Chi-Square Calculator

Chi-square works on frequencies, not means. For each cell you take the observed count minus the expected count, square it, and divide by the expected count; summing those gives the statistic. Squaring makes every deviation positive, and dividing by the expected count scales it — being 10 off when you expected 20 matters far more than being 10 off when you expected 1,000.

The deviations are +10, −10, and 0. Each of the first two contributes 100 ÷ 40 = 2.5, and the third contributes nothing, so chi-square is 5.0 on 2 degrees of freedom. That gives p = 0.0821 — not significant at the 0.05 level, despite one category being 25% above expectation. With only 100 observations, a gap that size is not yet hard to explain by chance.

A goodness-of-fit test compares one set of counts against a theoretical distribution, with degrees of freedom equal to categories minus one. A test of independence asks whether two categorical variables are related, using a contingency table, with degrees of freedom of (rows − 1) × (columns − 1) — so just 1 for a 2 × 2 table. Same statistic, different question and different degrees of freedom.

The chi-square distribution is a continuous approximation to a discrete problem, and it only holds when the counts are reasonably large. The standard requirement is that every expected count is at least 5; some texts allow a single cell down to 1 provided no more than 20% fall below 5. Below that, use Fisher's exact test, which computes the probability directly rather than approximating it.

A significant result says the observed pattern is unlikely under independence — it does not say how strong the association is or which direction it runs. For that, inspect the individual cell contributions and report an effect size such as Cramér's V or an odds ratio. And because the statistic grows with sample size, a very large table will flag trivial departures as significant, so always look at the size of the deviations, not just the p-value.