Testing hypotheses concerning frequency tables using the χ ² distribution

Abstract

The distribution of χ ² (chapter 7) has many uses in statistics. In the case where a continuous variate X follows a normal distribution (chapter 5), the χ ²distribution may be used to test hypotheses or to determine confidence intervals exactly about the parametric variance σ ²_x of the population (chapter 10). When the hypothesis that within-groups variances are equal must be tested in an analysis of variance, Bartlett’s criterion follows the χ ²distribution approximately (section 12.7). Moreover, the hypotheses that the distribution of a set of data has the same skewness index (γ ₁= 0) and the same peakedness index (γ ₂= 0) as a normal distribution can be tested jointly by using the χ ²distribution (section 13.5). But the most common application of the χ ²distribution, which was discovered by Karl Pearson (1900), is perhaps its approximate use to test hypotheses about frequency tables (also called contingency tables),one of the oldest among the so-called nonparametricmethods.