Associations in Categorical Data

5 min

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.

Student Task

The 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.

  1. 10% (from 10100=0.1\frac{10}{100} = 0.1)
  2. 67% (from 180270≈0.67\frac{180}{270} \approx 0.67)
  3. 30% (from 90300=0.3\frac{90}{300} = 0.3)

Sample Response

Sample responses:

  1. Out of the 100 people who prefer English, 10% of them are left-handed.
  2. Out of the 270 people who are right-handed, 67% of them prefer math.
  3. 30% of those surveyed are both right-handed and prefer English.
Activity Synthesis (Teacher Notes)

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.)
Anticipated Misconceptions

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.

Standards
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.
  • S-ID.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.
  • S-ID.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.
  • S-ID.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.

15 min

15 min