Free tool · Data & statistics
Effect size and power calculator
Sample size or achieved power for t-tests, ANOVA and correlations, with the effect size explained.
This calculator tells you how many participants you need to detect an effect of a given size with a two-group t-test, a one-way ANOVA or a correlation. Enter the effect size, the significance level and the power you want, and it gives the minimum sample, matching G*Power for the t-test and ANOVA. If your sample is already fixed, it works the other way and tells you the power that sample gives.
A power analysis is what examiners expect when a thesis tests hypotheses rather than describing a population. It also answers the DC member who asks why you stopped at 120 respondents. If you don’t yet have an effect size, the helper under the tool works one out from the means and standard deviations, or the eta squared, reported in an earlier study.
Work out the effect size
Method
How this tool works
Four numbers are tied together in a power analysis: the significance level, the power, the effect size and the sample size. Fix any three and the fourth follows. By default this tool fixes the first three and finds the sample. Switch it to “The power of the sample I have” and it takes the sample size instead and finds the power, which is useful when the number of respondents is set by access, such as the 90 nurses in one hospital. Enter the effect size you expected when planning, not the one you found in your own data.
The significance level (α) is the risk you accept of finding an effect that isn’t there, usually 5%. Power is the chance of detecting a real effect of the size you assumed, usually 80%, which leaves a 20% risk of missing it. The effect size is how big the difference or relationship is, in standard units: Cohen’s d for two means (the difference divided by the pooled standard deviation), Cohen’s f for three or more means, and the correlation r itself.
Cohen (1992) proposed conventional values for small, medium and large effects. The table shows them, with the sample each needs here at α = .05 and power .80.
| Design and effect size | Small | Medium | Large |
|---|---|---|---|
| Two groups, Cohen’s d | 0.20 (394 per group) | 0.50 (64 per group) | 0.80 (26 per group) |
| Three groups, Cohen’s f | 0.10 (969 in total) | 0.25 (159 in total) | 0.40 (66 in total) |
| Correlation, r | 0.10 (783) | 0.30 (85) | 0.50 (30) |
Look at how fast the numbers grow as the effect shrinks. That is why the effect size is the input examiners question most. Use one from earlier studies close to yours, or the smallest effect that would matter in practice. Fall back on Cohen’s conventions only if nothing better exists, and say that you did. The sample size guide covers where each kind of effect size comes from.
For the t-test and ANOVA the tool uses the exact noncentral F distribution, the same method G*Power 3.1 uses. A two-group t-test is solved as a two-group ANOVA with f = d ÷ 2, which gives identical answers. At d = 0.5, α = .05 two-tailed and power .80, it gives 64 per group, and at f = 0.25 with three groups it gives 159 in total (53 per group), both as G*Power reports them. Totals are rounded up so every group is the same size.
For correlations it uses Fisher’s z approximation: N = ((z₁₋α/₂ + z₁₋β) ÷ atanh r)² + 3, rounded up. For r = 0.3 that gives 85. G*Power’s exact method gives 84, so for correlations this tool can come out one higher. We chose the approximation because you can check it by hand, and the extra participant errs on the safe side. In power mode, the power for a correlation comes from the same approximation.
The helpers use two standard conversions. Cohen’s d from two groups is the difference in means divided by the pooled standard deviation, √(((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)). Cohen’s f from eta squared is √(η² ÷ (1 − η²)), so an η² of 0.0588 gives f ≈ 0.25. Everything is calculated in your browser.
Limits
What this tool can’t do
FAQ
Frequently asked questions
Where do I get the effect size?
In order of preference: from published studies close to yours (same construct, similar population), from your own pilot study, or from the smallest effect that would matter in practice. Many papers report means, standard deviations and group sizes, or eta squared, without an effect size, and the helper under the tool converts those for you. Pilot studies are small, so treat their effect sizes with caution. Cohen’s small, medium and large values are a last resort.
Is 0.80 power enough?
It is the convention Cohen suggested and what most Indian examiners and journals accept. It still means a one-in-five chance of missing a real effect of the size you assumed. If a missed effect would be costly, or recruitment is easy, 0.90 is a stronger choice. Whatever you pick, state it and the reason.
Why does G*Power give one fewer for correlation?
G*Power’s default for correlation uses the exact distribution of r, while this tool uses Fisher’s z approximation, which is slightly conservative. For r = 0.3 at α = .05 and power .80, G*Power gives 84 and this tool gives 85. Either figure is defensible. Report the one from the software you name in your methodology chapter.
Should I use this or Cochran’s formula?
It depends on what the study does. If it estimates something about a population, such as the percentage of teachers who use a method, Cochran’s formula in our sample size calculator is the right tool. If it tests a hypothesis, such as whether two groups differ, you need a power analysis. Many theses do both; then use the larger number. The statistical test guide helps you pin down which test your hypotheses need.
What do I write in the methodology chapter?
Name the test, the effect size and where it came from, the significance level, the power, and the resulting sample, then cite the method. For example: “An a priori power analysis for an independent-samples t-test (two-tailed, α = .05, power = .80), assuming d = 0.5 from Author (Year), gave a required sample of 64 per group (Faul et al., 2007).” Then say how many you will approach to allow for non-response.
Can I run the power analysis after collecting data?
You can use the power mode for that, with one condition: enter the effect size you expected or the smallest one that matters, not the one you found. “Observed power” worked out from your own result adds nothing, because it simply restates the p-value. A power analysis belongs at the planning stage. If you are writing up after the fact, report the power your sample had for the effect you expected, and present that as a limitation or a strength.
Keep reading
Related guides and services
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Cochran, power analysis and SEM rules of thumb, and how to justify the number you choose.
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Cochran’s formula, the small-population correction and the regression rules of thumb, with every step shown so you can defend the number.
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Match your question and data type to the right test, including non-parametric options.
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