Chi-Square Test Statistic Calculator

Example Data Table

Formula Used

The calculator uses this statistic for every supported design:

χ² = Σ ((O - E)² / E)

O is the observed count. E is the expected count. For a contingency table, E equals row total times column total, divided by the grand total.

Goodness-of-fit degrees of freedom equal categories minus one minus estimated parameters. Independence degrees of freedom equal rows minus one, multiplied by columns minus one.

The p-value is the right-tail probability from the chi-square distribution. The critical value is the cutoff where the right-tail probability equals alpha.

How to Use This Calculator

1. Choose goodness-of-fit or independence table.
2. Enter your observed counts.
3. Enter expected counts, proportions, or table dimensions.
4. Set alpha, such as 0.05.
5. Press Calculate to show results below the header.
6. Review warnings, residuals, p-value, and effect size.
7. Use CSV or PDF download for records.

Chi-Square Test Statistic Guide

A chi-square test checks how far observed counts move from expected counts. It works with categories, not averages. The statistic grows when the gap is large. It stays small when data follows the expected pattern. This calculator helps you run two common designs.

Goodness-of-Fit Testing

Use goodness-of-fit when one variable has several categories. You enter observed counts from your sample. Then enter expected counts, expected proportions, or choose equal expectation. The tool scales proportions to the observed total. It then adds each squared difference divided by its expected count. The degrees of freedom usually equal categories minus one. Reduce it when expected values were estimated from the same data.

Independence Testing

Use an independence test for a contingency table. Rows may show groups. Columns may show outcomes. The calculator finds row totals, column totals, and the grand total. Expected counts come from row total times column total, divided by grand total. The statistic sums every cell contribution. Degrees of freedom equal rows minus one, times columns minus one.

Reading Results

The p-value estimates the chance of seeing a statistic this large, assuming the null hypothesis is true. A small p-value means the observed pattern is unusual under that assumption. Compare it with alpha. Many users choose 0.05, but your field may require another level. The critical value gives a second decision check.

Practical Checks

Expected counts should not be zero. Very small expected counts can weaken the approximation. The calculator warns when expected counts are below five. For a two by two independence table, you may apply Yates correction. That option makes the statistic more conservative. It is often used with small samples.

Reporting Tips

Report the test type, statistic, degrees of freedom, p-value, alpha, and conclusion. Include effect size too. Goodness-of-fit uses Cohen's w. Independence uses Cramer's V. These values show practical strength, not only significance. Export the report when you need a clean record for notes, assignments, audits, or peer review.

Keep inputs numeric and nonnegative. Do not mix percentages with counts. If your expected proportions do not sum to one, the calculator normalizes them. Review residuals to see which categories or cells drove the final statistic most strongly during your final analysis.