How this calculator works
Under an alternative mean, the z statistic is normally distributed with unit variance and a non-zero mean shift. Integrate that shifted normal distribution over the rejection region to obtain power. The standardised effect divides the mean difference by the known population SD; a paired design uses the SD of differences. Two independent equal groups use shift d√(n/2). Sample and effect modes invert this normal power calculation. For raw means, planning sample size is (zalpha + zpower)²(s1² + s2²/r)/difference², with n2/n1 = r. Binary planning uses separate null-pooled and alternative variance terms. Relative modes derive the alternative from a baseline. These large-sample planning formulas require power at least 50%, round both groups upward and do not include a continuity correction or t-distribution correction.
Formula
Shift δ = d√n for one sample, d√(n/2) for two equal groups. Right-tail power = 1 - Φ(c - δ). Two-sided power = 1 - Φ(c - |δ|) + Φ(-c - |δ|).
Worked example
For a one-sided right test at alpha 0.05, n = 1 and effect equal to z0.95 ≈ 1.644854, power is 50%. At effect zero, either a one- or two-sided test has power equal to alpha.
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Frequently asked questions
Is n per group?
In standardised mode, n is per group and allocation is equal. Raw endpoint modes show n1, n2 and their total, using the requested enrollment ratio.
Can I use this for an estimated-variance t test?
It is a normal planning approximation. Exact t-test power uses a noncentral t distribution and can need more observations.
What effect sign should I enter?
Power mode uses the signed effect for one-sided alternatives. Sample-size mode requires an effect in the selected direction; effect mode reports a positive magnitude.