---
title: "Exact binomial test (one-sample proportion)"
method_id: SM.DIF.PRP.1SAM.BINOM
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/binomial_test.md
source: Clarus method library
---

# Exact binomial test (one-sample proportion)

> 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:** binomial test; exact test of a single proportion; Clopper-Pearson test; one-sample proportion exact test.

## Hypotheses

- **Null (H0):** p = {p_null} (the population success probability equals the reference value)
- **Alternative (H1):** p != {p_null} (two-sided; the population success probability differs from the reference value)

## When to use it

Use this method when your goal is to:
- test a single proportion against a reference value
- compare observed success rate to a hypothesized probability

## Data it expects

- **dependent:** 1 dichotomous (binary) outcome — success/failure on each independent trial
- **independent:** none (a single reference proportion p0)
- **pairing:** independent (each trial is a separate Bernoulli observation)

## Assumptions Clarus checks

- **independence of trials** (severity: critical; on violation: warn_only)
- **constant success probability** (severity: warning; on violation: warn_only)

## Effect size reported

- **proportion** (observed proportion p-hat = k/n with reference p_null)

## Honest limitations

- An exact binomial test describes a single proportion against a reference value; it is never evidence of causation unless the data come from a confirmed randomized experiment.
- With small n the test has low power and the confidence interval for the proportion is wide — a non-significant result does not prove the proportion equals p_null.
- With very large n even a trivially small departure from p_null becomes statistically significant; read the size of the proportion difference and its confidence interval.
- The result is only valid if trials are independent with a constant success probability; clustered or repeated observations on the same units violate the binomial model.
- p_null must be a genuine external reference, not a value estimated from the same data being tested.
- Results generalize only to the population from which the trials were sampled.

## Primary sources

- Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404-413. https://doi.org/10.2307/2331986
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Conover, W. J. (1999). Practical Nonparametric Statistics (3rd ed.), Ch.3. Wiley.
- Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). 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, "Exact binomial test (one-sample proportion)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/binomial_test.md

**APA**

Clarus. (2026). Exact binomial test (one-sample proportion) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/binomial_test.md

**BibTeX**

```bibtex
@misc{clarus-binomial-test,
  author       = {Clarus},
  title        = {Exact binomial test (one-sample proportion)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/binomial_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.1SAM.BINOM · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library