Unit 2 · Topic 2.2 Beta

Summary Statistics for Two Categorical Variables

From one two-way table you can find three kinds of share.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

A state park wants to know who comes to fish so it can plan bait shops and boat rentals. Rangers ask 250 visitors two questions: is this your first visit, and what did you come to do? The same table of counts answers three different questions, depending on which total you divide by.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. In a two-way table, where is the grand total?

  1. In the corner, where the Total row and Total column meet
  2. At the end of the first row
  3. At the top of the first column
  4. In the largest cell
Show the answer

The grand total counts every individual and sits where the totals meet.

  • Correct: In the corner, where the Total row and Total column meet:
  • At the end of the first row:
  • At the top of the first column:
  • In the largest cell:

2. A relative frequency is

  1. a count divided by a total
  2. a count multiplied by 100
  3. the largest count in a table
  4. a total minus a count
Show the answer

A relative frequency is a share: count over total.

  • Correct: a count divided by a total:
  • a count multiplied by 100:
  • the largest count in a table:
  • a total minus a count:

Part 4 · See it

See it first

The two-way table of 250 park visitors by visit type and activity, with three cells marked. The joint cell, first visit and hiking, holds 45 of all 250 visitors (0.18). The marginal total for hiking is 105 of 250 (0.42). The conditional share of hikers among first visits is 45 of the 100 first visits (0.45).
One table, three kinds of share: a cell over the grand total (joint), a total over the grand total (marginal), and a cell over its row or column total (conditional). LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Divide one cell by the grand totala joint relative frequency: the share of everyone in that pairing (45 of 250 are first visits who hiked, 0.18)
  2. Divide a row or column total by the grand totala marginal relative frequency: the share of everyone in one category (105 of 250 hiked, 0.42)
  3. Divide one cell by its row or column totala conditional relative frequency: the share within one group (45 of the 100 first visits hiked, 0.45)
  4. Compare conditional relative frequencies across groupsif they differ, the variables are associated

Part 6 · Key ideas

Key ideas

  • Joint relative frequency: one cell over the grand total.
  • Marginal relative frequency: a row or column total over the grand total. The marginal distribution of a variable lists them for every category.
  • Conditional relative frequency: one cell over its row or column total. Read "of those who ..." to find the group, and divide by that group's total.
  • Conditional relative frequencies for one group add to 1. If they are the same for every group, there is no association.

Part 7 · Misconception

A common mistake

The wrong idea: The share of course takers who passed and the share of passers who took the course are the same number.

What actually happens: They have different denominators: 60 of the 80 course takers passed (0.75), but 60 of the 126 passers took the course (0.476). The group named first is the denominator.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Park visitors by visit type and activity

Rangers at a state park asked a random sample of 250 visitors whether this was their first visit and what their main activity was that day (invented data).
Visit type and main activity for 250 park visitors
Visit typeHikingFishingPicnicTotal
First visit451540100
Returning605436150
Total1056976250

1. What proportion of the 250 visitors were on a first visit and hiking?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

A joint relative frequency divides one cell by the grand total: 45250=0.18\frac{45}{250} = 0.18.

  • Answer: 0.18

2. What is the marginal relative frequency of fishing?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

A marginal relative frequency uses a total on the edge of the table: 69 of the 250 visitors fished, 69250=0.276\frac{69}{250} = 0.276.

  • Answer: 0.276

3. Of the returning visitors, what proportion came to fish?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

"Of the returning visitors" means divide by that row's total: 54150=0.36\frac{54}{150} = 0.36.

  • Answer: 0.36

Table

Driver's education and the road test

A state records, for a random sample of 200 new drivers, whether each took a driver's education course and whether each passed the road test on the first try (invented data).
Course and first-try result for 200 new drivers
CoursePassedFailedTotal
Took the course602080
No course6654120
Total12674200

4. What proportion of the drivers who took the course passed on the first try?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

Start from the 80 course takers: 6080=0.75\frac{60}{80} = 0.75.

  • Answer: 0.75

5. A school ad says, "Most drivers who pass on the first try took a driver's education course." Do these data support the ad?

  1. No: of the 126 drivers who passed, only 60 (about 48%) took the course
  2. Yes: 75% of the drivers who took the course passed, which is most of them
  3. Yes: 60 course takers passed, which is most of the group of 80
  4. No: the course takers are 80 of 200 drivers, which is 40% of the sample
Show the answer

Start from the drivers who passed: 60 of the 126 took the course, 60126≈0.476\frac{60}{126} \approx 0.476, less than half.

  • Correct: No: of the 126 drivers who passed, only 60 (about 48%) took the course: Right: "of drivers who pass" means divide by the 126 who passed.
  • Yes: 75% of the drivers who took the course passed, which is most of them: True, but that answers a different question, about course takers rather than about drivers who passed.
  • Yes: 60 course takers passed, which is most of the group of 80: The ad is about drivers who passed, so the group to start from is the 126 who passed.
  • No: the course takers are 80 of 200 drivers, which is 40% of the sample: That is the marginal share who took the course, not the share of passers who did.

6. A table of 500 voters shows party membership and support for a school bond. 120 voters are Party A members who support the bond. What is 120500\frac{120}{500}?

  1. A joint relative frequency
  2. A marginal relative frequency
  3. A conditional relative frequency
  4. The percent of Party A members who support the bond
Show the answer

One cell over the grand total is a joint relative frequency: 24% of all 500 voters are Party A members who support the bond.

  • Correct: A joint relative frequency: Right: one cell divided by the grand total.
  • A marginal relative frequency: A marginal uses a row or column total on top, not one cell.
  • A conditional relative frequency: A conditional divides by a row or column total, not by 500.
  • The percent of Party A members who support the bond: That would divide by the number of Party A members.

Part 9 · Summary

Summary

From one two-way table you can find three kinds of share. Joint: a cell over the grand total. Marginal: a row or column total over the grand total. Conditional: a cell over the total of its group. Compare groups with conditional relative frequencies; if they differ, the variables are associated.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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