Confidence Interval for a Proportion Calculator

Frequently Asked Questions

How is a confidence interval for a proportion built?

From the sample proportion p̂ = x/n. The Wald interval is p̂ ± z·√(p̂(1−p̂)/n); the Wilson score interval is more accurate, especially for small n or extreme p̂.

How is this different from the Confidence Interval Calculator?

That tool is for a population mean (using z or t). This is for a proportion/percentage, which uses a different standard error and formula.

Why prefer the Wilson interval?

The Wald interval performs poorly when p̂ is near 0 or 1 or n is small (it can even fall outside 0–1). Wilson stays valid and is recommended by statisticians.

What sample size do I need?

Margin of error shrinks with √n. Halving the margin needs about four times the sample. Use a sample-size calculator to plan before collecting data.

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Estimates for informational purposes only.

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This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

A confidence interval for a proportion estimates the unknown population proportion p from a sample of n trials with x successes. The point estimate is p̂ = x ÷ n. The classic Wald interval is p̂ ± z × √[p̂(1 − p̂) ÷ n], where z is the standard-normal critical value for the chosen confidence level (z ≈ 1.96 for 95%), found here with the Beasley–Springer/Moro inverse-normal approximation. Because Wald behaves badly near 0 or 1 and for small n, this tool also reports the Wilson score interval, which inverts the score test and stays inside [0, 1] with far better coverage. Its center is (p̂ + z²÷2n) ÷ (1 + z²÷n) with a matching adjusted half-width. The margin of error reported is the Wald half-width.

Worked Example: x = 45, n = 200, 95%

Here p̂ = 45 ÷ 200 = 0.225. The standard error is √[0.225 × 0.775 ÷ 200] ≈ 0.02952. With z ≈ 1.95996 the margin of error is 1.96 × 0.02952 ≈ 0.0579, giving a Wald interval of about [0.1671, 0.2829]. The Wilson interval is slightly shifted toward 0.5 and a touch narrower - roughly [0.1727, 0.2873] - which is the more trustworthy interval, especially when p̂ is far from 0.5 or n is modest.

When to Use It

Survey results: the percentage who favor an option, with its margin of error.

A/B testing: a conversion rate and how precisely it is pinned down.

Quality & defect rates: the fraction defective in a production lot.

Polling & epidemiology: prevalence or approval estimates from a sample.

This is for a single proportion - distinct from a confidence interval for a population mean.

Common Mistakes & Assumptions

The Wald interval assumes the normal approximation to the binomial, which needs roughly np̂ ≥ 5 and n(1 − p̂) ≥ 5; with few successes it can extend below 0 or above 1, which is nonsensical - prefer Wilson there. The sample must be a simple random sample of independent observations; cluster or convenience sampling understates the true uncertainty. A 95% interval does not mean a 95% probability that p lies inside this particular interval; the parameter is fixed, the interval is random.

Interpretation & Limitations

Correctly read, "95% confidence" means that over many repeated samples, 95% of the constructed intervals would contain the true p. A wider interval signals less precision - quadrupling n only halves the margin of error, since it shrinks with √n. For very small samples or proportions near the boundary, the exact Clopper–Pearson interval is more conservative than either method shown. All critical values here are approximations accurate to about four decimals, which is ample for reporting margins of error to typical survey precision.

Confidence Interval for a Proportion Calculator

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