Power: the probability your test would notice
Before collecting data, one question decides whether the study is worth running: if the effect is really there, at the size you care about, how likely is this design to detect it? That probability is the test's power. Its complement, β, is the risk of the cruellest outcome in statistics: a real effect, studied carefully, and missed.
A one-sample test hunting a medium effect (d = 0.5) at α = 0.05, two-tailed, with n = 32: power ≈ 0.81, just over the conventional 0.80 planning line. Halve the sample to 16 and power slumps to about 0.57: a coin-flip study wearing a lab coat. Run the calculation backwards and the same page sizes the study: d = 0.5 at 80% power demands n = 32, not 31, because sample sizes round up or the promise breaks.
The four-way tug of war
- Effect size: bigger effects are easier to see; halving d quadruples the required n.
- Sample size: the lever you actually control, at square-root exchange rates.
- α: stricter significance demands more data for the same power.
- Power target: 0.80 is convention, not physics; expensive misses argue for 0.90.
This calculator uses the normal approximation to the noncentral t, the planner's standard arithmetic, good to about a percentage point. Exact engines (G*Power, R's pwr) refine the second decimal.
"Observed power" computed from your finished study's effect size. It is a restatement of the p-value wearing a costume and adds no information. Power is a design tool: use it before the data exist.
Frequently asked questions
Why 0.80 power as the standard?
Convention, traceable to Cohen: it prices a missed effect at four times the cost of a false alarm (β = 0.20 against α = 0.05). Your study may rationally choose differently, but choose before collecting data.
What effect size should I plan for?
The smallest effect that would change a decision: not the effect you hope for, and not the one a small pilot exaggerated.