Researchers often have to deal with frequencies instead of means and binomial proportions because they have their cases categorized in bins. The chi-squared distribution allows you to compare observed frequencies with expected frequencies. The chi-distribution works with qualitative variables with data based on categories rather than measurements. Until now, you have only dealt with quantitative variables.

Chi-tests are usually based on tables with a two-way structure. Each cell in a two-way table contains counts. However, the chi-test has one important condition: Each cell count must be at least 5. Could you use percentages instead? No, because percentages disregard sample size. For instance, "80 out of 100" is statistically better than "8 out of 10" but in either case, the percentage would be 80%.

The basic idea behind a chi-test is that the observed frequencies have to be tested against the expected frequencies. In other words, you need to create a copy of the observed table and replace the frequencies with expectations. How do you do this?

Figure 5.45 shows a situation in which the chi-test is an appropriate choice. You test the effect of a certain pill on the recurrence of estrogen-fed tumors versus the effect of placebos, and you end up with four categories and their frequency counts. Because of the frequencies, the chi-test is called for. The table of observed frequencies (A3:D6) has total calculations in the end row and column. The table of expected frequencies (A11:D14) is an exact replica of the table above it, except for the observed...