Confidence Interval Calculator

Methodology. Critical values use the inverse normal (Acklam, 2003) for z* and the inverse t-distribution (computed via the regularized incomplete beta function) for t*. The proportion interval uses the standard Wald form p̂ ± z* · √(p̂(1−p̂)/n); for very small n or extreme p̂, prefer the Wilson or Agresti–Coull interval. More on methods ↓

One-sample mean (σ unknown). Uses x̄ ± t* · (s/√n) with df = n − 1. Assumes the data are approximately normal or n is large enough for the central limit theorem to apply (a common rule of thumb is n ≥ 30, though smaller n is acceptable when the population is roughly symmetric and free of strong outliers).

One-sample mean (σ known). Uses x̄ ± z* · (σ/√n). The σ-known case is rare in practice — it appears in introductory courses and in problems where σ is fixed by design (instrument calibration, simulated populations).

One-sample proportion. Uses p̂ ± z* · √(p̂(1−p̂)/n). The Wald interval requires np̂ ≥ 10 and n(1−p̂) ≥ 10 to be reliable. When p̂ is close to 0 or 1, or when n is small, the Wilson score and Agresti–Coull intervals are preferred (Agresti & Coull, 1998; Brown, Cai, & DasGupta, 2001).