The Chi-Square Distribution

Summary :

This chapter introduces the chi-square distribution through examples such as whether a coffee machine dispenses a consistent amount each time, framing it as the tool behind three major hypothesis tests. It covers the facts and notation of the chi-square distribution, the test of a single variance, and the goodness-of-fit test.

Three applications of one distribution

The chapter introduces the chi-square distribution as the basis for three distinct hypothesis tests: the goodness-of-fit test, which checks whether data fit a particular distribution, such as evenly distributed lottery numbers; the test of independence, which checks whether two categorical variables such as movie preference and age group are related; and the test of a single variance, illustrated with whether a coffee machine's dispensed amount is consistent.

Facts about the chi-square distribution

The chi-square distribution is nonsymmetrical and skewed to the right, with a different curve for each value of its degrees of freedom, and the chapter notes that its test statistic is always greater than or equal to zero and that, for a sufficiently large number of degrees of freedom, the chi-square curve begins to approximate the normal distribution.

Testing a single variance and goodness of fit

Where earlier chapters focused on means and proportions, the chi-square test of a single variance shifts attention to variability itself, while the goodness-of-fit test determines whether an observed distribution of categorical data is consistent with a claimed or expected distribution, both worked through with the degrees of freedom appropriate to each specific use.