How this instrument works
Statistical power is the probability that a hypothesis test correctly detects a real effect when one truly exists — formally, power = 1 − beta, where beta is the probability of a Type II error, a false negative. This calculator estimates one-tailed, one-sample z-based power: given the size of the effect you're trying to detect (delta), the population standard deviation (sigma), your planned sample size (n), and your chosen significance level (alpha), it returns the probability your test would actually flag a significant result if that effect genuinely exists.
Power depends on all four inputs working together. A bigger effect (larger delta) is easier to detect and raises power. A larger sample size (n) also raises power, since it shrinks the standard error and makes real effects easier to distinguish from noise. A smaller, stricter significance level (alpha) does the opposite — tightening the threshold for declaring significance also makes it harder to detect a real effect, lowering power for the same effect size and sample size.
Power analysis is most useful before collecting data, as a way to decide how large a sample is actually needed for a reasonable chance of detecting an effect you care about — an underpowered study risks missing a real effect and wrongly concluding there's nothing there. Common convention treats 80% power as a widely used minimum target, though the right target depends on the cost of missing a real effect versus the cost of collecting more data.
- Enter the effect size to detect into delta — the difference from the null hypothesis considered practically meaningful.
- Enter the population standard deviation into sigma.
- Enter the planned sample size into n.
- Choose the one-tailed significance level (alpha) from the dropdown.
- Read Power — the estimated probability of correctly detecting an effect of this size at this sample size and significance level.
Worked example — a small effect at n = 25
You're testing for an effect of delta = 5, against a population standard deviation sigma = 15, with a planned sample size of n = 25, at a one-tailed alpha of 0.05 (z = 1.6449). Enter those four values.
zBeta = (5 × sqrt(25) / 15) − 1.6449 = (25/15) − 1.6449 = 1.666667 − 1.6449 = 0.021767. Power = CDF(0.021767) ≈ 0.5087 — barely better than a coin flip. This small sample size gives roughly a 51% chance of actually detecting an effect of this size, meaning nearly half the time a real effect this small would go undetected purely due to insufficient sample size.
Doubling the effect size to delta = 10, keeping sigma, n and alpha the same, changes zBeta to 1.688433, and power jumps to about 95.4% — illustrating how much easier a larger effect is to detect at a fixed sample size, compared to the barely-better-than-chance result for the smaller effect above.
Questions
What happens to power when there's no real effect at all (delta = 0)?
When delta = 0, zBeta collapses to exactly -z(alpha), so power = CDF(-z(alpha)) = alpha exactly. When there's genuinely nothing to detect, the test still 'succeeds,' rejects the null hypothesis, at precisely its own false-positive rate — for a 0.05 alpha, that means about 5% power, exactly what you'd expect: a test can't do better than its own Type I error rate when there's no real effect to find.
How does sample size affect power?
Larger sample sizes increase power, because a bigger n makes the standard error, sigma/sqrt(n), smaller, which makes a real effect of a given size easier to distinguish from random noise. This is the main lever researchers pull during study design: if a power calculation shows insufficient power for a planned sample size, increasing n is usually the most direct way to raise it, assuming delta, sigma, and alpha stay fixed.
Why does choosing a stricter alpha lower power?
A stricter, smaller, alpha means stronger evidence is needed before declaring a result significant, which reduces the Type I error rate, false positives, but simultaneously raises the Type II error rate, false negatives, for the same data — that trade-off is exactly what lowers power. There's no way to reduce both error types simultaneously without changing something else, like collecting a larger sample.
What's a typical target for statistical power?
80% power is the most commonly cited minimum target across many fields, meaning the test has an 80% chance of detecting the effect size specified if it truly exists, a 20% chance of missing it. Some fields, especially where a missed effect is especially costly, target 90% or higher. There's no single universal standard — the right target balances the cost of an underpowered study against the cost of collecting a larger sample.