What is the purpose of a chi-square test of independence?

• A. To determine if a single category fits a theoretical distribution.
• B. To assess how well two distributions fit.
• C. To determine if there is an association between two categorical variables in a contingency table.
• D. To estimate effect size for a binary outcome.

Answer: (C)

Two categorical variables are being examined to see if they are related rather than independent. The test asks whether the observed counts in the contingency table differ from what would be expected if the variables did not influence each other. By calculating expected frequencies under the assumption of independence and measuring how far the observed counts deviate (the chi-square statistic), we assess whether any association is unlikely to occur by chance. A small p-value indicates that the pattern of counts is unlikely under independence, suggesting an association between the variables. This is different from a goodness-of-fit test (which checks if a single category fits a theoretical distribution) and from estimating an effect size for a binary outcome; the chi-square test of independence identifies whether a relationship exists, while effect size measures like Cramér’s V can quantify its strength.