Power Analysis
How many participants you need, or what power you already have, for five common tests.
to reach 80% power
2 groups
whole numbers overshoot the target slightly
Power against sample size
The dashed line is the conventional 80%. Notice how flat the curve becomes past it — chasing 95% costs far more participants than reaching 80% did.
Power is the chance of detecting an effect that is really there. Fix any three of power, effect size, α and sample size and the fourth follows. Comparing two groups on a medium effect at 80% power needs 64 participants per group.
The question a power calculation actually answers
A significance test controls one kind of mistake: calling an effect real when it is not. Power is about the other kind — missing an effect that is genuinely present. A study at 50% power is a coin flip: if the effect exists at the size you assumed, you have an even chance of coming back with nothing. That is why a non-significant result from an underpowered study says almost nothing, and why funders and ethics committees ask for the calculation before the data is collected rather than after.
Four quantities are locked together: the effect size you care about, the significance level α, the sample size, and the power. Fix any three and the fourth is determined. Designing a study means choosing the first two, picking a target for the fourth — 80% by long convention — and reading off the third.
Why the noncentral distribution matters
Under the null hypothesis a t statistic is centred on zero. Under the alternative it is not: it follows a noncentral t distribution, shifted by an amount that depends on both the effect size and the sample size. Power is the area of that shifted distribution beyond the critical value. Shorter treatments replace it with a normal approximation, which is fine at large n and noticeably wrong at the small samples people actually ask about — this page uses the noncentral distributions throughout.
for a t-test the noncentrality is δ = d·√(n/2) with two groups of n, or d·√n with one sample
- 1 Choose the effect size you would not want to miss. Not the one you hope for — the smallest that would matter. Cohen’s d of 0.5 is the conventional medium.
- 2 Set α and the direction. 0.05 two-tailed is the default. A one-tailed test needs fewer participants but commits you to a direction before seeing the data.
- 3 Pick a target power. 80% is conventional; 90% is increasingly expected for confirmatory work.
- 4 Read off the sample size. For two groups, d = 0.5, α = 0.05 two-tailed and 80% power: 64 per group, 128 in total.
- 5 Sanity-check it against what you can recruit. If the number is impossible, the honest responses are to target a larger effect, accept lower power explicitly, or not run the study — not to proceed and hope.
Participants per group, two independent groups
α = 0.05, two-tailed. Computed from the noncentral t distribution.
| Effect size d | 80% power | 90% power | 95% power |
|---|---|---|---|
| 0.2 — small | 394 | 527 | 651 |
| 0.5 — medium | 64 | 86 | 105 |
| 0.8 — large | 26 | 34 | 42 |
| 1.2 — very large | 12 | 16 | 20 |
Where power analyses go wrong
The commonest mistake is running one after the data is in, using the effect size the study happened to observe. This “observed power” carries no information: it is a deterministic function of the p-value, so a non-significant result always yields low observed power, and reporting it as though it explained the null result is circular. Power analysis belongs to the design stage, with an effect size chosen from previous work or from what would be practically meaningful.
The second is optimism about the effect. Published effect sizes are inflated by publication bias — significant results are likelier to appear — so planning around them systematically under-powers the replication. Taking a conservatively small effect costs participants but is the difference between a study that can answer its question and one that cannot.
The third is not noticing how expensive extra power is. On the same medium effect, going from 80% to 90% power costs 22 more participants per group; 95% costs 41 more; 99% costs 84 more. The curve on this page flattens visibly past 80%, which is the real reason that convention has stuck rather than any deep property of the number.