The distribution of χ ² (chi squared)

Abstract

The distribution of χ ² (chi squared) is a continuous and asymmetrical distribution which ranges from 0 to + ∞ and is followed by a sum of squares of independent standardized normal variates. The χ ² distribution and its application to frequency tables (chapter 15) were discovered by the British biometrician Karl Pearson (1857–1936), who was considered as the “father of biometry” and was one of the founders of the periodical Biometrika. The χ ²distribution is exact when hypotheses must be tested or confidence intervals must be determined (chapter 10) concerning the parametric variance σ ² of a continuous variate X which follows a normal distribution (chapter 5). Moreover, the χ ²distribution may be used as an approximation in many cases, including Bartlett’s test of the homogeneity of variance (section 12.7), tests of hypotheses concerning frequency tables (chapter 15), tests of goodness of fit (chapters 16, 17 and 18), the binomiality (chapter 17) and Poissonianity (chapter 18) tests, etc.. Paradoxically, the approximate utilizations of the χ ² distribution are perhaps better known than the exact ones.