Plan a study before you collect data. Choose a t-test, a proportion test or a correlation, enter the effect size you expect, α and either the sample size (to get power) or the target power (to get the sample size you need). A power curve shows how quickly power grows with n.
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Test
Calculate
Effect size
Difference in means ÷ SD. 0.2 small, 0.5 medium, 0.8 large
Group 1 (or true) proportion
Group 2 (control) proportion
0.1 small, 0.3 medium, 0.5 large
Design
1 = equal groups
%
Enter your values to see the result.
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How to use the Statistical Power Calculator
Pick the test you plan to run: two-sample t, one-sample/paired t, two proportions, one proportion or correlation.
Choose what to calculate: the required sample size, or the power for a sample size you already have.
Enter the effect size: Cohen's d, the expected proportions or the expected correlation.
Set α, the alternative and the target power (80% is the usual minimum) and press Calculate.
Read the result and the power curve, and use the tables to see how the answer changes with power or effect size.
Power formulas used
All tests use the critical value for your α (split over both tails for a two-sided test) and then ask how often the test statistic lands beyond it when the true effect is the one you entered.
Test
Effect size
Power calculation
Two-sample t-test
d = (μ₁ − μ₂) / σ
Noncentral t with df = n₁ + n₂ − 2, δ = d / √(1/n₁ + 1/n₂)
One-sample or paired t-test
d = (μ − μ₀) / σ (paired: mean difference / SD of differences)
For a required sample size the calculator searches for the smallest whole n whose power reaches your target, so the actual power is always at or slightly above it.
Sample sizes at a glance (two-sided, α = 0.05)
Design
Effect
80% power
90% power
Two-sample t-test (per group)
d = 0.2
394
527
Two-sample t-test (per group)
d = 0.5
64
86
Two-sample t-test (per group)
d = 0.8
26
34
Paired / one-sample t-test
d = 0.5
34
44
Correlation
ρ = 0.3
85
113
Tips for a sound power analysis
Do it before collecting data. Power computed from the observed effect after a study ("observed power") adds nothing beyond the p-value.
Plan for drop-out. If you expect 15% of participants to leave, divide the required n by 0.85.
Be conservative with effect sizes. Published effects tend to be overestimated; halving a pilot estimate roughly quadruples the sample you need, so check the effect-size table.
Check the assumptions. The t-test results assume roughly normal data with equal variances; the proportion formulas rely on the normal approximation, which is reliable once each group expects at least 10 successes and 10 failures.
Frequently asked questions
What is statistical power?
Power is the probability that a test rejects the null hypothesis when a real effect of the size you specified exists. It equals 1 − β, where β is the Type II error rate (missing a real effect). Most studies aim for at least 80% power, and many grant reviewers expect 90%.
What four quantities are linked in a power analysis?
Significance level α, effect size, sample size and power. If you fix any three, the fourth is determined. A priori power analysis fixes α, effect size and power to find the sample size; post hoc or sensitivity analysis fixes n to find power or the detectable effect.
How many participants do I need for a medium effect?
For a two-sided two-sample t-test with α = 0.05 and Cohen's d = 0.5, you need 64 participants per group (128 in total) for 80% power and 86 per group for 90% power. For a small effect (d = 0.2) the requirement jumps to 394 per group for 80% power.
Which effect size should I enter?
Use the smallest effect that would matter in practice, or an estimate from pilot data or published studies. Cohen's conventions are d = 0.2, 0.5 and 0.8 for small, medium and large mean differences, and ρ = 0.1, 0.3 and 0.5 for correlations. Avoid using an inflated effect from a single small study.
How is the power of a t-test calculated?
The calculator uses the noncentral t distribution. With df = n − 1 (one sample) or n₁ + n₂ − 2 (two samples) and noncentrality δ = d·√n or δ = d / √(1/n₁ + 1/n₂), power is the probability that a noncentral t variable exceeds the critical t value. This matches G*Power and similar software.
Does a one-sided test need fewer participants?
Yes, because all of α sits in one tail, so the critical value is smaller. Only use a one-sided test if an effect in the other direction would genuinely be treated the same as no effect, and decide this before seeing the data.
Why do unequal groups need more participants?
For a fixed total, the standard error of a difference is smallest when the groups are equal. With a 2:1 ratio you need roughly 12% more participants in total to keep the same power, and the penalty grows quickly for more lopsided designs.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.