the population). Choose your alternative: H₁: τ ≠ 0 (two-tailed, standard), H₁: τ > 0 (one-tailed, positive monotonic), or H₁: τ < 0 (one-tailed, negative monotonic). Set α before examining the data — typically α = 0.05 for social and health research.
Step 2 — Rank Both Variables
Assign ranks 1 through n to observations on each variable independently, from smallest to largest. When two observations tie on a variable, assign each the average of the tied ranks. Software handles ties automatically; manual ranking is straightforward for small n.
Step 3 — Count Concordant (C) and Discordant (D) Pairs
For every pair of observations (i, j) where i < j: the pair is concordant if xi < xj and yi < yj, or xi > xj and yi > yj — the rank order matches on both variables. The pair is discordant if the rank order reverses. Ties are counted separately and do not contribute to C or D.
Step 4 — Compute Kendall's Tau
For data without ties (Tau-a):
τ ranges from −1 (perfectly discordant ranks) to +1 (perfectly concordant ranks). τ = 0 indicates no monotonic tendency. The denominator n(n−1)/2 is the total number of possible pairs from n observations.
Step 5 — Look Up τ Critical
Find the row in the table above that matches your sample size n. Read across to the column for your chosen α and test direction. That value is τcrit. For n not in the table, use the next smaller n (the conservative choice). For n > 60, use the normal approximation: z = 3τ√[n(n−1)] / √[2(2n+5)], then compare to z = 1.96 (α = 0.05, two-tailed).
Step 6 — Compare |τ| vs τ Critical and Report
If |τ| < τcrit → Fail to reject H₀ → Not statistically significant at α
Report τ, n, test direction, α, and your conclusion. Example: τ = 0.47, n = 18, p < 0.05 (two-tailed) — a statistically significant positive monotonic relationship between variables X and Y.
Two-Tailed vs One-Tailed Kendall's Tau Test
The test direction must be decided before examining the data. Using the data to choose the direction inflates the Type I error rate and invalidates the test.
Two-Tailed — Standard Choice
Tests whether τ differs from zero in either direction (positive or negative monotonic). Used in most research when no directional prediction exists before data collection. The critical region splits across both tails of the distribution.
One-Tailed — Directional Hypothesis
Used only when prior theory or research strongly predicts the direction before data collection. The one-tailed τcrit is lower, giving more statistical power — but misapplying it to catch a result after looking at the data is a statistical error.
Concordant and Discordant Pairs: A Worked Example
Scenario: A clinical researcher records patient anxiety scores and sleep quality ratings (both ordinal, scale 1–10) for n = 6 patients. She wants to know whether higher anxiety is associated with lower sleep quality, using Kendall's Tau at α = 0.05 (two-tailed).
Data and Ranks
| Patient | Anxiety Score | Rank X | Sleep Quality | Rank Y |
|---|---|---|---|---|
| A | 3 | 1 | 8 | 6 |
| B | 5 | 2 | 7 | 5 |
| C | 6 | 3 | 6 | 4 |
| D | 7 | 4 | 4 | 3 |
| E | 8 | 5 | 3 | 2 |
| F | 10 | 6 | 2 | 1 |
Calculation — Step by Step
| Step | Action | Result |
|---|---|---|
| 1 | State hypotheses | H₀: τ = 0; H₁: τ ≠ 0; α = 0.05, two-tailed |
| 2 | Total pairs | n(n−1)/2 = 6×5/2 = 15 |
| 3 | Count concordant pairs (C) | C = 0 (every higher X is paired with lower Y) |
| 4 | Count discordant pairs (D) | D = 15 |
| 5 | Compute τ | τ = (0 − 15) / 15 = −1.000 |
| 6 | Look up τcrit | n=6, α=0.05, two-tailed → τcrit = 0.733 |
| 7 | Compare |τ| vs τcrit | |−1.000| = 1.000 > 0.733 → Reject H₀ |
Interpretation
With τ = −1.00, n = 6, p < 0.05 (two-tailed), there is a statistically significant perfect negative monotonic relationship: patients with higher anxiety scores consistently had lower sleep quality ratings. A τ of −1.00 is a clear teaching case; in practice, τ values between −0.3 and −0.6 represent moderate negative monotonic associations across most research contexts.
How Sample Size Affects Kendall's Tau Critical Values
The same τ value can be statistically significant in one sample and not in another, depending entirely on n. The table below shows this effect directly.
| Computed τ | n = 8 (τcrit = 0.643) | n = 15 (τcrit = 0.467) | n = 30 (τcrit = 0.314) |
|---|---|---|---|
| 0.25 | Not sig. | Not sig. | Sig. ✓ |
| 0.48 | Not sig. | Sig. ✓ | Sig. ✓ |
| 0.65 | Sig. ✓ | Sig. ✓ | Sig. ✓ |
All comparisons use two-tailed α = 0.05. τcrit values taken from the table above.
Why τ Critical Falls as n Grows
Larger samples produce more reliable estimates of the population monotonic relationship. With more pairs counted, even a modest τ becomes distinguishable from sampling noise. This is the same statistical principle at work in the Spearman rank correlation table and in parametric tests like the t-distribution table — more information means lower thresholds for significance.
As with all significance tests, reaching the threshold does not mean the relationship is practically large or meaningful. A τ of 0.15 can be statistically significant with n = 100 while representing a very small monotonic tendency. Always report τ alongside the significance decision.
Kendall Tau-a, Tau-b, and Tau-c: Which to Use?
Three variants of Kendall's Tau exist, each appropriate for different data conditions. The critical value table in this article applies to Tau-a (the basic formula). Software typically reports Tau-b or Tau-c for real data with ties.
| Variant | Formula | When to Use | Software |
|---|---|---|---|
| Tau-a | (C − D) / [n(n−1)/2] | No ties; theoretical derivations | Manual |
| Tau-b | (C − D) / √[(P+Q+X₀)(P+Q+Y₀)] | Ties present; square tables | R, Python, SPSS |
| Tau-c | 2(C − D) / [n²(m−1)/m] | Rectangular tables; unequal categories | SPSS, SAS |
X₀ = ties on X variable, Y₀ = ties on Y variable, m = min(rows, columns). For most practical research with tied ranks, Tau-b is the default.
Kendall's Tau vs Spearman's Rho vs Pearson r: When to Use Each
Choosing the right correlation measure depends on data type, distribution, and research question. The comparison below guides that decision.
| Feature | Kendall's Tau | Spearman's Rho | Pearson r |
|---|---|---|---|
| Data type | Ordinal, ranked | Ordinal, ranked | Continuous, interval |
| Distribution assumption | None | None | Bivariate normal |
| Relationship type tested | Monotonic | Monotonic | Linear |
| Handles ties well? | Yes (Tau-b/c) | With correction | Yes |
| Preferred for small n? | Yes | Acceptable | Not ideal |
| Probabilistic interpretation | Direct (P(C) − P(D)) | Indirect | Linear via r² |
| Generalize to population? | Yes (τ population) | Approximate | Yes (ρ population) |
For the Spearman rank correlation critical values, see the Spearman Correlation Table. For Pearson r, see the Pearson Correlation Critical Value Table.
Computing Kendall's Tau in R, Python, and SPSS
For samples beyond n = 60, or whenever an exact p-value is preferred over a table lookup, software handles the computation directly. Each tool returns τ and a p-value; the critical value table is most useful for manual verification and classroom work.
R — cor.test()
Python — SciPy kendalltau()
SPSS
Analyze → Correlate → Bivariate → select Kendall's tau-b checkbox. SPSS reports Tau-b with a two-tailed p-value and flags significance at α = 0.05 and α = 0.01. For the equivalent nonparametric tests, also see the Mann-Whitney U Table and Wilcoxon Signed-Rank Table.
Applied Examples Across Research Fields
The following examples show how the Kendall's Tau critical value table is applied in different research contexts. Each uses a different sample size to show how n affects the significance threshold.
Education: Teacher Rating vs Student Performance
An education researcher asks n = 12 teachers to rank students by predicted academic potential, then collects end-of-year rankings by actual performance. With C = 53 and D = 13, τ = (53−13)/66 = 0.606. At n = 12, τcrit = 0.497 (α = 0.05, two-tailed). Since 0.606 > 0.497, teacher predictions correlated significantly with actual performance.
Healthcare: Pain Score vs Medication Dose Rank
A clinical team ranks n = 18 post-operative patients by pain score and records their ranked analgesic dose. They compute τ = −0.44 (higher dose, lower pain). At n = 18, τcrit = 0.401 (α = 0.05, two-tailed). Since |−0.44| = 0.44 > 0.401, the negative monotonic association between dose rank and pain rank is statistically significant.
Psychology: Job Satisfaction and Peer Cooperation Ratings
An organizational psychologist ranks n = 25 employees on self-reported job satisfaction and supervisor ratings of peer cooperation. The computed τ = 0.31. At n = 25, τcrit = 0.347 (α = 0.05, two-tailed). Since 0.31 < 0.347, the result does not reach significance at the 5% level. With n = 30, the same τ = 0.31 would be significant (τcrit = 0.314).
Market Research: Product Preference Rank and Price Rank
A market researcher surveys n = 30 consumers who rank 30 products by preference and by perceived price fairness. With τ = 0.38, and τcrit = 0.314 at n = 30 (α = 0.05, two-tailed), the positive monotonic association between preference and price fairness is significant. Products rated more fairly were consistently preferred.
Common Mistakes When Using the Kendall's Tau Critical Value Table
These are the most frequent errors in student and researcher work. Each leads to an incorrect significance decision if not corrected.
| Mistake | What Goes Wrong | Correction |
|---|---|---|
| Using n(n−1) instead of n(n−1)/2 | τ is halved, underestimating the association | Denominator is always n(n−1)/2, not n(n−1) |
| Choosing test direction after seeing data | Type I error rate doubles; result is not valid | Set direction (one- or two-tailed) before collecting data |
| Using Tau-a when ties are present | τ inflated; significance overstated | Use Tau-b (or Tau-c for rectangular tables) |
| Confusing Kendall's Tau with Spearman's rho values | τ and rho are not interchangeable; different scales | Use the Spearman table for rho; use this table for τ |
| Treating significance as practical importance | A small τ can be significant with large n | Report τ (effect size) alongside the significance decision |
Symbol and Concept Glossary
Every symbol used in Kendall's Tau hypothesis testing is defined below, with the most common misinterpretation noted for each.
| Symbol | Name | Definition | Common Error |
|---|---|---|---|
| τ | Kendall's Tau coefficient | Sample measure of monotonic association; −1 to +1 | Treating τ as equivalent to Pearson r or Spearman rho |
| τcrit | Critical value | Minimum |τ| for significance at chosen α and n | Looking up wrong n row or wrong test direction |
| C | Concordant pairs | Pairs where rank order agrees on both X and Y | Counting tied pairs as concordant |
| D | Discordant pairs | Pairs where rank order reverses between X and Y | Counting tied pairs as discordant |
| n | Sample size | Number of paired ranked observations | Using number of variables rather than observations |
| α | Significance level | Probability of Type I error; determines table column | Setting α after observing the data |
| H₀ | Null hypothesis | τ = 0; no monotonic association in population | Failing to reject H₀ as proof that τ = 0 in population |
| z | Normal approximation statistic | Used for n > 60: z = 3τ√[n(n−1)] / √[2(2n+5)] | Applying normal approximation to very small n |
Quick Reference: Most Common Kendall Tau Critical Values
The values below are the most frequently used in student assignments, exam problems, and published research — two-tailed α = 0.05 and α = 0.01.
| n | τcrit (α=0.05, 2-tail) | τcrit (α=0.01, 2-tail) | τcrit (α=0.05, 1-tail) |
|---|---|---|---|
| 5 | 0.800 | 1.000 | 0.700 |
| 10 | 0.511 | 0.644 | 0.422 |
| 15 | 0.390 | 0.505 | 0.333 |
| 20 | 0.326 | 0.421 | 0.274 |
| 25 | 0.285 | 0.368 | 0.240 |
| 30 | 0.256 | 0.333 | 0.218 |
| 50 | 0.196 | 0.257 | 0.167 |
References and Sources
Kendall, M.G. (1948). Rank Correlation Methods. Charles Griffin & Company, London. — The original monograph introducing τ, its distribution, and the basis for all critical value tables.
Conover, W.J. (1999). Practical Nonparametric Statistics, 3rd ed. Wiley, New York. — The standard textbook reference for Kendall Tau-a, Tau-b, exact distributions, and normal approximations. Used widely in social science and engineering statistics courses.
Hollander, M., Wolfe, D.A., & Chicken, E. (2014). Nonparametric Statistical Methods, 3rd ed. Wiley. — Covers Kendall's Tau in the context of rank-based inference, with exact tables and asymptotic results.
NIST/SEMATECH e-Handbook of Statistical Methods (2013). Section 5.4.3: Rank Correlation. National Institute of Standards and Technology. itl.nist.gov — U.S. government reference for nonparametric correlation and rank-based testing methodology.
Mann, H.B. (1945). "Nonparametric Tests Against Trend." Econometrica, 13(3), 245–259. — Introduced the Mann-Kendall test for trend detection, which uses the same concordant-discordant pair logic as Kendall's Tau.
Related Statistical Tables & Resources
Understanding What the Kendall's Tau Table Tells You
Why τcrit Decreases as n Increases
With more observations, the number of possible pairs n(n−1)/2 grows rapidly, giving the test far more information about the population monotonic relationship. Even a modest τ becomes statistically detectable. A sample of n = 50 can reliably identify τ ≈ 0.20; a sample of n = 10 needs τ ≥ 0.511 to meet the same threshold. See the power of a test for a deeper look at sample size and detection ability.
Kendall's Tau and Probabilistic Interpretation
Kendall's Tau has a direct probability meaning that Spearman's rho lacks. If τ = 0.40, the estimated probability that a randomly selected pair of observations will be concordant exceeds the probability of being discordant by 0.40. This makes τ particularly useful in areas where the ordinal relationship has a natural interpretation, such as rankings in survey research or clinical assessments.
When to Use Exact vs Approximate p-values
For n ≤ 40, the exact permutation distribution should be used — this is what the table provides. For n > 40, the normal approximation is sufficiently accurate and is what R and Python use by default. Always report whether you used an exact or approximate p-value when publishing, so readers can assess the precision of the inference. The parametric vs nonparametric tests guide explains when each approach applies.