In statistics, the phi coefficient, or mean square contingency coefficient, denoted by φ or r_φ, is a measure of association for two binary variables.

In machine learning, it is known as the Matthews correlation coefficient (MCC) and used as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975.

Introduced by Karl Pearson, and also known as the Yule phi coefficient from its introduction by Udny Yule in 1912 this measure is similar to the Pearson correlation coefficient in its interpretation.

In meteorology, the phi coefficient, or its square (the latter aligning with M. H. Doolittle's original proposition from 1885), is referred to as the Doolittle Skill Score or the Doolittle Measure of Association.

A Pearson correlation coefficient estimated for two binary variables will return the phi coefficient.

Two binary variables are considered positively associated if most of the data falls along the diagonal cells. In contrast, two binary variables are considered negatively associated if most of the data falls off the diagonal.

If we have a 2×2 table for two random variables x and y

where n₁₁, n₁₀, n₀₁, n₀₀, are non-negative counts of numbers of observations that sum to n, the total number of observations. The phi coefficient that describes the association of x and y is