What is the Chi-Square Test of Independence?

The Chi-Square Test of Independence is also known as Pearson’s Chi-Square and has two major applications: 1) goodness of fit test and 2) test of independence.

First, the Chi-Square Test can test whether the frequencies of a categorical variable are equal across categories.

Second, the Chi-Square Test can be used to test of independence between two categorical variables.  Specifically, it tests whether the frequencies of one categorical variable differ across levels of another categorical variable.  In other words, it tests whether or not a statistically significant relationship exists between the two variables.

Typical questions answered with the Chi-Square Test of Independence are as follows:

  • Medicine – Are children more likely to get infected with a virus than adults?
  • Marketing – Are women more likely than men to buy a product online?
  • Economy – Are white-collar workers more likely to quit their jobs than blue-collar workers?


Pearson’s Chi Square Test of Independence is an approximate test.  This means that the distribution of test statistics produced by this analysis only approximate the Chi-Square distribution.  This approximation improves with large sample sizes.  However, it poses a problem with small sample sizes, such as when expected cell sizes are below five.

Taking this into consideration, Fisher developed an exact test for contingency tables with small samples.  Exact tests do not need to approximate a theoretical distribution, such as the Chi-Square distribution.  In Fisher’s exact test, the exact distribution of possible outcomes is known, so the p-value produced by the test represents the exact probability of obtaining an outcome as extreme as the observed result under the null hypothesis that the two nominal variables are independent.

A rule of thumb is to use exact tests with sample sizes less than ten. Both Fisher’s exact test and Pearson’s Chi-Square Test of Independence can be easily calculated with statistical software such as Intellectus Statistics.