Chi Square Calculator

The chi-square goodness-of-fit statistic sums, over every category, the squared difference between an observed and an expected count divided by that expected count. Scaling by the expected value matters: a shortfall of 10 against an expectation of 20 is strong evidence, while the same shortfall against an expectation of 2,000 is noise, and the division encodes that intuition because the variance of a Poisson or binomial count grows with its mean. Under the null hypothesis the statistic follows, approximately, a chi-square distribution with k-1 degrees of freedom for a goodness-of-fit test on k categories, losing one degree of freedom because the totals are constrained to match. The distribution is right-skewed and strictly positive, so the test is inherently one-tailed on the upper side even though the underlying question is nondirectional. Common critical values at alpha = 0.05 are 3.841 for 1 degree of freedom, 5.991 for 2, 7.815 for 3 and 9.488 for 4; exceeding them rejects the hypothesis that observed counts came from the expected distribution. Biological uses include testing Mendelian segregation against a 3:1 monohybrid or 9:3:3:1 dihybrid ratio, and testing a biallelic locus for Hardy-Weinberg equilibrium, which carries 1 degree of freedom because two allele frequencies are estimated from three genotype counts. Three limitations bind the test. It must be run on raw counts, never on percentages or proportions. The chi-square approximation degrades when expected counts are small, so Cochran's guidance requires no expected count below 1 and at most a fifth below 5, with Fisher's exact test as the alternative. And with very large samples any trivial departure becomes significant, so an effect size such as Cramer's V should accompany the p-value.

Karl Pearson introduced the statistic in a 1900 paper in the Philosophical Magazine, written at University College London, asking whether a system of deviations from expectation could reasonably have arisen by random sampling. The underlying distribution was not entirely new, having been derived by the German geodesist Friedrich Helmert in the 1870s in the context of sums of squared normal deviates, but Pearson turned it into a general test of fit and applied it to problems ranging from roulette outcomes at Monte Carlo to biometric data. It was the first widely usable significance test for categorical data and remains one of the most heavily used procedures in applied statistics. The degrees of freedom were wrong as Pearson stated them for contingency tables, and Ronald Fisher demonstrated in a 1922 paper that one degree of freedom must be subtracted for every parameter estimated from the data, giving (r-1)(c-1) for an r by c table. Pearson rejected the correction and the dispute between the two men lasted for years, but Fisher's formulation prevailed and he embedded the test in genetics through Statistical Methods for Research Workers in 1925. In 1936 he used it in reverse, arguing that the chi-square values from Gregor Mendel's pea data fitted expectation too closely to be plausible sampling outcomes. Frank Yates proposed his continuity correction for 2 by 2 tables in 1934, and William Cochran published the minimum expected-count guidelines still cited in the early 1950s.