Lesson 23
Looking for Associations
Let’s look for associations in data.
23.1: Notice and Wonder: Bar Association
What do you notice? What do you wonder?
23.2: Card Sort: Matching Representations
Your teacher will hand out some cards.
Some cards show twoway tables like this:
has cell phone  does not have cell phone  total  

10 to 12 years old  25  35  60 
13 to 15 years old  40  10  50 
16 to 18 years old  50  10  60 
total  115  55  170 
Some cards show bar graphs like this:
Some cards show segmented bar graphs like this:
The bar graphs and segmented bar graphs have their labels removed.

Put all the cards that describe the same situation in the same group.

One of the groups does not have a twoway table. Make a twoway table for the situation described by the graphs in the group.

Label the bar graphs and segmented bar graphs so that the categories represented by each bar are indicated.

Describe in your own words the kind of information shown by a segmented bar graph.
One of the segmented bar graphs is missing. Construct a segmented bar graph that matches the other representations.
23.3: Building Another Type of TwoWay Table
Here is a twoway table that shows data about cell phone usage among children aged 10 to 18.
has cell phone  does not have cell phone  total  

10 to 12 years old  25  35  60 
13 to 15 years old  40  10  50 
16 to 18 years old  50  10  60 
total  115  55  170 

Complete the table. In each row, the entries for “has cell phone” and “does not have cell phone” should have the total 100%. Round entries to the nearest percentage point.
has cell phone does not have cell phone total 10 to 12 years old 42% 13 to 15 years old 100% 16 to 18 years old 17% This is still a twoway table. Instead of showing frequency, this table shows relative frequency.
 Twoway tables that show relative frequencies often don’t include a “total” row at the bottom. Why?
 Is there an association between age and cell phone use? How does the twoway table of relative frequencies help to illustrate this?
A pollster attends a rally and surveys many of the participants about whether they associate with political Party A or political Party B and whether they are for or against Proposition 3.14 going up for vote soon. The results are sorted into the table shown.
for  against  

party A  832  165 
party B  80  160 
 A news station reports these results by saying, “A poll shows that about the same number of people from both parties are voting against Proposition 3.14.”
 A second news station shows this graphic.
 Are any of the news reports misleading? Explain your reasoning.
 Create a headline, graphic, and short description that more accurately represents the data in the table.
Summary
When we collect data by counting things in various categories, like red, blue, or yellow, we call the data categorical data, and we say that color is a categorical variable.
We can use twoway tables to investigate possible connections between two categorical variables. For example, this twoway table of frequencies shows the results of a study of meditation and state of mind of athletes before a track meet.
meditated  did not meditate  total  

calm  45  8  53 
agitated  23  21  44 
total  68  29  97 
If we are interested in the question of whether there is an association between meditating and being calm, we might present the frequencies in a bar graph, grouping data about meditators and grouping data about nonmeditators, so we can compare the numbers of calm and agitated athletes in each group.
Notice that the number of athletes who did not meditate is small compared to the number who meditated (29 as compared to 68, as shown in the table).
If we want to know the proportions of calm meditators and calm nonmeditators, we can make a twoway table of relative frequencies and present the relative frequencies in a segmented bar graph.
meditated  did not meditate  

calm  66%  28% 
agitated  34%  72% 
total  100%  100% 
Glossary Entries
 relative frequency
The relative frequency of a category tells us the proportion at which the category occurs in the data set. It is expressed as a fraction, a decimal, or a percentage of the total number.
For example, suppose there were 21 dogs in the park, some white, some brown, some black, and some multicolor. The table shows the frequency and the relative frequency of each color.
color frequency relative frequency white 5 \(\frac{5}{21}\) brown 7 \(\frac{7}{21}\) black 3 \(\frac{3}{21}\) multicolor 6 \(\frac{6}{21}\)  segmented bar graph
A segmented bar graph compares two categories within a data set. The whole bar represents all the data within one category. Then, each bar is separated into parts (segments) that show the percentage of each part in the second category.
 twoway table
A twoway table provides a way to compare two categorical variables.
It shows one of the variables across the top and the other down one side. Each entry in the table is the frequency or relative frequency of the category shown by the column and row headings.
A study investigates the connection between meditation and the state of mind of athletes before a track meet. This twoway table shows the results of the study.
meditated did not meditate total calm 45 8 53 agitated 23 21 44 total 68 29 97