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Statistical Power: Means and Binary Rates

Compare explicit two-sided power models for one known-variance mean, two independent means or independent binary rates with continuous alpha.

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Inputs that matter
Sample size, Standardized planned difference, Two-sided significance level (%), Group A sample size, and 5 more
Output to expect
Modeled two-sided power, Approximate two-sided power
  • Check the units and required inputs before comparing results.
  • Keep the assumptions with a copied result so you can reproduce the calculation later.
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Reference & details

How it works

One mean: known-variance normal power

δ = d√n; power = 1−Φ(z(1−α/2)−δ) + Φ(−z(1−α/2)−δ) A two-sided one-mean z model with known variance. Both rejection tails and the continuous alpha value are included; the output is not artificially capped at 99%. This is not power for binary conversions, paired means, clustered experiments or multiple comparisons. Post-hoc power computed from the observed effect does not establish the reliability of a completed experiment.

Two independent means: normal-approximation power

δ = d/√(1/nA+1/nB); power = 1−Φ(z(1−α/2)−δ) + Φ(−z(1−α/2)−δ) Normal approximation for a two-mean comparison with a common SD. Both rejection tails and the continuous alpha value are included; the output is not artificially capped at 99%. This is not power for binary conversions, paired means, clustered experiments or multiple comparisons. Post-hoc power computed from the observed effect does not establish the reliability of a completed experiment.

Two independent binary rates: approximate power

p̄ = (nA pA+nB pB)/(nA+nB); c = z(1−α/2)√[p̄(1−p̄)(1/nA+1/nB)]; s = √[pA(1−pA)/nA+pB(1−pB)/nB]; power ≈ 1−Φ((c−Δ)/s)+Φ((−c−Δ)/s) Two-proportion normal planning approximation. Uses both rejection tails and explicit allocation; no continuity correction is applied. Expected successes and failures below 10 are flagged as a fragile normal approximation, not as a definitive validity boundary. No sequential monitoring, clustering, repeated observations, covariate adjustment or multiplicity is modeled.

Updated: September 2026

Example Scenarios

Compute two-sided z-test power for one mean with a known population standard deviation and a prespecified standardized alternative.

Plan a two-sided comparison of independent means using an explicit common standard deviation and standardized difference. This is a normal approximation, not a small-sample t test.

Plan a comparison of two independent binary rates using pooled null variance and unpooled alternative variance, with explicit sample counts in each group.

Common Mistakes to Avoid

Applying one mean: known-variance normal power outside its stated assumptions

Both rejection tails and the continuous alpha value are included; the output is not artificially capped at 99%. This is not power for binary conversions, paired means, clustered experiments or multiple comparisons. Post-hoc power computed from the observed effect does not establish the reliability of a completed experiment.

Applying two independent means: normal-approximation power outside its stated assumptions

Both rejection tails and the continuous alpha value are included; the output is not artificially capped at 99%. This is not power for binary conversions, paired means, clustered experiments or multiple comparisons. Post-hoc power computed from the observed effect does not establish the reliability of a completed experiment.

Applying two independent binary rates: approximate power outside its stated assumptions

Uses both rejection tails and explicit allocation; no continuity correction is applied. Expected successes and failures below 10 are flagged as a fragile normal approximation, not as a definitive validity boundary. No sequential monitoring, clustering, repeated observations, covariate adjustment or multiplicity is modeled.

FAQ

Choose a target from the decision costs and feasibility; the calculator does not impose a universal adequacy grade.

Under this model it returns the chosen Type I error probability, subject to numerical precision.

Enter both assumed rates directly; 5 and 6 represent a 1 percentage-point difference.

No. Report the effect and uncertainty from a suitable analysis; planned power belongs to a prespecified alternative.

About Statistical Power: Means and Binary Rates

Compare explicit two-sided power models for one known-variance mean, two independent means or independent binary rates with continuous alpha. Choose the mode that matches your measurements or study design, enter the stated units and keep the method and limits with the result.