Calculator
Formula used
Goodness of fit degrees of freedom: df = k - 1.
Independence degrees of freedom: df = (rows - 1) × (columns - 1).
Noncentrality parameter: λ = N × w².
Critical value: χ² critical = inverse central χ² at 1 - α.
Power: Power = P(χ² with df and λ is greater than χ² critical).
The calculator searches upward for the smallest whole N that reaches the target power.
How to use this calculator
- Choose the test design that matches your planned analysis.
- Enter categories, table dimensions, or custom degrees of freedom.
- Enter Cohen’s w, alpha, and desired power.
- Add expected proportions when categories are uneven.
- Add a loss allowance for missing or unusable data.
- Press Calculate and review the result above the form.
- Use CSV or PDF to save the result.
Example data table
| Design | df | w | Alpha | Power | Estimated N |
|---|---|---|---|---|---|
| Goodness of fit | 3 | 0.3 | 0.05 | 0.8 | 122 |
| Goodness of fit | 5 | 0.2 | 0.05 | 0.8 | 321 |
| Independence table | 2 | 0.3 | 0.05 | 0.9 | 141 |
| Custom plan | 8 | 0.15 | 0.01 | 0.8 | 918 |
Planning a Chi Square Study
A chi square test checks whether observed counts differ from expected counts. It can support a goodness of fit study, an independence study, or a custom degrees of freedom plan. Sample size matters because small tables may miss real differences. Very large tables may waste time and budget. This calculator links the planned effect size, alpha level, power target, and degrees of freedom. It then estimates the minimum analyzable sample size for the test.
Why Effect Size Matters
Cohen’s w describes the gap between expected and alternative category patterns. A small w needs more observations. A large w needs fewer observations. The calculator accepts any positive value, so you can model pilot results, published studies, or conservative planning assumptions. Typical reference values are 0.10, 0.30, and 0.50. They are only guides. Real studies should use subject knowledge.
Power and Significance
Power is the chance of detecting the planned effect. Alpha is the false positive risk you accept before testing. A common target is 80 percent power with 0.05 alpha. Higher power raises the needed sample size. Smaller alpha also raises the sample size. The tool solves this tradeoff using a noncentral chi square approach. It reports the critical value and noncentrality value for review.
Expected Cell Counts
Chi square methods also need reasonable expected counts. Many basic guides use five as a practical minimum. The calculator estimates the smallest expected cell count from your category proportions. Equal proportions are used when no proportions are entered. Uneven categories can require a larger practical sample, even when the power result looks acceptable.
Using Results Carefully
The final number is an analyzable sample size. If missing data, screening failures, or incomplete surveys are expected, add a loss allowance. The recruitment sample shown here adjusts for that allowance. Treat the result as a planning estimate. Confirm important studies with a statistician, especially for sparse tables, complex sampling, clustering, or regulatory work.
Interpreting the Output
Use the analyzable sample for completed records. Use the recruitment sample for field planning. Check the warning message before finalizing. If the minimum expected count is low, revise categories or increase the sample to protect the test and final reporting decisions.
FAQs
What is a chi square sample size?
It is the minimum number of analyzable observations needed to reach the planned power for a chi square test, given alpha, effect size, and degrees of freedom.
What effect size should I enter?
Use Cohen’s w from a pilot study, prior research, or a meaningful planning target. Reference values are often 0.10, 0.30, and 0.50.
What does alpha mean?
Alpha is the planned false positive rate. A value of 0.05 means you accept a 5 percent risk of rejecting the null when it is true.
What does target power mean?
Power is the chance of detecting the planned effect. Many studies use 0.80, but important studies may use 0.90 or higher.
Can I use this for independence tests?
Yes. Select the independence option. Then enter row and column counts. The calculator uses the table degrees of freedom.
Why enter expected proportions?
Expected proportions help estimate the smallest expected count. Uneven category shares can create sparse cells and may require a larger practical sample.
What is the recruitment sample?
It is the larger starting sample after adding your loss allowance. It helps cover missing data, screening failures, or unusable responses.
Is this a final study design?
No. It is a planning calculator. Complex surveys, clustered samples, sparse tables, or regulated studies should be checked by a qualified statistician.