# Lesson 3

Associations in Categorical Data

• Let’s look for relationships between categorical variables.

### 3.1: Cake or Pie

The table displays the dessert preference and dominant hand (left- or right-handed) for a sample of 300 people.

prefers cake prefers pie total
left-handed 10 20 30
right-handed 90 180 270
total 100 200 300

For each of the calculations, describe the interpretation of the percentage in terms of the situation.

1. 10% from $$\frac{10}{100} = 0.1$$
2. 67% from $$\frac{180}{270} \approx 0.67$$
3. 30% from $$\frac{90}{300} = 0.3$$

### 3.2: Associations in Categorical Data

1. The two-way table displays data about 55 different locations. Scientists have a list of possible chemicals that may influence the health of the coral. They first look at how nitrate concentration might be related to coral health. The table displays the health of the coral (healthy or unhealthy) and nitrate concentration (low or high).

low nitrate concentration high nitrate concentration total
healthy 20 5 25
unhealthy 8 22 30
total 28 27 55
1. Complete the two-way relative frequency table for the data in the two-way table in which the relative frequencies are based on the total for each column.

low nitrate concentration high nitrate concentration
healthy
unhealthy
total 100% 100%
2. When there is a low nitrate concentration, which had a higher relative frequency, healthy or unhealthy coral?

3. When there is a high nitrate concentration, is there a higher relative frequency of healthy or unhealthy coral?

4. Based on this data, is there a possible association between coral health and the level of nitrate concentration? Explain your reasoning.

5. The scientists next look at how silicon dioxide concentration might be related to coral health. The relative frequencies based on the total for each column are shown in the table. Based on this data, is there a possible association between coral health and the level of silicon dioxide concentration? Explain your reasoning.

low silicon dioxide concentration high silicon dioxide concentration
healthy 44% 46%
unhealthy 56% 54%
total 100% 100%
2. Jada surveyed 300 people from various age groups about their shoe preferences. The two-way table summarizes the results of the survey.
prefers sneakers without laces prefers sneakers with laces prefers shoes that are not sneakers total
4–10 years old 21 12 3 36
11–17 years old 21 48 39 108
18–24 years old 15 54 87 156
total 57 114 129 300

Jada concludes that there is a possible association between age and shoe preference. Is Jada’s conclusion reasonable? Explain your reasoning.

3. The two-way table summarizes data on writing utensil preference and the dominant hand for a sample of 100 people.
left-handed right-handed total
prefers pen 7 82 89
prefers pencil 6 5 11
total 13 87 100

Is there a possible association between the dominant hand and writing utensil preference? Explain your reasoning.

The incomplete two-way table displays the results of a survey about the type of sports medicine treatment and recovery time for 33 student athletes who visited the athletic trainer.

 returned to playing in less than 2 days returned to playing in 2 or more days treated with ice 8 4 treated with heat
1. What 2 values could you use to complete the two-way table to show that there is an association between returning to playing in less than 2 days and the treatment (ice or heat)? Explain your reasoning.

2. What 2 values could you use to complete the two-way table to show that there is no association between returning to playing in less than 2 days and the treatment (ice or heat)? Explain your reasoning.

3. Which 2 values were easier to choose, the 2 values showing an association, or the 2 values showing no association? Explain your reasoning.

### 3.3: Associating Your Own Variables

1. Work with your group to identify a pair of categorical variables you think might be associated and another pair you think would not be associated.
2. Imagine your group collected data for each pair of categorical variables. Create a two-way table that could represent each set of data. Invent some data with 100 total values to complete each table. Remember that one table shows a possible association, and the other table shows no association.
3. Explain or show why there appears to be an association for the first pair of variables and why there appears to be no association for the other pair of variables.
4. Prepare a display of your work to share.

### Summary

An association between two variables means that the two variables are statistically related to each other. For example, we might expect that ice cream sales would be higher on sunny days than on snowy days. If sales were higher on sunny days than on snowy days, then we would say that there is a possible association between ice cream sales and whether or not it is sunny or snowing. When dealing with categorical variables, row or column relative frequency tables are often used to look for associations in the data.

Here is a two-way table displaying ice cream sales and weather conditions for 41 days for a particular creamery.

sunny day snowy day total
sold fewer than 50 cones 8 7 15
sold 50 cones or more 22 4 26
total 30 11 41

Noticing a pattern in the raw data can be difficult, especially when the row or column totals are not the same for different categories, so the data should be converted into a row or column relative frequency table to better compare the categories. For the creamery, notice that the number of days with low sales is about the same for the two weather types, which contradicts our intuition. In this case, it makes sense to look at the percentage of days that sold well under each weather condition separately. That is, consider the column relative frequencies.

sunny day snowy day
sold fewer than 50 cones 27% 64%
sold 50 cones or more 73% 36%
total 100% 100%

From the column relative frequency table, it is clear that most of the sunny days resulted in sales of at least 50 cones (73%), while most of the snowy days resulted in fewer than 50 cones sold (64%). Because these percentages are quite different, this suggests there is an association between the weather condition and the number of cone sales. A bakery might wonder if the weather conditions impact their muffin sales as well.

sunny day snowy day
sold fewer than 50 muffins 32% 35%
sold 50 muffins or more 68% 65%
total 100% 100%

For the bakery, it seems there is not an association between weather conditions and muffin sales, since the percentage of days with low sales are very similar under the different weather conditions, and the percentages are also close on days when they sold many muffins.

Using row or column relative frequency tables helps organize data so that columns (or rows) can be easily compared between different categories for a variable. This comparison can be accomplished using a two-way table or a two-way relative frequency table, but it requires you to account for the differences in the number of data values in a given category.

### Glossary Entries

• association

In statistics we say that there is an association between two variables if the two variables are statistically related to each other; if the value of one of the variables can be used to estimate the value of the other.

• categorical variable

A variable that takes on values which can be divided into groups or categories. For example, color is a categorical variable which can take on the values, red, blue, green, etc.

• relative frequency table

A version of a two-way table in which the value in each cell is divided by the total number of responses in the entire table or by the total number of responses in a row or a column.

The table illustrates the first type for the relationship between the condition of a textbook and its price for 120 of the books at a college bookstore.

$10 or less more than$10 but less than $30$30 or more
new 0.025 0.075 0.225
used 0.275 0.300 0.100
• two-way table

A way of organizing data from two categorical variables in order to investigate the association between them.

has a cell phone does not have a cell phone
10–12 years old 25 35
13–15 years old 38 12
16–18 years old 52 8
• variable (statistics)

A characteristic of individuals in a population that can take on different values.