Two-Way Frequency Tables

A two-way frequency table is a visual tool used to organize and analyze data that belongs to two different categorical variables. It helps us see the relationship between the two categories and calculate probabilities.

Structure of a Two-Way Table

Imagine we surveyed 100 students to see if they studied for a test and whether they passed or failed.

	Passed	Failed	Total
Studied	40	10	50
Did Not Study	5	45	50
Total	45	55	100

• Joint Frequencies: These are the numbers in the interior or middle cells of the table. They represent the intersection of both categories. For example, $40$ students both studied AND passed.
• Marginal Frequencies: These are the numbers in the Total row and Total column. They represent the totals for a single category. For example, $50$ students studied in total, and $45$ students passed in total.

Calculating Probabilities

You can use the table to find the probability of a specific event happening.

Example: Finding a Joint Probability If you randomly select a student, what is the probability they are a student who studied and passed?

• Look at the joint frequency for "Studied" and "Passed": $40$
• Divide by the grand total: $100$
• Probability = $\frac{40}{100} = 0.40$ (or $40\%$)

(Note: This is the exact same logic you would use to find the probability that a randomly selected student is a "senior who drives" in a grade vs. transportation table!)

Conditional Relative Frequencies

A conditional relative frequency compares a joint frequency to a marginal frequency. Instead of looking at the whole group, you only look at a specific row or column (a specific "condition").

Example: Conditional Frequency of Passing Given Studying What is the probability that a student passed, given that they studied?

Because we are given that the student studied, we only look at the "Studied" row.

• Total number of students who studied: $50$
• Number of those specific students who passed: $40$
• Conditional Relative Frequency = $\frac{40}{50} = 0.80$ (or $80\%$)

By restricting our focus to a single row or column, we can easily determine how one variable might affect the other.