Example Data Table
| Sample Size |
p0 |
p1 |
Alpha |
Alternative |
Expected Use |
| 40 |
0.50 |
0.65 |
0.05 |
Greater |
Small pilot comparison |
| 80 |
0.40 |
0.55 |
0.05 |
Greater |
Product response study |
| 120 |
0.30 |
0.22 |
0.01 |
Less |
Error reduction test |
| 150 |
0.50 |
0.60 |
0.05 |
Two-sided |
General proportion change |
Formula Used
The calculator uses the exact binomial distribution. For a fixed sample size n, success count X follows Binomial(n, p).
Probability mass: P(X = x) = C(n, x) × px × (1 - p)n - x
Power: Power = P(reject H0 | p = p1)
Beta: Beta = 1 - Power
Actual alpha: Actual alpha = P(reject H0 | p = p0)
For one-sided tests, the rejection region uses exact binomial tail probabilities. For two-sided tests, outcomes with exact probability not greater than the observed probability are counted in the exact p-value rule.
How to Use This Calculator
Choose fixed sample power when your sample size is already known. Choose find sample size when you need the minimum n for a target power.
Enter the null proportion p0. This is the value assumed by the null hypothesis. Enter the true proportion p1. This is the practical value you want the study to detect.
Select alpha and the alternative hypothesis. Add observed successes only when you also want an exact p-value for collected data. Press calculate. The result appears above the form.
Understanding Binomial Test Power
What Power Means
Power is the chance that a test detects a real effect. In a binomial test, the effect is a difference between the null proportion and the true proportion. A high power value means the study has a better chance of rejecting a false null hypothesis. Many studies target power near 0.80 or higher.
Why Exact Calculation Matters
The binomial distribution is discrete. This means possible outcomes are whole counts. Because of this, the actual alpha may not equal the chosen alpha exactly. An exact method checks every possible success count. It then builds the rejection region from exact binomial probabilities. This avoids hidden normal approximation errors.
Choosing p0 and p1
The null proportion p0 is the benchmark value. It may come from a rule, previous study, claim, defect rate, or historical conversion rate. The true proportion p1 is the effect you want to detect. Select p1 with practical meaning. Very small differences often need large samples.
One-Sided and Two-Sided Tests
A greater test is useful when only an increase matters. A less test is useful when only a decrease matters. A two-sided test is better when either direction matters. Two-sided tests usually need more evidence because they protect both tails of the distribution.
Sample Size Planning
The sample size search checks values from the minimum n to the maximum n. It stops when the exact power reaches the target. If the target is not reached, increase the maximum search limit or accept lower power. Large samples can be required when p0 and p1 are close.
Reading the Results
The critical region lists success counts that reject the null hypothesis. Power shows the chance of falling inside that region when p1 is true. Beta shows the chance of missing the effect. Actual alpha shows the real false positive rate created by the exact rejection rule.
FAQs
What is a binomial test power calculator?
It estimates the chance that an exact binomial test rejects the null hypothesis when a chosen true proportion is correct.
What is p0?
p0 is the null hypothesis proportion. It is the benchmark value tested against the observed or planned result.
What is p1?
p1 is the true or expected proportion. Power is calculated assuming this value is the real population proportion.
Why is actual alpha different from alpha?
Binomial outcomes are discrete. The rejection area cannot always match the requested alpha exactly, so actual alpha may be smaller.
Should I use a one-sided test?
Use a one-sided test only when the opposite direction is not scientifically or practically important before seeing the data.
What does beta mean?
Beta is the chance of not rejecting the null hypothesis when p1 is true. It equals one minus power.
How does sample size affect power?
Larger samples usually increase power. They make smaller differences easier to detect with the same alpha level.
Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report download.