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The phi-coefficient is the unit for the strength of cohesion between two dichotomy variables, so only for a 2x2-table.

The formula of the phi-coefficient

There are two formulas for the phi-coefficient:

Formula of the phi-coefficient
Formula phi-coefficient

Although both formulas look very differently computations with the same data give the same outcome.

The second formula is similar to the formula of Cramérs V. Therefore the phi-coefficient can be seen as a special case of Cramérs V.

How to interpret the phi-coefficient?

The outcome of this formula is always between 0 and 1. The value 0 means there is no cohesion between the two variables. The value 1 means a perfect correspondence: only the cells on the diagonal are filled.

Because we are dealing with variables measured at a nominal level, the rows or columns can be reversed without any problem. Due to this reversibility it is not important if the cells in the diagonal are filled from left above to right down or from left down to right above. The sign (that is minus (-) or plus (+)) is not important.

Testing the strenght of the phi-coefficient

The phi-coefficient is tested with the chi square test. This is a little bit odd, because the chi square value is also part of the formula for computing the phi-coefficient too.

An example of computing the phi-coefficient

One might wonder if women more often take a course on self-defence than man. Now two variables are needed: gender (woman = 1, man = 2) and did take the course (no = 0, yes = 1). For 98 youngsters between 15 and 30, these numbers were found:

An example in claculating the phi-coefficient

Without the percentages the differences are not easily seen. It really helps to interpret the result.

Continu to learn more about ...

  • Pearson product moment correlation 
  • Spearman rank correlation 
  • Kendall’s tau 
  • Cramérs V 
  • Variable 
    • Nominal data

Deepen your knowledge and read our manual about ...

  • How to find the correct statistical test
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