Phi Coefficient

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Definition: Phi Coefficient

Phi coefficient is a measure of association for two binary variables, it is denoted by φ. A Pearson’s correlation coefficient extended for binary variables will result in phi coefficient. It was designed for the comparison of two dichotomous distributions i.e distributions which have only two points on the scale and indicate some measurable attribute.  Examples are attributes like living or dead, success or failure etc.

Consider the following table


Attribute 1

Attribute 2










a,b,c,d – represent the frequencies of observation then phi is determined as the following:

φ = (ad-bc)/sqrt((a+b)(c+d)(a+c)(b+d))

The significance of phi is tested by computing the chi square value from phi for corresponding degree of freedom. The relationship used is as follows:

Φ2 = χ2/N

Phi coefficient is used in psychological testing.

Example :

43 persons were asked if they believed that there was any truth in horoscopes or in the existence of UFOs. The results were as follows:

A= 14, b=10, c= 6,d=13

Using formula: φ=0.266, and χ2 =3.04 , which can be tested for significance.

Hence, this concludes the definition of Phi Coefficient along with its overview.

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