What the Statistical Power Calculator does
This power analysis calculator solves for any one of sample size, statistical power or the smallest detectable effect, given the other two, for t-tests, two proportions, correlations and one-way ANOVA. It uses the noncentral t and F distributions for the t-tests and ANOVA, shows a power curve, and states the method, so the numbers can go into a grant proposal, ethics application or pre-registration.
Power is the probability that your study detects an effect of the size you specify, if that effect is real. Too little power wastes participants on a study that will probably come back inconclusive; far too much spends money detecting differences nobody cares about.
How to use it
- Pick the test your analysis will actually use: a one-sample or paired t-test, a t-test between two independent groups, two proportions, a correlation or a one-way ANOVA.
- Choose alpha (usually 0.05) and, for t-tests, proportions and correlations, whether the test is one- or two-tailed. Use two-tailed unless you pre-register a direction and would ignore an effect the other way.
- Choose what to solve for. Most planning solves for sample size; solve for power to check a fixed sample, or for effect size to say what a study of that size can realistically detect.
- Enter the effect size. Base it on the smallest effect that would matter in practice or on earlier studies; the small, medium and large buttons fill in Cohen's conventions as a last resort.
- Read the result, check the power curve, and add an allowance for drop-out before you recruit.
Reading the results
For two-group designs the sample is per group; the total is shown alongside. For ANOVA it is per group times the number of groups.
Achieved power is slightly above your target because people come in whole numbers. The power curve shows how quickly power falls if recruitment comes up short.
Solving for effect size answers "what could this study detect?" - useful for reporting an existing sample honestly rather than calculating post-hoc power from an observed effect, which adds no information beyond the p-value.
Worked example: a two-group trial with a medium effect
A researcher plans to compare a new revision method with the usual one on exam scores, using an independent-samples t-test. Earlier work suggests a standardised difference of about d = 0.5. She wants 80% power at alpha 0.05, two-tailed.
The calculator returns 64 students per group, 128 in total, with achieved power of 80.15% - the same figure G*Power reports for this design. With 63 per group power would fall just under 80%.
She expects 15% of students to miss the final exam, so she recruits 128 / 0.85 = 150.6, rounded up to 151 - in practice 76 per group. Had she assumed a large effect (d = 0.8) she would have planned for only 26 per group, and a true medium effect would then have been detected barely half the time.
Formulas and scoring rules
- Two independent groups (t)
df = n1 + n2 - 2; delta = d x sqrt(n1 n2 / (n1 + n2)); power = P(|T'(df, delta)| > t crit)T' is the noncentral t distribution, integrated numerically. One-sample and paired designs use df = n - 1 and delta = d sqrt(n).- Two proportions
h = 2 asin(sqrt p1) - 2 asin(sqrt p2); power = Phi(|h| sqrt(n1 n2 / (n1 + n2)) - z crit)Cohen's arcsine approximation.- Correlation
power = Phi(|atanh r| x sqrt(n - 3) - z crit)Fisher's z approximation; exact methods can differ by one or two participants.- One-way ANOVA
lambda = f^2 x N; power = 1 - F'(F crit; k - 1, N - k, lambda)F' is the noncentral F distribution, summed as a Poisson mixture of incomplete beta functions.- Rounding
n = smallest whole number with power >= targetPowers are shown to two decimals of a percentage point.
Choosing an effect size you can defend
Cohen's small, medium and large labels were offered as a fallback when nothing better is known, and reviewers increasingly ask for more. Better sources are the smallest effect that would change a decision (a minimal clinically or practically important difference), or a meta-analysis of similar studies. Published single studies tend to overstate effects, so shrinking a reported effect before planning is a sensible habit.
Remember that a small change in effect size moves the sample a lot: sample size scales with one over the effect squared, so halving d roughly quadruples the participants.
Limitations: what the result does not prove
- The calculations assume the test's own conditions: independent observations, roughly normal data for t-tests and ANOVA, and equal variances for the pooled designs. Clustered, repeated-measures or multilevel designs need specialised methods.
- The two-proportion and correlation results are normal approximations. For very small samples or proportions near 0 or 100%, use an exact method.
- Power only concerns random error. It cannot fix a biased design, poor measurement or an effect size chosen to make the sample look affordable.
- Several outcomes or comparisons need an adjusted alpha; lower alpha here to match your correction.
Privacy: where your data goes
Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.
Standards and sources
- NIST/SEMATECH e-Handbook - sample sizes and power - checked 19 Sep 2026
- Faul et al. (2007) - G*Power 3, Behavior Research Methods
Frequently asked questions
Why is 80% power the usual target?
It is a convention from Cohen, balancing a 1 in 5 chance of missing a real effect against the cost of more participants. Many funders now prefer 90%, especially for confirmatory trials, because a missed effect wastes the whole study.
What is the difference between power and significance?
Alpha is the chance of a false positive when there is no effect. Power is the chance of a true positive when the effect is as large as you assumed. Planning fixes both before you collect data.
Should I calculate power after my study using the observed effect?
No. Observed or post-hoc power is a direct function of the p-value and adds nothing. Instead, report confidence intervals, or use this tool to state the smallest effect your sample could reliably detect.
How do I convert eta squared into Cohen's f for ANOVA?
Use f = sqrt(eta squared / (1 - eta squared)). An eta squared of 0.06 gives f of about 0.25, Cohen's medium effect. Partial eta squared from a published factorial design is not directly comparable to a one-way design.
Does unequal group size reduce power?
For a fixed total, yes: a 1:1 split is most efficient for two groups with equal variances. The ratio field shows the cost; a 2:1 split needs about 12% more people in total for the same power.
Why might my result differ slightly from other software?
Tools differ in whether they use exact noncentral distributions or normal approximations and in how they round. For t-tests and ANOVA this calculator matches the published G*Power results; the approximations for proportions and correlations can differ by a participant or two.
Last reviewed by the A2Z.Tools team against the sources listed above.
