When is the chi-square test appropriate, and what does it assess?

• A. For continuous data; tests correlation
• B. For categorical data; tests association between nominal variables or goodness-of-fit to expected distributions
• C. To compare means of two groups
• D. For ordinal data only

Answer: (B)

The main idea is that chi-square is used with categorical data to see if what you observe matches what would be expected under a null assumption. There are two common ways it’s used. First, goodness-of-fit checks whether the observed counts in each category follow a specified distribution, such as equal proportions or a theory-driven pattern. For example, you might test whether a six-faced die is fair by comparing observed roll counts in each face to the 1/6 expectation for each face. Second, the test of independence looks at a contingency table to determine whether the distribution of one categorical variable is related to another—i.e., are the categories independent or not. In both cases you compare observed frequencies to expected frequencies and use the chi-square statistic, calculated as the sum over categories of (observed minus expected) squared divided by the expected, to obtain a p-value. Chi-square is not appropriate for continuous data, and it isn’t about comparing means. It’s specifically about counts in categories and assessing either fit to a distribution or association between categorical variables.