Kendall's Tau-b

Kendall's tau-b takes every two observations in the data set as a pair and asks whether that pair runs the same way on both variables or the opposite way. A pair that follows the same order on both counts as concordant, one whose order flips counts as discordant, and the coefficient expresses the surplus of concordant pairs relative to the pairs available. Tau therefore ignores the distances between rank positions entirely and works from order alone, which is what lets it be read in the language of probability: it states how much more likely a randomly drawn pair is to run with the association than against it. The tau-b form adjusts its denominator for observations that share a value, which keeps the coefficient from shrinking artificially on short rating scales. YouReply Analyze reports the tau value, the p value, the number of observations entering the analysis and a strength label derived from the absolute magnitude.

• Do seniority level and perceived decision authority line up in the same direction?
• Do a three-point difficulty rating and a four-point support-need rating move together?
• Do two independent coders assign overlapping importance orders to the same open-ended answers?
• Is there a monotonic association between expertise level and risk perception in a small expert sample?

• When both variables sit on short ordinal scales and a large share of observations share the same value.
• When the sample is small, where pair counting yields a steadier p value than a coefficient built on rank differences.
• When you want to interpret the coefficient as a probability statement about how pairs line up.
• When the question is how far two ranking lists overlap, as with the importance orders of two coders.

• Both variables must be orderable: an ordinal scale, a ranking list, or a continuous measurement that can be turned into ranks.
• Ordinal categories must be coded numerically in a way that preserves their order; numbering labels alphabetically produces the wrong order.
• Rows with a valid value in both columns enter the analysis; a row with one half of the pair missing stays out.
• The minimum sample check matches the panel's other association methods: at least three observations are required, and that technical floor is not a statement about power.

Independence of pairs

Every row is its own unit of observation and rows do not influence each other. Stacking repeated measurements from the same person into the same column pair biases the coefficient.

At least ordinal measurement

Calling a pair concordant or discordant requires being able to say which of two values is larger. On categorical codes without order the coefficient is meaningless.

Monotonic association

The coefficient assumes the association does not change direction. In a pattern that rises and then falls, concordant and discordant pairs cancel out and tau lands near zero.

Enough observations

Because the p value rests on pair counts, very small samples leave only a handful of possible outcomes and significance moves in coarse steps.

• Independence of pairs: The panel does not check this assumption automatically; the researcher evaluates it.
• At least ordinal measurement: The method picker asks for numeric columns in the fields to be correlated; marking the measurement level as ordinal on the Variable tab puts the method in the right place among the suggestions.
• Monotonic association: The panel does not check whether the pattern turns; reading the scatter plot, when scatter data is present, is the only way to judge that.
• Enough observations: The engine will not run with fewer than three valid observations, and it performs no power calculation beyond that.

1. 1Upload your data file and check on the Data tab that the ordinal categories carry the numeric codes you expect.
2. 2On the Variable tab mark the measurement level of both variables as ordinal and declare your missing value codes.
3. 3Choose Kendall's tau-b in the method picker on the Analysis tab; the data requirements box shows which columns can be correlated.
4. 4Name the two columns to correlate in the parameter form and run the analysis.
5. 5Read the tau value, the p value, the number of observations and the strength label from the sections that open, and use the scatter plot, if one is drawn, to confirm the pattern does not turn.
6. 6Save the correlation bar chart as a PNG and the result table as Excel; the computation credits card names the call and its version.

Tau-b coefficient

Ranges from minus one to plus one, coming out positive when concordant pairs dominate and negative when discordant pairs do.

p value

The significance value computed under the hypothesis that no monotonic association exists in the population.

Number of observations

How many rows carried a valid value in both columns, a figure expected alongside the coefficient in any report.

Strength and direction label

A verbal label derived from the absolute magnitude, written together with the direction and using the same thresholds as the panel's other association methods.

Association charts

The correlation bar chart reserved for association methods is drawn, with a scatter plot added when the result carries scatter data.

The tau value itself

The coefficient is unitless and reads directly as an effect size. The panel's label thresholds run on the absolute value: below 0.2 negligible, below 0.3 weak, below 0.5 moderate, below 0.7 strong and above that very strong. Those thresholds are shared across every association method in the panel, whereas tau comes out structurally smaller than a coefficient built on rank differences for the same data, so treat the labels as a somewhat conservative reading when applied to tau.

No confidence interval comes back: tau, the p value and the number of observations are point estimates. An interval would have to come from a resampling calculation, which the panel does not perform. For the same reason the difference between tau values from two subgroups is not tested in the output either.

The numbers below are a representative example, not data from a real study or a real user.

Research question

In a representative employee study, is there a monotonic association between a four-point seniority level and a five-point perception of decision authority?

Variables

First variable: seniority level on a four-point ordinal scale · Second variable: perceived decision authority on a five-point ordinal scale

Example result

214 respondents answered both items and entered the analysis. Tau-b = 0.34, p < 0.001, and the strength label points to a moderate positive association. Both scales being short, the answers spread across 20 possible combinations of steps and a great many pairs share a value.

Interpretation

As seniority rises, perceived decision authority tends to rise with it. The coefficient says that a randomly drawn pair of observations is clearly but far from uniformly more likely to line up with the association than against it: the share of pairs running the other way is not small. The number says nothing about how much a one-step rise in seniority moves the perception, because pair counting does not carry the unit of the scale. Causation cannot be drawn either, since something that changes with seniority, such as the type of role, may be moving both measurements. The figures are illustrative.

Research question: Do the five level satisfaction scale and the recommendation score rank in the same direction?

First variable

satisfaction_level

Second variable

nps_score

Computation credits: scipy 1.18.0 · statsmodels 0.14.6 · scikit-learn 1.9.0 · numpy 2.5.1 · pandas 3.0.5 · semopy 2.3.11 · 89fc29a · Data seed: 20260914

Seniority level was moderately and positively associated with perceived decision authority, Kendall's tau-b = .34, p < .001, n = 214.

An example sentence close to APA style; the numbers are representative.

• Not for patterns that change direction, where concordant and discordant pairs cancel and the coefficient misses the association.
• Meaningless for categorical variables without order, where a test of independence is what the data calls for.
• Because the coefficient carries no unit of the scale, it cannot translate a one-step change or serve prediction.
• It comes out smaller than a coefficient based on rank differences for the same data, so placing the two side by side and comparing magnitudes misleads.
• It establishes no causation, and a third variable moving both measurements cannot be separated out.
• On large data sets the number of pairs grows quadratically, so the computation is heavier than a procedure based on rank differences.

• Spearman Rank CorrelationWhy: It measures the same monotonic association through rank differences and, being reported more widely in the literature, may suit longer scales better.
• Pearson CorrelationWhy: When both variables are continuous and the association is linear; the coefficient uses raw values and can be read as variance explained.
• Chi-Square Test of IndependenceWhy: When one variable is a categorical measurement without order, turning the question into independence rather than direction of association.

• Spearman Rank Correlation
• Pearson Correlation
• Chi-Square Test of Independence

How do I choose between Kendall's tau and Spearman's rho?

Both measure monotonic association and usually reach the same verdict on the same data. Tau behaves more steadily on short scales and in small samples, because it counts pairs and is not sensitive to squared rank differences. Rho is reported more widely in the literature, which gives it an edge where reviewer expectations matter. Either way, it is enough to name the coefficient you report.

Why is my tau smaller than my rho?

That is expected and signals nothing wrong. The two coefficients rest on different definitions: tau ratios the surplus of pairs running with the association, while rho is a product-moment correlation applied to rank positions. For a moderate association tau typically lands around two thirds of rho. Keep that gap in mind when applying one threshold table to both.

Why tau-b rather than tau-a?

The function the engine calls produces tau-b, the form whose denominator is adjusted for observations sharing a value. Tau-a leaves ties out of account, which drags the coefficient down artificially on short rating scales and keeps its absolute value from ever approaching one. Ties being unavoidable in survey data, the adjusted form is the right choice.

Can I use this coefficient for inter-coder agreement?

It can be used, and commonly is, to measure how far the orders two coders assign to the same units overlap. The coefficient looks only at agreement in order, though: if two coders produce the same order while coding systematically at different levels, tau comes out high although absolute agreement is poor. A question about absolute agreement needs a separate measure such as kappa or an intraclass correlation.

• Kendall, M. G. (1990). Rank Correlation Methods
• Agresti, A. (2010). Analysis of Ordinal Categorical Data
• Gibbons, J. D. (1993). Nonparametric Measures of Association
• scipy.stats.kendalltau documentation