Calculator Inputs
This calculator uses a practical normal approximation for the noncentral chi square distribution.
Formula Used
Noncentrality parameter: λ = n × w²
Approximate critical value: χ²crit ≈ df × [1 − 2/(9df) + z1−α√(2/(9df))]³
Noncentral chi square moments: mean = df + λ, variance = 2(df + 2λ)
Approximate power: Power ≈ 1 − Φ[(χ²crit − mean) / √variance]
Cohen's w summarizes how far observed category proportions depart from the null model. Larger values produce larger λ values and raise power.
For planning, common effect size references are about 0.10 for small, 0.30 for medium, and 0.50 for large categorical effects.
How to Use This Calculator
- Choose whether you want estimated power, required sample size, or minimum detectable effect.
- Select the test context and enter alpha, degrees of freedom, and category count.
- Provide either sample size and effect size, or the target power, depending on your selected mode.
- Submit the form and review the result panel shown above the calculator inputs.
Example Data Table
| Scenario | df | Alpha | Effect Size w | Sample Size | Approx. Power |
|---|---|---|---|---|---|
| Small effect survey | 3 | 0.05 | 0.10 | 500 | 0.5502 |
| Medium effect market test | 4 | 0.05 | 0.30 | 200 | 0.9579 |
| High df behavior table | 8 | 0.01 | 0.20 | 350 | 0.8624 |
Frequently Asked Questions
1. What does chi square power mean?
It is the probability that your chi square test will detect a real categorical effect when that effect actually exists at the specified size.
2. What is Cohen's w?
Cohen's w is a standardized effect size for chi square tests. It measures how strongly observed proportions differ from expected proportions.
3. Why do degrees of freedom matter?
Degrees of freedom shape the null distribution and critical threshold. More degrees of freedom usually require more information to achieve the same power.
4. Can I use this for independence tests?
Yes. The power logic is the same once you know the degrees of freedom and the anticipated effect size for the contingency table.
5. Is the result exact?
No. This tool uses a practical approximation to the noncentral chi square distribution, which is helpful for planning but not a substitute for specialized software.
6. What alpha should I enter?
Most studies use 0.05, while stricter projects may use 0.01. Lower alpha reduces false positives but usually lowers power.
7. What target power is common?
A target power of 0.80 is common for many applied studies. Higher targets improve sensitivity but increase sample requirements.
8. When should I increase sample size?
Increase sample size when power is too low, alpha is strict, the effect is subtle, or the table has many categories.