Confidence Interval for a Proportion Calculator
Estimate a confidence interval for a population proportion (a percentage or rate) from your successes and sample size using the Wald method.
How it works
CI = p̂ ± z · √[ p̂·(1 − p̂) ÷ n ] · where p̂ = x ÷ n
A proportion is just a probability estimated from counts — the share of a sample that has some trait. Its standard error is √[p̂(1 − p̂) ÷ n], which is largest when p̂ is near 50% and shrinks toward the extremes. Multiplying by the z critical value gives the margin of error, and the interval is clamped to the valid 0–100% range. This Wald interval works well when both x and n − x are at least about 10.
Worked example
In a poll, 320 of 500 people (p̂ = 0.64) approve. The standard error is √[0.64 · 0.36 ÷ 500] ≈ 0.02146, so the 95% margin is 1.96 · 0.02146 ≈ 0.042. The interval is 64% ± 4.2%, or about 59.8% to 68.2%.
Sources: Binomial proportion confidence interval — Wikipedia
Frequently asked questions
How is this different from a confidence interval for a mean?
A proportion is a mean of 0/1 outcomes, so it uses the same idea but its standard error comes from p̂(1 − p̂) ÷ n rather than a separate standard deviation. You only need the count of successes and the sample size.
When does the Wald interval break down?
When the proportion is very close to 0% or 100%, or the sample is small, the interval can be too narrow or spill past 0–100%. A rule of thumb is to have at least 10 successes and 10 failures; otherwise a Wilson or exact interval is safer.
Why is the margin largest around 50%?
The term p̂(1 − p̂) peaks at p̂ = 0.5, meaning a near-even split carries the most uncertainty. That is also why sample-size planning uses p = 0.5 as the worst-case assumption.
Related calculators
For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.