{{ toc.name }}
{{ toc.signature }}
{{ toc.name }} {{ 'ml-btn-view-details' | message }}
{{ stepNode.name }}
{{ 'ml-toc-proceed' | message }}
Lesson
Exercises
Recommended
Tests
An error ocurred, try again later!
arrow_back
eCourses /
Algebra 1 /
Chapter {{ article.chapter.number }}
{{ article.number }}. 

{{ article.displayTitle }}

{{ article.intro.summary }}
{{ 'ml-btn-show-less' | message }} {{ 'ml-btn-show-more' | message }} expand_more
{{ 'ml-heading-abilities-covered' | message }}
{{ ability.description }} {{ ability.displayTitle }}

{{ 'ml-heading-lesson-settings' | message }}

{{ 'ml-lesson-show-solutions' | message }}
{{ 'ml-lesson-show-hints' | message }}
{{ 'ml-lesson-number-slides' | message : article.intro.bblockCount}}
{{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount}}
{{ 'ml-lesson-time-estimation' | message }}
Image Credits expand_more
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 students if they enjoy watching dramas. Paulina also noted if her classmates were years and under or older than 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 participants were asked if they have a driver's license and if they own a car.

Two-way table
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.
two-way table
The sum of the Total row and the Total column, which in this case is equals the sum of all joint frequencies. This is called the grand total. A joint frequency of shows that people have a driver's license and own a car. A marginal frequency of shows that 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.

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

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

1
Determine the Categories
expand_more

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.

two-way table

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

2
Fill the Table With Given Data
expand_more

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

two-way table
3
Find Any Missing Frequencies
expand_more
Using the given frequencies, more information can potentially be found by reasoning. For instance, because out of the males prefer berets, the number of males who prefer top hats is equal to the difference between these two values.
Therefore, there are males who prefer top hats. Since there are females who prefer top hats, the number of participants who prefer this type of hat is the sum of these two values.
It has been found that participants prefer top hats. Continuing with this reasoning, the entire table can be completed.
two-way table

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 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 and or older. Here is part of what was gathered.

  • people age or older said they are early birds.
  • people younger than participated in the survey.
  • 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.

two way table
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 or older who participated in the survey. To do so, calculate the difference between the grand total and the number of participants younger than That would be divided by

Solution
Start by finding the missing marginal frequency of the last column of the table, labeled A. Note that people participated in the survey and of them are younger than Therefore, the number of participants who are or older can be found by calculating the difference between these two values.
This information can be added to the table.
two way table
With this information, the joint frequency that represents the number of night owls aged or older can be calculated. Of the participants aged or older, are early birds. Therefore, the number of night owls aged or older is the difference between these two values.
This information can also be added to the table.
two way table
The missing marginal frequency in the last row will now be calculated. Of the participants, said they are night owls. To find the number of early birds, the difference between these two values will be calculated.
One more cell can be filled in!
two way table
Finally, the missing joint frequencies and in the first row can be found.
The table can be completed with this information! Click on each cell to see its interpretation.
two way table

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.

two-way table

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

two-way table

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,

finding the joint and marginal relative frequencies

The table below shows the joint and marginal relative frequencies.

two-way table showing 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 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


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.

two-way table

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

two-way table

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 of them own a car.
  • Out of all the participants with a driver's license, about of them do not own a car.
  • Out of all the participants without a driver's license, about of them own a car.
  • Out of all the participants without a driver's license, about 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.

two way table

Zain will use the row totals to make the calculations.

two way table
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 and the joint frequencies in the second row must be divided by

Solution

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

two way table

The table below shows the conditional relative frequencies.

two way table

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.

Two-way table

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, of people with a driver's license own a car, while 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.

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

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 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.

Survey Results

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
a
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.
Two-way table with the data from the survey
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.
Marginal frequencies are added to the two-way table

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

Conditional relative frequencies are identified

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

Joint relative frequencies are divided by marginal frequencies according to the age column

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

Marginal frequencies are added to the two-way table

Using the marginal frequencies of and make the necessary calculations.

Joint Frequncies are divided by Marginal relative frequencies of yes or no rows

The results indicate that there are associations between watching dramas and a student's age. It is seen that of the students under watch dramas. Whereas, of students 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.

{{ grade.displayTitle }}