# Lesson 17: Interpreting Two-Way Tables

## Lesson 17: Interpreting Two-Way Tables

### Objective:

Students will calculate conditional, marginal, and joint frequencies and explain what they mean in the context of the data.

### Materials:

1. Poster paper

2. Markers

3. Analyzing Categorical Variables (LMR_1.18)

Advanced preparation required (see step 19 below)

4. Interpreting Categorical Variables (LMR_1.19)

### Essential Concepts:

Essential Concepts:

Marginal (relative) frequencies tell us about the distribution of a single variable. Conditional relative frequencies tell us about the distribution of one variable when "subsetting" the other.

### Lesson:

1. Time Use Campaign Data Collection Monitoring:

1. Display the IDS Campaign Monitoring Tool, found at https://portal.idsucla.org

2. Inform students that you will be monitoring their data collection again today.

1. See User List and sort by Total. Ask: Who has collected the most data so far?

2. Click on the pie chart. Ask: How many active users are there? How many inactive users are there?

3. See Total Responses. How many responses have been submitted?

4. Using TPS, ask students to think about what can they do to increase their data collection.

2. Remind students that this is the last day to collect data.

3. Ask student teams to take out the 2 questions and the two-way table that they created in the previous day’s lesson.

4. Before teams ask the class their questions, ask them to strategize about how they will collect and record their data, because they can only ask the 2 questions.

5. Students in the class will respond to each question by raising their hands.

6. In a Whip Around, have each team ask their 2 questions. Pause briefly between teams so that the asking team has time to collect and record their data.

7. Students will use their frequency tables before the end of the lesson.

8. Recall that in the previous lesson, students learned to calculate relative frequencies. Now it’s time to look at other ways of understanding a two-way frequency table.

9. Display the Cat Ownership and Instruments table:

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
10. Suppose that we want to know the following information (display questions):

1. How many students own a cat? 16

2. What is the proportion of students who own a cat? 16/35 ≈ 0.46

3. What is the proportion of students who do not play an instrument? 18/35 ≈ 0.51

11. In teams, discuss where on the table you would find this information and how you would calculate it. The specific answers are not important; but what is important is to know how to obtain the information. Possible answer: You would find the proportion of students who do not play an instrument by dividing the number in the “Total” row that is in the "Does not play instrument" column by the total number of students (35).

12. After a few minutes, ask a team to volunteer a response. Mark up the margins on the table to show that the cells with the initial total counts are called marginal frequencies. Note: 10 b and c are marginal relative frequencies.

13. Now suppose that we want to know the following information (display questions):

1. How many students own a cat and play an instrument? 7

2. What is the proportion of students that own a cat and play and instrument? 7/35=0.2

3. What is the proportion of students who do not own a cat and play an instrument? 10/35 ≈ 0.286

14. In teams, discuss where on the table you would find this information and how you would calculate it. The specific answers are not important; but what is important is to know how to obtain the information. Possible answer: You would find the answers in the cells that make up the body of the table. The value for each proportion is the frequency for each cell over the total number of observations.

15. After a few minutes, ask a team to volunteer a response. Mark up the cells in the body of the table to show that the cells with the initial counts are called joint frequencies. Note: 13 b and c are joint relative frequencies.

16. Finally, suppose that we wanted to answer the question: Do a greater proportion of students in Rosa’s art class who do not own cats prefer to play an instrument than those who do own cats?

17. In teams, discuss where on the table you would find this information and how you would calculate it. The specific answers are not important; but what is important is to know how to obtain the information.

18. After a few minutes, ask a team to volunteer a response. Encourage students to agree or disagree with the explanations provided. Lead students to see that the total for the “No cats” row is important because we are only concerned with that subset of the group. Mark up “No cats” row on the table to show that we have conditioned, or are bound, by this variable. Compare the values that show the conditional relative frequency for the row. More non-cat owners slightly prefer to play an instrument (display table below).

Cat Ownership and Instruments

Conditional Relative Frequencies by Row

Plays an instrument Does not play instrument Total
Owns a cat 7/16 ≈ 0.44 9/16 ≈ 0.56 16/16 ≈ 1.00
No cats 10/19 ≈ 0.53 9/19 ≈ 0.47 19/19 ≈ 1.00
Total 17/35 ≈ 0.49 18/35 ≈ 0.51 35/35 = 1.00

Note: This is a conditional relative frequency by row. We can also calculate conditional relative frequencies by column if we were interested in knowing the difference in cat preference for those who play instruments versus those who don’t.

19. Distribute one full set of cards from the Analyzing Categorical Variables file (LMR_1.18) to each student team.

Print the Analyzing Categorical Variables file (LMR_1.18). The handouts can then be cut into a total of 20 cards (12 visuals, 8 numerical summaries). You will need enough sets of the cards for each student team to share one full set. For example, if there are 5 student teams in a class, then 5 copies of the file will need to be printed so that each team gets all 20 cards.

20. Distribute LMR_1.19_Interpreting Categorical Variables to each student team.

21. Each student team will work together and decide which visualization(s) and numerical summaries can be used to answer each statistical question. They will then answer each statistical question, citing a numerical summary as evidence.

Note: Student teams may tape or glue visuals and numerical summaries onto LMR_1.19, or they can simply write the plot letter and table number in the appropriate box. The blank column is for student teams to write a statistical question than can be answered with a visual and a numerical summary that was not used.

22. After student teams have been allotted ample time to complete LMR_1.19, lead a class discussion to go over the answers. It is extremely important to have students justify their answers by referring to their visuals and tables. For example, the statistical question “How many students neither own a cat or play an instrument?” can be answered with Plot E, Plot G, Plot K, Plot I, and with Tables 1 and 7.

23. Ask students to refer back to the two-way frequency tables they created earlier. Have each team create one poster that shows their two-way frequency table. Then, ask each team to ask 4 questions about the data in their table that must be answered by a:

1. marginal frequency

2. marginal relative frequency

3. joint relative frequency

4. conditional relative frequency (either by row or column)

24. If time permits, pair teams up and ask them to present their findings to each other.

### 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 & Next Day

Using the data below, generate 2 questions: one must be answered with a marginal relative frequency and the other must be answered by a conditional relative frequency.

Gender and the Color Red

Which emotion do you most relate with the color red?

Love Anger Fear Total
Male 7 11 5 23
Female 12 15 10 37
Total 19 26 15 60

Lab 1G: What’s the FREQ?

Complete Lab 1G prior to the Practicum.