M3 · Study design

Sample size & power

Four quantities locked in one equation — sample size, effect, significance and power. Fix any three and read the fourth, and see the trade-off move.

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Sample size per group

Total sample size

Enter the three known quantities to solve for the fourth.

Power vs sample size

00%25%50%75%100%n per group →

Power rises steeply, then plateaus — buying the last few percent costs many more participants.

Required n vs effect size

Effect size →

Small effects are expensive — halving the effect roughly quadruples the sample size.

How it's computed

z(α) = Φ⁻¹(1 − α/2) = Φ⁻¹(1 − /2) =
z(β) = Φ⁻¹(power) = Φ⁻¹() =
d = δ/σ = / =
n/group = (1+1/k)·(z(α)+z(β))² / d² = (1+1/)·(+)² / ² =
Total = + =