Test Statistic — Pearson's Chi-Square

Degrees of Freedom

Standardized Residual (per cell)

Effect Size — Cramér's V (one-variable case)

Yates' Continuity Correction (only for df = 1)

Technical Notes

• The test statistic follows a χ² distribution with df = k − 1 − m under the null hypothesis.
• The p-value is computed from the upper tail: P(χ² > observed | df).
• If any expected count Eᵢ < 5 in more than 20% of cells, the χ² approximation may be unreliable; consider Fisher's exact test or Monte Carlo simulation.
• Yates' correction is applied only for df = 1 and is conservative — it reduces Type I error but lowers power.

This free chi-square goodness of fit test calculator is designed for testing whether observed frequencies in a single categorical variable match a hypothesized distribution. It answers the question: does my sample come from the population I think it does?

Decision Checklist

• You have one categorical variable with two or more mutually exclusive categories
• Your data are frequencies (counts), not means or proportions
• Each observation belongs to exactly one category (mutually exclusive)
• Observations are independent of each other
• Expected count in each category is at least 5 (or 1 in ≤20% of cells)
• Do NOT use if you have two categorical variables → use Chi-Square Test of Independence
• Do NOT use if your data are continuous → use Kolmogorov-Smirnov or Anderson-Darling
• Do NOT use if observations are paired/repeated on the same subjects → use McNemar's test
• Do NOT use if expected counts are very small (<1 in any cell) → use Fisher's exact test

Real-World Examples

1. Genetics — Mendelian inheritance: A geneticist crosses two heterozygous pea plants and counts 315 round-yellow, 101 wrinkled-yellow, 108 round-green, 32 wrinkled-green offspring. Test whether these counts match the theoretical 9:3:3:1 ratio.
2. Quality control — die fairness: A casino auditor rolls a die 600 times and records frequencies for faces 1–6. Test whether the die is fair (each face expected at 1/6 = 100 times).
3. Marketing — consumer preference: A company tests five flavors of ice cream. From 500 surveyed customers, observe how many prefer each flavor. Test whether preferences are equally distributed (uniform null) or whether some flavors are preferred.
4. Wildlife ecology — habitat selection: A camera-trap study records 240 detections of leopards across four habitat types (forest, grassland, scrub, agriculture). Test whether detections deviate from random use proportional to habitat availability.
5. Education — multiple choice answer keys: A teacher analyzes 100-item answer keys to test whether correct answers (A, B, C, D) are uniformly distributed, as a check for test-construction bias.

Sample Size Guidance

Minimum recommended total N: at least 5 × k (so each category's expected count ≥ 5). For small effect detection (V ≈ 0.10), aim for N ≥ 200; for medium (V ≈ 0.30), N ≥ 90; for large (V ≈ 0.50), N ≥ 40.

Decision Tree — Choosing the Right Test

One categorical variable → THIS TEST (Chi-Square Goodness of Fit)
                         → Small expected counts (<5 in many cells) → Fisher's exact / Monte Carlo
Two categorical variables → Chi-Square Test of Independence
                         → 2×2 small N → Fisher's exact test
                         → Paired binary outcomes → McNemar's test
Continuous data → Normality? → Shapiro-Wilk / Anderson-Darling / Kolmogorov-Smirnov
        

The following references support the statistical methods used in this chi-square goodness of fit test calculator, covering p-value interpretation, effect size reporting, and best practices in hypothesis testing with categorical data.

1. 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. https://doi.org/10.1080/14786440009463897
2. Yates, F. (1934). Contingency tables involving small numbers and the χ² test. Supplement to the Journal of the Royal Statistical Society, 1(2), 217–235. https://doi.org/10.2307/2983604
3. Cochran, W. G. (1954). Some methods for strengthening the common χ² tests. Biometrics, 10(4), 417–451. https://doi.org/10.2307/3001616
4. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
5. Cramér, H. (1946). Mathematical methods of statistics. Princeton University Press.
6. Agresti, A. (2018). An introduction to categorical data analysis (3rd ed.). John Wiley & Sons.
7. Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
8. Howell, D. C. (2013). Statistical methods for psychology (8th ed.). Cengage Learning.
9. McHugh, M. L. (2013). The chi-square test of independence. Biochemia Medica, 23(2), 143–149. https://doi.org/10.11613/BM.2013.018
10. Sharpe, D. (2015). Chi-square test is statistically significant: Now what? Practical Assessment, Research & Evaluation, 20(8), 1–10. https://doi.org/10.7275/tbfa-x148
11. Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science. Frontiers in Psychology, 4, 863. https://doi.org/10.3389/fpsyg.2013.00863
12. American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). APA. https://doi.org/10.1037/0000165-000
13. R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/
14. NIST/SEMATECH. (2013). e-Handbook of statistical methods — Chi-square goodness of fit test. https://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm
15. Virtanen, P., Gommers, R., Oliphant, T. E., et al. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods, 17, 261–272. https://doi.org/10.1038/s41592-020-0772-5