Kendall's Tau (τ) Equation, Derivation & Rearrangements

Q: What are common mistakes with the Kendall's Tau (τ) formula?

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.

Q: What is a real-world example of the Kendall's Tau (τ) formula?

When assessing the agreement between two judges ranking contestants in a competition, Kendall's Tau (τ) is used to calculate Kendall's Tau from Number of Concordant Pairs, Number of Discordant Pairs, and Number of Observations. The result matters because it helps judge uncertainty, spread, or evidence before making a conclusion from the data.

Q: What are some study tips for the Kendall's Tau (τ) formula?

Ensure your data is ordinal or can be meaningfully ranked. Understand that a positive tau indicates concordant rankings, while a negative tau indicates discordant rankings. Be aware of how ties are handled; the formula provided is for Kendall's Tau-a, which does not adjust for ties. More complex versions (Tau-b, Tau-c) exist for data with ties. Interpret the magnitude: values closer to ±1 indicate stronger associations, while values near 0 suggest weak or no association.

Core idea

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.

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.

Symbols

Variables

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

C

Number of Concordant Pairs

pairs

D

Number of Discordant Pairs

pairs

n

Number of Observations

observations

τ

Kendall's Tau

Variable

Walkthrough

Derivation

Formula: Kendall's Tau (τ)

Kendall's Tau measures the strength of association between two ranked variables by comparing concordant and discordant pairs.

Data consists of at least ordinal measurements for both variables.
Observations are independent.
The formula provided (Tau-a) assumes no tied ranks. For data with ties, Kendall's Tau-b or Tau-c should be used.

1

Define Concordant and Discordant Pairs:

For any two pairs of observations ( $x_{i}$ , $y_{i}$ ) and ( $x_{j}$ , $y_{j}$ ), they are concordant if ( $x_{i}$ - $x_{j}$ ) and ( $y_{i}$ - $y_{j}$ ) have the same sign (i.e., if $x_{i}$ > $x_{j}$ and $y_{i}$ > $y_{j}$ , or $x_{i}$ < $x_{j}$ and $y_{i}$ < $y_{j}$ ). They are discordant if ( $x_{i}$ - $x_{j}$ ) and ( $y_{i}$ - $y_{j}$ ) have opposite signs. Pairs with ties are handled differently in variations of Tau.

2

Calculate Total Possible Pairs:

For a sample size 'n', the total number of unique pairs of observations that can be formed is given by the combination formula 'n choose 2', which is `n(n-1)/2`. This represents the maximum possible number of concordant and discordant pairs combined, assuming no ties.

Total Pairs = \frac{1}{2} n (n - 1)

3

Define Kendall's Tau (τ):

Kendall's Tau is defined as the difference between the number of concordant pairs (C) and discordant pairs (D), divided by the total number of possible pairs. This ratio normalizes the measure to a range between -1 and +1.

τ = \frac{Number of Concordant Pairs - Number of Discordant Pairs}{Total Possible Pairs}

4

Substitute and Final Formula:

Substituting the definitions of C, D, and the total possible pairs yields the final formula for Kendall's Tau (Tau-a).

τ = \frac{C - D}{\frac{1}{2} n ( n - 1 )}

Result

τ = \frac{C - D}{\frac{1}{2} n ( n - 1 )}

Source: Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). Sage. (Chapter on Non-parametric Correlation)

Free formulas

Rearrangements

Solve for $C$

Kendall's Tau: Make C the subject

C = τ \cdot \frac{1}{2} n (n - 1) + D

To make C (Number of Concordant Pairs) the subject, first multiply both sides by the denominator, then add D (Number of Discordant Pairs).

Difficulty: 2/5

Solve for $D$

Kendall's Tau: Make D the subject

D = C - τ \cdot \frac{1}{2} n (n - 1)

To make D (Number of Discordant Pairs) the subject, first multiply both sides by the denominator, then subtract the result from C, and finally multiply by -1.

Difficulty: 2/5

Solve for $n$

Kendall's Tau: Make n the subject

n = \frac{1 + 1 + \frac{8 ( C - D )}{τ}}{2}

Making n (Number of Observations) the subject involves isolating the `n(n-1)` term, which leads to a quadratic equation that must be solved using the quadratic formula.

Difficulty: 4/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a complex curve where the correlation strength is inversely proportional to the square of the number of observations, causing the value to approach zero as the sample size grows. For a psychology student, this means that as the number of observations increases, a fixed difference between concordant and discordant pairs results in a weaker statistical association. The most important feature of this curve is that the value never reaches zero for any finite number of observations, meaning that even a small difference between ranked pairs maintains a measurable correlation regardless of how large the sample becomes.

Graph type: other

Why it behaves this way

Intuition

Imagine plotting all data points and then drawing lines between every possible pair. Kendall's Tau counts how many lines 'slope' in the same direction for both variables (concordant)

τ

The Kendall rank correlation coefficient, quantifying the strength and direction of monotonic association between two ranked variables.

A direct measure of how consistently two sets of rankings agree or disagree with each other.

Number of Concordant Pairs

The count of pairs of observations where the relative order of the two variables is identical.

Pairs where if one variable increases, the other also tends to increase (or if one decreases, the other also tends to decrease).

Number of Discordant Pairs

The count of pairs of observations where the relative order of the two variables is inverse.

Pairs where if one variable increases, the other tends to decrease (or vice versa).

n

The total number of observations or data points in the sample.

The size of the dataset being analyzed.

\frac{1}{2} n (n - 1)

The total number of unique pairs that can be formed from 'n' observations.

The maximum possible number of comparisons between distinct data points.

Signs and relationships

Number of Concordant Pairs - Number of Discordant Pairs: This difference determines the sign and magnitude of Kendall's Tau. A positive difference indicates more agreement in rankings, resulting in a positive τ.

Free study cues

Insight

Canonical usage

Kendall's Tau (τ) is a dimensionless statistic used to quantify the strength and direction of monotonic association between two ranked variables.

Common confusion

Students sometimes mistakenly attempt to assign units to correlation coefficients or other statistical measures of association, despite their dimensionless nature as ratios or indices.

Dimension note

Kendall's Tau is a dimensionless statistic, representing a ratio of the difference between concordant and discordant pairs to the total number of possible pairs.

Unit systems

$n$ dimensionless · Represents the number of observations or data pairs, and is therefore a count without physical units.

Ballpark figures

Quantity:

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.

Where it shows up

Real-World Context

When assessing the agreement between two judges ranking contestants in a competition, Kendall's Tau (τ) is used to calculate Kendall's Tau from Number of Concordant Pairs, Number of Discordant Pairs, and Number of Observations. The result matters because it helps judge uncertainty, spread, or evidence before making a conclusion from the data.

Study smarter

Tips

Ensure your data is ordinal or can be meaningfully ranked.
Understand that a positive tau indicates concordant rankings, while a negative tau indicates discordant rankings.
Be aware of how ties are handled; the formula provided is for Kendall's Tau-a, which does not adjust for ties. More complex versions (Tau-b, Tau-c) exist for data with ties.
Interpret the magnitude: values closer to ±1 indicate stronger associations, while values near 0 suggest weak or no association.

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.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Kendall's Tau measures the strength of association between two ranked variables by comparing concordant and discordant pairs.

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.

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.

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.

When assessing the agreement between two judges ranking contestants in a competition, Kendall's Tau (τ) is used to calculate Kendall's Tau from Number of Concordant Pairs, Number of Discordant Pairs, and Number of Observations. The result matters because it helps judge uncertainty, spread, or evidence before making a conclusion from the data.

Ensure your data is ordinal or can be meaningfully ranked. Understand that a positive tau indicates concordant rankings, while a negative tau indicates discordant rankings. Be aware of how ties are handled; the formula provided is for Kendall's Tau-a, which does not adjust for ties. More complex versions (Tau-b, Tau-c) exist for data with ties. Interpret the magnitude: values closer to ±1 indicate stronger associations, while values near 0 suggest weak or no association.

References

Sources

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

Kendall's Tau (τ)

Overview

Variables

Derivation

Define Concordant and Discordant Pairs:

Calculate Total Possible Pairs:

Define Kendall's Tau (τ):

Substitute and Final Formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Spearman's Rank Correlation Coefficient (ρ)

Pearson's Product-Moment Correlation Coefficient (r)

Frequently Asked Questions

Sources