2.2 Summary Statistics for Two Categorical Variables

Use the grade-and-sport table from 2.1 (200 students):

The phrase after "among," "of the" or "given" tells you which total goes in the denominator.

• Joint relative frequency: one cell divided by the grand total. Freshmen who play a sport: 72/200 = 0.36.
• Marginal relative frequency: a row or column total divided by the grand total. All sport players: 100/200 = 0.50. All freshmen: 120/200 = 0.60. These come from the margins of the table, hence the name.
• Conditional relative frequency: a cell divided by its own row total or column total. You restrict attention to one category, then ask what share of it falls in a cell. Among freshmen, the proportion who play a sport is 72/120 = 0.60. Among sport players, the proportion who are freshmen is 72/100 = 0.72.

"Proportion of freshmen who play a sport" (72/120 = 0.60) and "proportion of sport players who are freshmen" (72/100 = 0.72) use the same cell but different denominators, so they answer different questions. Read the wording slowly.

Compare the conditional distribution of one variable across the categories of the other. If they're about the same, there's no association. If they differ, there is.

Here, 60% of freshmen play a sport, compared with 35% of seniors (28/80). Those conditional proportions are quite different, so grade and sport participation are associated for these students.

A conditional distribution is the full set of conditional proportions for one group, and it adds to 1. For freshmen: 60% sport, 40% no sport. For seniors: 35% sport, 65% no sport. Comparing whole conditional distributions, not single numbers, is the complete way to check for association.

Another way to see it: overall, 50% play a sport. If grade didn't matter, both grades would be near 50%. They aren't.

How different is different enough? For now, judge by size. In topics 3.14 and 3.15 you'll test whether a difference like this could be just chance.