Chi-Square and Contingency Tables

Abstract

This chapter introduces methods to explore the relationship between two categorical variables (either measured at the nominal or ordinal levels) using the British Crime Survey data. We will cover how to tabulate and visualize this kind of relationships using a two-way contingency table (also referred to as cross tabulation or cross tabs), but also review the chi-square test (χ² test), which is the statistical significance test used to infer association in these cases. The chapter also covers Fisher’s exact test and calculation of residuals (difference between observed and expected frequencies).

Key Terms

Chi-square statistic

The test statistic resulting from applying the chi-square formula to the observed and expected frequencies for each cell. This statistic tells us how much the observed distribution differs from that expected under the null hypothesis.

Contingency table

A tabular way of viewing the relationship between categorical variables (also referred to as cross tabs).

Expected frequency

The number of observations one would predict for a cell in a contingency table if the null hypothesis were true.

Marginal

The value in the margin of a table that totals the scores in the appropriate column or row.

Observed frequency

The observed result of the study, recorded in a cell of a contingency table.

Residual

An index of the relative deviation of the observed frequency from the expected frequency for a cell of a contingency table. It is useful for guiding the interpretation of an association between two nominal variables.