---
title: "Two-proportion z-test"
method_id: SM.DIF.PRP.2IND.ZPROP
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/two_proportion_z_test.md
source: Clarus method library
---

# Two-proportion z-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:** two-sample z-test for proportions; test of two proportions; z-test for difference in proportions; comparison of two independent proportions.

## Hypotheses

- **Null (H0):** p_{g1} = p_{g2}
- **Alternative (H1):** p_{g1} != p_{g2}

## When to use it

Use this method when your goal is to:
- compare two independent proportions
- compare success rate between two groups

## Data it expects

- **dependent:** 1 binary/dichotomous
- **independent:** 1 nominal/2 levels
- **pairing:** independent

## Assumptions Clarus checks

- **large sample normal approximation** (gate: proportion_normal_approx; severity: warning; on violation: reroute)
- **independence of observations** (severity: error; on violation: reroute)

## Effect size reported

- **cohen h** (Cohen (1988) Ch.6)

## Honest limitations

- Large samples can make a trivial difference in rates statistically significant — read Cohen's h and the size of the percentage-point difference.
- A significant difference between groups is an association, not a causal effect, unless subjects were randomly assigned to the groups.
- When any expected count is small, the z-test's normal approximation is unreliable; Fisher's exact test is preferred.
- Results generalize only to the sampled population.

## Primary sources

- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Agresti, A. (2018). An Introduction to Categorical Data Analysis (3rd ed.). Wiley.
- Fleiss, J. L., Levin, B., & Paik, M. C. (2003). Statistical Methods for Rates and Proportions (3rd ed.). Wiley.

## 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, "Two-proportion z-test", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/two_proportion_z_test.md

**APA**

Clarus. (2026). Two-proportion z-test (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/two_proportion_z_test.md

**BibTeX**

```bibtex
@misc{clarus-two-proportion-z-test,
  author       = {Clarus},
  title        = {Two-proportion z-test},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/two_proportion_z_test.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.PRP.2IND.ZPROP · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library