Why is 0.8 the conventional target for power?

Cohen proposed 0.80 as a reasonable default, reasoning that a Type II error (missing a real effect) is roughly a quarter as costly as a Type I error (a false positive) at α = 0.05. Like the 0.05 threshold itself, it is a convention rather than a derived constant.

How accurate is the normal approximation?

For t-tests it typically lands within one or two participants of the exact answer from software using the non-central t distribution, such as G*Power. The gap is widest at very small sample sizes, where you should treat this tool's answer as a lower bound and round up.

Can I compute power after running my study?

You can, but observed power calculated from your own observed effect size adds nothing beyond the p-value and is widely criticised. Power analysis is most useful before data collection, using an effect size from prior work or the smallest effect you would care about.

Statistical Power & Sample Size Calculator

⚠️ This calculator uses the normal-approximation method (the standard Cohen 1988 approach), reusing the same Z-score math as the rest of the site, rather than the exact non-central t/F distribution. It's accurate enough for most purposes, but required-sample-size results may differ by 1–2 from dedicated software (e.g. G*Power) that uses the exact non-central distributions.

Test type Significance level α Tail

Effect size (Cohen's d)

Sample size (n)

Target power (0–1, commonly 0.8)

Common uses

Estimating how many samples a study needs before running it
Checking whether an existing sample size is enough to detect an expected effect
Writing the methods-section power analysis for homework or a paper

What power analysis answers

Statistical power is the probability that your study will detect an effect, given that the effect is real and of a particular size. It ties together four quantities, and fixing any three determines the fourth:

Effect size — how big the difference you want to detect is.
Significance level α — your false-positive tolerance, conventionally 0.05.
Sample size n — how many observations you collect.
Power (1 − β) — your chance of finding the effect if it is there, conventionally targeted at 0.80.

The two useful directions are: before a study, fix effect size, α and power to get the sample size you need; and while planning within constraints, fix the sample size you can afford to see what effect size you could realistically detect. This tool does both.

n ≈ 2(z_1−α/2 + z_1−β)² ÷ d² (per group, two-sample test)

Worked example

You want to detect a medium effect (d = 0.5) in a two-group comparison, at α = 0.05 two-tailed, with 80% power.

z_0.975 = 1.96 and z_0.80 = 0.84, so n ≈ 2 × (1.96 + 0.84)² ÷ 0.25 = 2 × 7.84 ÷ 0.25 = 63 per group, or 126 in total.

Now halve the effect size to d = 0.25: n ≈ 2 × 7.84 ÷ 0.0625 = 251 per group. Halving the effect quadruples the sample required — sample size scales with 1/d². This inverse-square relationship is the single most useful thing to internalise about study design.

Why underpowered studies are worse than they look

The obvious cost of low power is missing real effects. The less obvious cost is more damaging: among the results an underpowered study does find significant, a large share will be false positives, and the true positives will have inflated effect sizes.

The mechanism is straightforward. If your study can only reach significance when the observed effect is large, then every significant result you get is, by construction, a large observed effect — whether or not the underlying effect is large. This is the "winner's curse" in the replication literature, and it explains a good deal of why headline findings from small studies so often shrink or vanish on replication.

The practical consequence: a small pilot study is fine for checking that your procedure works, and close to useless for estimating an effect size to power the real study with. Take your planning effect size from meta-analysis, from prior published work, or from the smallest effect that would be worth acting on.

Common mistakes

Post-hoc power computed from your own observed effect. "Observed power" is a deterministic function of the p-value — it adds no information and is roundly criticised in the methodological literature. If a result was non-significant, the useful question is what effect size your design could have detected, not what power you had for the effect you happened to see.
Powering for the effect you hope for. Choosing d = 0.8 because it makes n comfortably small guarantees an underpowered study for any realistic effect.
Forgetting attrition. The n here is analysable observations. If you expect 15% dropout, recruit accordingly.
Ignoring multiple comparisons. Testing several outcomes means correcting α, which raises the sample size needed for each.
Reading n as a total. For the two-sample test the figure is per group. Double it for the study total.

The method used here, and its limits

This tool uses the normal-approximation method — the standard textbook approach set out in Cohen (1988). It substitutes the normal distribution for the non-central t and F distributions that describe the exact sampling behaviour under the alternative hypothesis.

That approximation is good but not exact. Compared against software using the exact non-central distributions, such as G*Power, results here typically differ by one or two participants per group. The discrepancy is largest at small sample sizes and low power, where the normal approximation is least appropriate. Two practical implications:

Always round up, never down.
For a grant application, an ethics submission, or a pre-registration where the number will be scrutinised, verify the figure in G*Power or an equivalent tool before submitting. This tool is built for quick planning estimates.

Power for the correlation test uses Fisher's z transformation, which is well-behaved except at very small n or r very close to ±1.

Sources

Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
Hoenig, J. M. & Heisey, D. M. (2001). The abuse of power: the pervasive fallacy of power calculations for data analysis. The American Statistician, 55(1), 19–24.
Button, K. S. et al. (2013). Power failure: why small sample size undermines the reliability of neuroscience. Nature Reviews Neuroscience, 14, 365–376.
Faul, F., Erdfelder, E., Lang, A.-G. & Buchner, A. (2007). G*Power 3. Behavior Research Methods, 39, 175–191.