AP Statistics Section 4.2

Relationships Between Categorical 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. To analyze categorical data, we use groups or classes of individuals

that fall into various categories.

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

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.

493,3589,1904,1 494,3

rounding

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. The distribution of a categorical variable says how often each

outcome occurred. Usually it is advantageous to look at percents as

opposed to counts.

marginal distributions

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

%9.

16639

150

15.8% 21.0% 62.3%

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. How can we describe the relationship between age and sex of college students? To

describe relationships among categorical variables, calculate appropriate percents from the

counts given.

Example 2: Complete the tables below which give the conditional distribution of sex, given age.

54.7%

10365

5668

63.1%

2630

1660

45.3%

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%

2418 1715

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. How

discouraging.

But wait.

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 helicopter carries

patients who are more likely to die. The seriousness of the accident

was the _______ variable.lurking

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