Kendall's Tau Correlation

What Is Kendall's Tau?

Kendall's Tau is a measure of the strength and direction of association between two variables that are measured on an ordinal (ranked) scale. In plain language, it tells you whether two sets of rankings tend to move in the same direction. If students who rank highly on a maths test also tend to rank highly on a physics test, that is a positive association. If high rankings on one variable tend to go with low rankings on the other, that is a negative association. Kendall's Tau gives you a single number that captures how consistent this pattern is.

Why Do We Need It?

Many variables in the social sciences are not measured with precise numerical values. Satisfaction ratings, preference rankings, and Likert-scale responses (for example, "strongly agree" to "strongly disagree") produce ordinal data — data where the order matters but the exact distance between values is unclear. Standard correlation measures like Pearson's r assume that data are measured on an interval or ratio scale. Kendall's Tau is designed specifically for situations where all you can rely on is the rank order of your observations.

Concordant and Discordant Pairs

The key idea behind Kendall's Tau is surprisingly intuitive. Imagine you have data from ten students, each with a rank on Variable A and a rank on Variable B. You take every possible pair of students and ask a simple question: do these two students appear in the same order on both variables? If Student 1 ranks higher than Student 2 on both Variable A and Variable B, that pair is called concordant — the rankings agree. If Student 1 ranks higher on Variable A but lower on Variable B, the pair is called discordant — the rankings disagree.

Kendall's Tau is essentially the proportion of concordant pairs minus the proportion of discordant pairs. When almost all pairs are concordant, Tau is close to +1. When almost all are discordant, Tau is close to −1. When concordant and discordant pairs are roughly equal, Tau hovers around 0, indicating no consistent association.

Tau-b and Tau-c

You will sometimes encounter two variants: Tau-b and Tau-c. Tau-b adjusts for tied ranks — cases where two or more observations share the same rank on one or both variables. This makes Tau-b the most commonly used version in practice, since tied values are extremely common in survey data and rating scales. Tau-c (sometimes called Stuart's Tau-c) is designed for rectangular tables where the number of rows and columns differ. If you are working with a square table or continuous ranks, Tau-b is typically the better choice. If your data form a table that is clearly not square, Tau-c may be more appropriate.

When to Use Kendall's Tau Instead of Spearman's Rho

Both Kendall's Tau and Spearman's Rho measure rank-based correlation, so students often wonder which one to choose. There are a few practical guidelines. First, Kendall's Tau tends to be more robust — it is less sensitive to errors or outliers in the data, because it is based on counting pairs rather than on the magnitude of rank differences. Second, for small sample sizes, Kendall's Tau has better statistical properties: its sampling distribution converges to normality more quickly, which means significance tests tend to be more accurate with fewer observations. Third, Kendall's Tau has a more direct interpretation as a probability: a Tau of 0.4 means that if you pick a random pair of observations, the probability that they are concordant is 0.4 greater than the probability that they are discordant.

A Concrete Example

Imagine a researcher studying whether judges at a baking competition agree with each other. Judge A and Judge B each rank eight cakes from best to worst. The researcher computes Kendall's Tau-b and obtains a value of 0.71. This tells us there is a strong positive association between the two judges' rankings — they tend to agree about which cakes are better and which are worse. A significance test shows p < 0.05, so we can be confident this agreement is not simply due to chance.

Interpreting the Values

Kendall's Tau ranges from −1 to +1. A rough guide for interpretation is:

• 0.00 to 0.10: negligible association
• 0.10 to 0.30: weak association
• 0.30 to 0.50: moderate association
• 0.50 and above: strong association

These thresholds apply to both positive and negative values — a Tau of −0.45 indicates a moderate negative association, just as a Tau of +0.45 indicates a moderate positive one. Keep in mind that Kendall's Tau values tend to be somewhat lower than Spearman's Rho values for the same data, so do not be alarmed if the number looks smaller than you expected.

Key Assumptions

• Both variables should be measured on at least an ordinal scale.
• Observations should be independent — each data point should come from a different individual or unit.
• There should be a monotonic relationship between the two variables (as one increases, the other consistently increases or consistently decreases), though the relationship does not have to be linear.

If your data meet these conditions, Kendall's Tau provides a reliable and interpretable measure of association that works well even with small samples and data that contain many tied values.