Python | Kendall Rank Correlation Coefficient (original) (raw)

Last Updated : 12 Jul, 2025

**What is correlation test?

The strength of the association between two variables is known as the correlation test. For instance, if we are interested to know whether there is a relationship between the heights of fathers and sons, a correlation coefficient can be calculated to answer this question. For know more about correlation please refer

this.

**Methods for correlation analysis:

There are mainly two types of correlation:

• **Parametric Correlation - Pearson correlation(r) : It measures a linear dependence between two variables (x and y) is known as a parametric correlation test because it depends on the distribution of the data.
• Non-Parametric Correlation - Kendall(tau) and **Spearman(rho): They are rank-based correlation coefficients, are known as non-parametric correlation.

Kendall Rank **Correlation Coefficient formula:

\tau=\frac{\text { Number of concordant pairs-Number of discordant pairs }}{n(n-1) / 2}

where,

• **Concordant Pair: A pair of observations (x1, y1) and (x2, y2) that follows the property
  • x1 > x2 and y1 > y2 or
  • x1 < x2 and y1 < y2
• **Discordant Pair: A pair of observations (x1, y1) and (x2, y2) that follows the property
  • x1 > x2 and y1 < y2 or
  • x1 < x2 and y1 > y2
• **n: Total number of samples

**Note:

The pair for which

**x1 = x2

and

**y1 = y2

are not classified as concordant or discordant and are ignored.

**Example:

Let's consider two experts ranking on food items in the below table.

The table says that for item-1, expert-1 gives rank-1 whereas expert-2 gives also rank-1. Similarly for item-2, expert-1 gives rank-2 whereas expert-2 gives rank-3 and so on.

**Step1:

At first, according to the formula, we have to find the number of concordant pairs and the number of discordant pairs. So take a look at item-1 and item-2 rows. Let for expert-1,

_x1 = 1

and

x2 = 2

. Similarly for expert-2,

_y1 = 1

and

_y2 = 3

. So the condition

_x1 < x2

and

_y1 < y2

satisfies and we can say item-1 and item-2 rows are concordant pairs. Similarly take a look at item-2 and item-4 rows. Let for expert-1,

_x1 = 2

and

_x2 = 4

. Similarly for expert-2,

_y1 = 3

and

_y2 = 2

. So the condition

_x1 < x2

and

_y1 > y2

satisfies and we can say item-2 and item-4 rows are discordant pairs. Like that, by comparing each row you can calculate the number of concordant and discordant pairs. The complete solution is given in the below table.

**Step 2:

So from the above table, we found that, The number of concordant pairs is: 15 The number of discordant pairs is: 6 The total number of samples/items is: 7 Hence by applying the Kendall Rank Correlation Coefficient formula

tau = (15 - 6) / 21 = 0.42857

This result says that if it's basically high then there is a broad agreement between the two experts. Otherwise, if the expert-1 completely disagrees with expert-2 you might get even negative values.

**kendalltau() :

Python functions to compute Kendall Rank Correlation Coefficient in Python

Syntax: kendalltau(x, y)

• x, y: Numeric lists with the same length

**Code:

Python program to illustrate Kendall Rank correlation

Python `

Import required libraries

from scipy.stats import kendalltau

Taking values from the above example in Lists

X = [1, 2, 3, 4, 5, 6, 7] Y = [1, 3, 6, 2, 7, 4, 5]

Calculating Kendall Rank correlation

corr, _ = kendalltau(X, Y) print('Kendall Rank correlation: %.5f' % corr)

This code is contributed by Amiya Rout

`

**Output:

Kendall Rank correlation: 0.42857