Transcript Section 1.1

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Chapter 1: Exploring Data
Section 1.1
Analyzing Categorical Data
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE
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Chapter 1
Exploring Data
 Introduction:
Data Analysis: Making Sense of Data
 1.1
Analyzing Categorical Data
 1.2
Displaying Quantitative Data with Graphs
 1.3
Describing Quantitative Data with Numbers
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Section 1.1
Analyzing Categorical Data
Learning Objectives
After this section, you should be able to…
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CONSTRUCT and INTERPRET bar graphs and pie charts
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RECOGNIZE “good” and “bad” graphs
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CONSTRUCT and INTERPRET two-way tables
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DESCRIBE relationships between two categorical variables
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ORGANIZE statistical problems
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Individuals vs Variables
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Individuals: the objects
described by a set of data
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Variables: any characteristic of
an individual

Individuals can be people,
animals or things
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Variables can take different
values for different individuals
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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.
Example, page 8
Frequency Table
Format
Variable
Values
Relative Frequency Table
Count of Stations
Format
Percent of Stations
Adult Contemporary
1556
Adult Contemporary
Adult Standards
1196
Adult Standards
8.6
Contemporary Hit
4.1
Contemporary Hit
569
11.2
Country
2066
Country
14.9
News/Talk
2179
News/Talk
15.7
Oldies
1060
Oldies
Religious
2014
Religious
Rock
869
Spanish Language
750
Other Formats
Total
1579
13838
7.7
14.6
Rock
6.3
Count
Spanish Language
5.4
Other Formats
11.4
Total
99.9
Percent
Analyzing Categorical Data
Variables place individuals into one of
several groups or categories
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 Categorical
Variables: take on numerical
 What’s
the difference between categorical
and quantitative variables?
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“who” is being measured vs. “what” is being
measured
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Do we ever use numbers to describe the
values of a categorical variable?
Analyzing Quantitative Data
values
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 Quantitative
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The distribution of a variable tells us what
values the variable takes and how often it
takes these values.
Distribution
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Who are the individuals in the data set?
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What variables are measured? Identify each as
categorical or quantitative. In what units were the
quantitative variables measured?
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Describe the individual in the first row.
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Here is information about 10 randomly selected US
residents from the 2000 census.
Example
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Categorical: state, gender, marital status
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Quantitative: number of family members, age in
years, total income in dollars, travel time to work in
mins.
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This person is a 61 year old married female from
Kentucky who drives 20 minutes to work and has a
total income of $21,000. She has 2 family
members in her household.
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Individuals: the 10 randomly selected U.S.
residents from the 2000 census.
Answers
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categorical data
Frequency tables: displays the counts of each category
Relative Frequency table: shows the percents of each category
Frequency Table
Format
Relative Frequency Table
Count of Stations
Format
Percent of Stations
Adult Contemporary
1556
Adult Contemporary
Adult Standards
1196
Adult Standards
8.6
Contemporary Hit
4.1
Contemporary Hit
569
11.2
Country
2066
Country
14.9
News/Talk
2179
News/Talk
15.7
Oldies
1060
Oldies
Religious
2014
Religious
7.7
14.6
Rock
869
Rock
6.3
Spanish Language
750
Spanish Language
5.4
Other Formats
Total
1579
13838
Other Formats
11.4
Total
99.9
Analyzing Categorical Data
 Displaying
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What is the difference between a frequency table and
a relative frequency table? When is it better to use
relative frequency tables?
Analyzing Categorical Data
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Frequency tables can be difficult to read. Sometimes
is is easier to analyze a distribution by displaying it
with a bar graph or pie chart.
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categorical data
Analyzing Categorical Data
 Displaying
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When is it inappropriate to use a pie chart?
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Hint: categorical vs. quantitative
What are some common ways to make a misleading
graph?
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What is the most important thing to remember when
making pie charts and bar graphs? Why do statisticians
prefer bar graphs?
Analyzing Categorical Data
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Good and Bad
Our eyes react to the area of the bars as well as
height. Be sure to make your bars equally wide.
Avoid the temptation to replace the bars with pictures
for greater appeal…this can be misleading!
Alternate Example
This ad for DIRECTV
has multiple problems.
How many can you
point out?
Analyzing Categorical Data
Bar graphs compare several quantities by comparing
the heights of bars that represent those quantities.
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 Graphs:
1. Choose or generate a question that will result in a range of
categorical data.
Examples: What is your favorite ice cream flavor?, If you could be
a superhero, what would your super power be?, etc.
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2. Survey at least 25 people to gather data to answer your
question. Record your responses.
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3. Organize your data into a frequency table.
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4. Create a relative frequency table of the results.
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5. Create a bar graph of your data. (Don’t forget labels!)
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6. Write a brief explanation as to why or why not a pie chart
would be appropriate for your data.
Classwork: Frequency Tables
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Reminder:
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Explanatory Variable:Any variable that explains the response
variable. Often called an independent variable or predictor
variable.
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Response Variable: The outcome of a study. A variable you
would be interested in predicting or forecasting. Often called a
dependent variable or predicted variable.
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In the past, we have looked at data with one categorical
variable. Now we will look at data with more than one
categorical variable.
Analyzing Categorical Data
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When a dataset involves two categorical variables, we begin by
examining the counts or percents in various categories for one
of the variables.
Definition:
Two-way Table – describes two categorical
variables, organizing counts according to a row
variable and a column variable.
Example, p. 12
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 twoway table?
How many young
adults were surveyed?
How many females
were surveyed?
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Tables and Marginal Distributions
Analyzing Categorical Data
 Two-Way
Definition:
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.
Why use the marginal distribution?: Percents 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 percents) of the row or column totals.
2)Make a graph to display the marginal distribution.
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Tables and Marginal Distributions
Analyzing Categorical Data
 Two-Way
Example, p. 13
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
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.
Hint: Marginal distributions are
calculated in the margins! If
there is no “total” row or
column, make one!
Chance of being wealthy by age 30
Percent
Percent
Response
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Tables and Marginal Distributions
35
30
25
20
15
10
5
0
Almost
none
Some
50-50
Good
chance
chance
chance
Survey Response
Almost
certain
Analyzing Categorical Data
 Two-Way
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Problem: Marginal distributions tell us nothing about the
relationship between two variables.
Definition:
A Conditional Distribution of a variable
describes the values of that variable among
individuals who have a specific value of another
variable.
To examine or compare conditional distributions,
1)Select the row(s) or column(s) of interest.
2)Use the data in the table to calculate the conditional
distribution (in percents) of the row(s) or column(s).
3)Make a graph to display the conditional distribution.
• Use a side-by-side bar graph or segmented bar
graph to compare distributions.
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Between Categorical Variables
Analyzing Categorical Data
 Relationships
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Tables and Conditional Distributions
Analyzing Categorical Data
 Two-Way
Example, p. 15
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
Male
Female
Almost no chance
98/2459 =
4.0%
96/2367 =
4.1%
286/2459 =
11.6%
426/2367 =
18.0%
720/2459 =
29.3%
696/2367 =
29.4%
Some chance
A 50-50 chance
A good chance
Almost certain
758/2459 =
30.8%
663/2367 =
28.0%
597/2459 =
24.3%
486/2367 =
20.5%
Examine the relationship
between gender and
opinion.
Chance of being wealthy
Chance
wealthy by
by age
age30
30
100%
90%
80%
Percent
Percent
Response
Calculate the conditional
distribution of opinion
among males.
70%
35
60%
30
25
50%
20
40%
15
10
30%
5
20%
0
Almost certain
Good chance
Males
Males
50-50 chance
10%Almost no
chance
0%
Some
chance
Males
50-50
chance
Good
chance
chance
Females
OpinionOpinion
Opinion
Females
Some chance
Almost
Almost
certain
certain
Almost no chance
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Why are they good to use?
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They are easy to compare!
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Forces you to use percents
Chance of being wealthy by age 30
100%
90%
80%
Percent
70%
60%
Almost certain
50%
Good chance
40%
30%
50-50 chance
20%
Some chance
10%
Almost no chance
0%
Males
Females
Opinion
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Segmented Bar Graph: For each category of one variable,
there is a single bar divided into categories of the other
variable.
Segmented Bar Graph
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What does it mean for two variables to have an
association?
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Knowing the value of one variable helps you predict the
value of the other variable. (Think about explanatory and
response)
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The whole point of analyzing more than one categorical
variable at the same time is to see if they are
associated.
Association
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Example: A sample of 200 children from the United
Kingdom ages 9–17 was selected from the
CensusAtSchool website. The gender of each student
was recorded along with which super power they would
most like to have: invisibility, super strength, telepathy
(ability to read minds), ability to fly, or ability to freeze
time.
Female
Male
Total
Invisibilty
17
13
30
Super
Strength
3
17
20
Telepathy
39
5
44
Fly
36
18
54
Freeze
Time
20
32
52
Total
115
85
200
Would you say there is
association between the
variables by looking at
the two-way table? In
other words, does
gender explain super
power preference?
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Let’s create the conditional distribution:
Female
Invisibility
.15
Male
Invisibility
.15
Super Strength .03
Super Strength .20
Telepathy
.34
Telepathy
.06
Fly
.31
Fly
.21
Freeze Time
.17
Freeze Time
.38
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a) Explain what it would mean if there was no association
between gender and superpower preference?

(b) Based on this data, can we conclude there is an
association between gender and super power preference?
Justify.
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Section 1.1
Analyzing Categorical Data
Summary
In this section, we learned that…

The distribution of a categorical variable lists the categories and gives
the count or percent of individuals that fall into each category.

Pie charts and bar graphs display the distribution of a categorical
variable.

A two-way table of counts organizes data about two categorical
variables.

The row-totals and column-totals in a two-way table give the marginal
distributions of the two individual variables.

There are two sets of conditional distributions for a two-way table.
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Section 1.1
Analyzing Categorical Data
Summary, continued
In this section, we learned that…

We can use a side-by-side bar graph or a segmented bar graph
to display conditional distributions.

To describe the association between the row and column variables,
compare an appropriate set of conditional distributions.

Even a strong association between two categorical variables can be
influenced by other variables lurking in the background.

You can organize many problems using the four steps state, plan,
do, and conclude.
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Looking Ahead…
In the next Section…
We’ll learn how to display quantitative data.
Dotplots
Stemplots
Histograms
We’ll also learn how to describe and compare
distributions of quantitative data.