Confidence Interval Calculator

A confidence interval puts a plausible range around a sample estimate. For a chosen confidence level C, the two-tailed tail area is α = 1 − C, split into α/2 in each tail. Every interval here has the same shape, point estimate ± (critical value × standard error), but the critical value and standard error change with the method, which the tool selects from your inputs.

Mean, population SD known (z-interval). When the population standard deviation σ is genuinely known, the standard error is SE = σ / √n and the critical value is the standard-normal quantile z = Φ⁻¹(1 − α/2) (1.96 at 95%). The interval is x̄ ± z·SE (NIST §1.3.5.2).

Mean, SD estimated (t-interval). When σ is estimated from the sample, the usual case, the standard error is SE = s / √n and the critical value is the Student's t quantile t = T⁻¹(1 − α/2, n − 1). This tool inverts the t distribution numerically through the regularized incomplete beta function, so any degrees of freedom resolve exactly rather than being read off a rounded table.

Proportion, Wilson score interval (default). With p̂ = x/n and z = Φ⁻¹(1 − α/2), the interval centres on (p̂ + z²/2n) / (1 + z²/n) with half-width [z/(1+z²/n)] · √(p̂(1−p̂)/n + z²/4n²), clamped to [0, 1] (Wilson 1927; NIST §8.4.1.3). Because it is asymmetric the tool shows both the Wilson centre and the sample p̂.

Proportion, Wald interval (optional). The simpler p̂ ± z·√(p̂(1−p̂)/n) is offered for comparison, with a warning: it under-covers and can leave [0, 1] when n is small or p̂ is near 0 or 1, which is exactly why Wilson is the default. As a built-in check, each critical value is pushed back through its own CDF to confirm it recovers the confidence level you asked for.

• NIST/SEMATECH e-Handbook §1.3.5.2: Confidence Limits for the Mean (z and t)
• NIST/SEMATECH e-Handbook §7.2.4: Confidence Interval on a Proportion
• E. B. Wilson (1927), Probable Inference… JASA 22(158):209–212: the Wilson score interval
• Abramowitz & Stegun, Handbook of Mathematical Functions §26.5, §26.7 (incomplete beta, Student's t)

The z, t and Wilson formulas on this page were last cross-checked against the NIST e-Handbook on 2026-06-11. Critical values are computed numerically (Acklam's inverse-normal approximation for Φ⁻¹; incomplete-beta inversion for the t quantile), so any confidence level and degrees of freedom resolve exactly to the displayed precision.