AP Stats 2A-2C Visual Study Guide: Two-Way Tables & Probability RulesAP Stats 2A-2C Visual Study Guide: Two-Way Tables & Probability Rules

Two-Way Tables & Probability Rules

2A · Marginal / joint / conditional 2A · Segmented bar charts 2A · Association

2B · Long-run relative frequency 2C · Probability models 2C · Complement & addition rules

2C · Venn diagrams FRQ frames Cheat sheet

The one big idea: a two-way table is a probability machine. In 2A you turn its counts into proportions (and the

denominator you choose decides what kind). In 2B you learn that a probability is just a long-run proportion. In 2C

you go back to the same tables (plus Venn diagrams) and calculate probabilities with three rules: complement,

addition for mutually exclusive events, and the general addition rule. Most quiz mistakes come from using the

wrong denominator or forgetting to subtract the overlap.

Relationships Between Two Categorical Variables

2 0 - S E C O N D S U M M A R Y

A two-way table counts individuals by two categorical variables at once. Divide by the grand total to get marginal

(one variable) or joint (both variables) relative frequencies. Divide by a row or column total to get a conditional

relative frequency ("of the people in group X…"). Compare conditional distributions with a segmented bar chart: if

the segments differ between groups, there's an association. Association is never proof of causation.

The Yellowstone table (from the slides)

A sample of 1526 visitors to Yellowstone was asked whether they belong to an environmental club and how they use

snowmobiles. Both variables are categorical.APS TAT ISTICS · UNIT2 PA RT1 · SECTIONS2A – 2 C · TPS8 EEverything on the Quiz 2A–2C: relative frequencies from two-way tables (2A), what a probability actually means (2B), and the probability rules: complement, addition, two-way tables and Venn diagrams (2C). Built from the

three lecture slide decks and the Practice 2A–2C key, and it uses their examples. 2D (conditional probability and

independence) comes after this quiz and isn't in here.SECTION2 ATwo-way tables · marginal, joint and conditional relative frequencies · segmented bar charts and mosaic plots · association

Snowmobile use

Environmental club member?

Total

No Yes

Never used 445 212 657

Snowmobile renter 497 77 574

Snowmobile owner 279 16 285

Total 1221 305 1526

Check the totals like the slides do: 445 + 212 + 497 + 77 + 279 + 16 = 1526, 657 + 574 + 285 = 1526, 1221 + 305 = 1526. Blue = values in the margins. Orange = values inside the table where a row and a column meet. Three kinds of relative frequency

Kind What you divide Yellowstone example Memory hook

Marginal A row or column total ÷

the grand total. Tells you

about one variable only.

Club members: 305/1526 = 0.200

(20.0%). Never used a snowmobile:

657/1526 = 0.431.

Comes from the

margins of the table.

Joint One inside cell ÷ the

grand total. Involves both

variables ("A and B").

Members who own snowmobiles:

16/1526 = 0.0105. Non-members

who never used one: 445/1526 =

0.292.

A joint is where two

bones meet; a joint

cell is where a row and

column meet.

Conditional One inside cell ÷ that

row's or column's total.

Only counts individuals

who meet a condition.

Of snowmobile renters, the

proportion who are not members:

497/574 = 0.866.

"Of the …" names the

condition, and the

condition is the

denominator.

TRAP: THE "OF" TELLS YOU THE DENOMINATOR

The slides' rule: the denominator is the group named right after "of." "What percentage of Beatles songs reached #1?" divides by all 205 songs. "What percentage of songs written by Paul McCartney alone reached #1?" divides by Paul's 75. "What proportion of number-one songs were written by Paul alone?" divides by the 20 #1 songs. Same 12 on top, three different answers.

Worked example: the Beatles (slides, Alternate Example 1)

George Harrison John Lennon Paul McCartney Ringo Starr Multiple writers Total

#1: Yes 0 5 12 0 3 20

#1: No 25 69 63 2 26 185

Total 25 74 75 2 29 205

Question Work Kind

(a) What percentage of Beatles songs reached #1? 20/205 =

9.76%

Marginal

(b) What proportion of Beatles songs were written by Paul alone and

reached #1?

12/205 =

0.0585

Joint

(c) What percentage of songs written by Paul alone reached #1? 12/75 = 16% Conditional (given Paul)

(d) What proportion of #1 songs were written by Paul alone? 12/20 = 0.60 Conditional (given

#1)

TRAP: (C) AND (D) SOUND ALIKE BUT AREN'T

The slides call this out. (c) asks about Paul's songs, so 75 goes on the bottom. (d) asks about the #1 songs, so 20 goes on the bottom. Flipping the condition changes the answer from 16% to 60%. Before you divide, underline the group after "of" (or after "given," or "among").

Proportion vs. percentage: a proportion is between 0 and 1. A percentage is between 0 and 100. Multiply a proportion by 100 to get a percentage. Answer in whichever form the question uses.

Comparing groups: conditional distributions

To look at a relationship, marginal and joint frequencies aren't enough (the slides say they "don't tell us much about the relationship"). Instead, find the conditional distribution of the response variable within each group of the explanatory variable. Each group's percentages should add to 100%, give or take rounding.

Snowmobile use Not a member (n = 1221) Member (n = 305)

Never used 445/1221 = 36.4% 212/305 = 69.5%

Rented 497/1221 = 40.7% 77/305 = 25.2%

Owned 279/1221 = 22.9% 16/305 = 5.2%

Total 100% 99.9% (rounding)

Snowmobile use by environmental club membership

0%

25%

50%

75%

100%

Percent

36.4%

40.7%

22.9%

Not a member

69.5%

25.2%

Member

Never used

Rented

Owned

Segmented bar chart. One bar per explanatory group,

each bar is 100%, split into the conditional distribution.

Members are far more likely to never use a snowmobile.

Mosaic plot: same data, bar width = group size

Not a member (1221) Member (305)

Never used

Rented

Owned

Wide bar = 80% of the sample are non-members; narrow bar = 20% are members.

Mosaic plot. Like a segmented bar chart, but each bar's

width shows how big that group is.

Worked example: no rest for the weary (Alternate Examples 2 and 3)