Cramer’s V is a measure of the strength of association between two .

It ranges from 0 to 1 where:

• 0 indicates no association between the two variables.
• 1 indicates a perfect association between the two variables.

It is calculated as:

Cramer’s V = √(X²/n) / min(c-1, r-1)

where:

• X²: The Chi-square statistic
• n: Total sample size
• r: Number of rows
• c: Number of columns

How to Interpret Cramer’s V

The following table shows how to interpret Cramer’s V based on the degrees of freedom:

The following examples show how to interpret Cramer’s V in different situations.

Example 1: Interpreting Cramer’s V for 2×3 Table

Suppose we want to know if there is an association between eye color and gender so we survey 50 individuals and obtain the following results:

We can use the following code in R to calculate Cramer’s V for these two variables:

library(rcompanion)

#create table
data = matrix(c(6, 9, 8, 5, 12, 10), nrow=2)

#view table
data

     [,1] [,2] [,3]
[1,]    6    8   12
[2,]    9    5   10

#calculate Cramer's V
cramerV(data)

Cramer V 
  0.1671

Cramer’s V turns out to be 0.1671.

• df = min(#rows-1, #columns-1)
• df = min(1, 2)
• df = 1

Referring to the table above, we can see that a Cramer’s V of 0.1671 and degrees of freedom = 1 indicates a small (or “weak”) association between eye color and gender.

Example 2: Interpreting Cramer’s V for 3×3 Table

Suppose we want to know if there is an association between eye color and political party preference so we survey 50 individuals and obtain the following results:

We can use the following code in R to calculate Cramer’s V for these two variables:

library(rcompanion)

#create table
data = matrix(c(8, 2, 4, 5, 8, 6, 6, 3, 8), nrow=3)

#view table
data

     [,1] [,2] [,3]
[1,]    8    5    6
[2,]    2    8    3
[3,]    4    6    8

#calculate Cramer's V
cramerV(data)

Cramer V 
  0.246

Cramer’s V turns out to be 0.246.

The degrees of freedom would be calculated as:

• df = min(#rows-1, #columns-1)
• df = min(2, 2)
• df = 2

Referring to the table above, we can see that a Cramer’s V of 0.246 and degrees of freedom = 2 indicates a medium (or “moderate”) association between eye color and political party preference.

The following tutorials explain how to calculate Cramer’s V in different statistical software: