Chisquared

Chi-squared refers to both a family of statistics and a probability distribution used in categorical data analysis. It is commonly employed to test how observed frequencies compare with those expected under a null hypothesis, and to assess the fit of a data set to a specified distribution.

The chi-squared distribution is the distribution of a sum of squares of independent standard normal variables. It is defined by its degrees of freedom, which determine the shape of the distribution. As the degrees of freedom increase, the distribution becomes more symmetric and approaches a normal shape.

Pearson's chi-squared statistic is the standard form used in hypothesis testing. For goodness-of-fit, it is χ² = sum_i (O_i − E_i)² / E_i, where O_i are observed counts and E_i are expected counts under the null. In a contingency table, expected counts are E_ij = (row_total_i × column_total_j) / grand_total, and the degrees of freedom are (r − 1)(c − 1) for an r by c table. The test yields a p-value by comparing the statistic to the chi-squared distribution with the appropriate degrees of freedom.

Assumptions include data from a multinomial (or Poisson-like) model, independence of observations, and sufficiently large expected counts (often at least 5). When counts are small, alternatives such as Fisher’s exact test or Yates’ continuity correction for 2x2 tables may be preferred. The likelihood-ratio chi-square (G²) statistic is another related form.

Interpretation centers on the p-value: a large chi-squared statistic indicates a poor fit or association, leading to rejection of the null hypothesis. Uses include goodness-of-fit tests and tests of independence or homogeneity between categorical variables.