Chi-Square Calculator

In short A chi-square test compares counts you observed with counts you would expect, and sums (observed − expected)² ÷ expected across every category. Rolling a die 60 times and getting 16, 8, 10, 10, 8, 8 gives χ² = 4.80 on 5 degrees of freedom and p = 0.4408 — nothing unusual, despite one face appearing 60% more often than it should. That is the useful lesson: eyeballing counts overstates how surprising they are. There are two tests here. Goodness of fit checks one set of counts against a hypothesis, with degrees of freedom one less than the number of categories. Independence checks whether two variables in a contingency table are related, with (rows − 1) × (columns − 1) degrees of freedom; the expected counts come from the row and column totals, and they reproduce those totals exactly, which is what "expected" means. Three cautions. The statistic grows in direct proportion to the sample size, so the same percentages with ten times the data give ten times the χ² — a significant result is not by itself an important one, which is why Cramér's V is printed as the effect size. The approximation needs reasonably large expected counts; the usual rule is at least 5, and for a small 2×2 table Fisher's exact test is the honest choice. And a significant result says the rows and columns are not independent. It does not say which way the effect runs, and it certainly does not say one causes the other.