Achieved power for a chi-square test

When to use it

Assess the sensitivity of a test of association in an r × c table, or of a goodness-of-fit test against stated category proportions.

Not for

With small expected cell counts the chi-square approximation is poor; an exact test is then safer than its power calculation suggests.

Notation

$N$: Total participants.
$w$: Cohen’s w, the effect size for categorical data.
$\nu$: Degrees of freedom: (r − 1)(c − 1) for independence, categories − 1 for goodness of fit.
$\lambda$: Noncentrality, Nw².

Formulas

Effect size from cell proportions

w=\sqrt{\sum_i \frac{(p_{1i}-p_{0i})^2}{p_{0i}}}, \qquad \lambda = N w^2

p₀ are the cell proportions under the null and p₁ those expected. Cohen’s benchmarks are w = 0.10, 0.30, and 0.50 [1].

Power of the chi-square test

\text{Power}=P\left(\chi^2_{\nu,\lambda} > \chi^2_{1-\alpha;\,\nu}\right)

The alternative is a noncentral chi-square distribution [1, 2].

Parameters

Total sample size $N$

Values: Integer, at least 2

Enter all participants classified in the table.

Degrees of freedom $\nu$

Values: Integer, at least 1

(rows − 1) × (columns − 1) for a test of independence; categories − 1 for goodness of fit.

Effect size (Cohen’s w) $w$

Values: Positive

From expected and null cell proportions, as in the formula; 0.10, 0.30, and 0.50 are small, medium, and large.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Assumptions

Observations are independent and each falls in exactly one cell; expected cell counts are large enough for the chi-square approximation.
F and chi-square tests reject only in the upper tail, so there is no one- or two-sided choice.

Worked examples

A worked input set

For 100 participants in a 2 × 3 table (ν = 2) and a medium effect, w = 0.30, at α = 0.05: λ = 100 × 0.30² = 9.

Sample size: 100

Open this example in the calculator

Validation

Power is the exact noncentral chi-square tail probability. It agrees with pwr::pwr.chisq.test in R and with G*Power, and at the size the test-of-independence calculator returns it reaches that calculator’s target power.

How to cite

StatWisely. Achieved power for a chi-square test. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-chi-square/

Cite the original references for the formulas your design uses as well.

References

1.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.
2.Agresti A. Categorical Data Analysis. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2013.