Binomial Sample Size Calculator

Formulas Used

Estimation (Wald): n₀ = z² · p(1−p) / E². If population size N is known, finite population correction (FPC) gives n = n₀ / (1 + (n₀ − 1)/N).

Estimation (Wilson score): Solve for the smallest n for which the Wilson half-width H(n) = z · √( p(1−p)/n · f + z²/(4n²) ) / (1 + z²/n) is ≤ E. Here f is the FPC factor f = (N − n)/(N − 1) if finite population is provided; otherwise f = 1.

Power (one-sample z test for proportion): For baseline p₀, alternative p₁, two-sided α, and power 1−β, n = [ z1−α/2·√(p₀(1−p₀)) + z1−β·√(p₁(1−p₁)) ]² / (p₁ − p₀)². For one-sided tests use z1−α. (Approximate; continuity and exact methods typically require iteration.)

How to Use This Calculator

1. Select the mode: Estimate for CI width or Power to detect a difference.
2. Enter inputs. For estimation, set confidence, margin of error, proportion, and optional population.
3. For power, set α, desired power, baseline p₀, and target p₁; choose one- or two-sided.
4. Click Calculate. Review results for normal (Wald) and Wilson methods and FPC variants.
5. Export your table via Download CSV or Download PDF.

Tip: Use p = 0.5 for the most conservative (largest) sample size when unsure.

Reference: Z-values for Common Confidence Levels

Reference: Sample Size vs. Margin (p=0.5, 95% CL)

Finite population correction example for E=0.03: N=1,000 → n≈517; N=5,000 → n≈880; N=10,000 → n≈965; N=50,000 → n≈1045.