Binomial Test (one-proportion exact)

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Independently verified. Every statistic this test reports has been re-derived against an independent reference — never the library the pipeline itself calls — the rendered output was read back in a browser, and the result is locked with a committed regression suite.

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Tests whether a single observed proportion (k successes out of n Bernoulli trials) differs from a pre-specified null proportion p₀.

Uses the EXACT binomial distribution P(X | n, p₀) rather than the normal approximation, so it is valid at any sample size. Reports the exact two-sided p, observed proportion k/n with Clopper-Pearson 95% CI, and the difference from p₀. The small-sample-safe one-proportion test; companion to the asymptotic z-test which is faster but unreliable at small n or extreme p̂.

Worked example

Is a success rate different from chance?

The exact binomial test compares an observed number of successes against a hypothesised probability.

Result

22 successes in 30 trials differed from a .50 rate, exact p = .016.

How you'd report it (APA)

An exact binomial test showed the success rate (22/30) exceeded .50, p = .016.

When to use it

  • One-sample binary outcome with fixed n
    A diagnostic kit is claimed to have 90% accuracy; the researcher tests it on 100 known cases and counts how many are correctly classified, then compares to p₀ = 0.90.
  • Fairness test (coin or single binary outcome with p₀ = 0.5)
    60 wine-tasters compare two wines and 38 prefer A.

When NOT to — use instead

Hypotheses

H₀: the population proportion of successes equals p₀.
Hₐ: the population proportion differs from p₀ (two-tailed) — or is greater / less than p₀ for one-tailed variants.

Parameter tested: population proportion of successes p

Assumptions (and what to do if they fail)

Sample size n is adequate for stable inference — the exact binomial test is valid for any n, but larger n narrows the confidence intervallow

Check: Verify n ≥ 10 for a useful CI width; smaller samples produce very wide intervals.

If violated: With very small n, the CI is extremely wide and most effect sizes cannot be detected.

The expected proportion p₀ represents a theoretically or empirically motivated null valuehigh

Check: Cite the source of p₀ in the pre-registration or methods section.

If violated: A poorly chosen p₀ makes the test uninformative even if it reaches significance.

The definition of 'success' vs 'failure' is unambiguous and applied consistentlymedium

Check: Verify that 'success' has a single operational definition.

If violated: Inconsistent classification biases the observed proportion and invalidates the test.

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