Chi-Square Test: Independence and Goodness of Fit, With Examples
Reviewed by Prof. Dr. Alexander Holtermann · Updated 16 October 2026 · 5 min read
The chi-square test checks whether two categorical variables are related (independence) or whether frequencies match an expected distribution.
Key takeaways
- The chi-square test of independence analyzes a contingency table of two nominal or ordinal variables, for example study group membership × passing the exam.
- It compares observed with expected frequencies:
χ² = Σ (O − E)² / E, whereE = row total × column total / N. - Degrees of freedom: (rows − 1) × (columns − 1) for independence; categories − 1 for goodness of fit.
- Assumption: expected frequencies of at least 5 in (almost) all cells. Otherwise use Fisher’s exact test.
- Report φ (2 × 2 tables) or Cramér’s V as the effect size: .10 small, .30 medium, .50 large for df = 1.
Two chi-square tests
The name chi-square test (χ² test, Pearson’s chi-square) covers two related tests for categorical data, i.e. counts of how many cases fall into each category.
- Data: frequencies (counts), never means or percentages.
- Each person appears in exactly one cell. Repeated measurements of the same people need McNemar’s test instead.
- The test ignores the order of categories; for ordinal variables there are more powerful trend tests.
| Test | Question | Example | df |
|---|---|---|---|
| Test of independence | Are two categorical variables related? | Is passing related to joining a study group? | (r − 1)(c − 1) |
| Goodness-of-fit test | Does one variable follow an expected distribution? | Do students choose all three exam dates equally often? | k − 1 |
Worked example: test of independence (2 × 2)
One hundred students are asked whether they joined a study group and whether they passed the statistics exam.
| Passed | Failed | Total | |
|---|---|---|---|
| Study group | 40 (E = 32.5) | 10 (E = 17.5) | 50 |
| No study group | 25 (E = 32.5) | 25 (E = 17.5) | 50 |
| Total | 65 | 35 | 100 |
Larger tables and Cramér’s V
The same logic works for any r × c table. For tables larger than 2 × 2 the effect size is Cramér’s V: V = √(χ² / (N · (k − 1))), where k is the smaller of the number of rows and columns.
| Preferred exam preparation | Flashcards | Summaries | Practice exams | Total |
|---|---|---|---|---|
| Bachelor’s students | 30 | 20 | 10 | 60 |
| Master’s students | 20 | 25 | 35 | 80 |
| Total | 50 | 45 | 45 | 140 |
Worked example: goodness-of-fit test
Sixty students can choose one of three exam dates. If there were no preference, you would expect 20 per date. Observed: 18, 30, 12.
χ² = (18 − 20)² / 20 + (30 − 20)² / 20 + (12 − 20)² / 20 = 0.2 + 5.0 + 3.2 = 8.40
df = 3 − 1 = 2, critical value 5.99, p = .015. The dates are not chosen equally often; the second date is clearly preferred.
Assumptions and small samples
The chi-square distribution is only an approximation, and it works well when the expected frequencies are not too small.
- Expected frequencies: all E ≥ 5 is the classic rule; a common relaxed version allows up to 20% of cells with E < 5 as long as none is below 1.
- Fisher’s exact test computes the exact p-value and is the standard alternative for small 2 × 2 tables.
- Yates’ continuity correction for 2 × 2 tables makes the test more conservative. In the study-group example it gives χ² = 8.62, p = .003. Many statisticians consider it unnecessary today; whichever you use, say so.
- Independence of observations: each person is counted once. Do not enter multiple answers of the same person as separate cases.
- Do not merge categories after looking at the data just to obtain significance; merge only for substantive reasons or to meet the expected-frequency rule.
Sample size and power
For a 2 × 2 table, the power analysis tool estimates the sample size from the effect size w (equal to φ for 2 × 2): to detect a medium effect (w = .30) with α = .05 and power .80 you need 88 people in total; for a small effect (w = .10), 785.
How to report a chi-square test in APA style
Give the chi-square value with degrees of freedom and sample size in parentheses, the p-value and an effect size. Report the percentages that show the direction of the association, because χ² itself has no direction.
Doing this in SPSS, R or JASP
All programs compute the same Pearson chi-square. Note that R applies Yates’ continuity correction to 2 × 2 tables by default, so its value can differ slightly from the uncorrected statistic in SPSS.
| Program | How to run it |
|---|---|
| SPSS | Analyze → Descriptive Statistics → Crosstabs; Statistics: Chi-square, Phi and Cramér’s V; Cells: Expected counts |
| R | chisq.test(table(df$a, df$b)); without Yates’ correction: correct = FALSE |
| JASP / jamovi | Frequencies → Contingency Tables; tick Phi and Cramér’s V and expected counts |
| Stata | tabulate a b, chi2 V |
Frequently asked questions
When do I use a chi-square test?
When both variables are categorical (nominal or ordinal with few categories) and you have counts: for example gender × choice of major, or yes/no answers in two groups.
What is the difference between chi-square and Fisher’s exact test?
The chi-square test uses an approximation that needs expected frequencies of about 5 or more. Fisher’s exact test computes the exact probability and is preferred for small samples, especially in 2 × 2 tables.
How do I calculate expected frequencies?
For each cell: row total × column total / overall total. In the example, 50 × 65 / 100 = 32.5.
Can I use percentages in a chi-square test?
No. The test needs absolute counts. Percentages hide the sample size, and the result would change with the scale.
What effect size goes with chi-square?
φ (phi) for 2 × 2 tables and Cramér’s V for larger tables. For df = 1, values of .10, .30 and .50 are considered small, medium and large.
Sources and further reading
- Agresti, A., & Finlay, B. (2018). Statistical Methods for the Social Sciences (5th ed.). Pearson.
- Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.). Wadsworth Cengage Learning.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
Your university’s examination regulations and style guide always take precedence.
Related articles
- Levels of measurementVariables are nominal, ordinal, interval or ratio. The level decides which averages, charts and statistical tests you are allowed to use.
- Effect sizeAn effect size tells you how large a difference or relationship is, independent of sample size. Cohen’s d of 0.2, 0.5 and 0.8 is small, medium, large.
- Choosing a statistical testYour research question, the level of measurement of your outcome, the number of groups and whether samples are paired decide which statistical test fits.
- Logistic regressionLogistic regression predicts a binary outcome (yes/no) from one or more predictors. Coefficients are log-odds; exp(b) gives the odds ratio.
- Reporting statistics in APAAPA style reports test statistic, degrees of freedom, exact p-value and effect size, e.g. t(13.46) = 3.42, p = .004, d = 1.71. Examples for every test.
- Statistics software comparedSPSS and Stata are menu- or command-based commercial packages, R and Python are free code, JASP and jamovi are free point-and-click. Choose by field and task.
