Power Analysis Calculator

How the four quantities trade off

• A larger sample raises power. For a medium effect (d = 0.5, α = .05, two-tailed), 64 per group gives power 0.80 and 86 per group gives 0.90.
• A smaller effect needs far more people. At d = 0.2 you need 394 per group; at d = 0.8 only 26.
• A stricter α lowers power unless n grows. A one-tailed test at the same settings needs 51 per group instead of 64, but only if the direction is fixed in advance.
• The conventional target is power 0.80 with α = .05, a convention rather than a law. Grant applications and clinical studies often ask for 0.90.

How to choose an effect size

The effect size is the hardest input and the one that matters most. Three sources are defensible. First, a meta-analysis or a closely matched earlier study in your field, ideally shrunk a little because published effects tend to be inflated. Second, a pilot study, with the caution that a small pilot gives a noisy estimate. Third, the smallest effect that would be meaningful in practice, such as the smallest drop in blood pressure a clinician would care about. Cohen’s small, medium and large conventions are a last resort because the right benchmark depends on the field. If you only have raw means and standard deviations, the effect size calculator converts them to d and related measures.

Why post hoc power is not recommended

Observed (post hoc) power is computed from the effect you actually found. It is a direct function of the p-value, so observed power is at most about 50% whenever p is above α and the number adds no new information (Hoenig and Heisey, 2001). A better use of the same tools is a sensitivity analysis: given the sample you have, which effect sizes could you have detected with 80% power? Report that range, together with a confidence interval for the effect you found.

Multiple comparisons and dropout

If you plan several tests and use a Bonferroni correction, enter the corrected α. For three comparisons at an overall 0.05, α = 0.0167, and the medium-effect example grows from 64 to 86 per group. For dropout, divide the required size by (1 − dropout) and round up. The calculator does this for you and shows the recruitment number next to the analysis number.

Worked example: three groups

A teacher compares three teaching methods and expects a medium effect, f = 0.25. With α = .05 and power 0.80, the one-way ANOVA tab returns 53 students per group, 159 in total, using the noncentral F distribution with 2 and 156 degrees of freedom. For a paired design with dz = 0.3 and the same α and power you would need 90 pairs, and for two proportions of .30 and .20 you would need 292 per group.

Worked example: an animal experiment

A mouse study expects a large difference of d = 1.2 between control and treated animals. At α = .05, two-tailed, power 0.80, the independent t tab gives 12 mice per group. If you expect 10% of animals to be lost, divide by 0.9 and round up: 14 per group. Because small groups make the estimate of d very uncertain, ethics boards often expect the effect size to be justified from earlier data rather than assumed to be large.