6.3.1: Looking for Associations

Lesson

Let's look for associations in data.

Exercise \(\PageIndex{1}\): Notice and Wonder: Bar Association

What do you notice? What do you wonder?

Figure \(\PageIndex{1}\): Double bar graphs in blue and yellow stripes. Horizontal labeled plays sports and no sports. Vertical labeled 0 to 40. Blue represents watches TV. Yellow stripes represents not much TV.

Exercise \(\PageIndex{2}\): Card Sort: Matching Representations

Your teacher will hand out some cards.

Some cards show two-way tables like this:

Table \(\PageIndex{1}\)

	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:

Figure \(\PageIndex{2}\): Double bar graph in blue and yellow stripes. Horizontal labeled 10 to 12 years old, 13 to 15 years old and 16 to 18 years old. Vertical labeled 0 to 60. Blue section means has cell phone. Yellow stripes means has no cell phone.

Some cards show segmented bar graphs like this:

Figure \(\PageIndex{3}\): Stacked bar graph in blue and yellow stripes. Horizontal labeled 10 to 12 years old, 13 to 15 years old and 16 to 18 years old. Vertical labeled 0 to 100. Blue section means has cell phone. Yellow stripes means has no cell phone.

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.

Are you ready for more?

One of the segmented bar graphs is missing. Construct a segmented bar graph that matches the other representations.

Exercise \(\PageIndex{3}\): 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.

Table \(\PageIndex{2}\)

	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.

Table \(\PageIndex{3}\)

	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?

Are you ready for more?

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.

Table \(\PageIndex{4}\)

	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.

Figure \(\PageIndex{4}\)

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.

Table \(\PageIndex{5}\)

	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.

Figure \(\PageIndex{5}\): Double bar graph in blue and red. Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 50. Blue represents calm. Red represents agitated.

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.

Table \(\PageIndex{6}\)

	meditated	did not meditate
calm	66%	28%
agitated	34%	72%
total	100%	100%

Figure \(\PageIndex{6}\): Stacked bar graph in blue and red. Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 100. Blue represents calm. Red represents agitated.

Glossary Entries

Definition: 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.

Table \(\PageIndex{7}\)

color	frequency	relative frequency
white	\(5\)	\(\frac{5}{21}\)
brown	\(7\)	\(\frac{7}{21}\)
black	\(3\)	\(\frac{3}{21}\)
multi-color	\(6\)	\(\frac{6}{21}\)

Definition: 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.

Figure \(\PageIndex{7}\)

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.

Definition: 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.

Table \(\PageIndex{8}\)

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

Practice

Exercise \(\PageIndex{4}\)

A scientist wants to know if the color of the water affects how much animals drink. The average amount of water each animal drinks was recorded in milliliters for a week and then graphed. Is there evidence to suggest an association between water color and animal?

Table \(\PageIndex{9}\)

	cat intake (mL)	dog intake (mL)	total (mL)
blue water	210	1200	1410
green water	200	1100	1300
total	410	2300	2710

Figure \(\PageIndex{8}\)

Exercise \(\PageIndex{5}\)

A farmer brings his produce to the farmer’s market and records whether people buy lettuce, apples, both, or something else.

Table \(\PageIndex{10}\)

	bought apples	did not buy apples
bought lettuce	14	58
did not buy lettuce	8	29

Make a table that shows the relative frequencies for each row. Use this table to decide if there is an association between buying lettuce and buying apples.

Exercise \(\PageIndex{6}\)

Researchers at a media company want to study news-reading habits among different age groups. They tracked print and online subscription data and made a 2-way table.

Table \(\PageIndex{11}\)

	internet media	print media
18-25 year olds	151	28
26-45 year olds	132	72
46-65 year olds	48	165

1. Create a segmented bar graph using one bar for each row of the table.
2. Is there an association between age groups and the method they use to read articles? Explain your reasoning.

Exercise \(\PageIndex{7}\)

Using the data in the scatter plot, what is a reasonable slope of a model that fits this data?

Figure \(\PageIndex{9}\)

1. \(-2.5\)
2. \(-1\)
3. \(1\)
4. \(2.5\)

(From Unit 6.2.4)