# Associations Between Categories

Often there are two different ways of splitting up a collection into categories, and we want to know if those ways are related. We will study a few different ways of dividing the states of the United States into categories, and see if we can determine the relationship between these divisions.

## Frequencies

A map of the United States is drawn to the left, with the Mississippi River marked in blue. Of the 50 states in the U.S., 26 are east of the Mississippi, and 24 are west of the Mississippi. (We count Minnesota and Louisiana — the only two states that the Mississipi passes through in addition to bordering — as west of the Mississippi. We also count Alaska and Hawaii as west of the Mississippi.)

The number of objects in a category is called a frequency. So the number of states on each side of the Mississippi is also called the frequency of states on that side of the Mississippi.

The fraction of states which are east of the Mississippi is \$\$26/50 = 52%\$\$. The fraction (or percentage) of elements that have some property is called a relative frequency. So 52% is the relative frequency of states which are east of the Mississippi.

 What is the relative frequency of states west of the Mississippi?

## Location versus national parks

We would like to understand how state location is related to other ways of splitting up the states, such as by whether or not they contain certain features. We’ll start by comparing each state’s location to whether or not the state contains a national park. The map to the left draws states with national parks in green.

The table below the map to the left gives the number of states both east and west of the Mississippi, divided up by whether or not they contain at least part of a national park (as of 2018). For example, there are 6 states west of the Mississippi which do not have a national park (Louisiana, Missouri, Iowa, Nebraska, Kansas, and Oklahoma). This is referred to as the joint frequency of the two categories (being west of the Mississippi and not having a national park).

 How many states east of the Mississippi do not have a national park? That is, what is the joint frequency of being east of the Mississippi and not having a national park?

A table like the one on the left, giving the joint frequencies of two categorizations, is called a two-way frequency table.

The total number of states is 50. This means that the fraction of states which are west of the Mississippi and have no national parks is \$\$6/50\$\$, or 12%. A joint frequency divided by the total number of data elements like this is called a joint relative frequency, so 12% is the joint relative frequency of being west of the Mississippi and not having a national park.

What is the joint relative frequency of each of the other three groups of states shown in the table?

Relative
frequency
National parkNo national park

We want to find the answer to questions like: “Where are states more likely to have national parks, east of the Mississippi or west of the Mississippi?”

Of the 26 states east of the Mississippi, 9 of them have national parks. So the fraction of states east of the Mississippi which have national parks is \$\$9/26≈34.6%\$\$ (rounded to one decimal place). This is called the conditional relative frequency of states with national parks among states east of the Mississippi.

 What is the conditional relative frequency of states with national parks among states west of the Mississippi?
 Which half of the country has a larger conditional relative frequency of national parks: the half east of the Mississippi or the half west of the Mississippi?
 Are national parks more common among states east of the Mississippi or west of the Mississippi?
 Because western states joined the United States later, they tend to be larger. This means there is more space in them for national parks. Does this fit with your answer to the last question?

## Location versus NBA teams

The map to the left draws states with NBA basketball teams in purple (as of 2018). The table below the map shows the number of states both east and west of the Mississippi, divided up by whether or not they have an NBA team. (Some states have more than one NBA team, and two NBA teams are not in any U.S. state — the Toronto Raptors are in Canada, and the Washington Wizards are in the District of Columbia.)

Fill in the table below, giving the joint relative frequencies corresponding to the table on the left.

Relative
frequency
NBA teamNo NBA team
 What is the conditional relative frequency of states with NBA teams among states east of the Mississippi? Give your answer as a fraction and as a percentage, rounded to the nearest tenth of a percent.
 What is the conditional relative frequency of states with NBA teams among states west of the Mississippi?
 Which half of the country has a larger conditional relative frequency of NBA teams: the half east of the Mississippi or the half west of the Mississippi?
 Are NBA teams more common among states east of the Mississippi or west of the Mississippi?

The conditional relative frequencies of both having a national park and having an NBA team, among states east and west of the Mississippi, are shown in the table below.

Conditional relative
frequency
National parkNBA team
East of the Mississippi
West of the Mississippi

As you have seen, national parks are more common west of the Mississippi (where states are larger) and NBA teams are more common east of the Mississippi (where there are more big cities).

 Which is larger: the difference between the conditional relative frequencies for national parks, or that difference for NBA teams?
 Which are more concentrated in states on one side of the Mississippi: national parks or NBA teams?
 Which are more evenly spread across states on both sides of the Mississippi: national parks or NBA teams?

Based on these computations, we would say that there is a strong association between which side of the Mississippi a state is on and whether it has a national park: states west of the Mississippi are significantly more likely to have a national park. On the other hand, there is not a particularly strong association between which side of the Mississippi a state is on and whether it has an NBA team: states on either side of the Mississippi are about equally likely to have an NBA team.