Statistical Power Analysis Calculator

Formula

How this is calculated

Statistical power is the probability that a hypothesis test correctly detects a true effect — i.e., it avoids a Type II error (false negative). Four linked quantities determine the result: effect size (Cohen's d, the expected group difference in standard-deviation units), sample size n per group, significance level α (Type I error rate, typically 0.05), and power (1 − β, typically 0.80 or 0.90).

This calculator uses the two-sample z-approximation, the standard planning formula for continuous outcomes when σ is estimated from pilot data or prior literature. Required n per group is n = ⌈2 × ((z_α/2 + z_β) / d)²⌉, where z_α/2 is the two-tailed critical value for α (e.g., 1.960 for α = 0.05) and z_β = Φ⁻¹(power) (e.g., 0.842 for 80% power). With d = 0.5 and α = 0.05 at 80% power this gives about 64 participants per group. The power mode reverses the formula: given n, power = Φ(d√(n/2) − z_α/2), where Φ is the standard normal CDF. The power curve shows how power grows with n for your chosen effect size and α.

Assumptions and limitations: the z-approximation is accurate for moderate-to-large samples; very small samples benefit from a non-central t distribution approach (slightly different answers). The formula assumes equal group sizes, a two-tailed test, and that Cohen's d is known exactly — in practice d is estimated from prior data, introducing uncertainty. Post-hoc power calculations (computing power after an experiment using the observed effect) are generally discouraged in statistics as they are circular and uninformative.