Binomial Test

Test whether the proportion of a yes/no outcome differs from a hypothesised value.

The binomial test compares the proportion of one outcome in a binary variable — pass or fail, yes or no, correct or incorrect — with a hypothesised proportion. The classic question is whether a coin is fair (proportion 0.5), but the hypothesised value can be anything: whether more than 80% of patients respond, or whether a pass rate differs from last year's 0.62.

Because it is exact, the binomial test makes no large-sample approximation. It calculates the probability of a result at least as extreme as the one observed directly from the binomial distribution. That makes it the right choice for small samples, where a chi-square or z-test for one proportion can be inaccurate.

StatRyx reports the number of successes and failures, the expected counts under the hypothesis, the observed proportion with its 95% confidence interval, and the exact two-tailed p-value, together with the one-tailed probabilities. You can name which category counts as a success, change the hypothesised proportion, or dichotomise a numeric variable with a threshold (for example, the share of scores below 50).

The observations must be independent, and each must fall into exactly one of the two categories. The test answers a question about one group; to compare two groups' proportions use a chi-square test or Fisher's exact test instead.

When to use the Binomial Test

You have a binary outcome and want to test its proportion against 0.5
You want to test a proportion against a known benchmark, such as a previous year's rate
Your sample is small, so an exact p-value is preferable to an approximation
You want to know whether a share of scores above or below a cut-off differs from a target
You need a confidence interval around a single proportion

When not to use it

You are comparing proportions between two independent groups — use the chi-square test or Fisher's exact test instead
The same people were measured twice on a yes/no outcome — use McNemar's test instead
Your variable has three or more categories — use the chi-square goodness of fit test instead

Worked example

Measure	Observed	Expected
Passed	64	50.0
Failed	36	50.0
Proportion	.64	.50
Exact two-tailed p	.007

How to report it (APA 7)

An exact binomial test showed that the proportion of students who passed (.64, 95% CI [.54, .73]) was higher than the hypothesised .50, p = .007, N = 100.

Frequently asked questions

Is the binomial test free on StatRyx?

Yes. You can run it without an account (3 free analyses), and a free account adds 10 more — a one-time allowance, no payment needed. Every result includes the APA 7 write-up. Paid plans (Student from $9/month) add more analyses and Word, PDF and Excel export.

How do I report a binomial test in APA 7?

Give the observed proportion with its confidence interval, the hypothesised proportion, the exact p-value and N: "The pass rate (.64, 95% CI [.54, .73]) was higher than .50, p = .007, N = 100."

When should I use a binomial test instead of a chi-square test?

Use the binomial test for one binary variable compared with a hypothesised proportion, especially with small samples. Use chi-square when you compare groups or have three or more categories.

Can I test against a proportion other than 0.5?

Yes. Name the proportion in your request, for example 'test whether the response rate differs from 0.8'.

Is the p-value one-tailed or two-tailed?

StatRyx reports the exact two-tailed p-value as the main result and also gives both one-tailed probabilities, so you can use a directional hypothesis you stated in advance.

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