Chi-square calculator: independence and goodness of fit

The chi-square (χ²) test compares what you observed with what you would expect by pure chance. With a contingency table it answers “are these two variables related?” (does the treatment affect recovery? does gender affect preference?). With the counts of a single variable it answers “is it distributed as I expected?” (a die, a genetic ratio).

Enter your counts and you get the statistic, the p value, the effect size, the expected frequencies and the residuals that show which cells drive the result. For 2×2 tables it adds Yates’ correction, Fisher’s exact test and the odds ratio. Everything runs in your browser.

For each cell it works out how many cases you would expect if the variables were independent (row total × column total ÷ N) and adds up the squared difference between observed and expected, divided by expected. The larger that χ², the further the data are from independence.

The p value comes from the chi-square distribution with (rows − 1) × (columns − 1) degrees of freedom. Because χ² grows with sample size, the effect size (Cramér’s V, or Cohen’s w for goodness of fit) tells you whether the difference is large or merely detectable.

What the chi-square test checks

The test compares observed counts (frequencies) with those you would get under a hypothesis: that two variables are independent, or that one variable follows a specific distribution. It always works with **categorical** variables (yes/no, groups, categories), never with means or loose percentages.

The two tests

• **Independence**: a contingency table (rows × columns). It asks whether the split on one variable depends on the other.
• **Goodness of fit**: the counts of a single variable against expected proportions (equal, or whatever you specify).

How to read the result

• **p value**: the probability of seeing differences this large or larger if there were really no relationship. Below 0.05 is usually called significant.
• **Cramér’s V**: the effect size, from 0 to 1. Cohen’s thresholds (0.1 small, 0.3 medium, 0.5 large) are divided by the square root of the smaller of (rows − 1) and (columns − 1) in tables bigger than 2×2.
• **Adjusted standardised residuals**: show which cells contribute most. Beyond ±2 they stand out; beyond ±3, strongly.

When it is reliable

The approximation needs enough **expected** frequencies. Cochran’s rule: none below 1 and no more than 20% below 5. If that fails, merge categories or, for a 2×2 table, use Fisher’s exact test, which does not rely on the approximation.

Yates and Fisher for 2×2 tables

Yates’ correction subtracts 0.5 from each difference to allow for counts being discrete; it is conservative and many authors no longer recommend it when expected frequencies are high. With small samples, Fisher’s exact test is the reference.

Common mistakes

• Entering percentages or means instead of counts.
• Using chi-square with paired data (before and after on the same people): that calls for McNemar’s test.
• Counting the same person more than once: observations must be independent.
• Taking “significant” for “large”: with many cases, a tiny association comes out significant.
• Keeping only the p value and forgetting the effect size and the residuals.

Limits of this calculator

• It does not do McNemar, Cochran-Armitage or Fisher’s test on tables larger than 2×2.
• Fisher’s exact test is for 2×2 tables of up to 100,000 cases; it uses the “probabilities less than or equal” criterion.
• The odds ratio of a 2×2 table with a zero cell is computed by adding 0.5 to every cell (Haldane); the interval uses the Woolf approximation.
• It accepts tables up to 20 × 20 and up to 200 categories in goodness of fit.
• Cramér’s V carries no bias correction.

Notice

This is an estimate. For publication, check the figures with your statistics package and report χ², the degrees of freedom, the sample size and the effect size.