Kendall’s Tau: The Rank Correlation You Should Know (But Probably Don’t)

What Is Kendall’s Tau?

Kendall’s Tau (written as τ, the Greek letter tau) is a nonparametric rank correlation coefficient proposed by Maurice Kendall in 1938.¹ It measures the strength and direction of the ordinal association between two variables — without assuming your data is normally distributed, without caring whether your variables are measured on an interval scale, and without being destabilised by outliers the way Pearson’s r often is.

The core idea is beautifully simple: take every pair of observations and ask — are they ordered the same way in both variables, or do they flip? Pairs that agree are called concordant; pairs that disagree are called discordant. Tau is essentially the net balance between them, scaled between −1 and +1.

The Math — Made Readable

For a dataset of n observations, consider every possible pair (i, j) where i < j. A pair is concordant if both x and y move in the same direction; discordant if they move in opposite directions. Ties are handled separately (more on that in the next section).

Let’s make this concrete. Suppose five patients are ranked by their pain score (X) and their recovery speed (Y):

Comparing all \(\binom{5}{2} = 10\) pairs: 7 are concordant, 3 are discordant. So: \(\tau_a = \frac{7 - 3}{10} = 0.40\). A moderate positive association — higher pain is somewhat linked to slower recovery.²

Tau-b vs. Tau-c: Which One?

Real-world data has ties. The basic Tau-a formula doesn’t handle them well, which is why two refined versions exist:

Tau-b (the default in most software)

Tau-c (Stuart’s Tau-c)

Tau-c is designed for rectangular tables — when X and Y have a different number of categories. It applies a correction based on the smaller of the two dimensions. In practice, Tau-b is the safe default unless your table is clearly rectangular.³

Kendall’s Tau vs. Spearman’s rho

Both are rank-based correlation coefficients. Both are nonparametric. But they have real differences that matter in practice:

The key conceptual advantage of Tau is that its value has a direct probabilistic interpretation: τ = 0.40 means that a randomly selected pair of observations is 40 percentage points more likely to be concordant than discordant. Spearman’s rho doesn’t offer that clean a reading.

Example 1: Medical Research

Setting: An oncologist wants to know whether tumour staging (Stage I–IV) at diagnosis correlates with quality-of-life scores (0–100) at six months post-treatment. Both variables are ordinal or at best pseudo-interval. Pearson’s r assumes linearity and normality — unreasonable here.

Using Kendall’s Tau-b on a dataset of 48 patients, the team finds τ_b = −0.54 (p < 0.001). This is a moderately strong negative correlation: higher tumour stage is associated with lower quality of life. Because Tau is based on pair comparisons, the result is interpretable even with the non-normal distribution of quality-of-life scores and the presence of ceiling effects near 100.⁴

Example 2: Social Science Research

Setting: A political scientist surveys 120 respondents on their level of political trust (1 = none, 5 = complete) and their frequency of civic participation (1 = never, 4 = often). Both are Likert-type ordinal scales — classic Kendall’s Tau territory.

The calculated Tau-b = +0.31 (p = 0.003) suggests a statistically significant, moderate positive relationship: people with higher political trust tend to participate more in civic activities. Importantly, reporting Tau here is more defensible than reporting Pearson’s r, since treating ordinal Likert data as continuous interval data is a contested methodological choice.⁵

In cross-national comparisons — where you’re ranking countries by governance indices — Tau also appears frequently. The World Values Survey, for instance, often produces Tau-based correlations across country-level rankings of inequality and social trust.

Interpretation Guide

Unlike Pearson’s r, there is no single universally accepted benchmark for “strong” Tau, since Tau values are systematically lower in magnitude than rho. The table below offers a pragmatic guide used across several methodological handbooks:⁶

Code Snippets

Most major statistical environments implement Kendall’s Tau out of the box. Here are ready-to-run examples:

Python (SciPy)

from scipy.stats import kendalltau

x = [1, 2, 3, 4, 5]   # e.g., tumour stage ranks
y = [2, 1, 4, 3, 5]   # e.g., quality-of-life ranks

tau, p_value = kendalltau(x, y)
print(f"Kendall's Tau-b: {tau:.4f}")
print(f"P-value:         {p_value:.4f}")

# Output:
# Kendall's Tau-b: 0.6000
# P-value:         0.1417

R

x <- c(1, 2, 3, 4, 5)
y <- c(2, 1, 4, 3, 5)

# Tau-b (default)
result <- cor.test(x, y, method = "kendall")
cat("Kendall's Tau-b:", result$estimate, "\n")
cat("P-value:", result$p.value, "\n")

# Tau-b via Kendall package (for ties & larger datasets)
# install.packages("Kendall")
library(Kendall)
Kendall(x, y)

SPSS (Syntax)

NONPAR CORR
  /VARIABLES = pain_rank recovery_rank
  /PRINT = KENDALL TWOTAIL SIG
  /MISSING = PAIRWISE.

Try It: Free Online Calculator

No software? No problem. The calculator below computes Kendall’s Tau-b (and Spearman’s rho simultaneously) from your raw data — no installation, no account required.