Kendall Tau-b Calculator
Enter two aligned numeric series to calculate Kendall Tau-b from every unordered observation pair. Concordant, discordant, X-only ties, Y-only ties and joint ties remain visible so the denominator can be audited without a p-value, agreement grade, significance conclusion, forecast or trading verdict.
Enter two aligned series for pair comparison
Every unordered row pair is classified once; exact ties are retained rather than broken arbitrarily.
Enter 5 to 500 finite values without percent signs; one complete aligned observation per row.
The Y row count and original order must match X exactly.
Concordance and tie arithmetic
Entered Rank and Lag Correlation 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Pair category | Count | Share of all pairs |
|---|
How Kendall Tau-b is calculated
n(n − 1) ÷ 2 = P + Q + T + U + J
For each unordered pair of rows, the model compares the direction of the X difference with the direction of the Y difference. Matching nonzero directions are concordant; opposite nonzero directions are discordant.
An exact X tie with different Y values enters T, a Y tie with different X values enters U, and a pair tied in both enters J. Joint ties reconcile the total pair count but do not enter either single-variable tie correction.
The numerator is concordant minus discordant pairs. The denominator adjusts separately for X-only and Y-only ties. A zero denominator is withheld instead of converted to zero, and no asymptotic or exact p-value is produced.
Worked example from the audited fixture
The audited 12-row fixture creates 66 unordered pairs. The enumeration produces 62 concordant pairs, 1 discordant pair, 2 ties only in X, 1 tie only in Y and no joint ties.
- The five categories sum exactly to 66. The two Tau-b denominator counts are 65 and 64, preserving the unequal tie adjustments.
- Substituting the counts gives Tau-b = 0.94576485. This is a descriptive concordance coefficient for the entered pairs, not an agreement grade or significance verdict.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- A positive coefficient means concordant pairs outnumber discordant pairs; a negative coefficient means the reverse.
- Tau-b is bounded from minus one to one and corrects for ties in both variables, but its scale should not be converted automatically into trading labels.
- The pair-count table exposes how much of the sample evidence comes from concordance, discordance and ties.
- Spearman and Kendall can differ because average-rank correlation and pairwise concordance weight the entered ordering differently.
- A high entered coefficient can still reflect a selected window, shared trend, duplicated observations or one preprocessing choice.
Assumptions and limits
- Enter 5 to 500 complete aligned observations; the O(n squared) pair enumeration remains bounded by that input cap.
- Exact numeric equality defines a tie. Rounding before entry can therefore change the tie counts and Tau-b value.
- Tau-b describes ordinal concordance and discards original distance between observations.
- The calculator cannot verify independence, alignment, stationarity, missing data, source completeness or cost treatment.
- A coefficient does not establish an economic mechanism, causal sequence, stable hedge or future persistence.
- No p-value, significance, agreement grade, predictive edge, forecast, signal, position instruction or recommendation is generated.
Which relationship diagnostic answers which question?
Average-rank association, pairwise concordance, signed-lag Pearson correlation and contemporaneous Pearson uncertainty describe different evidence. The comparison keeps ties, original magnitude, row order and inference boundaries visible. A bounded coefficient is not a universal dependence or trading score.
| Tool | Evidence entered | Primary output | Main boundary |
|---|---|---|---|
| Spearman Rank Correlation | Two aligned numeric or ordinal series | Pearson correlation of average ranks | Magnitude information is discarded. |
| Kendall Tau-b | Two aligned numeric or ordinal series | Tie-corrected concordant-minus-discordant balance | Pairwise agreement is not a significance verdict. |
| Lagged Cross-Correlation | Two ordered equal-frequency numeric series | Pearson coefficient at each signed displacement | Lag peaks do not establish lead, cause or prediction. |
| Pearson Confidence Interval | Two contemporaneous numeric series | Original-value Pearson coefficient plus Fisher limits | Linear inference remains assumption-sensitive. |
Frequently asked questions
- It describes the balance of concordant and discordant unordered observation pairs while correcting separately for ties in X and Y.
- Two rows are concordant when their nonzero X difference and nonzero Y difference have the same direction.
- Two rows are discordant when their nonzero X and Y differences have opposite directions.
- The model counts ties only in X, ties only in Y and ties in both separately, then applies the published Tau-b denominator.
- A constant required series makes the tie-corrected denominator zero, so the calculator fails closed instead of returning zero.
- No. Tau-b uses pairwise concordance counts, while Spearman applies Pearson correlation to average ranks.
- No. The coefficient is conditional on the entered rows and can change with the sample, rounding, alignment and preprocessing.
- No. It generates no p-value, agreement grade, significance decision, predictive edge, forecast, signal, position instruction or recommendation.
Sources and methodology
- NIST — Kendall’s Tau — Published concordant, discordant and Tau-b tie-adjustment definitions.
- SciPy — Kendall Tau — Official Tau-b implementation used for independent fixture reconciliation.
Version 1.0.0 was locked only after the rank, tie and lag conventions were checked against the cited NIST and SciPy material. Canonical fixtures were independently recomputed with SciPy 1.13.1 before comparison with the browser engine. The calculator performs arithmetic locally and does not upload the entered observations.
Continue the concordance and correlation review
Verify the ordered evidence before interpreting relationships
Reconcile the exact statement window, sampling frequency, timestamps, missing rows, realized P&L, spread, commission, financing, currency conversion, rounding and preprocessing across every series. Correct rank, pair-count or lag arithmetic cannot repair mismatched, selected or cost-inconsistent evidence.
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