Bayesian statistics

Bayesian Binomial Test

Test a proportion against a chance or benchmark value with a Bayes factor, and get the posterior distribution of the true proportion with its 95% credible interval and highest density interval.

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Updated 13 October 2026 · By Bibin Xavier

Data 90 × 12

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Example data 90 rows, 12 variables. Data stays in this browser tab.

IDtext · 90 levels	Grouptext · 3 levels	Gendertext · 2 levels	Agenumeric	Hours_Studynumeric	Anxietynumeric	Motivationnumeric	Pre_Scorenumeric	Post_Scorenumeric	Exam_Scorenumeric	Passednumeric · coded	Satisfactionnumeric · coded
S001	Control	Male	23	11.2	29	4.6	60	62	67	1	2
S002	Workshop	Female	23	3.9	28	3.5	54	70	54	0	2
S003	Online	Female	18	10.1	32	4	55	57	66	0	3
S004	Control	Male	26	13.6	41	3.5	52	55	59	0	3
S005	Workshop	Female	22	9.7	27	3.6	56	63	65	0	4
S006	Online	Female	20	7.9	32	3.8	65	82	65	0	2
S007	Control	Female	20	6.3	28	3	72	83	61	0	3
S008	Workshop	Male	22	8	29	3.1	64	72	62	1	3

Variables and options

Successes

Trials

Test value p₀

Alternative hypothesis

Prior α

Prior β

Results update automatically when you change an option.

Results

Calculated in your browser

Table 1

Bayesian binomial test

Hypothesis	Observed proportion	BF₁₀	BF₀₁	Evidence
p ≠ 0.5	34 / 50 = 0.680	4.484	0.223	moderate evidence for H₁

Note. Beta(1, 1) prior on p under H₁. Classical exact binomial test: p = .015.

Table 2

Posterior of p

Posterior	Mean	Median	Mode	95% credible interval	95% HDI	P(p > 0.5 \| data)
Beta(35, 17)	0.673	0.675	0.680	[0.541, 0.792]	[0.546, 0.797]	0.995

Note. The posterior uses the two-sided prior. The HDI is the narrowest interval containing 95% of the posterior.

Prior and posterior of p

BF₁₀4.484

Posterior mean0.67395% CrI [0.541, 0.792]

P(p > 0.5)0.995

Interpretation

With 34 successes in 50 trials, BF₁₀ = 4.484, moderate evidence for H₁ against p = 0.5. The true proportion most likely lies between 0.541 and 0.792.

A Bayesian binomial test (Beta(1, 1) prior) gave BF₁₀ = 4.484; posterior mean p = 0.673, 95% CrI [0.541, 0.792].

How to cite this tool

Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.

APA 7

Xavier, B. (2026). Bayesian Binomial Test [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/bayesian-binomial-test/

BibTeX

@software{xavier_bayesianbinomialtest_2026,
  author = {Xavier, Bibin},
  title = {Bayesian Binomial Test},
  publisher = {FreeStatisticalTools.com},
  year = {2026},
  url = {https://freestatisticaltools.com/bayesian-binomial-test/},
  note = {Web application}
}

Suggested sentence for your methods section: “The analysis was carried out with the free Bayesian Binomial Test tool on FreeStatisticalTools.com (Xavier, 2026).”

What is a Bayesian binomial test?

The Bayesian binomial test compares the hypothesis that a proportion equals a test value p₀ with the hypothesis that it differs, using a Beta prior on the proportion. It returns a Bayes factor and the posterior Beta distribution of the true proportion with a credible interval and highest density interval.

When to use a Bayesian binomial test

Use it to test whether a success rate, pass rate or preference share differs from a benchmark such as 50%, or to estimate a proportion with a credible interval.

Assumptions

Independent trials with the same probability of success.

How to set up your data

The number of successes, the number of trials, and the test value.

How to use this calculator

Enter the values. Fill in successes, trials, test value p₀ and alternative hypothesis.
Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.

How to report a Bayesian binomial test in APA style

“34 of 50 participants preferred the new design; a Bayesian binomial test against p = .5 gave BF₁₀ = 7.9, posterior mean .67, 95% CrI [.54, .79].”

Frequently asked questions

What is a Beta-binomial model?

The number of successes follows a binomial distribution, and the unknown proportion has a Beta prior. The posterior is again a Beta distribution: Beta(α + successes, β + failures).

What is an HDI?

The highest density interval is the narrowest interval that contains 95% of the posterior, so every value inside it is more plausible than any value outside.

Which prior should I use?

Beta(1, 1), uniform on 0 to 1, is the default. Jeffreys’ prior Beta(0.5, 0.5) is another common reference prior.

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