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?

Double bar graphs.


 

23.2: Card Sort: Matching Representations

Your teacher will hand out some cards.

Some cards show two-way 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:

Double bar graph.

Some cards show segmented bar graphs like this:

Stacked bar graph.

The bar graphs and segmented bar graphs have their labels removed.

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

  2. One of the groups does not have a two-way table. Make a two-way table for the situation described by the graphs in the group.

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

  4. 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 Two-Way Table

Here is a two-way 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

  1. 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 two-way table. Instead of showing frequency, this table shows relative frequency.

  2. Two-way tables that show relative frequencies often don’t include a “total” row at the bottom. Why?
  3. Is there an association between age and cell phone use? How does the two-way 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.
    A bar graph.  Party A, the bar for 165 against is stacked on top of the bar for 832 for. Party B, the bar for 160 against is stacked on top of the bar for 80 for.
  1. Are any of the news reports misleading? Explain your reasoning.
  2. 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 two-way tables to investigate possible connections between two categorical variables. For example, this two-way 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 non-meditators, so we can compare the numbers of calm and agitated athletes in each group.

Double bar graph.


 

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 non-meditators, we can make a two-way 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%
Stacked bar graph.



 

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 multi-color. 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 \(\frac{3}{21}\)
    multi-color 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.

    a segmented bar graph

    This segmented bar graph shows the percentage of people in different age groups that do and do not have a cell phone. For example, among people ages 10 to 12, about 40% have a cell phone and 60% do not have a cell phone.

  • two-way table

    A two-way 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 two-way table shows the results of the study.

      meditated did not meditate total
    calm 45 8 53
    agitated 23 21 44
    total 68 29 97