Two-Way Tables and Relative Frequencies

Lesson 1 of 2: Summarizing Categorical Data

In this lesson:

• Build a two-way frequency table from raw data
• Compute joint and marginal relative frequencies

Learning Objectives for This Unit

By the end of this unit, you should be able to:

1. Construct a two-way table with all totals
2. Compute joint, marginal, and conditional frequencies
3. Interpret each as "out of whom?" in context
4. Recognize association by comparing conditionals

Does Grade Affect Learning Preference?

We surveyed 80 students about two things:

• Grade level: 9th or 10th
• Learning preference: online or in-person

Does grade level affect the preference? A list of 80 paired answers can't show it.

Why Eighty Pairs Are Unreadable

The raw data is 80 rows like "9th grade, prefers online."

• Scanning the list, can you see a pattern? No.
• The relationship between the two variables is hidden

We organize it into a table — rows for one variable, columns for the other.

Build the Two-Way Frequency Table

Rows = grade, columns = preference, cells = counts. We reuse this all lesson.

Name the Parts of the Table

• Joint counts — the inner cells (18 prefer 9th-and-online)
• Marginal counts — the row and column totals (40 ninth graders)
• Grand total — the corner (80 students)

Cells count a combination; margins count one variable; the corner is everyone.

Quick Check: Is the Table Consistent?

Verify the table you built:

• Each row sums to its total: 18 + 22 = 40 ✓
• Each column sums to its total: 18 + 22 = 40 ✓
• Margins sum to the grand total: 80 ✓

Always run this check.

Your Turn: Build a Table

50 people surveyed on pet ownership and housing:

• Pet + house: 20 · Pet + apartment: 5
• No pet + house: 10 · No pet + apartment: 15

Build the table, add all totals, check it.

Answer: Totals: 25 pets, 25 no, 30 house, 20 apartment, grand 50.

From Raw Counts to Percentages

A relative frequency is a count divided by a total, as a percent.

• It makes different-sized groups comparable
• The kind depends on which total you divide by

This lesson: the two that divide by the grand total — joint and marginal.

The Joint Relative Frequency Explained

22.5% of all students are 9th graders who prefer online.

The Joint Table Sums to 100%

The four joint relative frequencies:

• Every student falls in exactly one cell
• So the cell fractions account for the whole group

If they don't sum to 100%, check your arithmetic.

The Marginal Relative Frequency Explained

A marginal relative frequency divides a margin total by the grand total.

• It's about one variable — it ignores grade level entirely
• Same denominator as joint: the grand total

Joint and Marginal Share a Denominator

Both joint and marginal divide by the grand total (80).

• So both answer "what fraction of ALL students?"
• Joint asks it of a combination; marginal of one variable

Next lesson: the conditional breaks this by using a subgroup.

Your Turn: Joint and Marginal

From a table of 100 students:

• 12 students both walk and are seniors
• 70 students total are seniors

Compute the joint RF for "walks, senior" and the marginal RF for "senior."

Both divide by 100.

Answer: Joint = 12%; marginal = 70% of all students.

Full Task: Table, Joint, Marginal

60 commuters: drive-short 15, drive-long 20, transit-short 18, transit-long 7.

1. Build the table with all totals
2. Joint RF for "drives, long commute"
3. Marginal RF for "takes transit"

Interpret each.

Answer: Joint = 20/60 ≈ 33%; marginal transit ≈ 42%.

Key Takeaways From This Lesson

✓ A two-way table cross-classifies by two categories

✓ Joint in cells, marginal in margins

✓ Joint and marginal RFs both divide by the grand total

Both describe the whole group

Next: the conditional uses a subgroup

Next: "out of whom?" and detecting association.