Statistical power
Compare power at two sample sizes
Imagine planning a study where the true mean is half a population SD above the null value (d = 0.5). This is an assumed true effect, not an effect estimated after seeing data.
Keep the effect and 5% significance level fixed. Compare studies with 20 and 80 observations; look for the alternative distribution inside the rejection regions.
Null & specified alternative sampling densities
The null curve describes z scores when the null is true. The other curve describes z scores for the chosen true effect. The shaded area under that second curve is power. It is the probability of rejecting the null for that effect.
Power as sample size changes
Each point gives the chance of rejecting the null at that sample size, holding the true effect, significance level and alternative fixed.
Worked comparison
Follow the calculation
What the symbols mean
H₀ is the null hypothesis. d is the true difference between the population mean and the null mean, divided by the population standard deviation. Power is the probability of rejecting H₀ for that effect.
Assumptions and full calculation
The calculation assumes a one-sample z test with a known population standard deviation and independent normal observations. The effect d = (μ − μ₀)/σ describes the population. The z statistic has standard deviation 1 and mean d√n under that effect. The null distribution determines the rejection region. Power is the area of the alternative distribution inside that region. A one-sided test can have power below α when the true effect points in the opposite direction. These calculations apply to z tests. A t test needs a different power calculation.