Statistical Research

Binomial Test Guide: Steps, Examples, and Tools.

The binomial test is an exact statistical method used to determine whether an observed proportion in binary data differs from a specified population probability. It is appropriate when outcomes have two categories, observations are independent, the number of trials is fixed, and each trial has the same probability of success.

dataclue TeamJuly 13, 20269 min read

STATISTICS • HYPOTHESIS TESTING

Binomial Test

Definition, Formula, Assumptions, Examples, Interpretation, and Reporting

Quick definition

A binomial test checks whether an observed proportion is different from a specific expected probability when every observation has only two possible outcomes.

What Is a Binomial Test?

A binomial test is an exact statistical test used to compare an observed proportion with a specific expected probability. It is designed for data where each observation has only two possible outcomes.

The two outcomes are often called success and failure, but those labels are only convenient names. A success could mean a customer makes a purchase, a patient responds to treatment, a product passes inspection, or a participant answers a question correctly.

The test asks a simple question: if the probability stated in the null hypothesis were true, how unusual would the observed number of successes be?

Common Situations Where the Test Is Useful

Area	Binary outcome	Possible question
Marketing	Purchase / no purchase	Is the conversion rate higher than 10%?
Medicine	Response / no response	Does the response rate differ from 30%?
Manufacturing	Pass / fail	Is the defect rate below 2%?
Psychology	Correct / incorrect	Is performance better than chance?
Quality control	Defective / acceptable	Has the defect rate changed from the historical level?

What Is a Binomial Distribution?

The binomial distribution describes the number of successes that can occur in a fixed number of independent trials when the probability of success stays the same from one trial to the next.

For example, imagine flipping a fair coin 10 times. Each flip has two outcomes, the chance of heads is 0.50, and one flip does not affect another. The binomial distribution gives the probability of getting 0 heads, 1 head, 2 heads, and so on up to 10 heads.

Distribution vs. test

The binomial distribution is the probability model. The binomial test uses that model to test a claim about the true probability of success.

Assumptions of the Binomial Test

Use a binomial test only when the data match the conditions below. These conditions are important because the probability calculation depends on them.

Assumption	What it means	Simple example
Two outcomes	Each observation falls into one of two categories.	Yes / no, pass / fail, success / failure
Fixed number of trials	The total number of observations is known.	100 customers are observed
Independent trials	One result should not change another result.	One coin flip does not affect the next
Constant probability	Under H₀, the probability of success is the same for every trial.	Every fair coin flip has p = 0.50

Null and Alternative Hypotheses

A binomial test begins with a null hypothesis, written as H₀, and an alternative hypothesis, written as H₁.

Example hypothesis

H₀: p = 0.50
H₁: p ≠ 0.50

This two-sided test asks whether the true success probability is different from 50% in either direction.

If the research question is directional, the alternative can instead be p > 0.50 or p < 0.50. The direction should be decided before looking at the result.

One-Sided vs. Two-Sided Binomial Test

The choice depends on the question you want to answer. A one-sided test looks for a change in one specified direction. A two-sided test treats unusually high and unusually low results as evidence against the null hypothesis.

Figure 1. The shaded tail area shows which outcomes count as evidence against H₀.

Test	Typical question	Alternative hypothesis
One-sided, greater	Is the rate higher than 50%?	H₁: p > 0.50
One-sided, less	Is the rate lower than 50%?	H₁: p < 0.50
Two-sided	Is the rate different from 50%?	H₁: p ≠ 0.50

Binomial Test Formula

The probability of observing exactly x successes in n trials is calculated with the binomial probability formula:

In the formula, n is the total number of trials, x is the number of successes, p is the expected probability of success, and 1 − p is the probability of the other outcome.

The formula above gives the probability of exactly x successes. For a hypothesis test, the probabilities of the observed result and outcomes considered equally or more extreme are combined to obtain the p-value.

How to Perform a Binomial Test

Figure 2. A practical five-step workflow for an exact binomial test.

Step 1: Define the two outcomes

Decide which outcome will be counted as the success. For a conversion study, success might mean “purchase” and the other outcome is “no purchase.”

Step 2: Set the expected probability

Choose the probability stated by the null hypothesis. If the normal conversion rate is 10%, use p = 0.10.

Step 3: Count successes and trials

Suppose 100 people see a new advertisement and 16 make a purchase. Then x = 16 and n = 100. The observed rate is 16%.

Step 4: Choose the alternative hypothesis

If the question is whether the advertisement increases conversions, use H₁: p > 0.10. This is a one-sided test.

Step 5: Calculate and interpret the p-value

For x = 16, n = 100, p = 0.10, the exact one-sided p-value is about 0.0399. At α = 0.05, the result is statistically significant because 0.0399 < 0.05.

Interpretation for this example

The data provide evidence that the conversion rate is higher than 10%. The test does not prove the new rate is higher with absolute certainty; it measures how strongly the data conflict with the null hypothesis.

How to Interpret the P-Value

The p-value tells you how unusual the observed data would be if the null hypothesis were true. A smaller p-value means the observed result is less compatible with the probability stated by H₀.

Result	At α = 0.05	Careful interpretation
p-value < 0.05	Reject H₀	The data provide evidence against the null hypothesis.
p-value ≥ 0.05	Do not reject H₀	The data do not provide strong enough evidence against H₀ at this level.

Important

A p-value of 0.05 does not mean there is a 95% chance that the alternative hypothesis is true. It is a probability calculated under the assumption that H₀ is true.

Binomial Test Example: Is a Coin Fair?

Suppose a coin is flipped 100 times and produces 62 heads and 38 tails. If the coin is fair, the probability of heads should be 0.50.

· H₀: p = 0.50

· H₁: p ≠ 0.50

· Observed result: 62 heads out of 100

· Exact two-sided p-value: approximately 0.021

Figure 3. Under H₀, outcomes in the shaded tails are at least as extreme as the observed result in this symmetric example.

Because 0.021 is below 0.05, the result is statistically significant at the 5% level. The data provide evidence that the probability of heads differs from 0.50.

The statistical test does not explain why the difference occurred. The experimental setup, the coin, and the way the flips were performed still matter when interpreting the result.

What Is an Exact Binomial Test?

The test is often called the exact binomial test because it uses probabilities calculated directly from the binomial distribution rather than relying on a large-sample approximation.

This exact calculation is useful with small samples or with probabilities close to 0 or 1, where a normal approximation may be less reliable. With large samples, a one-sample proportion z-test may also be suitable in some settings.

Binomial Test Compared With Other Tests

Method	Data type	Main question	Use it when
Exact binomial test	Binary	Does one proportion differ from a specified p?	Two outcomes and a known comparison probability
One-sample proportion z-test	Binary	Does one proportion differ from a specified p?	A large-sample approximation is appropriate
Chi-square test	Categorical	Do category counts differ, or are variables associated?	There are multiple categories or contingency tables
One-sample t-test	Numerical	Does a sample mean differ from a specified value?	The original outcome is quantitative, not binary

If numerical data are converted into only two categories, useful information can be lost. When the original measurements are continuous and the assumptions are suitable, a test of the mean may preserve more information.

How to Run a Binomial Test in R

R includes the binom.test() function for an exact binomial test. The basic structure is:

binom.test(x, n, p, alternative = "two.sided")

For example, to test whether 65 successes out of 100 are greater than an expected probability of 0.50:

binom.test(x = 65, n = 100, p = 0.5, alternative = "greater")

The one-sided p-value for this example is approximately 0.00176. The alternative option can be set to "greater", "less", or "two.sided".

Advantages and Limitations

Advantages	Limitations
Easy to explain for binary outcomes.	Not suitable when there are more than two meaningful outcomes.
Uses exact binomial probabilities.	Assumes independent observations.
Useful for small samples.	Assumes a constant success probability under H₀.
Works across many fields.	Turning continuous data into two categories can lose information.

How to Report a Binomial Test

A clear report should tell the reader what was observed, what probability was tested, which direction was used, and what the p-value showed. A confidence interval is also useful when your software provides one.

Report item	Example
Successes and total observations	65 successes out of 100 trials
Observed proportion	65%
Expected probability	50%
Direction	One-sided, greater
P-value	p = 0.0018
Conclusion	The observed rate is statistically higher than 50% at α = 0.05

Example report

Of the 100 trials, 65 were successful, giving an observed success rate of 65%. An exact one-sided binomial test showed that this rate was significantly greater than the expected probability of 50% (p = 0.0018).

Frequently Asked Questions

What does a binomial test tell you?

It tells you whether an observed proportion provides evidence of a difference from a specific expected probability.

What type of data is used in a binomial test?

Binary data. Each observation must fall into one of two outcome categories.

Why is it called a binomial test?

Because it is based on a model with two possible outcome categories and uses the binomial distribution.

Can a binomial test be one-tailed?

Yes. Use a one-sided test when the question is specifically whether the true probability is greater than or less than the expected value.

Is a binomial test non-parametric?

It is better described as an exact test based on a specific probability model. It does not require normally distributed continuous data.

What is the difference between a binomial test and a proportion test?

They can test similar hypotheses. The exact binomial test uses binomial probabilities directly, while some proportion tests use a normal approximation.

Final Thoughts

The binomial test is a practical method for simple yes-or-no data. It compares the success rate you observed with a probability you expected and measures whether the difference is larger than would normally be explained by random variation.

The test works best when there are two possible outcomes, the number of trials is fixed, observations are independent, and the probability of success is constant under the null hypothesis.

The main question to remember

Is the proportion we observed reasonably consistent with the proportion we expected?