Statistical Power Calculator

This calculator uses the normal (large-sample) approximation for a two-group test with equal group sizes — the same z-test model used by G*Power's z-test family and by most online power calculators. It is quick, transparent, and accurate to the third decimal of power. Write Φ for the standard-normal cumulative distribution function and Φ⁻¹ for its inverse. The steps are:

1. Fix the effect size (ES).For two means, ES is Cohen's d— the difference in means divided by the pooled standard deviation. For two proportions, ES is the absolute value of Cohen's h = 2·arcsin(√p₁) − 2·arcsin(√p₂), the arcsine transform that stabilises the variance so a given h means the same detectability across the 0–1 range.
2. Scale it by sample size. The noncentrality parameter for n per group is δ = ES · √(n / 2). This is ES divided by the standard error of the difference when the two groups are equal.
3. Compare against the critical value. The two-tailed critical value is z₁₋α/₂ (e.g. 1.959964 for α = 0.05); one-tailed uses z₁₋α. Power is then Φ(δ − z) + Φ(−δ − z) for a two-tailed test — the second term covers detecting the effect in the wrong direction and is negligible unless power is very low — or Φ(δ − z) for one-tailed.
4. Invert for planning. The sample size needed for a target power (1 − β) is n = 2·(z + z₁₋β)² / ES², rounded up. The minimum detectable effect at your current n is MDE = (z + z₁₋β)·√(2 / n).

Φ is evaluated with the Abramowitz & Stegun 7.1.26 error-function approximation (absolute error below 1.5×10⁻⁷), and the critical z-constants are cross-checked against the inverse-normal CDF (Acklam 2003) so the values are verified two independent ways. The formulas and the small-/medium-/large-effect conventions (d and h = 0.2 / 0.5 / 0.8) come from Cohen (1988) and the "Power Primer" (Cohen 1992); the required-n figures reconcile with that paper's tables to within the one-participant noncentral-t correction described in the FAQ.

• Cohen, J. (1992). A Power Primer. Psychological Bulletin, 112(1), 155–159 — the n-per-group tables used as the cross-check.
• Faul, Erdfelder, Lang & Buchner (2007). G*Power 3. Behavior Research Methods, 39, 175–191 — the reference implementation this tool matches.
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.) — power formulas and effect-size conventions.
• Abramowitz & Stegun (1964), eq. 7.1.26 — the erf approximation used for the normal CDF.

Formulas and constants were last cross-checked against these sources on July 13, 2026. The two worked examples reconcile to power 0.807 and 0.813 and to Cohen's 64-per-group table entry within the documented one-participant correction.