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 park | No 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 team | No NBA team
|
|---|
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 park | NBA 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.