One-Proportion Z-Test
Test whether a single sample proportion differs from a hypothesized value, such as a target defect rate or pass rate. Get the z-statistic, p-value and confidence interval, with the sample-size conditions checked for you.
Run One-Proportion Z-Test →What is a One-Proportion Z-Test?
A one-proportion z-test compares an observed sample proportion against a hypothesized population proportion. It answers questions such as whether a defect rate exceeds a target, whether a pass rate meets a standard, or whether a yes-response share differs from an expected value.
The test relies on the normal approximation to the binomial distribution, which is valid when the sample is large enough. The usual rule of thumb is that the expected number of successes and failures under the null hypothesis, n times p and n times (1 minus p), should each be at least about 10.
The z-statistic measures how many standard errors the observed proportion lies from the hypothesized one, and its p-value comes from the standard normal distribution. A small p-value indicates the sample proportion differs significantly from the hypothesized value.
In plain terms: You have a rate, like 12 percent defective, and a target, like 10 percent, and you want to know if the difference is real or just sampling luck. This test checks that for a single proportion, as long as your sample is big enough for the math to hold.
Key Points
Proportion vs Target
Tests one observed proportion against a hypothesized value, for one- or two-tailed alternatives.
Normal Approximation
Uses the normal approximation to the binomial, valid when successes and failures are each expected to be at least about 10.
Small Samples
When the sample is too small for the approximation, use an exact binomial test instead.
Key Formulas
Interpreting the Result
Compare the p-value to your significance level. A small p-value means the observed proportion is far enough from the hypothesized value that chance alone is an unlikely explanation, so you reject the null.
Report the confidence interval for the proportion alongside the test. It shows the plausible range for the true proportion and conveys the practical size of any difference, not just whether it is significant.
Assumptions & Validation
Random Sample
The data is a random sample of independent binary outcomes.
If violated: Address dependence or sampling bias.
Large Enough Sample
Expected successes and failures are each at least about 10 for the normal approximation.
If violated: Use an exact binomial test for small samples.
Binary Outcome
Each observation is a success or failure.
If violated: Use different methods for continuous or multi-category data.
⚠️ Check assumptions first
The one-proportion z-test depends on the normal approximation to the binomial, which breaks down for small samples or proportions near 0 or 1. Check that the expected counts of successes and failures are each at least about 10 before trusting the result. When they are not, use an exact binomial test, which is valid for any sample size.
When NOT to Use One-Proportion Z-Test
Small Samples
When expected successes or failures fall below about 10, use the exact binomial test.
Comparing Two Proportions
To compare proportions between two groups, use the two-proportion z-test.
Categorical Association
For association among categories, use a chi-square test.
Industry Applications
Defect-Rate Testing
Test whether an observed defect rate exceeds a target or specification.
Pass-Rate Verification
Check whether a pass or conformance rate meets a required standard.
Survey Proportions
Test whether the share choosing an option differs from an expected value.
Conversion Benchmarks
Compare an observed conversion or response rate against a benchmark.
Frequently Asked Questions
When should I use a one-proportion z-test?
Use it when you have a single sample of binary outcomes, such as pass or fail, and want to test whether the observed proportion differs from a specific hypothesized value, like a target defect rate. The sample must be large enough for the normal approximation to the binomial to hold. For comparing proportions between two groups, the two-proportion z-test is the correct tool instead.
What sample size does the one-proportion z-test require?
The usual rule of thumb is that the expected number of successes and the expected number of failures under the null hypothesis, calculated as the sample size times the hypothesized proportion and its complement, should each be at least about 10. This ensures the normal approximation to the binomial is reasonable. When the sample is smaller, an exact binomial test should be used instead.
What is the exact binomial test and when do I need it?
The exact binomial test computes probabilities directly from the binomial distribution rather than relying on the normal approximation, so it is valid for any sample size, including small ones. You need it when the expected counts of successes or failures fall below the rule-of-thumb threshold, or when the proportion is very close to zero or one, situations where the z-test approximation becomes unreliable.
How do I interpret the confidence interval for a proportion?
The confidence interval gives the plausible range for the true population proportion at your chosen confidence level. If the hypothesized proportion lies outside the interval, the two-tailed test is significant. The interval also communicates how precisely the proportion is estimated and how far the observed value is from the target, which helps judge practical importance beyond statistical significance.
What is the difference between a one-proportion z-test and a chi-square goodness-of-fit test?
A one-proportion z-test compares a single proportion against a hypothesized value and can be one- or two-tailed. A chi-square goodness-of-fit test compares observed counts across two or more categories against expected counts. For a binary outcome, the two-tailed one-proportion z-test and the chi-square test give equivalent results, but the z-test also supports directional, one-tailed hypotheses.
Can I use this test for A/B conversion rates?
For a single conversion rate compared against a fixed benchmark, yes. However, most A/B tests compare the conversion rates of two groups, such as a control and a variant, in which case the two-proportion z-test is appropriate because it accounts for the sampling variability in both groups. Use the one-proportion test only when comparing one observed rate to a known target.
Test a Proportion Against a Target
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