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Kendall's Tau (τ) Calculator

Measures the strength and direction of association between two ranked variables.

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Kendall's Tau

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Overview

Kendall's Tau (τ) is a non-parametric statistic used to measure the ordinal association between two measured quantities. It assesses the similarity of the ordering of data when ranked by each of the quantities. Unlike Pearson's r, it does not assume normality or linearity, making it suitable for data that is not normally distributed or when the relationship is not linear. It ranges from -1 (perfect negative association) to +1 (perfect positive association), with 0 indicating no association.

Symbols

Variables

C = Number of Concordant Pairs, D = Number of Discordant Pairs, n = Number of Observations, = Kendall's Tau

Number of Concordant Pairs
pairs
Number of Discordant Pairs
pairs
Number of Observations
observations
Kendall's Tau
Variable

Apply it well

When To Use

When to use: Use Kendall's Tau when you need to assess the monotonic relationship between two ordinal variables or when your data does not meet the assumptions for Pearson's correlation (e.g., non-normal distribution, small sample size). It is particularly useful for ranked data or when dealing with ties.

Why it matters: Kendall's Tau is crucial in fields like psychology, social sciences, and ecology for understanding relationships between variables without strict distributional assumptions. It allows researchers to quantify the agreement or disagreement in rankings, which is vital for validating survey results, assessing inter-rater reliability for ordinal scales, or analyzing trends in non-parametric data.

Avoid these traps

Common Mistakes

  • Confusing Kendall's Tau with Spearman's Rho; while both are non-parametric, they interpret rank differences differently.
  • Incorrectly calculating concordant and discordant pairs, especially with larger datasets.
  • Applying the formula to nominal data, where ranking is not meaningful.

One free problem

Practice Problem

A researcher is studying the relationship between two ranked variables in a sample of 10 participants. They identify 35 concordant pairs and 10 discordant pairs. Calculate Kendall's Tau for this dataset.

Number of Concordant Pairs35 pairs
Number of Discordant Pairs10 pairs
Number of Observations10 observations

Solve for: tau

Hint: First, calculate the total number of possible pairs using n.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Kendall rank correlation coefficient
  2. Nonparametric Statistics for the Behavioral Sciences by Sidney Siegel and N. John Castellan Jr.
  3. Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE Publications.
  4. Kendall rank correlation coefficient (Wikipedia article)
  5. Hollander and Wolfe Nonparametric Statistical Methods
  6. Howell Statistical Methods for Psychology
  7. Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). Sage. (Chapter on Non-parametric Correlation)