Sample Size Calculator for Chi-Square Test
Estimate a minimum sample using power, alpha, effect size, categories, contingency dimensions, proportions, and recruitment adjustments with transparent calculation steps.
Results
Enter values and calculateCalculator inputs
Expected and alternative probabilities
Each column must sum to 1. Cohen's w is calculated automatically.
| Cell/category | Expected under H₀ | Alternative under H₁ |
|---|---|---|
| Total | — | — |
Recruitment adjustment
Step-by-step calculation
Run the calculator to see the full statistical working.
Sensitivity analysis
These scenarios show how the required sample changes when power or alpha changes.
| Scenario | Alpha | Power | Required sample |
|---|---|---|---|
| Calculate to populate this table. | |||
Formula used
The calculator numerically searches for the smallest integer sample size whose noncentral chi-square power meets the requested target.
How to use this calculator
- Select the appropriate chi-square test type.
- Choose direct effect-size entry or probability-based calculation.
- Enter alpha, desired power, and category or table dimensions.
- Optionally add dropout, design effect, or oversampling adjustments.
- Review the minimum analyzable sample and recruitment target.
Understanding chi-square sample size planning
Why sample size matters
Chi-square tests compare categorical patterns rather than continuous means. Sample size planning helps ensure that a study has enough observations to detect a meaningful departure from the null hypothesis. A larger target power usually requires more observations. Smaller effect sizes also require larger samples because subtle differences are harder to distinguish from random variation. The significance level affects the evidence threshold, so stricter alpha values often increase the required sample. Planning should also reflect the smallest scientifically important difference, because an optimistic effect assumption can produce an unrealistically low recruitment target for the study.
Choosing the effect size
Cohen’s w summarizes the distance between null and alternative probability distributions. You may enter w directly when prior research already provides an estimate. You can also calculate w from expected and alternative probabilities. This second method is useful when study planning begins with concrete category proportions. The expected probabilities describe the null model, while alternative probabilities describe the pattern you want the study to detect.
Goodness-of-fit and contingency tests
A goodness-of-fit test examines one categorical variable against expected proportions. Independence and homogeneity tests use contingency tables. Their degrees of freedom depend on table dimensions. Larger tables generally have more degrees of freedom, which changes the chi-square distribution and required sample. Expected cell counts should also be reviewed because very sparse tables can weaken the usual asymptotic approximation.
Interpreting the result
The minimum analyzable sample is the smallest integer reaching the requested statistical power under the stated assumptions. The recruitment target then adjusts that number for design effect, anticipated dropout, and optional oversampling. Power calculations remain planning tools rather than guarantees. Researchers should justify effect-size assumptions, consider realistic response rates, and document any design adjustments. Simulation, exact methods, or specialist software may be preferable when probabilities are extremely small, tables are sparse, or the sampling design is complex.
Example planning table
| Planning choice | Example | Typical implication |
|---|---|---|
| Small effect | w = 0.10 | Usually needs a large sample. |
| Medium effect | w = 0.30 | Often needs a moderate sample. |
| Large effect | w = 0.50 | Often needs a smaller sample. |
| Higher power | 0.90 versus 0.80 | Raises the required sample. |
| Stricter alpha | 0.01 versus 0.05 | Usually raises the required sample. |
FAQ
What is Cohen's w?
Cohen's w is a standardized effect size for chi-square tests. It measures how far an alternative categorical distribution differs from the null distribution.
What power should I use?
A target of 0.80 is common, while 0.90 is often used when missing an important effect would be especially costly.
How are degrees of freedom calculated?
Goodness-of-fit typically uses categories minus one. Contingency-table tests use rows minus one multiplied by columns minus one.
Why does the calculator warn about expected counts?
Very small expected counts can make the usual chi-square approximation inaccurate. Combining categories, exact methods, or simulation may be preferable.
What does design effect do?
Design effect inflates the analyzable sample when clustering or another sampling design reduces effective information compared with simple random sampling.
Does the recruitment target equal the analyzed sample?
Not necessarily. The recruitment target includes allowances for dropout, nonresponse, design effect, and optional oversampling.