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Chapter 2
Exploring Data with Graphs and
Numerical Summaries

Learn ….
The Different Types of Data
The Use of Graphs to Describe
Data
The Numerical Methods of
Summarizing Data
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Section 2.1
What are the Types of Data?
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In Every Statistical Study:
 Questions
are posed
 Characteristics are observed
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Characteristics are Variables
A Variable is any characteristic that
is recorded for subjects in the study
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Variation in Data

The terminology variable highlights
the fact that data values vary.
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Example: Students in a
Statistics Class

Variables:
• Age
• GPA
• Major
• Smoking Status
•…
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Data values are called
observations

Each observation can be:
• Quantitative
• Categorical
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Categorical Variable

Each observation belongs to one of a set of
categories

Examples:
• Gender (Male or Female)
• Religious Affiliation (Catholic, Jewish, …)
• Place of residence (Apt, Condo, …)
• Belief in Life After Death (Yes or No)
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Quantitative Variable

Observations take numerical values

Examples:
• Age
• Number of siblings
• Annual Income
• Number of years of education completed
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Graphs and Numerical
Summaries

Describe the main features of a
variable

For Quantitative variables: key
features are center and spread

For Categorical variables: key feature
is the percentage in each of the
categories
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Quantitative Variables

Discrete Quantitative Variables
and

Continuous Quantitative Variables
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Discrete

A quantitative variable is discrete if its
possible values form a set of separate
numbers such as 0, 1, 2, 3, …
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Examples of discrete
variables



Number of pets in a household
Number of children in a family
Number of foreign languages spoken
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Continuous

A quantitative variable is continuous
if its possible values form an interval
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Examples of Continuous
Variables




Height
Weight
Age
Amount of time it takes to complete
an assignment
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Frequency Table

A method of organizing data

Lists all possible values for a variable
along with the number of
observations for each value
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Example: Shark Attacks
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Example:
Example: Shark
Shark Attacks
Attacks

What is the variable?

Is it categorical or quantitative?

How is the proportion for Florida
calculated?

How is the % for Florida calculated?
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Example: Shark Attacks

Insights – what the data tells us about
shark attacks
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Identify the following variable as
categorical or quantitative:
Choice of diet
(vegetarian or non-vegetarian):
a.
b.
Categorical
Quantitative
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Identify the following variable as
categorical or quantitative:
Number of people you have known who have
been elected to political office:
a.
b.
Categorical
Quantitative
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Identify the following variable as
discrete or continuous:
The number of people in line at a box office to
purchase theater tickets:
a.
b.
Continuous
Discrete
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Identify the following variable as
discrete or continuous:
The weight of a dog:
a.
Continuous
b.
Discrete
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Section 2.2
How Can We Describe Data Using
Graphical Summaries?
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Graphs for Categorical Data

Pie Chart: A circle having a “slice of
pie” for each category

Bar Graph: A graph that displays a
vertical bar for each category
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Example: Sources of Electricity Use
in the U.S. and Canada
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Pie Chart
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Bar Chart
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Pie Chart vs. Bar Chart


Which graph do you prefer?
Why?
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Graphs for Quantitative Data

Dot Plot: shows a dot for each
observation

Stem-and-Leaf Plot: portrays the
individual observations

Histogram: uses bars to portray the
data
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Example: Sodium and Sugar
Amounts in Cereals
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Dotplot for Sodium in Cereals

Sodium Data:
0 210 260 125 220 290 210 140 220 200 125
170 250 150 170 70 230 200 290 180
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Stem-and-Leaf Plot for
Sodium in Cereal
Sodium Data:
0 210
260 125
220 290
210 140
220 200
125 170
250 150
170 70
230 200
290 180
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Frequency Table
Sodium Data:
0 210
260 125
220 290
210 140
220 200
125 170
250 150
170 70
230 200
290 180
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Histogram for Sodium in Cereals
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Which Graph?

Dot-plot and stem-and-leaf plot:

Histogram
• More useful for small data sets
• Data values are retained
• More useful for large data sets
• Most compact display
• More flexibility in defining intervals
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Shape of a Distribution

Overall pattern
• Clusters?
• Outliers?
• Symmetric?
• Skewed?
• Unimodal?
• Bimodal?
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Symmetric or Skewed ?
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Example: Hours of TV Watching
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Identify the minimum and maximum
sugar values:
a.
2 and 14
c.
1 and 15
b.
d.
1 and 3
0 and 16
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Consider a data set containing IQ
scores for the general public:
What shape would you expect a histogram of
this data set to have?
a.
Symmetric
b.
Skewed to the left
c.
Skewed to the right
d.
Bimodal
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Consider a data set of the scores of
students on a very easy exam in which most
score very well but a few score very poorly:
What shape would you expect a histogram of
this data set to have?
a. Symmetric
b. Skewed to the left
c. Skewed to the right
d. Bimodal
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Section 2.3
How Can We describe the Center of
Quantitative Data?
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Mean

The sum of the observations
divided by the number of
observations
x 

x
n
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Median

The midpoint of the observations
when they are ordered from the
smallest to the largest (or from the
largest to the smallest)
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Find the mean and median
CO2 Pollution levels in 8 largest nations measured in
metric tons per person:
2.3 1.1 19.7 9.8 1.8 1.2 0.7 0.2
a.
b.
c.
Mean = 4.6
Mean = 4.6
Mean = 1.5
Median = 1.5
Median = 5.8
Median = 4.6
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Outlier

An observation that falls well above
or below the overall set of data

The mean can be highly influenced by
an outlier

The median is resistant: not affected
by an outlier
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Mode

The value that occurs most
frequently.

The mode is most often used with
categorical data
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Section 2.4
How Can We Describe the Spread of
Quantitative Data?
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Measuring Spread: Range

Range: difference between the largest
and smallest observations
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Measuring Spread: Standard
Deviation

Creates a measure of variation by
summarizing the deviations of each
observation from the mean and
calculating an adjusted average of these
deviations
s
( x  x )2
n 1
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Empirical Rule
For bell-shaped data sets:

Approximately 68% of the observations fall
within 1 standard deviation of the mean

Approximately 95% of the observations fall
within 2 standard deviations of the mean

Approximately 100% of the observations fall
within 3 standard deviations of the mean
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Parameter and Statistic

A parameter is a numerical summary of
the population

A statistic is a numerical summary of a
sample taken from a population
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