Confidence Interval Calculator for Proportion

What Is a Confidence Interval for a Proportion?

When you observe that 60% of a sample supports a policy, you want to estimate the true population proportion. A confidence interval gives a range: "We're 95% confident the true support is between 55% and 65%."

This interval captures both the estimate (60%) and the uncertainty from sampling (±5%).

The Methods

Wald Interval (Normal Approximation)

The simplest and most common method:

$CI = \hat{p} \pm z^* \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Where:

• $\hat{p}$ = sample proportion (successes/total)
• $z^*$ = critical z-value (1.96 for 95%)
• $n$ = sample size

Limitation: Performs poorly when p is near 0 or 1, or when n is small.

Wilson Score Interval

More accurate, especially for extreme proportions:

$\tilde{p} = \frac{\hat{p} + \frac{z^{*2}}{2n}}{1 + \frac{z^{*2}}{n}}$

$CI = \frac{\tilde{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n} + \frac{z^{*2}}{4n^2}}}{1 + \frac{z^{*2}}{n}}$

The Wilson interval is centered on $\tilde{p}$ (a modified estimate) rather than $\hat{p}$.

When to Use Which

• Wald: Quick calculations, large samples, p not extreme
• Wilson: Small samples, proportions near 0 or 1, computer-based calculation

This calculator uses Wilson by default for better performance across all scenarios.

Interpretation

"Support for the candidate was 52% (95% CI: 49% to 55%)"

The interval spans 49% to 55%, which includes 50%—meaning we can't confidently say the candidate leads. If the interval were 52% to 58%, the lead would be statistically significant.

Applications

Political Polling

"The poll shows 48% approval (margin of error ±3%)"

The margin of error is half the confidence interval width. True approval likely falls between 45% and 51%.

Quality Control

"Defect rate: 2.3% (95% CI: 1.8% to 2.9%)"

Management can see both the point estimate and plausible range.

Medical Studies

"Treatment success rate: 78% (95% CI: 71% to 84%)"

Clinicians need to know not just the observed rate but its uncertainty.

Market Research

"Brand awareness: 62% (95% CI: 57% to 67%)"

Marketing decisions should account for the range, not just the point estimate.

A/B Testing

"Conversion rate: 4.2% (95% CI: 3.8% to 4.7%)"

Compare intervals between variants to assess meaningful differences.

Sample Size Requirements

The normal approximation works reasonably when:

• $np \geq 5$
• $n(1-p) \geq 5$

For example, with p = 0.05, you need at least n = 100.

The Wilson interval works well even when these conditions aren't met.

Historical Context

Abraham de Moivre (1733) first derived the normal approximation for binomial proportions, establishing that sample proportions are approximately normally distributed for large samples.

Edwin Bidwell Wilson (1927) developed the Wilson score interval, addressing problems with the Wald interval near the boundaries. Despite being nearly 100 years old, Wilson's method is often underused.

Theodore Brown and colleagues in the 1990s-2000s renewed interest in interval methods, demonstrating that the Wald interval's poor performance near p = 0 or p = 1 was worse than previously appreciated.

Today, many statistical organizations recommend Wilson or similar adjusted intervals over the simple Wald method.

Why the Wilson Interval Is Better

The Boundary Problem

When the true proportion is near 0:

• Wald interval can include negative values (impossible)
• Wald interval can be too narrow
• Coverage probability drops below stated confidence level

Wilson interval avoids these issues by design.

Coverage Probability

The "stated" 95% CI should contain the true proportion 95% of the time. Studies show:

• Wald interval: Coverage can drop to 80-85% in extreme cases
• Wilson interval: Coverage stays close to 95%

Width Determinants

Interval width depends on:

1. Sample size: Larger n → narrower interval
2. Proportion value: p = 0.5 gives widest interval; extreme p gives narrower
3. Confidence level: Higher confidence → wider interval

The maximum width occurs at p = 0.5. If you're designing a study without a prior estimate, assume p = 0.5 for conservative planning.

Margin of Error

For proportions, margin of error is approximately:

$ME \approx z^* \times \sqrt{\frac{p(1-p)}{n}}$

At 95% confidence ($z^* = 1.96$) with p = 0.5:

$ME \approx \frac{1}{\sqrt{n}}$

This gives the rule of thumb: 1,000 respondents → about 3% margin of error.

Continuity Correction

For discrete proportions, a continuity correction can improve accuracy:

$CI = \hat{p} \pm \left[z^* \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} + \frac{1}{2n}\right]$

The $+\frac{1}{2n}$ adjustment accounts for the discrete nature of counts. This becomes negligible for large n.

Multiple Proportions

When estimating proportions for many categories:

• Each proportion has its own CI
• Multiple comparison adjustments may be needed
• Consider multinomial confidence regions

The CI for Difference in Proportions handles comparing two proportions.

Related Calculations

• Standard Error of Proportion — SE = √[p(1-p)/n]
• Margin of Error Calculator — Half the CI width
• Critical Value Calculator (z*) — Find z* values
• CI for Difference in Proportions — Compare two groups
• Sample Size Calculator — Plan for desired precision

Summary

Confidence intervals for proportions translate sample percentages into ranges of plausible population values. While the Wald (normal approximation) interval is simplest, the Wilson score interval provides more reliable coverage, especially for proportions near 0 or 1. Whether interpreting poll results, quality rates, or treatment success, the interval communicates not just the estimate but the uncertainty that accompanies any sample-based inference.