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Algebra 1 View details

4. Bivariate Categorical Data

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Algebra 1 /

Chapter 4

Bivariate Categorical Data

Bivariate categorical data provides insights into relationships between two categorical variables. Within this realm, understanding joint and marginal relative frequencies is crucial. These frequencies help analysts discern patterns and relationships in a dataset. The joint relative frequency measures the proportion of occurrences of a specific combination of two categories. On the other hand, marginal relative frequencies focus on a single category, disregarding the other variable. Moreover, conditional relative frequency comes into play when we're keen to understand the proportion of a category given that another specific category has occurred. With these tools, researchers and analysts can derive meaningful conclusions and make informed decisions.

Lesson Settings & Tools

	11 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 11

This lesson familiarizes the learner with how to summarize and display categorical data. Possible associations and trends will then be recognized by interpreting data from real-world examples.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Analyzing Data From a School Survey

Paulina loves dramas and wants to know what her classmates think about them. She decided to conduct a survey asking $50$ students if they enjoy watching dramas. Paulina also noted if her classmates were $16$ years and under or older than $16 .$ Paulina wrote her findings in a notebook.

a How can these results be displayed in a single table?

b Is there an association between classmates who enjoying dramas and their age?

Discussion

Two-Way Frequency Tables, Joint and Marginal Frequencies

A two-way frequency table, also known as a two-way table, displays categorical data that can be grouped into two categories. One of the categories is represented in the rows of the table, the other in the columns. For example, the table below shows the results of a survey where $100$ participants were asked if they have a driver's license and if they own a car.

Here, the two categories are car and driver's license. Both have possible responses of yes and no. The numbers in the table are called joint frequencies. Also, two-way frequency tables often include the total of the rows and columns — these are called marginal frequencies. Select any frequency in the table below to display more information.

The sum of the Total row and the Total column, which in this case is

100,

equals the sum of all joint frequencies. This is called the grand total. A joint frequency of

43

shows that

43

people have a driver's license and own a car. A marginal frequency of

53

shows that

53

people do not have a car. The rest of the numbers from the table can also be interpreted.

Discussion

Making a Two-Way Frequency Table

Organizing data in a two-way frequency table can help with visualization, which in turn makes it easier to analyze and present the data. To draw a two-way frequency table, three steps must be followed.

Determine the categories.
Fill the table with the given data.
Determine if there are any missing frequencies. If so, find those.

Suppose that $53$ people took part in an online survey, where they were asked whether they prefer top hats or berets. Out of the $18$ males that participated, $12$ prefer berets. Also, $15$ of the females chose top hats as their preference. The steps listed above will now be used to analyze and present the data.

Determine the Categories

First, the two categories of the table must be determined, after which the table can be drawn without frequencies. Here, the participants gave their hat preference and their gender, which are the two categories. Hat preference can be further divided into top hat and beret, and gender into female and male.

The total row and total column are included to write the marginal frequencies.

Fill the Table With Given Data

The given joint and marginal frequencies can now be added to the table.

Find Any Missing Frequencies

Using the given frequencies, more information can potentially be found by reasoning. For instance, because

12

out of the

18

males prefer berets, the number of males who prefer top hats is equal to the difference between these two values.

18 - 12 = 6

Therefore, there are

6

males who prefer top hats. Since there are

15

females who prefer top hats, the number of participants who prefer this type of hat is the sum of these two values.

6 + 15 = 21

It has been found that

21

participants prefer top hats. Continuing with this reasoning, the entire table can be completed.

Example

Night Owl or Early Bird?

Zain has a job leading backpackers on excursions in the High Sierras. To better understand what time of day to plan certain activities, Zain posed a question to $50$ backpackers about their sleep patterns: Are you a night owl or an early bird? Zain then categorized the participants by sleep pattern and age — younger than $30$ and $30$ or older. Here is part of what was gathered.

$11$ people age $30$ or older said they are early birds.
$23$ people younger than $30$ participated in the survey.
$28$ people, of any age, said they are night owls.

Zain made a two-way frequency table with the data they collected. Unfortunately, some of the data values got smudged and are unable to be read! The missing data values have been replaced with letters, for now.

Find the missing joint and marginal frequencies to help Zain complete the table. Zain's next excursion depends on it.

Hint

Begin by finding the number of people age $30$ or older who participated in the survey. To do so, calculate the difference between the grand total and the number of participants younger than $30 .$ That would be $50$ divided by $23 .$

Solution

Start by finding the missing marginal frequency of the last column of the table, labeled A. Note that

50

people participated in the survey and

23

of them are younger than

30 .

Therefore, the number of participants who are

30

or older can be found by calculating the difference between these two values.

50 - 23 = 27

This information can be added to the table.

With this information, the joint frequency

B

that represents the number of night owls aged

30

or older can be calculated. Of the

27

participants aged

30

or older,

11

are early birds. Therefore, the number of night owls aged

30

or older is the difference between these two values.

27 - 11 = 16

This information can also be added to the table.

The missing marginal frequency

C

in the last row will now be calculated. Of the

50

participants,

28

said they are night owls. To find the number of early birds, the difference between these two values will be calculated.

50 - 28 = 22

One more cell can be filled in!

Finally, the missing joint frequencies

D

and

E

in the first row can be found.

D : E : 22 - 11 = 11 28 - 16 = 12

The table can be completed with this information! Click on each cell to see its interpretation.

Discussion

Joint and Marginal Relative Frequencies

In a two-way frequency table, a joint relative frequency is the ratio of a joint frequency to the grand total. Similarly, a marginal relative frequency is the ratio of a marginal frequency to the grand total. Consider the following example of a two-way table.

Here, the grand total is $100 .$ The joint and marginal frequencies can now be divided by $100$ to obtain the $joint$ and $marginal$ relative frequencies. Clicking in each cell will display its interpretation.

Example

Joint and Marginal Relative Frequencies for Night Owls and Early Birds

Previously, Zain made a two-way frequency table about backpackers sleep patterns.

two-way table representing the data collected by Zain

Zain wants to dig deeper into the data for even more clear interpretations, so they plan to calculate the joint and marginal relative frequencies.

two-way table to be completed with the joint and marginal relative frequencies

Zain is beginning to feel a little tired themselves. Give them a hand and complete the table by matching each value with its corresponding cell.

Hint

Divide the joint and marginal frequencies by the grand total.

Solution

To calculate the joint and marginal relative frequencies, the joint and marginal frequencies must be divided by the grand total, $50 .$

The table below shows the joint and marginal relative frequencies.

One finding — of a variety — based on the joint and marginal relative frequencies, shows that about one-third of the participants who are $30$ or older are night owls. Additionally, Zain can see that the participants are almost equally distributed among the categories, as both pairs of marginal relative frequencies have values close to $50 - 50 .$

Discussion

Conditional Relative Frequency

A conditional relative frequency is the ratio of a joint frequency to either of its two corresponding marginal frequencies. Alternatively, it can be calculated using joint and marginal relative frequencies. As an example, the following data will be used.

Referring to the column totals, the left column of joint frequencies should be divided by $67$ and the right column by $33 .$ Furthermore, since the column totals are used, the sum of the conditional relative frequencies of each column is $1 .$

The resulting two-way frequency table can be interpreted to obtain the following information.

Out of all the participants with a driver's license, about $64 %$ of them own a car.
Out of all the participants with a driver's license, about $36 %$ of them do not own a car.
Out of all the participants without a driver's license, about $12 %$ of them own a car.
Out of all the participants without a driver's license, about $88 %$ of them do not own a car.

Example

Conditional Relative Frequencies for Night Owls and Early Birds

Using their two-way frequency table, Zain wants to continue improving the interpretation of their data by finding the conditional relative frequencies.

Zain will use the row totals to make the calculations.

Zain, really feeling close to being able to make some rock-solid interpretations, could still use a bit more help!

Hint

Since Zain uses the row totals, the joint frequencies in the first row must be divided by $23$ and the joint frequencies in the second row must be divided by $27 .$

Solution

Zain uses the row totals. Therefore, the joint frequencies in the first row must be divided by $23$ and the joint frequencies in the second row must be divided by $27 .$

The table below shows the conditional relative frequencies.

Extra

Applying Zain's Findings to an Actual Backpacking Trip

Zain interprets the various findings as reason to believe when planning night activities, like storytelling over a campfire, they could tailor the stories for an older generation. Interestingly, the older backpackers, as a whole, seem to prefer nights more than the younger backpackers. Zain can now plan according to these interpretations.

Discussion

Looking for Associations in Data

How to make two-way frequency tables and interpret the information presented in those tables has been shown. Next, how to recognize associations in data that come from a two-way table will be discussed.

Method

Recognizing Associations in Data

Studying the conditional relative frequencies of a two-way frequency table, it is possible to find potential associations in the data. As an example, the following survey results will be analyzed.

First, the conditional relative frequencies can be found by dividing each joint frequency by the corresponding column's marginal frequency.

Two-way table with conditional frequencies

As can be seen, $64 %$ of people with a driver's license own a car, while $88 %$ of people without a drivers license do not own a car. Therefore, an association between having a driver's license and owning a car might exist. On the other hand, finding the conditional relative frequencies using the row's marginal frequencies gives a slightly different result.

As can be seen, among car owners, almost everyone has a driver's license. Meanwhile, among the people who do not own a car, roughly half have a driver's license. This observation shows that car ownership is associated with having a driver's license, while not owning a car is not associated with not having a driver's license.

Car No car \Rightarrow Driver ’ s license ✓ \Rightarrow No driver ’ s license \times

Consider a different two-way frequency table that illustrates a stronger association when using the marginal frequencies of only one variable.

A person's bed time might be dependent on their age, but their age is not dependent on their bed time. Because of this, it is recommended to use the age's marginal frequencies when finding the conditional relative frequencies. This gives the distribution of bed time given a certain age span.

Two-way table for bedtime with conditional frequencies

The table illustrates a trend of going to bed after

9 : 30

pm the older a person becomes. Therefore, there is an association.

Closure

Determining Associations From a Survey's Data

The challenge at the beginning of this lesson showed the following information that Paulina gathered when she conducted a survey at her school.

The following challenge questions were then asked.

a How can these results be displayed in a single table?

b Is there an association between a student enjoying watching dramas and their age?

Answer

Two-way table with the data from the survey

b Yes, younger students are more likely to watch dramas.

Hint

a Use a two-way frequency table.

b Analyze the conditional relative frequencies.

Solution

a A two-way frequency table can be used to display the given information using a single table. Whether the students enjoy dramas or not and their age ranges can be chosen as the two main categories of the table. The data Paula collected can then be entered into the appropriate cells.

b The conditional relative frequencies of the data can be found and analyzed to help determine if the data values are related. Begin by calculating the marginal frequencies. Add the joint frequencies of each row and each column of the table to the Total's row and column, respectively.

First, focus on the cells related directly to the student's age. There are $20$ students under the age of $16$ and there are $30$ students who are $16$ or older. These marginal frequency totals occur regardless of the student's interest in dramas.

Next, divide each joint frequency by the corresponding marginal frequency related to age.

The table shows that $70 %$ of the students $16$ years old or younger watch dramas. Whereas, only $13 %$ of the students older than $16$ watch dramas. A similar analysis can be done with the marginal frequencies according to the Yes or No rows.

Using the marginal frequencies of $18$ and $32,$ make the necessary calculations.

The results indicate that there are associations between watching dramas and a student's age. It is seen that $80 %$ of the students under $16$ watch dramas. Whereas, $80 %$ of students $16$ or older do not watch dramas.

While Paula feels excellent about her interpretations of the data, she knows that these associations are limited to this survey's sample. Her interpretations apply only to her classmates who joined the survey, not how everyone at the school feels about dramas.

Level 1

Level 2

Level 3

Bivariate Categorical Data

Exercise 1.1

Heichi conducted a survey and asked $367$ students if they usually bring lunch from home. The results of the survey are shown in a two-way table.

How many sophomores were surveyed?

How many freshmen were surveyed?

How many students bring lunch from home?

We will begin by looking at the two-way table that shows the results of the survey. Here, we can identify the joint frequencies that correspond to sophomores.

To find how many sophomores were surveyed, we need to find the corresponding marginal frequency. A marginal frequency is the sum of the joint frequencies of a row or a column. To find how many sophomores were surveyed we will add the values in the Sophomore row, which is the second row of the table. Sophomores Surveyed: 62+ 119=181

As in Part A, to find how many freshmen were asked we will first identify the joint frequencies associated with the freshmen.

Now we need to find the freshmen marginal frequency. To do so we add the joint frequencies in the Freshman row, which is the first row of the table. Freshmen Surveyed: 82+ 104=186

Now we are asked to find the number of students that bring lunch from home. To do so, let's first identify the joint frequencies that correspond to the students that bring lunch from home.

We need to find the marginal frequency for the students that bring lunch. To do so, we need to add the joint frequencies of the first column. Students That Bring Lunch: 82+ 62=144

A survey was conducted in which people who exercise were asked if they prefer doing a team sport or going to the gym. The results are displayed in a two-way table.

two way table exercise preferences joint frequencies

What percent of people who exercise prefer going to the gym? Round the answer to one decimal place.

What percent of people who exercise are women and prefer doing a team sport? Round the answer to one decimal place.

To find the percent of people who exercise that prefer to go to the gym, we need to find the marginal frequencies by adding the joint frequencies of rows and columns.

To find the percent of each category, we divide the joint frequencies by the grand total, which is 358. By doing so we obtain the marginal and joint relative frequencies.

We can calculate the percent of people who exercise that prefer going to the gym by multiplying the corresponding marginal relative frequency by 100. 0.556 * 100 = 55.6 % Therefore, 55.6 % of people who exercise prefer going to the gym.

To find the percent of women that prefer practicing with a sport, we need to find the corresponding joint relative frequency from the table in Part A. This frequency is the intersection of the second row and the first column.

Now we can multiply this value by 100 to find the percent. 0.184 * 100 = 18.4 % Therefore, 18.4 % of people who exercise are women who prefer doing a team sport.

The two-way frequency table describes how many students passed a test and how many students reported that they had studied during the previous weekend.

Determine the contents of the cells labeled

A

through

E

The two-way frequency table describes how many students and teachers voted on building a new cafeteria.

Determine the contents of the cells labeled

A

through

E .

Examining the two-way frequency table, we see that in the row and column containing the totals we only have one missing value. This means we can solve for both B and E. B+10=50 &⇔ B= 40 E+38=50 &⇔ E= 12 Let's add this to the two-way frequency table.

In the row describing a passing grade, now A is the only unknown. Similarly, in the column describing how many students did not study before the test, only D is unknown. This means we can find both A and D. A+6=40 &⇔ A= 34 6+D=12 &⇔ D= 6 Let's add this to the diagram.

We can determine C, too. 34+C=38 ⇔ C= 4 Now we can complete the two-way frequency table.

Let's summarize what we have found. A &→ 34 B &→ 40 C &→ 4 D &→ 6 E &→ 12

Here, A and D are the only unknowns in the Yes and No columns. Therefore, we can determine their values. 56+10=D &⇔ D= 66 A+7=49 &⇔ A= 42 Let's add this to the diagram.

Now we have enough information to calculate B, C, and E. 56+42=B &⇔ B= 98 10+7=C &⇔ C= 17 66+49=E &⇔ E= 115 With this information we can complete our table.

Let's summarize. A &→ 42 B &→ 98 C &→ 17 D &→ 66 E &→ 115

Ignacio surveyed $190$ men and $160$ women passing through an airport and asked whether they washed their hands on a regular basis. Of these people, $143$ men and $130$ women reported that they washed their hands on a regular basis.

Find the survey's lowest joint frequency.

Find the survey's lowest marginal frequency.

The joint frequencies in a two-way frequency table are the entries that are not the totals.

We know that Ignacio surveyed 190 men and 160 women, and from these 143 men and 130 women reported that they wash their hands regularly. Let's add this information to the two-way frequency table.

Now we can determine the unknown joint frequencies by subtracting the number of men and women answering Yes from the totals in each of the corresponding rows. Let's label the number of men and women that answered No as X and Y, respectively. ccc 143+X=190 & ⇔ & X=47 130+Y=160 & ⇔ & Y=30 Let's add these results to the diagram.

We can see that the survey's lowest joint frequency is 30.

The marginal frequencies of a two-way frequency table are the sums of the rows and columns. In other words, we want to find the Total of each category.

From the given information, we can identify two of the marginal frequencies. The total number of men surveyed is 190, and the total number of women surveyed is 160.

Now, let's calculate the marginal frequencies corresponding to Washes hands. We can do this by adding the numbers in the Yes and No columns. 143+130&= 273 47+30&= 77 Let's add this to the information in our table.

We can see that lowest marginal frequency is 77.

The following two-way frequency table shows the results of a survey given to $100$ people, investigating whether or not they own a dog.

Two way frequency table representing the survey information

Determine the joint relative frequencies.

Determine the marginal relative frequencies.

In a two-way frequency table, the joint frequencies are the entries that are not the totals.

A joint relative frequency is the ratio of a joint frequency to the grand total. Therefore, to find the joint relative frequencies we need to divide each of the joint frequencies in the two-way frequency table by 100, which is the grand total. Joint Relative Frequencies [1em] A: 10/100=0.10 B: 24/100=0.24 [1em] D: 20/100=0.20 E: 46/100=0.46 [1em]

The marginal frequencies in a two-way frequency table are the totals of each row and column.

A marginal relative frequency is the ratio of the a marginal frequency to the grand total. Therefore, to find the marginal relative frequencies we need to divide each of the marginal frequencies in the table by 100, which is the grand total. Marginal Relative Frequencies [1em] G: 30/100=0.30 C: 34/100=0.34 [1em] H: 70/100=0.70 F: 66/100=0.66 [1em]

Bivariate Categorical Data

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