AP Statistics · Unit 2 · Topic 2.2

Joint, Marginal, and Conditional Relative Frequencies Study Lab

Every relative frequency from a two-way table is a count divided by a total. The only question is which total. A joint relative frequency divides by the grand total, a marginal one uses a row or column total over the grand total, and a conditional one divides by the total of the group you are looking within.

1. Pick a cell, find every relative frequency

Click a count in the table. Switch the table view to see the same data as joint or conditional relative frequencies.

Joint = cell ÷ grand total Marginal = row (or column) total ÷ grand total Conditional = cell ÷ group total

Preset tables

Table

Selected cell

Joint

Conditional, within row

Conditional, within column

Marginal

Practice: which total goes on the bottom?

Questions are written from the current table. Read the wording carefully: "of the …" or "given …" names the group you divide by.

2. Key ideas

Joint

"Both" questions

A joint relative frequency answers "what proportion of everyone is in this row and this column?" Divide one cell by the grand total. All the joint relative frequencies add up to 1.

Marginal

One variable only

A marginal relative frequency divides a row total or a column total by the grand total. It describes one variable and ignores the other.

Conditional

"Of the…" questions

A conditional relative frequency looks only inside one group. "Of the seniors, what proportion drive?" divides the senior-and-drive cell by the senior total.

Order matters

Two different conditionals

"Proportion of seniors who drive" and "proportion of drivers who are seniors" use the same cell but different denominators, so they usually give different answers.

Association

Compare conditionals

If the conditional relative frequencies are the same in every group, the variables are not associated. If they differ, they are associated. The difference between two conditional proportions is a simple summary statistic for the association.

Notation

Read the bar as "given"

Later in the unit, the conditional relative frequency of A within group B is written P(A | B), read "the probability of A given B."

3. Worked examples

A survey of 250 students: 90 juniors (54 have a pet, 36 do not) and 160 seniors (80 have a pet, 80 do not).

Question	Type	Work	Answer
What proportion of the students are juniors with a pet?	Joint	54 ÷ 250	0.216
What proportion of the students have a pet?	Marginal	(54 + 80) ÷ 250	0.536
Of the juniors, what proportion have a pet?	Conditional (within juniors)	54 ÷ 90	0.60
Of the pet owners, what proportion are juniors?	Conditional (within pet owners)	54 ÷ 134	about 0.403
Is class associated with having a pet?	Compare conditionals	0.60 for juniors vs. 80 ÷ 160 = 0.50 for seniors	Yes, a difference of 0.10

4. On the FRQ

How to earn the points

Write the fraction before the decimal: "54/90 = 0.60." The fraction shows the grader which total you used.
Describe the group in words: "Of the 90 juniors surveyed, 60% have a pet."
To show association, compute a conditional proportion for each group and compare them in one sentence with both numbers.
Keep at least three decimal places, or use percents with one decimal.

Common mistakes

Dividing by the grand total when the question says "of the juniors." That gives a joint relative frequency, not a conditional one.
Mixing up the direction of a conditional: "juniors who have pets" vs. "pet owners who are juniors."
Comparing counts across groups of different sizes. 80 seniors with pets is more than 54 juniors, but a smaller proportion.
Adding conditional relative frequencies from different groups. They each add to 1 within their own group.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 2.2. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.