Chi-square Test of Independence
Chi-square test of independence · Also known as: chi-squared test, Pearson's chi-square test, test of independence, ki-kare bağımsızlık testi
The chi-square test of independence is a nonparametric hypothesis test that examines whether two categorical variables are associated by comparing observed and expected frequencies in a cross-tabulation. It rests on the chi-square criterion introduced by Karl Pearson in 1900.
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When to use it
Use it to test the association between two categorical variables arranged in a contingency table, where observations are independent and each case contributes to exactly one cell. Two conditions should hold: every cell's expected frequency is at least 5, and the total sample is reasonably large (around n ≥ 30). When expected counts are low or the sample is small, Fisher's exact test is the appropriate alternative.
Strengths & limitations
- Distribution-free: it makes no normality assumption and works directly on frequency counts.
- Simple to compute and interpret, and available in every statistics package.
- Extends naturally from 2x2 tables to any r-by-c cross-tabulation.
- The chi-square approximation breaks down when expected cell frequencies fall below 5.
- It detects that an association exists but not its direction or pattern without follow-up analysis.
- The statistic grows with sample size, so very large samples can flag trivial associations as significant.
Frequently asked
What if some expected cell counts are below 5?
The chi-square approximation becomes unreliable when expected frequencies drop below 5 in any cell. In that situation, or when the total sample is small (under about 30), use Fisher's exact test, which computes the exact probability rather than relying on the asymptotic chi-square distribution.
Does a significant result tell me how strong the association is?
No. The chi-square statistic increases with sample size, so significance alone does not measure strength. Report an effect size such as Cramér's V, where values below 0.10 indicate a weak association and 0.30 or above a strong one.
Which cells are driving the result?
A significant overall test does not say where the association lies. Standardized (adjusted) residuals show which individual cells deviate most from their expected counts, identifying the contributing categories.
How is it different from McNemar's test?
The chi-square test of independence assumes the observations are independent. When the data are paired or matched — such as the same subjects measured before and after — McNemar's test is the correct choice instead.
Sources
- Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 50(302), 157–175. DOI: 10.1080/14786440009463897 ↗
- Agresti, A. (2007). An Introduction to Categorical Data Analysis (2nd ed.). Wiley. ISBN: 978-0471226185
How to cite this page
ScholarGate. (2026, June 1). Chi-square test of independence. ScholarGate. https://scholargate.app/en/statistics/chi-square-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Cramer's VStatistics↔ compare
- McNemar's testStatistics↔ compare