Unit 6, Lesson 3
Associations in Categorical Data
Integrated Math 1
3.1
Warm-up
Standards Alignment
Building On
Addressing
HSS-ID.B.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
The mathematical purpose of this activity is for students to interpret data in a two-way table using relative frequencies. Listen for students mentioning relative frequencies and how they are calculating it (using row, column, or overall totals).
Launch
Arrange students in groups of 2. Give students quiet think time to answer the first question and think about the others. Ask partners to compare and discuss their answers. Follow with a whole-class discussion.
Activity
NoneThe table displays the course preference and dominant hand (left- or right-handed) for a sample of 300 people.
| prefers English | prefers math | 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.
- 10% (from )
- 67% (from )
- 30% (from )
Student Response
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Building on Student Thinking
Students may be confused as to how to interpret the table. Encourage students to use the table to look for the specific numbers mentioned in the questions. Then students can describe the larger group as well as the subgroup being considered in each respective question.
Activity Synthesis
The goal is to make sure students understand that relative frequencies can be calculated using the overall total or the totals from the rows or the columns.
Ask students who were identified as mentioning relative frequency, “When you said relative frequency, what did you mean?” (I meant the frequency of a particular cell in the table occurring relative to the total. For example, 10% comes from 10 left-handed people who prefer English out of the 100 total people who prefer English.)
Here are some questions for discussion.
- “How does this activity relate to prior work with relative frequency tables?” (We did not make the table here, but we had to use the values that we could have found by creating relative frequency tables. For this activity, we would have needed three relative frequency tables.)
- “For the question about 67%, did you use the row totals or the column totals?” (We used the row totals.)
- “The overall total was 300. How many people would you expect to prefer English and be left-handed if the overall total was 600? How does this connect to the concept of relative frequency?” (20 people. It is connected to relative frequency because 10 out of 300 is the same ratio as 20 out of 600.)
3.2
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
The mathematical purpose of this activity is to look for associations in categorical data. Since the subgroups represented by rows or columns are not always equally represented in the data, associations are often best recognized by considering row or column frequencies and comparing across the categories. Students should determine that there is likely not an association between the categories if the percentages are similar, or that there is evidence of an association between the variables when the percentages are very different.
Monitor for students who use the term “association” in their explanations. Identify students who struggle with knowing whether to compare frequencies in the columns or the rows and students who do not know how to find relative frequencies.
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2.
Ask students what it might mean for two variables to be associated. Provide these examples, and ask students whether they think the variables might be associated:
- The color of a car’s seats and whether the owner returns to the dealership for maintenance or not (there is probably not an association)
- The color of a car’s seats and whether the owner is ever burned from the seats being hot (there is probably an association here since darker seats will probably get hotter than lighter-colored seats)
After students have had a chance to work through the first problem about coral health, pause the work to discuss what students noticed about possible associations. Ensure that the discussion concludes that when the column (or row) percentages are similar (like with the silicon dioxide concentration), there is probably no association between the different variables. When the percentages are not close (like with the nitrate concentration), there is likely an association between the variables.
Give students quiet think time to work through the remaining problems. Follow with a whole-class discussion.
MLR2 Collect and Display. Collect the language students use to talk about the first question. Display words and phrases, such as “the variables are related” or “connected.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading
Advances: Conversing, Reading
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each question. Use these intervals to clarify misconceptions before students continue to the next question.
Supports accessibility for: Attention, Social-Emotional Functioning
Supports accessibility for: Attention, Social-Emotional Functioning
3.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Tools for creating a visual displayActivity Narrative
The mathematical purpose of this activity is for students to understand what tables might look like when two categories are possibly associated and when they are not. Students invent their own pair of variables that they expect to be associated and another pair that they expect not to be associated. Then, students prepare a two-way table of invented data that shows (or does not show) evidence of an association. Then, students create a display of their information for the class to see.
Notice groups that create displays that communicate their mathematical thinking clearly, contain an error that would be instructive to discuss, or organize the information in a way that is useful for all to see.
For students struggling to think of examples, these are some suggestions:
Associated categories:
- Age groups and price of a movie ticket
- Region of the country (North, West, or South) and whether a person enjoys skiing
- Day of the week and wake-up time (before 7 a.m. or after 7 a.m.)
Unassociated categories:
- Eye color and whether or not a person sings in the chorus
- Saltwater or freshwater fish tank and whether or not there are decorations in the tank
- Favorite sport and whether or not a person eats broccoli
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Lesson Synthesis
The goal of this discussion is to make sure students know how to decide if there is an association between two categorical variables.
- “How do you know whether or not two variables are associated?” (First, you create a row or column relative frequency table. Second, you look for similarities in the columns or rows. If the relative frequencies in the rows or columns are similar, then there is probably no association. If they are very different, there is a possible association.)
- “What does it mean when we say two variables are associated? Provide an example.” (It means the variables are statistically related. For example, if we compared handedness (left or right) to messiness (neat or messy), we might find that left- and right-handed people are equally messy, and there would be no association. If we found out that the percentage who are messy was much greater for right-handed people than for left-handed people, then we would have evidence of an association between handedness and messiness.)
Student Lesson 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 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 cone 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, but the differences in the number of data values in a given category must be accounted for.