Phi Coefficient: Definition & Examples


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 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:

Phi Coefficient example calculation

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

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