Kendall's Tau (τ-b concordance / tie-aware)

Verified

Advanced & specialized

Independently verified. Every statistic this test reports has been re-derived against an independent reference — never the library the pipeline itself calls — the rendered output was read back in a browser, and the result is locked with a committed regression suite.

Run this test straight away on a free built-in teaching dataset — no data of your own needed — or bring your own. Either opens the guided workspace: variable setup, assumption diagnostics, results with effect sizes and confidence intervals, figures, and APA-ready reporting.

Loading teaching datasets…

Or use your own dataset

Loading your datasets…

Rank-based concordance test for two rankable variables.

Interprets as P(concordant pair) − P(discordant pair) on the underlying population: how much more likely is it that two randomly-chosen cases agree on rank order than disagree? Reports τ-b (tie-corrected, primary) and τ-a (no correction) with full concordance / discordance / tie counts. Cleaner small-sample distribution than Spearman ρ and arguably the most defensible non-parametric association measure.

Worked example

Do two judges rank the same entries similarly?

Two judges each ranked 15 competition entries; Kendall's tau measures how often their pairwise orderings agree — robust for small samples with ties.

Result

The judges' rankings agreed strongly, τ = .81, p < .001.

How you'd report it (APA)

Kendall's tau showed strong agreement between the two judges' rankings, τ = .81, p < .001.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

Ordinal × ordinal data with substantial ties
An education researcher correlates pupils' satisfaction with school (1–5) and engagement (1–5) for 600 pupils.
Small-sample rank correlation (n ≤ 30)
A field biologist ranks 12 study sites on biodiversity and on disturbance.
Inter-rater concordance on rankings
Two reviewers rank 25 grant applications by quality.

When NOT to — use instead

Two binary categorical variables
On a 2×2 cross-tab Kendall τ collapses to phi — use Fisher's exact / chi-square for the proper test. → Fisher's Exact Test
Rectangular ordinal tables (different category counts on each axis)
τ-b is bounded below 1 on rectangular tables; τ-c (Stuart's correction) gives a fairer ceiling. → Stuart's \u03c4-c (rectangular ordinal tables)
Asymmetric prediction (one variable predicts the other)
When the predict-from / predict-to direction matters, use Somers' d which is asymmetric by construction. → Somers' D (asymmetric ordinal, Dxy + Dyx)
Continuous variables with linear association
Pearson r is more powerful than rank-based τ when the relationship is genuinely linear and continuous. → Pearson Correlation (linear association)

Hypotheses

H₀: τ = 0 — no monotonic association between x and y in the population.

Hₐ: τ ≠ 0 — non-zero monotonic association (two-sided default).

Parameter tested: population Kendall's τ-b

Assumptions (and what to do if they fail)

Scatter of ranks shows a monotonic trend (no U-shape).high

Check: Inspect scatter; confirm τ-b and r share sign.

If violated: Non-monotonic relationships are under-represented by τ.

Ties are present but not so extreme that the asymptotic p is unreliable.low

Check: Report the largest tie block per variable.

If violated: Heavy ties reduce the effective number of comparable pairs and can destabilise the normal approximation to z.

Ready to run a Kendall's Tau (τ-b concordance / tie-aware) on your own data?

Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.

Run this test →

🔎 Worked example

✓ When to use it

✗ When NOT to — use instead

Hypotheses

Assumptions (and what to do if they fail)

Worked example

When to use it

When NOT to — use instead