Sample Size & Power Calculator.

Power Analysis Fundamentals

Statistical power analysis links four quantities:

• α (alpha) — significance level (Type I error rate), typically 0.05 • 1–β (power) — probability of detecting a true effect, typically 0.80 • Effect size — magnitude of the difference you want to detect • n (sample size) — number of observations per group

Given any three of these, the fourth can be computed. The most common use is to fix $\alpha$, power, and effect size, then solve for the required sample size. This is a *prospective* (a priori) power analysis and should be done before data collection begins.

For a two-sample t-test, the formula is:

where d is Cohen’s d (standardized mean difference), z_{α/2} is the critical value for the two-sided significance level, and z_β is the z-value corresponding to the desired power.

Choosing Effect Sizes

The effect size is the most difficult and most important input to a power analysis. Three approaches:

• Pilot data: Estimate from a small preliminary study. Be cautious — pilot studies typically overestimate effects. • Literature review: Use effect sizes reported in similar published studies. Prefer meta-analyses over individual studies. • Clinical significance: Define the smallest effect that would be practically meaningful. This is often the best approach for clinical trials.

Cohen’s conventions (small = 0.2, medium = 0.5, large = 0.8) should be used only as a last resort. A “medium” effect in one field may be unrealistically large in another. When in doubt, power for a smaller effect — the cost of over-sampling is usually less than the cost of an inconclusive study.