What this answers

This calculator answers "given the sample size I have, how likely am I to detect a difference of this size between these two rates?" the inverse question to this site's proportion sample-size planners, useful both when planning with a fixed budget and when evaluating an already-completed study's sensitivity.

How it is calculated

For the one-proportion design, this calculator uses the same normal approximation as the One-Proportion z-Test, computing the probability the test statistic exceeds its critical value under the assumed true rate rather than the benchmark. For two proportions, it uses the pooled standard error for the alpha-driven critical boundary and each group's own rate for the power-driven term, mirroring the construction already used in this site's Sample Size for Two Proportions planner.

Worked example

Feeding this calculator the exact per-group sample size this site's own Sample Size for Two Proportions calculator recommends for a .5 versus .65 comparison returns an achieved power at or just above 80%, the target that sample size was originally solved for, confirming the two engines are internally consistent.

Assumption audit

Calculated from your data: the achieved power given your entered rates, sample size, and alpha.

Evidence to review: how sensitive the achieved power is to your assumed rates; small changes in either rate can shift power substantially, especially near the extremes of the 0 to 1 range.

You must verify: which design matches your study, since the one- and two-proportion formulas use different standard errors and are not interchangeable.

What this result does not mean

Achieved power describes the probability of detecting an effect across many hypothetical repeated studies with the same true rates, not a guarantee for any single study. A high computed power does not make an already-observed nonsignificant result significant after the fact.

Common mistakes

Plugging in the observed rates from a study that already ran, then treating the resulting power as if it were known in advance, is a common and misleading form of post-hoc power analysis; power calculated this way tends to track the p-value mechanically and adds little genuine information beyond it. Power is most useful calculated before data collection, using externally justified assumed rates, not calculated backward from results you already have.

Limitations

This calculator uses a normal approximation throughout, standard practice for proportion designs at typical sample sizes, and assumes equal allocation for the two-proportion design.