Kendall Tau Calculator — Ordinal Association

The Kendall Tau Test. Kendall tau asks a pairwise question. Pick two observations: do X and Y order them the same way, or do their directions disagree? The two variables must remain paired row by row. Equal lengths, at least three independent pairs, and a meaningful ordinal scale come before any counting. When both differences share a sign, the observation pair is concordant. When one rises and the other falls, it is discordant. A tie in X or Y is neither concordant nor discordant. Keep X-only, Y-only, and joint ties visible because they affect the normalization differently. Tau-b divides the surplus C minus D by a denominator corrected for tied pairs in each variable. This keeps the coefficient between minus one and one. Without ties, tau-b reduces to tau-a and one plus tau over two is the concordant share. With ties, that shortcut is invalid. Choose the tail before seeing tau. Two-tailed asks whether association differs from zero; right means concordant order, and left means discordant order.

The page standardizes C minus D with a full tie-corrected null variance and uses a normal z reference. Its p-value is asymptotic, not exact. Below ten pairs, the calculator warns that the normal approximation is rough. It never switches to an exact permutation distribution, especially when ties are present. No confidence interval appears. The null variance is suitable for testing tau equal to zero, not a general sampling variance for nonzero tau. Strictly increasing transformations preserve every pair ordering. Changing positive X to log X leaves C, D, tau-b, z, and p exactly unchanged. Kendall uses order rather than raw distance, but it cannot repair wrong pairing, dependent units, clusters, selection effects, or a misspecified scientific question. A deterministic U-shape can have tau zero because rising and falling comparisons cancel. Zero ordinal association does not prove independence. The page scans all n choose two observation pairs, so work grows quadratically. It stops above five thousand pairs instead of silently becoming unusable.

Now use the same fourteen paired readings from the previous correlation examples, with a two-sided alpha of point zero five and ties in both variables. Fourteen observations create ninety-one unordered pairs. Classifying every comparison produces eighty-seven concordant pairs and only two discordant pairs. Two pairs are tied within X and one is tied within Y. One pair is jointly tied and contributes to neither variable-only tie term. The tie-adjusted denominator is eighty-nine point four nine nine. Dividing the concordance surplus eighty-five gives tau-b zero point nine four nine seven. Among comparable pairs only, eighty-seven divided by eighty-nine equals ninety-seven point eight percent concordant. That descriptive share is not tau-b. The tie-corrected null variance is three hundred thirty point six eight nine, giving z four point six seven four, beyond the two-sided critical magnitude one point nine six zero. The two-sided asymptotic p-value is about two point nine five times ten to the minus six. Reject zero population ordinal association at alpha point zero five.

Replace positive X by its logarithm and the complete comparison ledger stays fixed. Every calculator result is identical because order never changes. Negate Y and tau-b becomes negative zero point nine four nine seven. The two-sided p-value stays fixed while the directional tails swap. Heavy ties are handled honestly but reduce informative pair mass. Read the counts beside tau-b instead of treating a coarse scale like continuous data. On these identical fourteen pairs, Pearson r is about point nine nine one, Spearman rho point nine eight six, and Kendall tau-b point nine five zero. Their scales differ. Validation stops unequal lengths, fewer than three or more than five thousand pairs, constant variables, invalid alpha, and tie patterns with no statistic variance. Report the pairing design, n, tie profile, C, D, tau-b, chosen tail, asymptotic z and p, alpha, comparable-pair share if useful, and approximation caveats. Use Pearson for linear magnitude, Spearman for monotonic ranks, and exact or permutation methods for careful small samples. Run the Kendall Tau Test free at Distri Scope dot com.