Categorical Data and Two-Way Tables

You will be able to:

• Tell categorical data from measurement data
• Build a two-way frequency table
• Name joint, marginal, and grand-total frequencies

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Distinguish categorical data from measurement data
2. Explain why scatter plots fail for categories
3. Construct a two-way frequency table from data
4. Identify joint frequencies, marginal frequencies, and the grand total

Scatter Plots Only Work for Numbers

• Height vs. shoe size → a scatter plot works
• "Eat breakfast?" yes/no → you can't plot "yes"

A New Tool for Categories

When both variables are categories, organize the data in a two-way table.

The Two Fundamental Kinds of Data

• Measurement data: numbers where arithmetic makes sense — height, score
• Categorical data: labels or categories — yes/no, color, pet type
• The data type decides the display

Anatomy of a Two-Way Table

Rows and columns are the categories; the interior, margins, and corner each have a name.

Joint Frequency Counts Both Traits

• A joint frequency sits in an interior cell
• It counts subjects with both the row trait and the column trait
• Example: students who eat breakfast and feel tired

Marginal Frequency Counts One Trait

• A marginal frequency sits in a total row or total column
• It counts subjects with one trait, ignoring the other
• Example: all students who eat breakfast, tired or not

The Grand Total Counts Everyone

• The grand total is the corner cell
• It counts all subjects in the survey
• It equals the sum of any total row or total column

Quick Check: Name the Parts

In a two-way table, which number counts students with both traits — and which counts one trait?

Match "joint" and "marginal" to each before advancing.

Build From Raw Data: Frame and Tally

• Step 1: Draw the frame — rows, columns, totals
• Step 2: Tally each student into one cell

Step 3: Record the Joint Frequencies

Step 4: Verify the Totals

• Rows: , and
• Columns: , and
• Grand total: ✓

Verification Is a Built-In Check

• The margins must reconcile both ways
• Row totals sum = column totals sum = grand total
• Disagreement means a miscount somewhere

Build a Table From a Description

Quick Check: Draw the Skeleton

A survey asks 50 people: pet (dog / no dog) and home (house / apartment).

Sketch the two-way table skeleton — rows, columns, totals — before advancing.

Your Turn: Name and Explain

Using the breakfast table:

1. Point to one joint frequency and say what it means
2. Point to one marginal frequency and say what it means

Explain both in context before comparing.

Your Turn: Complete the Table

There are 60 students. 35 are girls. 20 girls play an instrument. 10 boys play an instrument.

Complete the full two-way table and verify the totals.

Your Turn: Build It Yourself

Raw data lists 24 students by (Walks to school?, Packs lunch?).

1. Draw the frame and tally each student
2. Record the counts and verify the totals

Build the whole table on your own first.

Same Question, but a New Tool

✓ Categorical data needs a two-way table, not a scatter plot
✓ Interior cells are joint; margins are marginal; corner is grand total
✓ Verify by reconciling the totals both ways

Coming Up Next: Finding Association

Next lesson, you'll turn these counts into percentages — relative frequencies — which is how you actually detect whether the two variables are associated.