Statistical Power Calculator

Statistical power (1−β) is the probability that a test correctly rejects the null hypothesis when a true effect exists. In other words, it is the likelihood of detecting a real difference if one is actually there. Conventional guidelines recommend a minimum power of 0.80 (80%) for most research studies.

Power is influenced by four key factors: effect size (larger effects are easier to detect), sample size (more participants increase power), significance level α (a higher α raises power but increases false positives), and the variability in your data. Increasing any of these — except variability — will boost power.

Cohen's conventions classify effect sizes as small (d = 0.2), medium (d = 0.5), and large (d = 0.8). If you don't have a prior estimate, a medium effect size (0.5) is a common default. Ideally, base your effect size on pilot data, a literature review, or the minimum clinically meaningful difference for your study.

A two-sided test checks whether the effect could go in either direction and is the standard default. A one-sided test is used when you have a strong directional hypothesis (e.g., 'Treatment A is better than B') — it provides more power but is only valid when the direction is pre-specified before data collection.

Alpha (α) is the probability of a Type I error — rejecting the null hypothesis when it is actually true (a false positive). A more lenient α (e.g., 0.10 instead of 0.05) increases power but also increases your false-positive rate. Most fields use α = 0.05 as the standard threshold.

The allocation ratio (n₂/n₁) represents the relative size of the two groups. A ratio of 1 means equal group sizes, which is most statistically efficient. You might deviate from 1 if one group is harder or more expensive to recruit, accepting a small power reduction in exchange for practical feasibility.

For a two-sample z-test, the non-centrality parameter λ = d × √(n / (1 + 1/k)), where d is Cohen's d and k is the allocation ratio. Power is then the probability that a standard normal variable exceeds the critical z-value minus λ. This calculator uses a normal approximation which is accurate for moderate to large sample sizes.

An underpowered study risks missing a real effect — a Type II error. This wastes resources, may lead to incorrect conclusions, and can be unethical in clinical research where real treatment benefits go undetected. Conversely, grossly overpowering a study wastes participants and money. Aiming for 80–90% power strikes the practical balance.