Chi-Square Test

Bivariate Tabular Analysis

Though one can apply the chi-square test to a single variable and judge whether the frequencies for each category are equal (or as expected), a chi-square is applied most commonly to frequency results reported in bivariate tables, and interpreting bivariate tables is crucial to interpreting the results of a chi-square test. Bivariate tabular analysis (sometimes called crossbreak analysis) is used to understand the relationship (if any) between two variables. For example, if a researcher wanted to know whether there is a relationship between the gender of U.S. undergraduates at a particular university and their footwear preferences, he or she might ask male and female students (selected as randomly as possible), “On average, do you prefer to wear sandals, sneakers, leather shoes, boots, or something else?” In this example, the independent variable is gender and the dependent variable is footwear preference. The independent variable is the quality or characteristic that the researcher hypothesizes helps to predict or explain some other characteristic or behavior (the dependent variable). Researchers control the independent variable (in this example, by sampling males and females) and elicit and measure the dependent variable to test their hypothesis that there is some relationship between the two variables.

To see whether there is a systematic relationship between gender of undergraduates at University X and reported footwear preferences, the results could be summarized in a table as shown in Table 1.

Each cell in a bivariate table represents the intersection of a value on the independent variable and a value on the dependent variable by showing [Page 145]how many times that combination of values was observed in the sample being analyzed. Typically, in constructing bivariate tables, values on the independent variable are arrayed on the vertical axis, while values on the dependent variable are arrayed on the horizontal axis. This allows one to read “across,” from values on the independent variable to values on the dependent variable. (Remember, an observed relationship between two variables is not necessarily causal.)

Reporting and interpreting bivariate tables is most easily done by converting raw frequencies (in each cell) into percentages of each cell within the categories of the independent variable. Percentages basically standardize cell frequencies as if there were 100 subjects or observations in each category of the independent variable. This is useful for comparing across values on the independent variable if the raw row totals are close to or more than 100, but increasingly dangerous as raw row totals become smaller. (When reporting percentages, one should indicate total N at the end of each row or independent variable category.)

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