Statistical power
Power is the probability that a test rejects the null hypothesis when a real effect of the size you specified exists. Its complement, β = 1 − power, is the probability of a Type II error — missing a real effect. The conventional target is a power of 0.8, meaning the study would miss such an effect one time in five.
This page answers the question the other way round from a sample size calculator: you already have a sample size, and it tells you how much power that buys. If the answer is low, the two honest options are to collect more data or to accept that only a larger effect is detectable — not to run the test anyway and read a non-significant result as evidence that there is no effect.
The three tests
Goodness of fit
Compares the counts you observed across k categories with the counts a hypothesis predicts. Under H0 the statistic follows a central χ² with df = k − 1; under H1 it follows a noncentral χ² with the same df and noncentrality λ = n·w². Because λ grows with n, the two curves separate as the sample grows — which is exactly what the power curve draws.
Independence (association)
Tests whether two categorical variables are related, using an r × c contingency table. The arithmetic is the goodness-of-fit arithmetic with different degrees of freedom: df = (r − 1)(c − 1), because the row and column totals are fixed by the data. Cohen's w means the same thing on both tests, so the same 0.1 / 0.3 / 0.5 conventions apply.
Test for a variance
Compares a sample variance with a hypothesised one. Here df = n − 1, and the effect is the variance ratio Var₁/Var₀. Under H1 the statistic divided by that ratio follows the same central χ², so the alternative curve is the null curve stretched along the axis rather than a different family. This is the one sub-test with a tail direction to choose; the other two are inherently right-tailed, because squaring the deviations folds every direction into the upper tail.
Expected counts: the assumption power cannot repair
The χ² statistic only approximately follows a χ² distribution, and the usual rule of thumb is that every expected count should be at least 5. Under an even null that needs about 5k observations for goodness of fit and about 5·r·c for an r × c table — so a 5 × 5 table wants roughly 125 observations, not 50. Below that the test's real error rate may not match α, and the power reported here is only as good as the approximation behind it. The page says so when your design falls short.
How to read the two charts
The power analysis chart draws both distributions on one axis. The red region is α, the chance of rejecting a true null; the amber region is β, the chance of missing a real effect; the blue region is the power. The power curve sweeps the sample size, marks the n you entered and the n the target power would need, and can be hovered for the exact figures at any point.