Confidence Interval for a Proportion Calculator

Q: What is the margin of error for 1,000 survey respondents at 95% confidence?

With p = 0.5 (most conservative), MoE ≈ 1.96 × √(0.25/1000) ≈ ±3.1%. This is why 'n = 1,000' is the standard polling sample size.

Q: Does the interval account for nonresponse bias?

No. The confidence interval only captures random sampling error. Bias from nonresponse, convenience sampling, or question wording is not reflected in the MoE. A biased poll with a tiny MoE can still be far from the true proportion.

Q: What is the Clopper-Pearson interval?

The Clopper-Pearson (exact) interval inverts the binomial CDF to guarantee at least the stated coverage probability. It is conservative (wider than Wilson or Wald) but never produces impossible values. Use it for small samples or extreme proportions.

Frequently Asked Questions

When should I use the Wilson interval instead of Wald?

Always prefer Wilson when p̂ is below 0.1 or above 0.9, or when n < 30. In those regions, the Wald interval can produce impossible values below 0 or above 1, and its actual coverage is far below the stated level.

What is the margin of error for 1,000 survey respondents at 95% confidence?

With p = 0.5 (most conservative), MoE ≈ 1.96 × √(0.25/1000) ≈ ±3.1%. This is why 'n = 1,000' is the standard polling sample size.

Does the interval account for nonresponse bias?

No. The confidence interval only captures random sampling error. Bias from nonresponse, convenience sampling, or question wording is not reflected in the MoE. A biased poll with a tiny MoE can still be far from the true proportion.

What is the Clopper-Pearson interval?

The Clopper-Pearson (exact) interval inverts the binomial CDF to guarantee at least the stated coverage probability. It is conservative (wider than Wilson or Wald) but never produces impossible values. Use it for small samples or extreme proportions.

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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 observations with x successes. The point estimate is p̂ = x ÷ n. This calculator reports two intervals:

Wald interval: p̂ ± z × √[p̂(1−p̂) ÷ n], where z is the standard-normal critical value (e.g. 1.96 for 95%).

Wilson score interval: center = (p̂ + z²÷2n) ÷ (1 + z²÷n), with half-width = z×√[p̂(1−p̂)/n + z²/(4n²)] ÷ (1 + z²/n). The Wilson interval is generally preferred because it has better coverage properties, especially when p̂ is near 0 or 1 or when n is small.

Wald vs. Wilson: Which to Use

The Wald interval is the simplest and most widely reported, but it has a well-documented failure mode: when p̂ is close to 0 or 1, or when n is small, the interval can extend below 0 or above 1 (impossible values) and its actual coverage probability falls below the stated confidence level.

The Wilson score interval always stays inside [0, 1] and maintains better coverage across the full range of p and n values. It is the preferred default in statistics literature. For moderate n and p̂ near 0.5, both intervals are nearly identical. When in doubt, use Wilson.

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

p̂ = 45 ÷ 200 = 0.225. SE = √(0.225 × 0.775 ÷ 200) ≈ 0.02952. With z ≈ 1.96, MoE = 1.96 × 0.02952 ≈ 0.0579. Wald interval: [0.225 − 0.058, 0.225 + 0.058] ≈ [0.167, 0.283].

Wilson interval: center = (0.225 + 1.96²/400) ÷ (1 + 1.96²/200) ≈ (0.225 + 0.0096) ÷ 1.0192 ≈ 0.2309. The Wilson interval is approximately [0.173, 0.287] - slightly shifted toward 0.5 compared to Wald.

Correct Interpretation

A 95% confidence interval does not mean there is a 95% probability that the true proportion p lies inside this specific interval. The true p is a fixed unknown constant; the interval is the random object. The correct interpretation: over many repeated samples constructed by the same procedure, 95% of the intervals would contain the true p.

In practice, you have one interval. It either contains p or it does not - you cannot know which. The confidence level describes the reliability of the procedure, not the probability for any single interval.

Sample Size and Precision

The margin of error shrinks with 1/√n. To halve the margin you must quadruple the sample. At 95% confidence with p̂ = 0.5 (worst case): n = 100 gives MoE ≈ ±9.8%, n = 400 gives MoE ≈ ±4.9%, and n = 1,600 gives MoE ≈ ±2.5%.

The Clopper-Pearson exact interval is the most conservative option for very small samples or proportions near the boundary. It guarantees at least the stated coverage but is wider than both Wald and Wilson. For samples with np̂ < 5 or n(1−p̂) < 5, neither Wald nor Wilson is reliable; use Clopper-Pearson or Fisher's exact method.

Confidence Interval for a Proportion Calculator

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