A **Phi Coefficient** (sometimes called a *mean square contingency coefficient*) is a measure of the association between two binary variables.

For a given 2×2 table for two random variables *x *and *y*:

The Phi Coefficient can be calculated as:

**Φ = (AD-BC) / √(A+B)(C+D)(A+C)(B+D)**

**Example: Calculating a Phi Coefficient**

Suppose we want to know whether or not gender is associated with political party preference. We take a simple random sample of 25 voters and survey them on their political party preference. The following table shows the results of the survey:

We can calculate the Phi Coefficient between the two variables as:

Φ = (4*4-9*8) / √(4+9)(8+4)(4+8)(9+4) = (16-72) / √24336 = **-0.3589**

**Note: **We could have also calculated this using the Phi Coefficient Calculator.

**How to Interpret a Phi Coefficient**

Similar to a Pearson Correlation Coefficient, a Phi Coefficient takes on values between -1 and 1 where:

**-1**indicates a perfectly negative relationship between the two variables.**0**indicates no association between the two variables.**1**indicates a perfectly positive relationship between the two variables.

In general, the further away a Phi Coefficient is from zero, the stronger the relationship between the two variables.

In other words, the further away a Phi Coefficient is from zero, the more evidence there is for some type of systematic pattern between the two variables.

**Additional Resources**

A Guide to the Pearson Correlation Coefficient

A Guide to Fisher’s Exact Test

A Guide to the Chi-Square Test of Independence

do you have a reference for the interpretation of the phi coefficient?