Power analysis for a χ² goodness-of-fit or independence test uses Cohen's w to solve for sample size or power.
Cohen's w = √(Σ (p₁ − p₀)² / p₀) summarises cell-by-cell departure from the null contingency table. Anchors: 0.1 small, 0.3 medium, 0.5 large. Required N grows with df, so larger tables need more data.
df + three of {N, w, α, power}.
Fourth quantity.
A small w (0.1) is what you'd expect from real-world contingency departures; planning for w = 0.3 'medium' often leaves studies under-powered.