### Key Concepts

- Construction of two-way frequency table.
- Interpretation of two-way frequency table.
- Construct and interpret a two-way frequency table.

**Introduction**:

### Two-way frequency table:

Information about people who are surveyed can be captured in two-way frequency tables. A two-way frequency table is a table of data that compares two variables (bivariate). It separates responses by a characteristic of the respondents.

It is a way of displaying the frequencies of categorical variables, usually displayed in a table like the one below.

### 4.4.1 Construction of Two-Way Frequency Table

**Example 1:**

Jeffrey owns a nail salon, and he is looking at some advertising options. He wants to know which gender and which service is the most popular so that he can target a particular audience. He looks at his records from the last month and found the following:

58 females – manicure

77 females – pedicure

31 males – manicure

27 males – pedicure

Construct a two-way frequency table that shows the relationship between the type of service and gender.

**Solution:**

**ONE WAY: **

Construct a two-way frequency table.

**ANOTHER WAY: **

Construct a different two-way frequency table.

### 4.4.2 Interpretation of Two-Way Frequency Table

**Example 2:**

Ann surveys students in grade 8 and grade 9 and asks each student which pet they prefer. Their responses are in the table below.

- How many 8
^{th}graders preferred fish?

- Which pet was the most popular among all the students surveyed?

- How many students of grade 8 are represented in the survey?

**Solution:**

- How many 8
^{th}graders preferred fish?

22 students from grade 8 preferred fish.

- Which pet was the most popular among all the students surveyed?

The most popular pet is a dog, as it is preferred by 117 students.

- How many students of grade 8 are represented in the survey?

127 students of grade 8 are represented in the survey.

### 4.4.3 Construct and Interpret a Two-Way Frequency Table

**Example 3:**

One hundred students were asked how they traveled to school. Of the girls, 18 rode in a car, 7 rode the bus, and 28 took the train. Of the boys, 13 took the train, 24 rode in a car, and 10 rode the bus. Construct a two-way frequency table. Then tell which mode of transportation is the most popular. Explain.

**Solution:**

The most popular mode of transportation is a car, as a total of 44 students travel by car.

## Exercise:

- A _________________ table displays the relationship between paired categorical data.
- The data that fall into categories and do not have an inherent order is called _______________.
- A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations.

4. Do the data in the two-way frequency table support the following statement? Explain. There are more middle school students who wear glasses than high school students who wear contacts.

5. You ask 70 of your classmates if they have any siblings. Complete the two-way frequency table to show the results of the survey.

6. A company surveyed 200 people and asked which car model they preferred. Complete the two-way frequency table to show the results of the survey.

7. Students at a local school were asked, “About how many hours do you spend on homework each week?” The two-way frequency table shows the results of the survey. Classify the statement below as true or false. Explain.

8. At one point last year, the local animal shelter had only cats and dogs. There were 74 animals in all. Of the cats, 25 were male, and 14 were female. Of the dogs, 23 were male, and 12 were female. Construct a two-way frequency table of the data.

9. Deshaun surveyed the seniors at his high school to find out who eats and doesn’t eat breakfast regularly. Their responses are in the table below.

- How many females “eat breakfast regularly”? _____

This is a ____________ frequency

- 260 is a _____________ frequency. It represents ______________________.

10. Looking at just the marginal frequencies (row & column totals), what can we conclude about the 3 activities?

- Dance is way more interesting.
- Sports is the least chosen activity.
- TV is the preferred activity.

### What have we learned:

■ Construct a two-way frequency table to show relationship between two variables.

■ Interpret a two-way frequency table and draw a conclusion.

■ Construct and interpret a two-way frequency table for the given scenario.

#### Concept Map:

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