Two-Way TablesNormal Distributions

Review: Categorical Variables place individuals into one of several groups or categories.

The values of a categorical variable are labels for the different categories.

The distribution of a categorical variable lists the count or percent of individuals who fall into each category.

Categorical Variables2

When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables.

Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.

Two-Way Table

3

Young adults by gender and chance of getting rich

Female

Male Total

Almost no chance 96 98 194

Some chance, but probably not

426 286 712

A 50-50 chance 696 720 1416

A good chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459

4826

What are the variables described by this two-way table?

How many young adults were surveyed?

Practice 1: Complete the following tables

1) Students are asked if they prefer to go swimming or to the gym.

2) Some people are asked about their favorite outdoor

sport.

Swimming Gym Total

Boys 25 45

Girls

Total 64 100

Hiking Canoeing Climbing TotalUnder 15’s 42 150

15 – 30 30 47 100Over 30 30 15

Total 75 300

Fill in the gaps

SexTOTA

LSMale Female

Eat breakfast on a regular

basis

Yes 190 300

No 165

TOTALS 320 275

110

130 295

595

Marginal Distribution

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The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table.

Note: Percent's are often more informative than counts, especially when comparing groups of different sizes.

To examine a marginal distribution:

1. Use the data in the table to calculate the marginal distribution (in percent's) of the row or column totals.

2. Make a graph to display the marginal distribution.

BPS Chapter 6 7

Example: Age and Education

Variables

Marginal distributions

“Age groups” is the categorical explanatory variable “Education level” is the categorical response variable

BPS Chapter 6 8

Example: Marginal Totals

Variables

Marginal totals

37,786 81,435 56,008

27,85858,07744,46544,828

BPS Chapter 6 9

Marginal Distributions

%100 totaltable

totalmarginal percent marginal

Marginal distributions are used as background information only.

They do not address association

BPS Chapter 6 10

Marginal Distribution, Row Variable

% not completed HS = 27,859 / 175,230 × 100% = 15.9%

% graduated HS = 58,077 / 175,230 × 100% = 33.1%

% finished 1-3 yrs col. = 44,465 / 175,230 × 100% = 25.4%

% finished ≥4 yrs col. = 44,828 / 175,230 × 100% = 25.6%

BPS Chapter 6 11

Marginal Distribution, Column Variable

% age 25–34 = 37,786 / 175,230 × 100% = 21.6%

% age 35–54 = 81,435 / 175,230 × 100% = 46.5%

% 55 and over = 56,008 / 175,230 × 100% = 32.0%

Marginal Distribution

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Young adults by gender and chance of getting rich

Female

Male Total

Almost no chance 96 98 194

Some chance, but probably not

426 286 712

A 50-50 chance 696 720 1416

A good chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459 4826

Response Percent

Almost no chance

194/4826 = 4.0%

Some chance 712/4826 = 14.8%

A 50-50 chance 1416/4826 = 29.3%

A good chance 1421/4826 = 29.4%

Almost certain 1083/4826 = 22.4%

Examine the marginal distribution of chance of getting rich.

Almost none

Some chance

50-50 chance

Good chance

Almost certain

05

101520253035

Chance of being wealthy by age 30

Survey Response

Perc

ent

Practice 2: Copy and complete the following two-way table about ways of eating potato and then answer the questions.

1) How many boys liked mashed?2) How many teachers preferred new potatoes?3) How many girls were asked?4) Out of the people who liked chips, how many were

boys?

New Chips Mashed Total

Boys 34 100

Girls 25 37

Teachers 12 22 50

Total 104 250

1) What is the probability the person picked is a boy?

2) What is the probability the person liked mash?

3) What is the probability the person was a teacher who preferred new potatoes?

4) What is the probability that, out of the girls, the person liked chips?

5) Out of the people who liked chips, what is the probability the person was a boy?

Practice 3: For the following two-way table answer the questions on probability. A person is picked at random from the sample.

New Chips Mash Total

Boys 15 51 34 100

Girls 25 37 38 100

Teachers 12 16 22 50

Total 52 104 94 250

TAILFor the following table, copy and complete it, then answer

the questions about types of coffee bought.

Next

Coffee is sold in three types and in three weights.

a) How many people bought 100g of powdered coffee?

b) How many people bought ground coffee?

c) Out of the packets weighing 200g, what is the probability the packet bought contained granules?

  100g 200g 300g Total

Ground   50   120

Powder 80 35 26  

Granules 40 45   

Total 135   135 400

Are you ready for the answers ?

Answers

(a) 80(b) 120(c) 45/130

  100g 200g 300g Total

Ground  15 50 55  120

Powder 80 35 26 141 

Granules 40 45 54  139

Total 135 130  135 400

Task One – Data Collection

Has an IpodTOTA

LSYes No

Has been to Myrtle Beach

Yes

No

TOTALS

What is the probability that…

• A student has an Ipod?• A student has been to Myrtle Beach?• A student does not have an Ipod?• A student has not been to Myrtle Beach?• A student has an Ipod and has been to the

Myrtle Beach?• A student has an Ipod or has been to Myrtle

Beach) • A student does not have an Ipod or has not

been to the Myrtle Beach?

Example Two

Brown HairTOTA

LSYes No

Hours watching TV

last night

Less than 2 hours

2 or more hours

TOTALS

What is the probability that…

• A student has brown hair?• A student spent less than 2 hours

watching TV last night?• A brunette student spent less than 2

hours watching TV last night?• A student who was watching TV for

more than 2 hours last night is brunette?

Example 3

Did HomeworkTOTA

LSMale Female

Has correct supplies for

class

Yes

No

TOTALS

Questions?

• Write your own set of at least 5 questions you could answer using the previous two-way table.

Homework