About the Statistical Power Calculator
This statistical power calculator answers the two questions every study plan needs: with the sample size I have, how likely am I to detect a real effect (power), and how many participants do I need to reach a target power such as 80% or 90%? It covers the four most common designs — two independent groups compared with a t-test, a one-sample t-test, a paired t-test, and a comparison of two proportions (conversion rates, cure rates, pass rates).
Researchers use power analysis when writing grant applications, ethics submissions and pre-registrations; students use it for dissertation planning; and product teams use the proportion option to size A/B tests. The power curve shows how quickly power rises with sample size, which makes it easy to discuss trade-offs between cost and certainty.
For t-tests the calculator uses the exact noncentral t distribution (the method used by G*Power and R's power.t.test), with the effect size entered as Cohen's d. For proportions it uses the standard normal-approximation formula with a pooled variance under the null hypothesis. Sample sizes are per group for two-group designs and are rounded up to whole participants; plan extra to allow for dropouts.
With the default inputs, the power at your sample size (1 − β) is 69.69%. Change any value above to recalculate instantly.
How to use the statistical power calculator
- 1Choose the test that matches your design: two independent groups, one sample, paired measurements or two proportions.
- 2Enter the expected effect: Cohen's d for t-tests, or the two expected proportions.
- 3Set the significance level α and whether the test is one- or two-tailed.
- 4Enter the sample size you have (per group) to see the power you would achieve.
- 5Set a target power (80% or 90% are common) to get the sample size you need.
- 6Use the power curve and table to weigh sample size against certainty.
Formula and method
Power is the probability that the test rejects the null hypothesis when the alternative is true. For t-tests the test statistic follows a noncentral t distribution with noncentrality δ: d√(n/2) with 2n − 2 degrees of freedom for two independent groups of n each, or d√n with n − 1 degrees of freedom for a one-sample or paired test. The calculator finds the critical value t꜀ from the central t distribution at your α and tails, then evaluates the noncentral t CDF F exactly (Lenth’s algorithm), so the results match G*Power and R.
For two proportions it uses the normal approximation: the null standard error is based on the pooled proportion p̄ = (p₁ + p₂)/2, the alternative standard error on p₁q₁ + p₂q₂. Required sample size is the smallest whole n whose power meets the target, found by searching the exact power function, so it is always rounded up.
- d
- Cohen's d: mean difference ÷ standard deviation
- n
- Sample size per group (or number of pairs)
- δ
- Noncentrality parameter of the t distribution
- t꜀, z꜀
- Critical value at significance level α (α/2 per tail if two-tailed)
- p₁, p₂
- Expected proportions in the two groups; q = 1 − p
- F, Φ
- Noncentral t CDF and standard normal CDF
Worked examples
Two groups, medium effect (d = 0.5), 50 per group
With 50 participants per group, the noncentrality is 0.5 × √25 = 2.5 on 98 degrees of freedom and the two-tailed critical t is 1.984. The probability of exceeding it is about 69.7%, short of the usual 80%. Searching upward, 64 per group (128 in total) is the smallest size that reaches 80% power — the classic figure for a medium effect.
Paired design, d = 0.3, 30 pairs, target 90%
For a paired test the noncentrality is 0.3 × √30 = 1.643 with 29 degrees of freedom, giving only about 35.5% power. A small effect like this needs 119 pairs to reach 90% power at α = 0.05 two-tailed.
A/B test: 10% vs 15% conversion, 500 per arm
The pooled proportion is 12.5%, so the null SD is √(2 × 0.125 × 0.875) = 0.4677 and the alternative SD is √(0.09 + 0.1275) = 0.4664. With 500 per arm, power = Φ((0.05√500 − 1.96 × 0.4677) ÷ 0.4664) = Φ(0.432) ≈ 66.7%. You need 686 visitors per arm (1,372 total) for 80% power.
Frequently asked questions
What is statistical power?+
Power is the probability that a study detects an effect when the effect really exists, written 1 − β. A power of 80% means that if the true effect is the size you assumed, 8 out of 10 identical studies would produce a statistically significant result.
Why is 80% power the standard?+
Jacob Cohen proposed 80% as a reasonable balance: a 20% risk of missing a true effect (β = 0.20) against a 5% risk of a false positive (α = 0.05). Many funders and clinical trials now ask for 90%, which needs roughly 30% more participants.
How do I choose the effect size?+
Base it on the smallest effect that would matter in practice, a pilot study, or published results in similar studies. Without other information, Cohen’s benchmarks are d = 0.2 (small), 0.5 (medium) and 0.8 (large). Using an inflated effect size is the most common reason studies end up underpowered.
Is the sample size per group or total?+
For the two-sample t-test and the two-proportion test, n is per group, and the total is twice that. For one-sample and paired tests, n is the number of subjects or pairs.
Why does this differ slightly from other calculators?+
Many simple calculators use a normal approximation for t-tests, which slightly understates the sample size for small samples. This tool uses the exact noncentral t distribution, matching G*Power and R’s power.t.test. Proportion results use the common pooled normal approximation without continuity correction.
Should I use a one-tailed test to reduce the sample size?+
Only if an effect in the opposite direction would be treated exactly the same as no effect and you commit to the direction before collecting data. Reviewers often question one-tailed tests chosen to cut sample size, so two-tailed is the safer default.