Two-way frequency tables
Two-Way Frequency Tables
Two-Way Frequency Tables is a topic in Probability & Statistics in the Common Core State Standards. It is aligned to Standard S-CP.4, which requires students to construct and interpret two-way frequency tables and use them to decide if events are independent.
A two-way frequency table organizes counts by two categories, yielding joint, marginal, and conditional probabilities.
Theory
A two-way frequency table shows counts for two categories, with row and column totals. From it:
- Joint: a single cell divided by the grand total.
- Marginal: a row or column total divided by the grand total.
- Conditional: a cell divided by its row or column total.
From counts in the table:
How to read a two-way table
- Find the relevant cell or total.
- Joint/marginal: divide by the grand total.
- Conditional: divide by the row or column total for the condition.
- Compare conditional and marginal to check for association.
Divide the cell count by the grand total.
| \(P\) | \(=\) | \(\dfrac{18}{50}=0.36\) |
Use the column total over the grand total.
| \(P\) | \(=\) | \(\dfrac{30}{50}=0.6\) |
Condition on the \(30\) who play a sport.
| \(P\) | \(=\) | \(\dfrac{18}{30}=0.6\) |
Compare \(P(\text{senior}\mid\text{sport})\) with \(P(\text{senior})\).
| \(P(\text{senior}\mid\text{sport})\) | \(=\) | \(0.6\) |
| \(P(\text{senior})\) | \(=\) | \(\dfrac{25}{50}=0.5\) |
\(0.6\neq0.5\), so they are not independent.
Common pitfalls
Frequently asked questions
What is a two-way frequency table?
A table of counts organized by two categories, with row and column totals.
What is a joint frequency?
The count in a single cell; its probability is the cell over the grand total.
What is a marginal frequency?
A row or column total; its probability is that total over the grand total.
How do you get a conditional probability from a table?
Divide a cell by its row or column total, not the grand total.