Confidence Interval Calculator
A confidence interval for a mean (t or z, from pasted data or summary figures) or a proportion (Wald and Wilson), with critical value and standard error shown.
Lower and upper bounds at your confidence level, the critical t or z used, the standard error, and a warning when the sample is too small for the method to be trusted.
Example: Eight readings 5.1, 4.9, 5.6, 5.8, 6.0, 5.2, 4.8, 5.5 have x̄ = 5.3625 and s = 0.4307; with t(7) = 2.365 the 95 % interval is 5.002 to 5.723. For 52 of 200, Wilson gives 0.204 to 0.325.
A range,
and how sure.
Which formula is used when, where the critical values come from, and what a confidence interval does and does not say.
Mean
The interval is x̄ ± (critical value) × s/√n. When the population standard deviation is unknown — the normal case, and always when you paste data — the critical value is Student's t with n − 1 degrees of freedom, computed here from the t distribution itself (checked against published tables: t(10) at 97.5 % = 2.2281). Tick “σ known” only when the standard deviation is a known population figure; the normal z is used then. Pasted values are split on spaces, commas, semicolons and line breaks; anything that is not a number is ignored and listed.
Proportion
Two intervals are reported. The Wald interval p̂ ± z √(p̂(1 − p̂)/n) is the one in most textbooks, but it is poor for small samples or proportions near 0 or 1 — it can even run past those limits. The Wilson score interval solves the same equation properly and is the one Brown, Cai and DasGupta recommend for general use; it is the figure to quote. The page flags the case of fewer than ten successes or failures.
What it means, and does not
A 95 % interval is a procedure that captures the true value in 95 % of repeated samples; it is not a 95 % probability that this particular interval contains it. The t interval assumes a roughly normal population when n is small and a simple random sample always. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.1 — confidence limits for the mean; §7.2.4 — proportions
- Brown, L. D., Cai, T. T., DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science 16(2), 101–133 — the Wilson interval recommendation
Last reviewed 19 September 2026. How results are checked: How we verify.