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What is a Chi-Square Test?

The chi-square (χ²) test is a non-parametric statistical test used to examine relationships between categorical variables. Unlike t-tests or ANOVA that compare means, the chi-square test works with frequency counts—how many observations fall into each category. It compares the frequencies you actually observed in your data to the frequencies you would expect if there were no relationship between the variables. When the difference between observed and expected frequencies is large enough, you can conclude the variables are significantly associated.

Chi-Square Test of Independence

Use the test of independence to determine whether two categorical variables are related. The data are arranged in a contingency table (cross-tabulation) where rows represent one variable and columns represent the other. For example, you might test whether there is a relationship between gender and product preference, or between treatment condition and recovery outcome. The null hypothesis states that the two variables are independent—knowing the value of one variable tells you nothing about the other.

Chi-Square Goodness-of-Fit Test

Use the goodness-of-fit test to determine whether observed frequencies for a single categorical variable differ from a set of expected frequencies. For example, testing if a die is fair by comparing observed rolls to the expected equal distribution (1/6 for each face), or testing whether customer visits are evenly distributed across days of the week. The null hypothesis states that the observed distribution matches the expected distribution.

When to Use Chi-Square vs. Other Tests

Choosing the right test depends on the type of data you have and the size of your sample. Use this guide to select the appropriate test:

Assumptions of the Chi-Square Test

Before interpreting your chi-square results, verify that these assumptions are met:

Understanding Cramer's V Effect Size

While the p-value tells you whether an association is statistically significant, Cramer's V tells you how strong the association is. This is critical because with large sample sizes, even trivial associations can reach statistical significance. Cramer's V ranges from 0 (no association) to 1 (perfect association), and its interpretation depends on the degrees of freedom (the smaller of rows − 1 or columns − 1):

*df* = min(rows − 1, columns − 1). For our worked example above (2×3 table), df* = 1, so V = .30 represents a medium effect.

How to Report Chi-Square Results in APA Format

According to APA 7th edition guidelines, chi-square results should include the chi-square statistic, degrees of freedom, sample size, p-value, and an effect size measure. Here is a template and a real example:

Note: Report χ² values to two decimal places. Report p-values to three decimal places, except use p < .001 when the value is below .001. Always include an effect size measure (Cramer's V for independence tests).

Common Mistakes to Avoid

• Using chi-square with small expected frequencies: When expected cell counts fall below 5, the chi-square approximation is unreliable. Use Fisher's exact test for 2×2 tables, or combine categories to increase expected counts in larger tables.
• Entering percentages instead of raw counts: The chi-square test requires actual frequency counts, not percentages or proportions. Using percentages will produce incorrect results because the test needs to know the actual sample size.
• Ignoring effect size: A statistically significant chi-square result with a tiny Cramer's V (e.g., .05) may not be practically meaningful. With large samples, even trivial associations become "significant." Always report and interpret Cramer's V.
• Violating independence of observations: Each participant should contribute to only one cell. If the same person appears in multiple cells (e.g., repeated measures), the chi-square test is invalid. Use McNemar's test for paired data.
• Confusing the two types of chi-square tests: The test of independence (two variables in a contingency table) and the goodness-of-fit test (one variable vs. expected proportions) answer different questions. Make sure you select the correct test for your research question.