Relative Frequencies and Association

You will be able to:

• Compute three kinds of relative frequency
• Use row percentages to detect association
• Explain why association is not causation

Learning Objectives for This Lesson

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

1. Compute relative frequencies for the grand total, rows, and columns
2. Use row relative frequencies to detect association
3. Interpret an association precisely in context
4. Explain why association does not imply causation

Raw Counts Can Easily Mislead You

5 breakfast-eaters felt tired; 9 skippers felt tired. But the rows are 18 vs. 12.

Divide by the Row Total

• Unequal group sizes make raw counts unfair
• Fix: divide each cell by its row total
• Now every row is on a 0-to-100% scale

Three Kinds of Relative Frequency

• Grand-total: divide by everyone — share of all subjects
• Row: divide by the row total — distribution within a row
• Column: divide by the column total — distribution within a column

Relative to the Grand Total

Relative to the Row Total

Compare Rows to Detect Association

28% of breakfast-eaters tired vs. 75% of skippers — a big gap means association.

Quick Check: Is This Association?

Row percentages for "tired": 28% for breakfast-eaters, 75% for skippers.

Does this gap suggest an association? Decide and say why.

Relative to the Column Total

Row and Column Are Not the Same

• "80% of curfew students have chores" — divide by curfew students
• "80% of chore students have curfew" — divide by chore students
• Different denominators → different numbers

Practice the Move: Sport and Subject

No Association Looks Like This

Small Differences Are Likely Chance

• 52% vs. 48% is a tiny gap — probably random
• Like 11 heads in 20 coin flips
• Look for substantial differences, not tiny ones

We Found an Association — Now What?

We saw breakfast and tiredness are associated.

But does skipping breakfast cause tiredness?

Association Is Not the Same as Causation

• Association: the variables are related in the data
• Causation: one variable directly produces the other
• A two-way table shows association, not causation

Lurking Variables Can Explain Both

• A hidden third factor can drive both variables
• It creates association with no direct cause between them

A Clear Example With Umbrellas

Interpret With Careful, Precise Language

Use this template:

"Students who [A] are more/less likely to [B] than those who [not A]. This suggests an association between [variable 1] and [variable 2]."

Your Turn: Row vs. Column

A table shows: 80% of curfew students have chores.

Does that mean 80% of chore-having students have a curfew? Explain.

Your Turn: Find the Causal Error

A headline says: "Owning a dog causes people to exercise more."

1. What is the logical error?
2. Propose a lurking variable

Fix the claim before advancing.

Your Turn: A Full Analysis

A two-way table of data is given.

1. Compute the row relative frequencies
2. Decide whether there is association
3. Write an interpretation sentence

Do all three on your own first.

Detect Association, Then Resist Causation

✓ Rescale rows to percentages, then compare a column across rows
✓ Different percentages → association; equal → none
✓ Association is not causation — mind lurking variables

Coming Up Next: Conditional Probability

In high school, this row comparison becomes conditional probability — the chance of one event given another — and the formal test for whether two events are independent.