Lesson 3: Two-Way Tables

Property

A two-way frequency table organizes data for two categorical variables. The categories of one variable form the rows, and the categories of the other form the columns.

• Joint Frequencies: The counts located in the body (inside cells) of the table. They represent data points that satisfy both the row AND column categories simultaneously.
• Marginal Frequencies: The counts located in the margins (the total column and total row). They represent the total count for a single category, regardless of the other variable.
• Missing Values: Because the joint frequencies in any row or column must perfectly sum up to its marginal frequency, you can find missing values using simple subtraction: $\text{Missing Value} = \text{Total} - \text{Known Values}$.

Examples

• Joint vs. Marginal: A table tracks 100 pet owners. The rows are "House" or "Apartment," and the columns are "Dog" or "Cat."

The number of people who live in an Apartment AND own a Cat is a Joint Frequency (it sits inside the table).
The total number of ALL Dog owners is a Marginal Frequency (it sits at the bottom of the "Dog" column).

• Finding Missing Values: A row for "9th Grade" shows 35 Dog owners and an unknown number of Cat owners. The Total for the 9th Grade row is 50.

The missing value is simply $50 - 35 = 15$ Cat owners.

Explanation

Think of a two-way table as a sorting grid. Every single person surveyed gets dropped into one specific box inside the grid based on their two answers. Those inner boxes are the "Joint" frequencies because two categories join together there. The totals on the outside margins are "Marginal" frequencies because they only care about one category at a time. Because the table is just a grid of basic addition, if a piece is missing, you can easily play Sudoku and subtract to find it!

Property

A two-way frequency table displays the counts (frequencies) of data for two different categorical variables to summarize the relationship between them.

To construct a two-way table from raw data:

1. Identify the two categorical variables.
2. Create rows for categories of one variable and columns for categories of the other.
3. Tally each data point in the appropriate cell based on its category combination.
4. Record the frequency count for each cell intersection.

Examples

• Analyzing Survey Data: A survey asked students about their gender and their favorite sport. The results are organized into rows (Male, Female) and columns (Football, Basketball, Baseball).

• Sorting Raw Data: Raw data shows: 9th-Soccer: 8 students, 9th-Basketball: 5 students, 10th-Soccer: 6 students, 10th-Basketball: 9 students. You create a table with Grade Level as rows and Sport as columns, filling in these exact frequencies into the intersecting cells.

Property

Raw counts are hard to compare, so we often convert them into percentages called relative frequencies.

To find a Joint or Marginal Relative Frequency, you ALWAYS divide by the Grand Total (the total number of all people surveyed, located in the bottom-right corner).

• Joint Relative Frequency = $\frac{\text{Joint Frequency}}{\text{Grand Total}}$
• Marginal Relative Frequency = $\frac{\text{Marginal Frequency}}{\text{Grand Total}}$