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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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Section 2.5
How Can Measures of Position
Describe Spread?
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Quartiles




Splits the data into four parts
The median is the second quartile, Q2
The first quartile, Q1, is the median of the lower
half of the observations
The third quartile, Q3, is the median of the
upper half of the observations
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Example: Find the first and third
quartiles
Prices per share of 10 most actively traded stocks on
NYSE (rounded to nearest $)
2 4 11 12 13 15 31 31 37 47
a.
b.
c.
d.
Q1 = 2
Q1 = 12
Q1 = 11
Q1 =11.5
Q3 =
Q3 =
Q3 =
Q3 =
47
31
31
32
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Measuring Spread: Interquartile
Range

The interquartile range is the distance
between the third quartile and first
quartile:
IQR = Q3 – Q1
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Detecting Potential Outliers

An observation is a potential outlier if
it falls more than 1.5 x IQR below the
first quartile or more than 1.5 x IQR
above the third quartile
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The Five-Number Summary

The five number summary of a
dataset:
• Minimum value
• First Quartile
• Median
• Third Quartile
• Maximum value
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Boxplot

A box is constructed from Q1 to Q3

A line is drawn inside the box at the median

A line extends outward from the lower end of
the box to the smallest observation that is not
a potential outlier

A line extends outward from the upper end of
the box to the largest observation that is not a
potential outlier
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Boxplot for Sodium Data
Sodium Data:
0 200
70 210
125 210
125 220
140 220
150 230
170 250
170 260
180 290
200 290
Five Number Summary:
Min: 0
Q1: 145
Med: 200
Q3: 225
Max: 290
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Boxplot 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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Z-Score

The z-score for an observation measures how far
an observation is from the mean in standard
deviation units
observatio n - mean
z
standard deviation

An observation in a bell-shaped distribution is a
potential outlier if its z-score < -3 or > +3
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