Confidence Interval Calculator

Type

Pick “Mean” for numeric data (x̄). Pick “Proportion” for yes/no outcomes (p̂).

Confidence Level (%)

Common choices: 90, 95, 99.

Sample mean (x̄)

Sample size (n)

Population σ (optional)

If provided and > 0, uses Z interval. Otherwise t interval with sample s.

Sample s (used when σ unknown)

Confidence Intervals – Quick Cheat Sheet

Mean, σ known (Z): SE = σ/√n, z* = Φ⁻¹(1−α/2), CI = x̄ ± z*·SE.
Mean, σ unknown (t): SE = s/√n, t* ≈ Cornish–Fisher(df=n−1), CI = x̄ ± t*·SE.
Proportion (Z): p̂ = x/n, SE = √[p̂(1−p̂)/n], z* = Φ⁻¹(1−α/2), CI = p̂ ± z*·SE.
Confidence level: 90% → α=0.10; 95% → α=0.05; 99% → α=0.01.
Notes: For small n or extreme p̂, Wald CI may underperform; Wilson/Agresti–Coull often better.

Edge cases & tips

Use Z only when population σ is known (rare). Otherwise use t with sample s.
Ensure n is adequate: for proportions, check n·p̂ ≥ 10 and n·(1−p̂) ≥ 10 for Wald CI.
Level can be any 50–99.999%; common choices: 90%, 95%, 99%.