Confidence Interval Calculator

Most calculators use the Wald interval: p ± z√(p(1−p)/n). It's the one in textbooks, and it breaks in exactly the cases people care about — small samples and results near 0% or 100%, where it can produce a lower bound below zero.

This uses the Wilson score interval, which is the standard recommendation and stays inside 0–100% by construction. At large samples with a mid-range result the two agree closely; at the edges Wilson is right and Wald isn't.

A proportion is a count out of a total: 240 of 400 respondents said yes, so p̂ = 0.60. The confidence interval gives the range of true population values consistent with that result at your chosen confidence level. The calculator above works only with proportions like this — yes/no answers, the share choosing an option, the share of top-two-box ratings.

It uses the Wilson score interval. With z from the confidence level (1.96 for 95%):

centre = (p̂ + z² ÷ 2n) ÷ (1 + z² ÷ n)

half-width = z × √(p̂(1−p̂) ÷ n + z² ÷ 4n²) ÷ (1 + z² ÷ n)

The interval is centre ± half-width. Compare the Wald interval taught in most introductory courses, p̂ ± z√(p̂(1−p̂) ÷ n), which is also what most margin-of-error calculators use.

p̂ = 0.60, n = 400, z² = 3.8416. The denominator is 1 + 3.8416 ÷ 400 = 1.0096. The centre is (0.60 + 0.0048) ÷ 1.0096 = 0.5991. Under the square root: 0.24 ÷ 400 + 3.8416 ÷ 640,000 = 0.000606, whose root is 0.02462; times 1.96 and divided by 1.0096 gives a half-width of 0.0478.

Wilson 95% interval: 55.1% to 64.7% — the calculator's default output. The Wald interval is 60% ± 4.8, or 55.2% to 64.8%. With a large sample and a mid-range result, the two methods agree to a tenth of a point. The differences show up elsewhere.

The problem with Wald is not only the impossible bounds. Brown, Cai and DasGupta showed in 2001 that its actual coverage — how often a nominal 95% interval really contains the true value — falls well short of 95%, erratically, even at sample sizes many people consider comfortable. They recommend the Wilson interval (or Agresti–Coull) instead, and the NIST engineering statistics handbook does the same, noting that a method which can produce a negative lower limit is inferior.

The practical rule: if n is in the hundreds and the result is between about 20% and 80%, both methods give the same answer and it does not matter. For small groups, rare events, or near-unanimous results — a subgroup of 15, a 2% complaint rate, 19 of 20 satisfied — use Wilson.

Higher confidence means a wider interval, because you are asking the range to catch the true value more often. For 240 of 400: 90% gives 55.9% to 63.9%, 95% gives 55.1% to 64.7%, and 99% gives 53.6% to 66.1%. The data did not change; only how sure you want to be. 95% is the convention in survey reporting.

Sample size has the bigger effect. The interval's width shrinks with the square root of n, so four times the responses gives half the width.

Working backwards: if you want an interval no wider than about ±5 points at 95% confidence, plan for roughly 385 completed responses in the group you will report on, and more for any subgroup. The sample size calculator does this for other targets. Collecting until the interval looks narrow enough and then stopping is not the same thing, because stopping when the result looks good biases it.

Averages — a mean rating, average spend, time on task — need a different interval: x̄ ± t × s ÷ √n, where s is the sample standard deviation and t comes from the t-distribution with n − 1 degrees of freedom. This calculator does not do means.

Illustration: 50 respondents give a mean satisfaction rating of 7.2 out of 10 with a standard deviation of 1.8. The standard error is 1.8 ÷ √50 = 0.255; t for 95% and 49 degrees of freedom is 2.010; the half-width is 0.51, so the interval is 6.69 to 7.71.

For rating scales, an alternative that works with this calculator is to report the share of top-box answers (for example, the share rating 9 or 10) as a proportion. Our Likert scale calculator summarises the full distribution.

Strictly, a 95% confidence level describes the method: if you repeated the survey many times, about 95% of the intervals built this way would contain the true value. Any single interval either contains it or does not.

An interval of 55% to 65% is about the population share, not about individual respondents. It says nothing about how varied the answers were.

Two groups' intervals can overlap slightly while the difference between them is still statistically clear, because the uncertainty of a difference is smaller than the sum of the two half-widths. Test the difference directly with the A/B test significance calculator.

It covers random sampling error only. Biased wording, a skewed list, or the people who chose not to answer can move the result outside the interval. AAPOR also notes that sampling-error intervals apply only to probability samples, not opt-in online surveys, so an interval computed from a publicly shared link overstates how much you know.

The calculator does not apply a finite population correction. If you surveyed 80 people out of a team of 100, the true uncertainty is much smaller than the interval shown, and at 100 of 100 there is no sampling error at all. The margin of error calculator accepts a population size.

A clean proportion needs a clear single-choice question (yes/no, or pick one option) and a known denominator. In Zunoform, responses export to CSV so you can count the group and the total yourself; one response per person stops repeat submissions from inflating either number; and conditional logic lets you ask a follow-up only of the people in the group you care about, without changing the base. Report the count and total alongside the interval ("240 of 400"), so readers can see the sample behind it.