Section 2.2
More Graphs and Displays
.
Section 2.2 Objectives
• Graph quantitative data using stem-and-leaf plots and
dot plots
• Graph qualitative data using pie charts and Pareto
charts (interpret)
• Graph paired data sets using scatter plots and time
series charts (interpret)
Constructing a Stem-and-Leaf Plot
Simple data set ( all values less than 100)
e.g. : 75, 83, 94, 90, 69, 98, 89, 85
Stem: left most digit
Leaf: right most digit
Key:
7 | 5  75
8 | 3  83
9 | 4  94
etc.
3
Constructing a Stem-and-Leaf Plot
Data set with larger values
e.g. : 232, 267, 221, 288, 208, 269 …
Stem: two left most digits
Leaf: right most digit
Key:
23 | 2  232
26 | 7  267
22 | 1  221
4
Constructing a Stem-and-Leaf Plot
Data set with non-integer values
e.g. : 54.3, 57.9, 55.5, 56.2 …
Stem: integer
Leaf: decimal (tenths) value
Key:
54 | 3  54.3
57 | 9  57.9
55 | 5  55.5
5
Graphing Quantitative Data Sets
Stem-and-leaf plot
• Each number is separated into a stem and a leaf.
• Similar to a histogram.
• Still contains original data values.
26
Data: 21, 25, 25, 26, 27, 28,
30, 36, 36, 45
2
3
1 5 5 6 7 8
0 6 6
4
5
Example: Constructing a Stem-and-Leaf
Plot
The following are the numbers of text messages sent
last week by the cellular phone users on one floor of a
college dormitory. Display the data in a stem-and-leaf
plot.
155 159
118 118
139 139
129 112
144
108
122
126
129
122
78
148
105 145 126 116 130 114 122 112 112 142 126
121 109 140 126 119 113 117 118 109 109 119
133 126 123 145 121 134 124 119 132 133 124
147
Solution: Constructing a Stem-and-Leaf
Plot
155 159
118 118
139 139
129 112
144
108
122
126
129
122
78
148
105 145 126 116 130 114 122 112 112 142 126
121 109 140 126 119 113 117 118 109 109 119
133 126 123 145 121 134 124 119 132 133 124
147
• The data entries go from a low of 78 to a high of 159.
• Use the rightmost digit as the leaf.
 For instance,
78 = 7 | 8
and 159 = 15 | 9
• List the stems, 7 to 15, to the left of a vertical line.
• For each data entry, list a leaf to the right of its stem.
Solution: Constructing a Stem-and-Leaf
Plot
Include a key to identify
the values of the data.
From the display, you can conclude that more than 50% of the
cellular phone users sent between 110 and 130 text messages.
Graphing Quantitative Data Sets
Dot plot
• Each data entry is plotted, using a point, above a
horizontal axis.
Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45
26
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Example: Constructing a Dot Plot
Use a dot plot organize the text messaging data.
155 159
118 118
139 139
129 112
144
108
122
126
129
122
78
148
105 145 126 116 130 114 122 112 112 142 126
121 109 140 126 119 113 117 118 109 109 119
133 126 123 145 121 134 124 119 132 133 124
147
• So that each data entry is included in the dot plot, the
horizontal axis should include numbers between 70 and
160.
• To represent a data entry, plot a point above the entry's
position on the axis.
• If an entry is repeated, plot another point above the
previous point.
Solution: Constructing a Dot Plot
155 159
118 118
139 139
129 112
144
108
122
126
129
122
78
148
105 145 126 116 130 114 122 112 112 142 126
121 109 140 126 119 113 117 118 109 109 119
133 126 123 145 121 134 124 119 132 133 124
147
From the dot plot, you can see that most values cluster
between 105 and 148 and the value that occurs the
most is 126. You can also see that 78 is an unusual data
value.
Graphing Qualitative Data Sets
Pie Chart
• A circle is divided into sectors that represent
categories.
• The area of each sector is proportional to the
frequency of each category.
Example: Constructing a Pie Chart
The numbers of earned degrees conferred (in thousands)
in 2007 are shown in the table. Use a pie chart to
organize the data.
Type of degree
Associate’s
Bachelor’s
Master’s
First professional
Doctoral
Number
(thousands)
728
1525
604
90
60
Solution: Constructing a Pie Chart
• Find the relative frequency (percent) of each category.
Type of degree
Associate’s
Bachelor’s
Master’s
First professional
Doctoral
Frequency, f
Relative frequency
728
728
 0.24
3007
1525
1525
 0.51
3007
604
604
 0.20
3007
90
90
 0.03
3007
60
60
 0.02
3007
Σf = 3007
Solution: Constructing a Pie Chart
• Construct the pie chart using the central angle that
corresponds to each category.
 To find the central angle, multiply 360º by the
category's relative frequency.
 For example, the central angle for associate’s
degrees is
360º(0.24) ≈ 86º
Solution: Constructing a Pie Chart
Type of degree
Relative
Frequency, f frequency
Central angle
Associate’s
728
0.24
360º(0.24)≈86º
Bachelor’s
1525
0.51
360º(0.51)≈184º
604
0.20
360º(0.20)≈72º
First professional
90
0.03
360º(0.03)≈11º
Doctoral
60
0.02
360º(0.02)≈7º
Master’s
Solution: Constructing a Pie Chart
Relative
frequency
Central
angle
Associate’s
0.24
86º
Bachelor’s
0.51
184º
Master’s
0.20
72º
First professional
0.03
11º
Doctoral
0.02
7º
Type of degree
From the pie chart, you can see that over one half of the
degrees conferred in 2007 were bachelor’s degrees.
Graphing Qualitative Data Sets
Frequency
Pareto Chart
• A vertical bar graph in which the height of each bar
represents frequency or relative frequency.
• The bars are positioned in order of decreasing height,
with the tallest bar positioned at the left.
Categories
Example: Constructing a Pareto Chart
In a recent year, the retail industry lost $36.5 billion in
inventory shrinkage. Inventory shrinkage is the loss of
inventory through breakage, pilferage, shoplifting, and
so on. The causes of the inventory shrinkage are
administrative error ($5.4 billion), employee theft
($15.9 billion), shoplifting ($12.7 billion), and vendor
fraud ($1.4 billion). Use a Pareto chart to organize this
data.
Solution: Constructing a Pareto Chart
Cause
$ (billion)
Admin. error
5.4
Employee
theft
15.9
Shoplifting
12.7
Vendor fraud
1.4
Millions of dollars
Causes of Inventory Shrinkage
20
15
10
5
0
Employee
Theft
Shoplifting Admin. Error
Cause
Vendor
fraud
From the graph, it is easy to see that the causes of inventory
shrinkage that should be addressed first are employee theft and
shoplifting.
Graphing Paired Data Sets
Paired Data Sets
• Each entry in one data set corresponds to one entry in
a second data set.
• Graph using a scatter plot.
 The ordered pairs are graphed as y
points in a coordinate plane.
 Used to show the relationship
between two quantitative variables.
x
Example: Interpreting a Scatter Plot
The British statistician Ronald Fisher introduced a
famous data set called Fisher's Iris data set. This data set
describes various physical characteristics, such as petal
length and petal width (in millimeters), for three species
of iris. The petal lengths form the first data set and the
petal widths form the second data set.
Example: Interpreting a Scatter Plot
As the petal length increases, what tends to happen to
the petal width?
Each point in the
scatter plot
represents the
petal length and
petal width of one
flower.
Solution: Interpreting a Scatter Plot
Interpretation
From the scatter plot, you can see that as the petal
length increases, the petal width also tends to
increase.
Graphing Paired (Time & Value) Data Sets
Quantitative
data
Time Series
• Data set is composed of quantitative entries taken at
regular intervals over a period of time.
 e.g., The amount of precipitation measured each
day for one month.
• Use a time series chart to graph.
time
Example: Constructing a Time Series
Chart
The table lists the number of cellular
telephone subscribers (in millions)
for the years 1998 through 2008.
Construct a time series chart for the
number of cellular subscribers.
Solution: Constructing a Time Series
Chart
• Let the horizontal axis represent
the years.
• Let the vertical axis represent the
number of subscribers (in
millions).
• Plot the paired data and connect
them with line segments.
Solution: Constructing a Time Series
Chart
The graph shows that the number of subscribers has been
increasing since 1998, with greater increases recently.