Type I error, Type II error, and power

Move the controls to see how the rejection threshold and the chance of detecting a specified effect change.

Type I error probability under H₀.
Population mean shift ÷ population SD.
Number of independent observations.
α · False positiveReject H₀ when H₀ is true.
β · False negativeFail to reject H₀ at the specified effect.
Power · Detect effectReject H₀ at the specified effect.
Sampling distributions under the null and a specified alternative
α: rejection area under H₀β: non-rejection area under H₁Power: rejection area under H₁

Power = 1 − β

Model: a one-sample z-test with known population standard deviation σ and independent normal observations. z = (sample mean − null mean) / (σ / √N), so z has no unit. Under H₀, z follows N(0, 1); at the specified alternative, it follows N(d√N, 1). These are distributions across repeated samples, not distributions of individual observations. For a t-test or ANOVA, exact power calculations use different distributions.