Lesson 16: Categorical Associations

Objective:

Students will learn to construct, interpret, and calculate the joint relative frequencies of two-way frequency tables.

Vocabulary:

two-way frequency table joint relative frequency

Essential Concepts:

A two-way table is a summary of the association/relationship between two categorical variables. Joint relative frequencies answer questions of the form "what proportion of the people/objects had this value on the first variable and this value on the second?".

Lesson:

Launch the lesson by displaying the following scenario:

Rosa has a theory that cat owners are also musical. To find out, she decided to collect data that would help her understand the relationship between cat ownership and instrument playing among the students in her art class. She conducted a survey and found that out of the 35 students in her art class, 16 owned a cat and out of those that owned a cat, 7 played an instrument. She also discovered that 9 owned a cat, but did not play an instrument. There were also 9 students who neither owned a cat nor played an instrument.
Inform students that Rosa asked two questions that provided the data for her two-way frequency table. What could those two questions be?

Possible Answer:

Question 1—Do you play an instrument?

Question 2—Do you own a cat?
What variables did Rosa collect? What were the values of those variables? Answer: Variable 1: Owns Cat - yes/no. Variable 2: Plays Instrument - yes/no.
In pairs, write out on paper what the original data must have looked like.

Answer: Variable 1: Owns Cat. Variable 2: Plays Instrument

Owns Cat     Plays Instrument

Yes              Yes      (There are 7 of these)

Yes             No       (9 of these)

No              Yes      (10 of these)

No              No       (9 of these)
Inform students that today they will be looking at associations in categorical variables.

Note: If necessary, review the difference between questions for categorical and numerical variables.
Explain that their task is to summarize Rosa’s findings in one table that shows totals. Remind students to use their knowledge of data structures from Lesson 2, especially organizing in rows and columns.
Allow time for teams to wrestle with how to organize their data in one table. As teams work, walk around monitoring their data tables.
Select a few data tables to display and share with the entire class.
Explain the Anonymous Author strategy to students (see Instructional Strategies in Teacher Resources).
Display the data tables and ask students to engage in the Anonymous Author strategy. You may want to start the discussion by asking about the total number of students Rosa surveyed.
Make sure the last data table you display correctly shows a two-way frequency table. A two-way frequency table displays the data that pertains to two categories from one group. One category is represented in rows and the other is represented in columns. In this exercise, the group is a class of art students.

Cat Ownership and Instruments

Plays an instrument Does not play instrument Total

Owns a cat 7 9 16

No cats 10 9 19

Total 17 18 35
Based on the Cat Ownership and Instruments table, ask student teams to generate questions that can be asked and answered by the data.
In a Whip Around, ask student teams to share one of their questions.
Explain that a two-way frequency table can show relative frequencies. A relative frequency is how often something occurs in relation to the total number of occurrences, and is expressed as a proportion or percentage of a total. For example, what is the relative frequency of those who own a cat and play an instrument? Answer: 7/35 or 0.2 or 20%.

Note: Review how to write a proportion and how to express a proportion as a percent.

Ask students to calculate the relative frequencies for the entire table. They may check their calculations with a partner.

Cat Ownership and Instruments

Relative Frequencies

	Plays an instrument	Does not play instrument	Total
Owns a cat	7/35=0.20	9/35 ≈ 0.26	16/35 ≈ 0.46
No cats	10/35 ≈ 0.29	9/35 ≈ 0.26	19/35 ≈ 0.54
Total	17/35 ≈ 0.49	18/35 ≈ 0.51	35/35 = 1.00

In teams, students will generate 2 questions about 2 categorical variables. Allow teams 3-5 minutes to generate their questions. Students should not choose two random categorical variables. Rather, they should choose two categorical variables that they predict might be associated.
The team will create a two-way table that corresponds to their categorical variables.

Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

Homework

Rosa posed this statistical investigative question:

What proportion of students did not play an instrument and did not own a cat? Answer: 0.29

Use what you know about two-way tables to answer her question.