---
title: "Chi-square goodness-of-fit test"
method_id: SM.DIF.CAT.1VAR.CHIGOF
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/chi_square_goodness_of_fit.md
source: Clarus method library
---

# Chi-square goodness-of-fit test

> A classical statistical method in the Clarus library (Difference family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** one-variable chi-square; chi-square test of a single proportion distribution; multinomial goodness-of-fit test; Pearson goodness-of-fit.

## Hypotheses

- **Null (H0):** The category proportions of {dependent} equal the specified expected proportions (p_1 = e_1, ..., p_k = e_k).
- **Alternative (H1):** At least one category proportion of {dependent} differs from its expected proportion.

## When to use it

Use this method when your goal is to:
- compare observed to expected distribution
- test categorical proportions against reference

## Data it expects

- **dependent:** 1 nominal/categorical variable (single classification)
- **independent:** none
- **pairing:** independent (each observation falls in exactly one category)

## Assumptions Clarus checks

- **expected frequency** (gate: expected_frequency; severity: critical; on violation: reroute)
- **independence of observations** (severity: critical; on violation: warn_only)

## Effect size reported

- **cohen w** (Cohen (1988))

## Honest limitations

- A goodness-of-fit test describes how a single variable is distributed; it is never evidence of causation.
- With very large N even a trivial departure from the expected distribution becomes statistically significant; read Cohen's w.
- Sparse categories (expected < 5) make the chi-square approximation unreliable; prefer an exact test or collapse categories.
- If p >= alpha, this is absence of evidence against the expected distribution, not proof that the data follow it exactly.
- Results generalize only to the population from which the sample was drawn, and depend on the expected distribution being a meaningful reference.

## Primary 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. https://doi.org/10.1080/14786440009463897
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Cochran, W. G. (1954). Some methods for strengthening the common chi-square tests. Biometrics, 10(4), 417-451. https://doi.org/10.2307/3001616
- Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.
- Sharpe, D. (2015). Your chi-square test is statistically significant: Now what? Practical Assessment, Research, and Evaluation, 20(8). https://doi.org/10.7275/tbfa-x148
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.), Ch.18. Sage.

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "Chi-square goodness-of-fit test", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/chi_square_goodness_of_fit.md

**APA**

Clarus. (2026). Chi-square goodness-of-fit test (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/chi_square_goodness_of_fit.md

**BibTeX**

```bibtex
@misc{clarus-chi-square-goodness-of-fit,
  author       = {Clarus},
  title        = {Chi-square goodness-of-fit test},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/chi_square_goodness_of_fit.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.DIF.CAT.1VAR.CHIGOF · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library