What is Somers’ D? (Definition & Example)

Somers’ D, short for Somers’ Delta, is a measure of the strength and direction of the association between an ordinal dependent variable and an ordinal independent variable.

An ordinal variable is one in which the values have a natural order (e.g. “bad”, “neutral”, “good”).

The value for Somers’ D ranges between -1 and 1 where:

  • -1: Indicates that all pairs of the variables disagree
  • 1: Indicates that all pairs of the variables agree

Somers’ D is used in practice for many nonparametric statistical methods.

Somers’ D: Definition

Given two variables, X and Y, we can say :

  • Two pairs (xi, yi) and (xj, yj) are concordant if the ranks of both elements agree.
  • Two pairs (xi, yi) and (xj, yj) are discordant if the ranks of both elements agree.

We can then calculate Somers’ D using the following formula:

Somers’ D = (NC – ND) / (NC + ND + NT)


  • NC: The number of concordant pairs
  • ND: The number of discordant pairs
  • NT: The number of tied pairs

The resulting value will always be between -1 and 1.

Somers’ D: Example in R

Suppose a grocery store would like to assess the relationship between the following two ordinal variables:

  • The overall niceness of the cashier (ranked from 1 to 3)
  • The overall satisfaction of the customer’s experience (also ranked from 1 to 3)

They collect the following ratings from a sample of 10 customers:

Somers' D example

To quantify the relationship between the two variables, we can calculate Somers’ D using the following code in R:

#enter data
nice <- c(1, 1, 1, 2, 2, 2, 3, 3, 3, 3)
satisfaction <- c(2, 2, 1, 2, 3, 2, 2, 3, 3, 3)

#load DescTools package

#calculate Somers' D
SomersDelta(nice, satisfaction)

[1] 0.6896552

Somers’ D turns out to be 0.6896552.

Since this value is fairly close to 1, this indicates that there is a fairly strong positive relationship between the two variables.

This makes sense intuitively: Customers who rate the cashiers as nicer also tend to rate their overall satisfaction higher.

Additional Resources

An Introduction to the Pearson Correlation Coefficient
An Introduction to Kendall’s Tau

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