Power Sample Size Calculator

Power is the probability that a study will reject the null hypothesis when the alternative is true, written 1 minus beta. For a two-group comparison of means the normal-approximation formula is n per group = 2 times (z at alpha plus z at beta) squared, divided by d squared, where d is the standardised difference between group means expressed in pooled standard deviations. At alpha = 0.05 two-tailed the critical value is 1.960, and the power multiplier is 0.842 for 80 percent power or 1.282 for 90 percent. Those constants produce familiar benchmarks: roughly 26 per group for d = 0.8, about 64 per group for d = 0.5, and near 394 per group for d = 0.2. Cohen's descriptive labels of 0.2, 0.5 and 0.8 for small, medium and large were never meant as design targets. A defensible d comes from a minimum clinically important difference divided by the standard deviation seen in pilot or published data, for instance 5 mmHg over an SD of 10 mmHg giving d = 0.5. The normal approximation slightly understates the requirement because it ignores the extra uncertainty in estimating the variance, so exact noncentral t calculations typically add one to three subjects per group. The formula also assumes equal allocation, independent observations and a continuous outcome, and it makes no allowance for dropout, which is usually handled by inflating n by 10 to 25 percent. Power computed after the fact from an observed effect adds no information.

The concept belongs to Jerzy Neyman and Egon Pearson, who in papers of 1928 and in their 1933 Philosophical Transactions article on the most efficient tests of statistical hypotheses introduced the alternative hypothesis, the second kind of error, and the idea of choosing a test to maximise the probability of detecting a real difference. Ronald Fisher never accepted this decision-theoretic framing, and for decades applied researchers used significance tests without any explicit consideration of the error rate for missed effects. Jacob Cohen changed that in applied science. His 1962 survey in the Journal of Abnormal and Social Psychology examined a full volume of published studies and found median power of roughly 0.48 for detecting medium-sized effects, meaning most experiments were close to a coin flip. His book Statistical Power Analysis for the Behavioral Sciences, first published in 1969 and revised in 1988, supplied the standardised effect size d, extensive tables, and the 0.80 convention, which he justified by treating a false positive as about four times more costly than a false negative given alpha at 0.05. Dedicated software followed, including the PS program of Dupont and Plummer in 1990 and G Power from Faul, Erdfelder and colleagues in 2007. The CONSORT reporting guidelines, introduced in 1996 and revised in 2010, made an explicit sample size justification a requirement for published trials.