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 power analysis calculator tells you how many participants you need to detect a given effect with a two-group t-test, one-way ANOVA or correlation. You enter the effect size, the significance level and the power you want. It gives the minimum sample, and for the t-test and ANOVA the answer matches G*Power. Sometimes your sample is already fixed. Then the tool works in reverse and tells you how much power that sample gives.
Your thesis is having hypotheses to test, like group A versus group B? Then examiner surely asks where is the power analysis. It is also your answer when a DC member asks why you stopped at 120 respondents. No effect size yet, then the helper under the tool works one out from an earlier study, using the means and standard deviations which that study reported, or its eta squared.
Work out the effect size
Method
How this tool works
A power analysis ties four numbers together. They are the significance level, the power, the effect size and the sample size. If you fix any three, the fourth one gets decided. By default this tool takes the first three and finds the sample. Change the setting to “The power of the sample I have” and it goes the other way, taking your sample size and finding the power. This is useful when access gives you fixed number of respondents, for example the 90 nurses in one hospital. For those 90 nurses also, put the effect size you had in mind at planning time, not the one from your data.
The significance level (α) is the risk you are ready to take of finding an effect that is not really there. Mostly it is kept at 5%. Power is the chance that you will detect a real effect of the size you assumed. Most people put 80% there. So one time in five you may miss a real effect. The effect size says how big the difference or relationship is, in standard units. For two means we take Cohen’s d. You get d by dividing the gap between the means by the pooled standard deviation. With three or more means you use Cohen’s f, and for a correlation the effect size is r itself.
Cohen (1992) gave conventional values for small, medium and large effects. You can see them in the table below, along with the sample each one needs in this tool 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) |
See how fast the required sample go up as the effect gets smaller. Because of this, examiners question the effect size more than any other input. Take it from earlier studies close to your own work, or use the smallest effect that would make a practical difference. Go to Cohen’s conventions only when nothing better is available, and mention in the thesis that you did. Where each kind of effect size comes from is explained in the sample size guide.
For the t-test and ANOVA, the tool works with the exact noncentral F distribution. G*Power 3.1 also uses this same method. A two-group t-test is solved as a two-group ANOVA with f = d ÷ 2, and the answers come out identical. At d = 0.5, α = .05 two-tailed and power .80, you get 64 per group. With f = 0.25 and three groups you get 159 in total, that is 53 per group. G*Power reports the same figures. The totals are rounded up so that all the groups have equal size.
Correlations are handled with Fisher’s z approximation. The formula is N = ((z₁₋α/₂ + z₁₋β) ÷ atanh r)² + 3, rounded up. For r = 0.3 this gives 85. The exact method in G*Power gives 84, so for a correlation this tool may show one participant more. Still we kept the approximation. Anybody can verify it on paper, and one more person in 85 is harmless. In power mode also, the power for a correlation is worked out from this same approximation.
The two helpers just use textbook conversions. For Cohen’s d from two groups, the tool divides the difference in means 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. All the calculation happens in your browser itself.
Limits
What this tool can’t do
FAQ
Frequently asked questions
Where do I get the effect size?
First preference is published studies close to yours, with the same construct and a similar population. Next comes your own pilot study, and after that the smallest effect that would matter in practice. Many papers report means, standard deviations and group sizes, or eta squared, but they don’t give an effect size. The helper under the tool converts these informations for you. Be careful with effect sizes from a pilot, because pilot studies are small. Cohen’s small, medium and large values are only the last option.
Is 0.80 power enough?
0.80 is the convention Cohen suggested, and most Indian examiners and journals accept it. But it still leaves a one-in-five chance that you miss a real effect of the size you assumed. Where missing the effect is a big loss, or students are easy to get, 0.90 makes more sense. In the thesis write the power value you used and one line on why you picked it.
Why does G*Power give one fewer for correlation?
By default, G*Power works out a correlation with the exact distribution of r. This tool uses Fisher’s z approximation, and that is slightly conservative. So for r = 0.3 at α = .05 and power .80, G*Power says 84 and this tool says 85. You can defend either figure in your viva-voce. Just report the number from whichever software you name in the methodology chapter.
Should I use this or Cochran’s formula?
That depends on what your study is doing. Some studies estimate something about a population, like the percentage of teachers using a method. For those, Cochran’s formula in our sample size calculator is the correct tool. When the study tests a hypothesis, for example whether two groups differ, a power analysis is needed. Many theses are doing both, and in that case you take the bigger number. To decide which test your hypotheses need, see the statistical test guide.
What do I write in the methodology chapter?
Write down the test, the effect size with its source, the significance level, the power and the sample you got, and cite the method. A sentence like this one will do. “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).” After that, mention how many people you will approach to cover non-response.
Can I run the power analysis after collecting data?
Yes, the power mode can be used for that. But enter the effect size you expected, or the smallest one that matters, and not the effect you actually found. That “observed power” figure in the output, ignore it, it is the p-value only in another form. Actually power analysis is a planning stage job. If you are writing up after data collection, report the power your sample had for the effect you expected. You can present that as a limitation or as a strength.
Keep reading
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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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