Confidence Intervals for a Single Proportion

The Wald interval, p̂ ± z·√(p̂(1−p̂)/n), is the one most textbooks teach first, and it's wrong often enough that you should stop using it. Brown, Cai & DasGupta (2001) showed Wald undercovers the nominal level chaotically across n and p, especially near 0 or 1, where it can produce bounds outside [0, 1]. Three successes in 10 at 95% confidence: Wald gives [0.016, 0.584]. One success in 10: Wald returns negative lower bounds. That isn't a small-n curiosity, it's a genuine failure mode.

Use Wilson's score interval as the default. It inverts a hypothesis test rather than approximating around p̂, stays bounded in [0, 1] by construction, and tracks the nominal coverage even at small n. Agresti-Coull (Wald with the +2/+4 adjustment) is fine too and easier to teach by hand. For very small n or extreme proportions, Clopper-Pearson is conservative but exact. The page returns all four side-by-side so you can see the disagreement and pick the one your reporting standard requires.

A confidence interval for a proportion starts with counting successes out of total trials. An e-commerce site might track 312 purchases from 4,200 visitors; a medical researcher might record 47 adverse events among 1,500 patients. The sample proportion p-hat equals successes divided by n. Plug those numbers into a formula alongside your chosen confidence level, and you get a range estimating the true population proportion.

Confidence level controls interval width. At 95%, the method captures the true proportion in 95 of every 100 repeated samples—not a statement that this specific interval has a 95% probability of containing the truth. Raising the level to 99% stretches the interval; dropping it to 90% tightens the bounds. Pick a level that matches the stakes: medical device clearances might demand 99%, while a quick marketing poll might tolerate 90%.

Sample size n drives precision. Doubling n shrinks the standard error by a factor of roughly 1.4, which narrows the interval. But sample collection costs money and time, so practical work involves balancing acceptable margin of error against budget. Pre-study power analysis helps decide n before data collection begins, but the calculator works with whatever sample you already have.

Successes must be whole numbers between zero and n. Zero successes or n successes require special handling—some methods produce degenerate intervals in these edge cases. If your count sits at an extreme, the calculator may flag a warning or switch methods automatically.

The Wald interval is the formula most textbooks introduce first: p-hat plus or minus z times the square root of p-hat times (1 − p-hat) divided by n. Simple to compute, easy to teach. But its coverage can fall short when p-hat is near zero or one, or when n is small. A study with 3 successes out of 10 trials can produce a Wald interval that dips below zero—impossible for a proportion.

Wilson's score interval fixes these problems by inverting a hypothesis test rather than relying purely on the normal approximation. The formula looks more complicated, but the payoff is intervals that stay bounded between zero and one and maintain actual coverage closer to the stated confidence level. Simulation studies show Wilson outperforms Wald almost universally, especially when p-hat is extreme.

When n is large—say, thousands of observations—and p-hat sits near 0.5, Wald and Wilson give almost identical results. The divergence grows as p-hat approaches the boundaries or n shrinks. A survey of 50 people finding 2 positive responses will see substantially different intervals depending on the method. If your software offers both, run them side by side on edge cases to understand how they differ.

Agresti-Coull is a compromise: add two pseudosuccesses and two pseudofailures before computing a Wald-like interval. This simple adjustment often achieves coverage comparable to Wilson without changing the formula structure. Some practitioners default to Agresti-Coull because it's easy to explain while avoiding the worst Wald pitfalls.

Proportion data are inherently discrete—you can observe 5 successes or 6, not 5.3. The normal distribution underlying Wald and Wilson is continuous, so there's a mismatch. Continuity correction adds 0.5 to or subtracts 0.5 from the success count before calculating, bridging discrete counts to a continuous curve. Some textbooks recommend it; others argue Wilson already handles the discreteness adequately.

Small samples amplify every approximation error. With n = 15, a single extra success can shift p-hat by nearly 7 percentage points. Interval width balloons because standard error is inversely proportional to the square root of n. Below about n = 30, many statisticians suggest exact binomial methods (Clopper-Pearson), which invert the cumulative binomial distribution rather than leaning on normal theory.

Clopper-Pearson intervals are wider than Wilson for the same data because they guarantee coverage at or above the stated level. That conservatism can feel frustrating when you want a tight estimate, but for high-stakes applications—like estimating the failure rate of a safety system—overestimating uncertainty beats underestimating it.

If your sample is small and p-hat is extreme, examine the calculator's output critically. A Wilson interval of [0.02, 0.35] might look wide, but it honestly reflects the data. Resist the temptation to pick whichever method gives the narrowest result—choose the method that best fits your assumptions and risk tolerance.

A 95% confidence interval for a proportion does not mean there is a 95% probability the true value lies inside. Once you compute the bounds, the parameter is either captured or not—probability is 0 or 1. The 95% refers to long-run coverage: repeat the survey many times, and about 95 of 100 intervals will contain the truth. This distinction matters when communicating results to decision-makers.

A poll reports 48% support with a margin of error of 3 points. Readers often think "the true support is somewhere between 45% and 51%." That's roughly correct for practical purposes, but technically the interval is a property of the method, not a probability envelope around the unknown parameter. Bayesian credible intervals do offer that probability interpretation, but they require specifying a prior distribution.

When comparing two proportions—say, conversion rates between landing pages—check whether the intervals overlap. Non-overlap suggests a statistically significant difference at roughly the stated confidence level, though a formal two-sample test is more precise. Overlapping intervals do not prove equality; the difference might still be real but smaller than your precision can detect.

Context shapes interpretation. An interval of [0.08, 0.12] for defect rate might be excellent in consumer electronics but alarming in aerospace. Always pair the statistical result with domain knowledge about acceptable thresholds and practical consequences of estimation error.

The binomial model underpins all standard proportion intervals. It assumes each trial is independent, with a constant probability of success. Drawing names from a hat without replacement violates independence once the pool shrinks noticeably. Surveying friends of friends can cluster responses, inflating apparent precision. Check whether your sampling design matches the model before trusting the output.

Normal approximation methods—Wald, Wilson, Agresti-Coull—work best when both np and n(1 − p) exceed about 5. Below that threshold, the binomial distribution is too skewed for a symmetric normal curve to mimic. The calculator may warn you, but vigilance on your part catches edge cases the software misses.

Cluster sampling, stratified sampling, and other complex designs need design-based variance estimators. Standard formulas assume simple random sampling. Using the wrong variance estimator can underestimate uncertainty by a factor of two or more, producing falsely narrow intervals that mislead stakeholders.

If you have paired or matched data—say, pre-and-post measurements on the same subjects—proportion intervals for independent samples do not apply. You would need McNemar's test framework or a paired-proportion interval, which accounts for within-subject correlation. Applying the wrong model leads to incorrect inference.