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Free statistical power calculator using the standard normal approximation. Set α, sample size, and Cohen’s d for one-sample or two-group (equal n) designs; get power, β, noncentrality λ, and the Z rejection rule.
Last updated: April 13, 2026
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Assumes known σ or large enough n that the test statistic is approximately standard normal under H₀. For small samples, t-based power can differ; pair with our t-test calculator for planning context.
Power (1 − β)
42.39%
β (Type II)
57.61%
λ (noncentrality)
1.7678
Rejection rule (standardized Z)
Reject H₀ if Z > 1.96 or Z < -1.96
Interpretation
Power rises with |d| and n. For d = 0, power equals α under this Z model for both one- and two-tailed tests. One-sided power depends on the sign of d relative to the tail you selected.
Uses Φ and Φ⁻¹ with a shifted normal under H₁. Best for intuition, large n, or σ treated as known.
d encodes how many σ apart means are. Combine with our effect size calculator when you start from raw summaries.
Reports power = 1 − β so you can compare false-positive control (α) with detection strength for the chosen d.
n is per group. The noncentrality uses √(n/2) paired with Cohen’s d for independent samples with common σ.
See the standardized cutoffs that define “reject H₀” for your α and tail choice—useful next to z-tables and slides.
ANOVA, regression, unequal n, and exact t power need richer tools. This page stays focused on a clear Gaussian core.
Two groups, n = 25 each, two-tailed α = 0.05, Cohen’s d = 0.5
Power ≈ 42.39%
Try raising n or |d| to see power move toward 100%.
Critical values come from Φ⁻¹ applied to (1 − α/2) or (1 − α) or α depending on tails. Power integrates the alternative normal centered at λ against those cutoffs using Φ. Details align with introductory mathematical statistics treatments; see more tools on our Math & Science calculators hub.
One sample: λ = d√nTwo groups (n per group): λ = d√(n/2)Pair with effect size and t-test calculators.
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