Conditional Frequencies and Association

Lesson 2 of 2: Summarizing Categorical Data

In this lesson:

• Compute conditional relative frequencies — "out of whom?"
• Detect association by comparing conditionals

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

Among Ninth Graders, How Many Online?

Last lesson: 18 of 80 students (22.5%) are 9th-and-online.

Now ask: among 9th graders, what fraction prefer online?

• This restricts to the 40 ninth graders
• So it's 18 out of 40, not 18 out of 80

Same cell, different denominator — a conditional relative frequency.

Same Cell, Three Different Denominators Compared

• Joint: of all students
• Row-conditional: of 9th graders

The Question: Out of Whom?

Before every computation, ask: out of whom?

• Out of everyone? → divide by the grand total
• Out of a subgroup? → divide by that subgroup's total

Name the denominator before you compute — every time.

Flip the Condition: Different Question

Among students who prefer online, what fraction are 9th graders?

• This conditions on online-preferrers — uses the column total
• "Among 9th graders, who's online?" started from grade

Different starting subgroups, different questions — the direction matters.

Row-Conditional Is Not the Column-Conditional

These are different questions with different denominators:

• Row: fraction of 9th graders who prefer online
• Column: fraction of online-preferrers who are 9th graders

Like "A given B" versus "B given A" — generally not the same number.

Build a Full Row-Conditional Distribution

The 9th-grade row (40 students), as percentages of 9th graders:

• 45% online, 55% in-person → sums to 100%

Each row becomes a distribution of preference within that grade.

Compare rows next — that's how we find association.

Quick Check: Three from One Cell

For the cell 18, compute all three:

• Joint: (of all)
• Row-conditional: (of 9th graders)
• Column-conditional: (of online-preferrers)

Same numerator, three denominators. Identify each, then advance.

Compare Rows to Find Association

We have a preference distribution for each grade. To find association:

• Compare the conditional distributions across grades
• 9th: 45% online vs 10th: 70% online → they differ

A substantial difference signals a possible association.

Association Versus No Association, Side by Side

• Differ (45% vs 70%) → possible association
• Match (50% vs 50%) → little or no association

When Conditionals Match: No Link

Matching conditional distributions mean no association:

• 50% of 9th graders and 50% of 10th graders prefer online
• Knowing the grade tells you nothing extra about preference

The variables are unrelated — conditioning doesn't shift the distribution.

Compare Conditionals, Not Raw Counts

Detect association from conditional distributions, never raw counts.

• Subgroups can be different sizes
• Raw counts reflect group size, not the relationship

Convert to percentages within each group, then compare.

Association Is Not the Same as Causation

A difference in conditional distributions shows association, not cause.

• Association = the variables tend to occur together in a pattern
• Causation requires far more — a controlled study (HSS.ID.C.9)

Keep the claim modest: a possible association, not a cause.

Your Turn: Detect a Possible Association

A crash study by seatbelt use:

• Seatbelt-wearers: 10% seriously injured
• Non-wearers: 40% seriously injured

Do the variables show a possible association?

Compare the conditionals, then advance.

Answer: Yes — the conditional distributions differ sharply (10% vs 40%), signaling a possible association.

Full Task: Conditionals and Association

A table relating study method (flashcards/rereading) to passing (pass/fail):

1. Build the row-conditional pass-rates
2. Compare them across methods
3. Decide: possible association?

Phrase it as association, not cause. Do it all, then advance.

Answer: Compare the two pass-rates; if they differ substantially, report a possible association.

Key Takeaways From This Lesson

✓ A conditional RF divides by a subgroup total — "out of whom?"

✓ Row and column conditionals are different questions

✓ Compare conditional distributions to detect association

Compare conditionals, not raw counts

Association is not causation (see C.9)

Next: probability and independence from the same table.