Lesson 9 Two-Way Streets Practice Understanding

Learning Focus

Organize two-variable categorical data.

Interpret two-way frequency tables.

What information does a two-way table reveal?

What information is highlighted when data is interpreted from relative frequency tables?

Open Up the Math: Launch, Explore, Discuss

Rashid is in charge of determining the upcoming after-school activity. To determine the type of activity, Rashid asked several students whether they prefer to have a dance or to play a game of soccer. As Rashid collected preferences, he organized the data in the following two-way frequency table:

	Girls	Boys	Total
Soccer	$14$	$40$	$54$
Dance	$46$	$6$	$52$
Total	$60$	$46$	$106$

Rashid is feeling unsure of the activity he should choose, based on the data he has collected, and is asking for help. To better understand how the data is displayed, it is useful to know that the outer numbers, located in the margins of the table, represent the total frequency for each row or column of corresponding values and are called marginal frequencies. Values that are part of the inner body of the table are created by the intersection of information from the column and the row, and they are called the joint frequencies.

1.

Using the data in the table, construct a viable argument, and explain to Rashid which after-school event he should choose. To support your argument, use the following sentence frame to get started.

I think the after-school event should be because:

out of prefer .

This means .

Two-way frequency tables allow us to organize categorical data in order to draw conclusions. For each set of data below, create a frequency table. When each frequency table is complete, write three sentences about observations of the data, including any trends or associations in the data.

2.

Data set: There are $45$ total students who like to read books. Of those students, $12$ of them like nonfiction and the rest like fiction. Four girls like nonfiction. Twenty boys like fiction.

	Fiction	Nonfiction	Total
Boys
Girls
Total

Observation 1: out of prefer .

Observation 2:

Observation 3:

3.

Data set: $35$ seventh graders and $41$ eighth graders completed a survey about the amount of time they spend on homework each night. $50$ students said they spent more than an hour. $12$ eighth graders said they spend less than an hour each night.

			Total
More than one hour
Less than one hour
Total

Observation 1: out of do .

Observation 2:

Observation 3:

Rachel and her mother have an ongoing disagreement about whether Rachel spends too much time on her phone and sends too many texts. To bolster their arguments, Rachel collected data from among her friends and her mother collected data from among her friends. We will learn later that this not a good sampling method, but that is a story for another day. Rachel thought about the data she and her mom collected for the average number of texts a person sends each day and started thinking that perhaps a two-way table of the data they collected would help convince her mom that she does not send an excessive number of texts for a teenager. The table separates the data by age (teenager and adult) and by the average number of texts sent (more than $100$ per day or less than $100$ per day).

	Average is more than $100$ texts sent per day	Average is less than $100$ texts sent per day	Total
Teenager	$20$	$4$	$24$
Adult	$2$	$22$	$24$
Total	$22$	$26$	$48$

4.

Write two observation statements from this two-way table.

To provide further evidence, Rachel decided to do some research. She found that only $43 %$ of people with phones send over $100$ texts per day. She was disappointed that the research data did not support her case and confused because it did not seem to match what she found in her survey.

5.

What questions do these statistics raise for you? What data should Rachel look for to support her case?

After looking more closely at the data, Rachel found other percentages within the same data that seemed more in line with the data she collected from her teenage friends.

6.

How might Rachel use the data in the two-way table to find percentages that would be useful for her case?

Once Rachel realized there are a lot of ways to look at a set of data in a two-way table, she was motivated to learn about relative frequency tables and conditional frequencies. When the data is written as a percent, this is called a relative frequency table. In this situation, the “inner” values represent a percent and are called conditional frequencies. The conditional values in a relative frequency table can be calculated as percentages of one of the following:

The whole table (relative frequency of table)
The rows (relative frequency of rows)
The columns (relative frequency of column)

Since Rachel wanted to emphasize that a person’s age makes a difference in the number of texts sent, the first thing she decided to do was to focus on the row of values, so she could write conditional statements about the number of texts a person is likely to send based on their age. This is called a relative frequency of row table.

7.

Fill in the percentage of teenagers for each of the conditional frequencies in the highlighted row:

	Average is more than $100$ texts sent per day	Average is less than $100$ texts sent per day	Total
% of Teenagers	$20$ $\underset{―}{} %$	$4$ $\underset{―}{} %$	$24$ $100 %$
% of Adults	$2$ $8 %$	$22$ $92 %$	$24$ $100 %$
% of Total	$22$ $46 %$	$26$ $54 %$	$48$ $100 %$

Average is more than $100$ texts sent per day

Average is less than $100$ texts sent per day

Total

% of Teenagers

$20$

$\underset{―}{} %$

$4$

$\underset{―}{} %$

$24$

$100 %$

% of Adults

$2$

$8 %$

$22$

$92 %$

$24$

$100 %$

% of Total

$22$

$46 %$

$26$

$54 %$

$48$

$100 %$

Since the percentages that were created focus on row values, all conditional observations are specific to the information in the row. Complete the following sentence for the relative frequency of row:

8.

Of all teenagers in the survey, $\underset{―}{} %$ average more than $100$ texts per day.

9.

Write another statement based on the relative frequency of row:

Below is the relative frequency of column, using the same data. This time, all of the percentages are calculated using the data in the column.

	Average is more than $100$ texts sent per day	Average is less than $100$ texts sent per day	Total
% of Teenagers	$20$ $91 %$	$4$ $15 %$	$24$ $50 %$
% of Adults	$2$ $9 %$	$22$ $85 %$	$24$ $50 %$
% of Total	$22$ $100 %$	$26$ $100 %$	$48$ $100 %$

Average is more than $100$ texts sent per day

Average is less than $100$ texts sent per day

Total

% of Teenagers

$20$

$91 %$

$4$

$15 %$

$24$

$50 %$

% of Adults

$2$

$9 %$

$22$

$85 %$

$24$

$50 %$

% of Total

$22$

$100 %$

$26$

$100 %$

$48$

$100 %$

10.

Write two conditional statements using the relative frequency of column.

This data represents the relative frequency of whole table:

	Average is more than $100$ texts sent per day	Average is less than $100$ texts sent per day	Total
% of Teenagers	$20$ $42 %$	$4$ $8 %$	$24$ $50 %$
% of Adults	$2$ $4 %$	$22$ $46 %$	$24$ $50 %$
% of Total	$22$ $46 %$	$26$ $54 %$	$48$ $100 %$

Average is more than $100$ texts sent per day

Average is less than $100$ texts sent per day

Total

% of Teenagers

$20$

$42 %$

$4$

$8 %$

$24$

$50 %$

% of Adults

$2$

$4 %$

$22$

$46 %$

$24$

$50 %$

% of Total

$22$

$46 %$

$26$

$54 %$

$48$

$100 %$

11.

Create two conditional distribution statements for the relative frequency of whole table.

12.

What information is highlighted when data is interpreted from relative frequency tables?

Ready for More?

Create a relative frequency of column table from the two-way table in problem 3.

			Total
% More than one hour
% Less than one hour
% Total

Takeaways

Two-way Frequency Table:

	Girls	Boys	Total
Soccer	$14$	$40$	$54$
Dance	$46$	$6$	$52$
Total	$60$	$46$	$106$

Relative Frequency Table:

	Average is more than $100$ texts sent per day	Average is less than $100$ texts sent per day	Total
% of Teenagers	$20$ $42 %$	$4$ $8 %$	$24$ $50 %$
% of Adults	$2$ $4 %$	$22$ $46 %$	$24$ $50 %$
% of Total	$22$ $46 %$	$26$ $54 %$	$48$ $100 %$

Average is more than $100$ texts sent per day

Average is less than $100$ texts sent per day

Total

% of Teenagers

$20$

$42 %$

$4$

$8 %$

$24$

$50 %$

% of Adults

$2$

$4 %$

$22$

$46 %$

$24$

$50 %$

% of Total

$22$

$46 %$

$26$

$54 %$

$48$

$100 %$

Vocabulary

conditional frequency
joint events
marginal frequency
relative frequency table (statistics)
two-way frequency and two-way relative frequency table
two-way table
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about two-way tables and relative frequency tables. Both types of tables are used for two-variable categorical data. Two-way tables show data counts, and relative frequency tables show percentages.

Retrieval

Find the explicit linear equation for the given information.

1.

2.

$x$	$y$
$- 5$	$3$
$2$	$24$

3.

Find the measure of central tendency (mean and median), the range, and the standard deviation for the data set.

$123$ , $134$ , $125$ , $130$ , $121$ , $131$ , $128$ , $127$ , $128$ , $125$ , $122$ , $125$ , $129$ , $125$ , $126$ , $129$ , $132$ , $125$