Kendall’s tau-b: what it means and how to read it

In the report: tau — Kendall’s tau-b: pairs of rows ordered the same way in both columns minus pairs ordered the opposite way, over the pairs not tied, from −1 to 1. Empty when a column has no spread or there are fewer than 2 common rows.

How to read it

The balance of agreeing and disagreeing pairs of rows, adjusted for ties: without ties, 0.10 means agreeing pairs outnumber disagreeing ones by a tenth of all pairs. Good for small samples and scales with many ties.

Its values usually run smaller than r or rho on the same data — for normally distributed data tau 0.3 corresponds to r about 0.45 — so do not read it with the r guide 0.1 / 0.3 / 0.5.

In a paper: τb = .10, p = .378.

In plain words

Two judges ranked 10 dishes, and a cook wanted to know how much the judges agree. With 10 dishes there are 45 pairs. In 33 pairs both judges agreed on which dish was better, and in 12 they disagreed. Tau came out about 0.47: agreeing pairs beat disagreeing ones by 21 of the 45 pairs. Put another way, the judges agree in about 73 pairs out of 100, where two judges tossing coins would agree in about 50. It works well for a few dishes and for ranks with many ties. Be careful: tau usually comes out smaller than other link numbers for the same dishes, so read it by its pairs, not by their guide.

How it is computed

τb = (C − D) / √((n₀ − Tx)(n₀ − Ty)), n₀ = n(n − 1)/2, Tx and Ty — the pairs tied in x and in y

Example

The cell with tau is selected in each report; point at a label to see its note.

— the sample data (50 rows) the examples use
— =STATS_KENDALL(Data!A1:T51, "x,x1,x2",0.95,"two-sided")

Where you see it

STATS_KENDALL — Kendall rank correlation (tau-b): the correlation of the order of two columns — for small samples and scales with many tied answers.

Partial correlation — The correlation of the two columns with the controls held constant: of what is left of both after the linear effect of the controls is removed. Empty when a control is constant or repeats another, or when there are fewer than k + 3 complete rows (k — the number of controls).
Cohen's d — The effect size: the difference in standard deviations, so it does not depend on the units of the data. Empty when there is no spread.
Phi — The strength of the relation, from the chi-square: for a 2 × 2 table with the sign of the relation (the sign depends on the order of the categories), for a larger table it equals the square root of χ² / N.
Cramér's V — The strength of the relation of two categorical variables, from 0 (none) to 1 (one determines the other).
Eta squared (η²) — The share of the total variance explained by the groups: 0 — none, 1 — all.

Get it in your sheet

One formula in a cell gives the whole report, Kendall’s tau-b included. ALS Stats is a free add-on for Google Sheets™.

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