I Unit 8: One Variable Statistics

t' TWO WAY TABLES

"I can ..." Statements• I can create a two-way frequency table showing the relationship between

two categorical variables.• I can find and interpret joint, marginal, and conditional relative frequencies.

Whatever way we spin it, statistics is about numbers. So obviously, it makes sensethat statisticians use a lot of numerical data (height, weight, age, etc.), but even thatgets too easy after a while. Data that isn't represented numerically is known ascategorical data (eye color, hair color, gender, etc.).

Although it may seem like there isn't much we can do with categorical data (after all,how can we analyze a person's brown eye color?), statisticians would beg to differ.Well, there's a 92% chance they'd beg to differ, anyway.

Example #1

200 people were surveyed (100 male and 100 female) about their favorite Avenger(Hulk, Thor, Iron Man or Captain America). The result of the survey is representedin the following table.

Hulk Thor Iron Man Cap. America TotalMale 34 28 22 16 100

Female 4 24 40 32 100Total 38 52 62 48 200

This table is a two-way frequency table because we can break the data down into 2categories: male or female (100 of them are male and 100 of them are female), or byfavorite Avenger (38 prefer Hulk, 52 prefer Thor, 62 prefer Iron Man and 48 preferCaptain America).

Practice ExercisesCreate a two-way freque

I Unit 8: One Variable Statistics

Vocabulary

The numbers in the middle of a two-way relative frequency table are called jointprobabilities because they depend on more than one category or event occurring atthe same time. In this case, we want to know if a person is male or female and whichAvenger they prefer. So written in math language, each entry in the table representsP(Gender & Avenger).

The marginal probabilities represent the probability of only one category,P(Gender) or P(Avenger). They're called marginal because they're on the margins ofthe table.

If we know the data for one category and not the other (say, we know the person ismale, but not which Avenger they prefer), we can calculate the probability that hisfavorite Avenger is Thor. This is called a conditional probability because it isconditional on knowing part of the data. We write this in math language asP(ThorIMale). (The I symbol means "given.")

Hulk I Thor I Iron Man I Cap.America TotalMale

'"The numbers in the middle are called joint probabilities because they f-- p .'6b~

depend on more than one category or event occurring at the same time. •.. ...(V1l..

Female :E

Total The marginal probabilities represent the probability of only one category. 200

I UnitB: One Variable Statistics

Example #2

Using the prior table we can create a two-way relative frequency table.

Hulk Thor Iron Man Cap.America TotalMale 34 ~= 0.14 220

20= 0.11 2

1060= 0.08 0.50200 = 0.17 200

Female 4 247:°0 = 0.20

32 0.50700 = 0.02 700 = 0.12 700 = 0.16

Total ~= 0.19 52 ~ = 0.31 2:80 = 0.24 1.0200 = 0.26200 200

More simply the chart can be written as the following:

Hulk Thor Iron Man Cap.America TotalMale 0.17 0.14 0.11 0.08 0.50

Female 0.02 0.12 0.20 0.16 0.50Total 0.19 0.26 0.31 0.24 1.0

-Notice each row and column should sum to the margin of that row or column. (forIron Man, 0.11 + 0.20 = 0.31, or for total 0.50 + 0.50 = 1.0)

Using the table, lets calculate some probabilities ...

a. P(Male) = 0.50 This is a marginal probability and is in the totalcolumn for male.

b. P(Hulk) = 0.19 This is a marginal probability and is in the totalcolumn for the Hulk.

c. P(Male & Hulk) = 0.17 This is a joint probability and is in the cell thatis both male and Hulk.

d. P(Female & Iron Man) = 0.20 This is a joint probability and is in the cell thatis both female and Iron Man.

e. P(HulkIMale) = 0.34 This is conditional probability. We can calculateP(HulkIMale) from the frequency table becausewe know 0.50 of the people surveyed weremen, and 0.17 of the people surveyed wereboth male and preferred Hulk. We can use bothofthese values to determine:

P(Hulk& Male) 0.17P(HulkIMale)= =-=0.34

P(Male) 0.50- - --

I Unit 8: One Variable Statistics

Practice Exercises B

A survey was taken of eye color. Create a two-way frequency table and a two-wayrelative frequency table.Of the men surveyed; 14 had brown eyes, 6 had green eyes and 24 had blue eyes. Ofthe women surveyed; 5 had green eyes, 14 had blue eyes and 25 had brown eyes.1. Two-way frequency table: I

Brown Eyes Blue Eyes Green Eyes TotalMale

FemaleTotal I t

2. Two-wa IJ relative frequency table:Brown Eyes Blue Eyes Green Eyes

Male •. , ") J1..--1 ,07Total

,,/"Female . "

3. P(Male) 4. P(Blue Eyes) 5. P(Female &Blue Eyes)

,?D .4-3 .I~

6. P(GreenIMale) 7. P(GreenIFemale) 8. P (Brown IMale)

, l'f 112- ,37-

Total • ~ ,.. . 4-3Find the probability of each of the following.

. \3

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