The chi-square test of independence is used when the independent variable (IV) and dependent variable (DV) are both categorical (nominal or ordinal). The chi-square test is a member of the family of nonparametric statistics, which are used when sampling distributions cannot be assumed to be normally distributed, as is the case when a DV is categorical. Chi-square thus sits in contrast to parametric statistics, which are used when DVs are continuous (interval or ratio) and sampling distributions are safely assumed to be normal. The t test, analysis of variance, and correlation are all parametric. Because they have continuous DVs, they can rely on normal (or at least relatively normal) sampling distributions such as the t curve. (There are exceptions to the use of parametric statistics on continuous data, such as when the data are severely skewed, but that is beyond our scope here. For present purposes, we will distinguish the two classes of statistics on the basis of level of measurement.) Before going into the theory and math behind the chi-square statistic, Research Example 10.1 for an illustration of a type of situation in which a criminal justice or criminology researcher would turn to the chi-square test.
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The Chi-square Test of Independence