Direct answer
Proportion intervals must respect binomial sampling behavior and the 0-to-1 parameter range; Wilson’s score interval generally behaves better than the elementary Wald interval, especially near boundaries or with modest n.
Visual explanation
Interval methods behave differently near boundaries
What this calculation tells you
A sample proportion estimates an unknown success probability under a defined binary outcome and sampling design. Different approximations translate count uncertainty into different interval endpoints.
Keep success count and total n visible, define success before observing results, and choose an interval whose coverage is appropriate for the design and sample size.
Where it is used
Surveys
Report a binary response estimate with sample-size context.
Quality
Estimate a defect proportion without hiding sparse counts.
Experiments
Present response-rate uncertainty alongside effect comparisons.
Education
Show why bounded parameters create asymmetric uncertainty.
When this guide helps
- Estimating a proportion from counts.
- Handling zero or all successes.
- Comparing Wilson and Wald behavior.
- Explaining why n matters beyond the displayed percentage.
Start with the statistical question
Keep success count and total n visible, define success before observing results, and choose an interval whose coverage is appropriate for the design and sample size.
Wilson and Wald intervals are drawn for identical x/n values near 0, 0.5, and 1. A boundary line shows when an unadjusted Wald endpoint leaves the possible parameter range.
Worked example
With 1 success in 10 trials, the sample proportion is 0.10. A simple normal interval behaves poorly because expected successes are sparse; a Wilson interval remains bounded and shifts asymmetrically.
Assumptions that carry the result
The ordinary binomial interval assumes independent equal-probability trials or a design supporting equivalent variance. Clustering, weights, finite sampling, repeated outcomes, and misclassification require other methods.
Interpret the result without overreaching
An interval does not validate the outcome definition, sample representativeness, nonresponse, or measurement quality. It estimates only the declared proportion under the model.
- Reporting only a percentage without x and n.
- Using an unbounded normal interval mechanically.
- Comparing intervals built from incompatible populations or outcome definitions.
Worked case: observed proportion
There are 120 successes in 200 trials.
Estimate = 0.60. A 95% Wilson interval is approximately 53.1% to 66.5%.
The interval describes uncertainty for the defined population proportion under the sampling model.
It does not predict every future sample.
Reproduce this worked caseOpen Confidence Interval for a Proportion Calculator
Worked case: same proportion, smaller n
Use 12 successes in 20 trials.
Estimate remains 60%, but the 95% Wilson interval is much wider, about 38.7% to 78.1%.
Equal point estimates can carry very different precision.
Small samples make method choice especially visible.
Reproduce this worked caseOpen Confidence Interval for a Proportion Calculator
proportion confidence intervals: compare assumptions, not just answers
Keep numerator/denominator and interval method with the result. Weighting, clusters and finite populations require design-aware methods.
| Case | Calculation focus | Interpretation |
|---|---|---|
| 120/200 | 60% | Wilson ≈ 53.1–66.5% |
| 12/20 | 60% | Wilson ≈ 38.7–78.1% |
proportion confidence intervals: calculation checklist
- Success definition fixed
- Trials eligible
- Method named
- Design handled
- No certainty claim
Practical questions
Frequently asked questions
Why not use p ± 1.96SE every time?
That Wald approximation can have poor actual coverage and impossible endpoints, particularly with small n or proportions near boundaries.
Can an interval include 0 or 1?
Yes when the data and method support an endpoint at the boundary; values below 0 or above 1 are impossible for the parameter.
Do overlapping proportion intervals prove no difference?
No. A direct interval or test for the difference addresses a different sampling quantity.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.