AP Statistics Section 4.2 Relationships Between Categorical Variables

Our concern in Chapter 3 and the first part of Chapter 4 was the relationship between two quantitative variables. We will now describe relationships between two or more categorical variables. Some variables, such as race or sex are categorical by nature. Other categorical variables are created by grouping values of a quantitative variable into classes.

The table below presents Census Bureau data describing the age and sex of college students. This is a _____-_____ table because it describes two categorical variables. (Age is categorical here because the students are grouped into age categories.) Age group is the ______ variable because each row in the table describes students in one age group. Sex is the ________ variable because each column describes one sex. The entries in the table are the counts of students in each age-by-sex class. two way row column row

Discrepancies may appear in tabular data Discrepancies may appear in tabular data. For example, the sum of entries in the “25 to 34” row is _________________. The entry in the total column is ______. The explanation is _________ error.

To best grasp the information contained in the table, first look at the distribution of each variable separately. The distributions of sex alone and age alone are called ___________________ because they appear at the right and bottom margins of the two-way table. marginal distributions

The distribution of a categorical variable says how often each outcome occurred. Usually it is advantageous to look at percents as opposed to counts.

Example 1: Determine the percent of college students in each age group.

Each marginal distribution from a two-way table is a distribution for a single categorical variable. We could use a pie graph or bar graph to display such a distribution.

The marginal distributions of sex and age do not tell us how the two variables are related. To describe relationships among categorical variables, calculate appropriate percents.

Example 2: Complete the tables below which give the conditional distribution of sex, given age. 54.7% 45.3% 63.1% 36.9%

When we compare the percent of women in two age groups we are comparing conditional distributions.   Comparing conditional distributions reveals the nature of the association between the sex and age of college students. Look at the bar graph at the right. The heights do not differ greatly but women are most common among the __________ age group. 35 or older

Example 3: Complete the tables below which give the conditional distribution of age, given sex. Male students are more likely to be __________ years old and quite a bit less likely to be _________. .01% 60.8% 20.4% 17.8% 0.8% 64.2% 21.7% 13.3%

CAUTION: No single graph (such as a scatterplot) portrays the form of the relationship between categorical variables. No single numerical measure (such as correlation) summarizes the strength of the association.

As is the case with quantitative variables, the effects of lurking variables can change or even reverse relationships between two categorical variables.

Example 4: Accident victims are sometimes taken by helicopter from the accident scene to a hospital. Helicopters save time. Do they also save lives? Complete the table at the right. 32% 24% 68% 76%

Notice that a greater percentage of helicopter patients died.

Here’s the data broken down by the seriousness of the accident Here’s the data broken down by the seriousness of the accident. Complete the tables at the right. 48% 60% 52% 40% 16% 20% 84% 80%

Among serious accident victims, the helicopter saves 52% compared with 40% for road transport. For less serious accidents, 84% of those transported by helicopter survive, versus 80% of those transported by road. Both groups have a higher survival rate when transported by helicopter.

The reason for the paradox in the data is

The lurking variable was

An association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group. This reversal is called _________ paradox. Simpson’s