MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 1
Learning Outcomes
After we cover Section 2.2, you should be able to:
1. Construct and interpret each of the following for a quantitative variable:
a. Frequency distribution and relative frequency distribution – grouped (using
classes) and ungrouped (using discrete values of the variable).
b. Histogram (grouped and ungrouped).
c. Stem-and-leaf plot and back-to-back stem-and-leaf plot.
d. Time-series graph (including variations such as a compound time-series graph or
normalized time-series graph).
2. Describe when each of these graphical methods is most appropriate for summarizing
information about a quantitative variable.
3. Describe what is meant by a data distribution.
4. Use a histogram or stem-and-leaf plot to describe the shape of a distribution (e.g.,
uniform, bell-shaped, left or right skewed, bimodal).
a. Describe what the shape says about the data distribution.
5. Recognize incorrectly constructed tables and charts (e.g., histograms with missing
and/or overlapping classes).
Descriptive Statistics for Quantitative Discrete Variables
Quantitative discrete variables have discrete values just like qualitative variables.
A frequency (or relative frequency) distribution for a quantitative discrete variable shows
the frequency (or relative frequency) of each value.
(Relative)
Frequency
distribution
EXAMPLE:
Number of
Siblings per Household
Quantitative discrete variable
Number of siblings living in
household
Frequency and
relative frequency
distribution
Raw data
3 2 8 3 2 0 1 0 1 0 4 1 0 0 2 4 0
1 0 1 2 3 0 1 3 3 2 2 3 8 3 2 0 1
1 3 0 2 0 0 1 3 2 3 2 1 0 5 3 0 1
Sorted data
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
2 3 3 3 3 3 3 3 3 3 3 3 4 4 5 8 8
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 26
geoffrey.krader@morton.edu
No. of
Siblings
0
1
2
3
4
5
6
7
8
Rel.
Freq. Freq.
14
0.27
11
0.22
10
0.20
11
0.22
2
0.04
1
0.02
0
0.00
0
0.00
2
0.04
51
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 2
Descriptive Statistics for Quantitative Discrete Variables (cont’d)
A histogram is a graph that uses bars to represent the frequency (or relative frequency) of each
discrete value. The values are listed in numerical order, as they would appear on a number line.
EXAMPLE: Number of Siblings at Home
Histogram
Quantitative discrete variable
Frequency and
relative frequency
distribution
No. of
Siblings
0
1
2
3
4
5
6
7
8

Rel.
Freq. Freq.
14
0.27
11
0.22
10
0.20
11
0.22
2
0.04
1
0.02
0
0.00
0
0.00
2
0.04
51
 Bars have equal widths.
 No gaps between bars unless there
are values with frequency = 0.
 Each bar is centered over the value it
represents.
Don’t skip any values, even if the frequency is zero!
MAT 141 (Sullivan 4e) - 2.2
Slide 4
GHK 01/2014
What observations can you make about the number of siblings in households of MAT
141 students?
Histograms vs. Bar Graphs
How a Histogram is Similar
to a Bar Graph
In a histogram and a bar graph:
 Use one bar for each value.
 Height of each bar corresponds to the
frequency or relative frequency of the
associated value of the variable.
 Bars are of equal width.
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How a Histogram is Different
than a Bar Graph
In a histogram:
 Values must be listed in order, from
smallest to largest, across the
horizontal axis.
 Show all values between the smallest
and largest observations, even if their
frequency is zero.
 Do not leave gaps between the bars
unless there are values with
frequency=0.
 Each bar is centered over the value it
represents.
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 3
How to Construct a Frequency Distribution or Relative Frequency Distribution for a
Quantitative Discrete Variable:



List each possible value of the variable between the smallest and largest observed
values.
o Do not skip any values, even if their frequency is zero.
Count the frequency (i.e., number of occurrences) of each value.
o The frequencies should add up to the total number of observations.
For a relative frequency distribution, determine the proportion of the total observations
represented by each value.
o The proportions should add up to 1 (except for rounding error).
How to Construct a Histogram for a Quantitative Discrete Variable:


List the discrete values of the variable in numerical order along the horizontal axis.
o List all possible values between the smallest and the largest observed values.
o Do not skip any values, even if their frequency is zero.
Draw one bar per discrete value.
o Height of each bar corresponds to the frequency or relative frequency of each
value.
o All bars should have the same width.
o Each bar is centered over the value it represents.
o Do not leave any gaps between the bars, unless there are values whose
frequency is zero.
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 4
Descriptive Statistics for Quantitative Continuous Variables
Unlike a discrete variable, which can only take on certain values, a continuous variable can take
on any value over an interval on the number line. For example, a person’s weight can be any
positive number.
It is impossible to list all possible values of a continuous variable and count the frequency of each
value. Therefore, we construct frequency distributions and histograms for continuous variables
by grouping the values into a finite number of classes and determining the frequency (or
relative frequency) of each class.
NOTE: Discrete variables with a large number of possible values may be summarized in the
same fashion as continuous variables, i.e., the values may be grouped into classes.
(Relative)
Frequencyofdistribution
EXAMPLE:
Five-Year Compensation
Selected Fortune 30 CEOs
Quantitative continuous variable
Too many values (and gaps) to make a
frequency distribution or histogram with
a frequency for each value.
5-year compensation
of Fortune 30 CEOs
Raw data
40.3
95.2
53.8
1.5
38.8
55.3
110.0
5.8
173.6
130.2
127.8
117.5
120.5
26.5
53.4
25.8
44.7
37.6
45.6
14.1
Sorted data
1.5
5.8
14.1
25.8
26.5
37.6
38.8
40.3
44.7
45.6
53.4
53.8
55.3
95.2
110.0
117.5
120.5
127.8
130.2
173.6
Defining classes of equal width
Lower class limits
are shown here
(e.g., 0.0, 25.0, 50.0, etc.)
Min value
MAT 141 (Sullivan
3e) - 2.1-2.2
Slide 28
1.5
Max value
0.0
25.0
50.0
75.0
100.0
125.0
150.0
175.0
[
)[
)[
)[
)[
)[
)[
)
Class width
Difference between consecutive
lower (or upper) class limits.
75.0 – 50.0 = 25.0
GHK 01/2012
173.6
Every data value lies in
exactly one class
Class 1 includes the values
0.0, 0.1, 0.2, … , 24.9
Class 2 includes the values
25.0, 25.1, 25.2, … , 49.9
Class limits are the lowest and highest values in each
class (e.g., 25.0 and 49.9).
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 30
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 5
How to Construct a Grouped Frequency Distribution
for a Quantitative Continuous Variable:
First, define the classes:
 Make a list of the observed values.
 Sort the list (manually or using technology) to determine the minimum and maximum
values.
 Choose an interval on the number line that includes all the observed values of the
variable.
o The interval may extend to the left of the minimum value and/or to the right of
the maximum value.
 Divide the interval into categories or classes of equal width. Each observed value must
clearly fall into exactly one of the classes.
o The classes must cover all the observed data values.
o The classes must not overlap (e.g., 50-60, 60-70, etc., are not valid classes,
because the value “60” does not fall into a single class).
o There should be no gaps between the classes, even if there are gaps in the
observed values (e.g., 50-57, 60-69, etc., are not valid classes because there is a
gap between 57 and 60).
o Use common sense in defining the classes. For example, there should not be too
many or too few classes (the “right” number depends on the data, though 5-15
classes is often reasonable).
 The class limits are the highest and lowest values within each class. The class width is
the difference between adjacent lower (or upper) class limits.
o Choose class limits that are easy to work with (e.g., if you were describing test
scores, the classes might be 50-59, 60-69, etc.).
o If the classes are 50-59, 60-69, etc., then the class width is 60 – 50 = 10
(not 59 – 50 = 9).
 Open-ended classes (e.g., “below 50”, “90 and above”) are sometimes used at either
end of the interval. If open-ended classes are used, they may have different widths than
the other classes.
After the classes have been defined:
 Count the frequency (i.e., number of occurrences) of the values in each class.
 For a relative frequency distribution, determine the proportion of the total observations
represented by each class.
o The proportions must add up to 1 (except for rounding error).
How to Construct a Histogram for a Quantitative Continuous Variable:


Divide the horizontal axis into classes, as defined above.
o Do not skip any classes, even if their frequency is zero.
Draw one bar per class.
o The height of each bar corresponds to the frequency or relative frequency of
each class.
o The width of each bar is the same as the class width (even if you have openended classes, draw all bars with equal width).
o Do not leave any gaps between the bars, unless there are classes whose
frequency is zero.
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 6
EXAMPLE: Five-Year Compensation of Selected Fortune 30 CEOs (cont’d)
Histogram
Quantitative continuous variable
 Bars have equal widths.
 No gaps between bars unless there
are values with frequency = 0.
Frequency and
relative frequency
distribution
Class
0.0 24.9
25.0 49.9
50.0 74.9
75.0 99.9
100.0 - 124.9
125.0 - 149.9
150.0 - 174.9
Freq.
3
7
3
1
3
2
1
Rel. Freq.
0.15
0.35
0.15
0.05
0.15
0.10
0.05
Lower class limits
shown on left side of
each bar.
0.0
25.0
50.0
75.0
100.0
125.0
150.0
MAT 141 (Sullivan 4e) - 2.2
Slide 10
175.0
GHK 01/2014
Using the TI-83/84 Calculator to Draw Histograms
Histogram
Quantitative continuous variable
Class
0.0 24.9
25.0 49.9
50.0 74.9
75.0 99.9
100.0 - 124.9
125.0 - 149.9
150.0 - 174.9
Freq.
Rel. Freq.
3
0.15
7
0.35
3
0.15
1
0.05
3
0.15
2
0.10
1
0.05
10
Frequency
Frequency and
relative frequency
distribution
Image from TI Calculator Screen.
(Labels were added manually.)
0.0
25.0
50.0
75.0
100.0
125.0
150.0
175.0
5-Year Compensation ($M)
MAT 141 (Sullivan 4e) - 2.2
Slide 11
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 7
Determining the “Right” Number of Classes
The “right number” of classes is the number of
classes that best conveys your message
The “right” number of classes is the number of classes that best conveys your message.
Too many classes may provide
too much detail, which may
obscure the message
Too few classes may not
provide enough detail.
4 classes
0.0-199.9
Width = 50

6 classes
0.0-179.9
Width = 30
7 classes
0.0-174.9
Width = 25
12 classes
0.0-179.9
Width = 15
How does each of the four histograms describe the data?
MAT 141 (Sullivan 4e) - 2.2
Slide 12
GHK 01/2014
Visit: www.rossmanchance.com/applets/Histogram.html for an app which shows how the class
width affects the shape of the histogram.
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MAT 141 – Statistics
Section
2.2 (Sullivan Frequency
4e)
(Relative)
distribution
EXAMPLE:
Age of Fortune
30 CEOs
Quantitative
continuous
variable
Page 8
Age of Fortune 30 CEOs
Raw data
 The range includes 30
possible values:
60 58 54 64 52 55 80 62 51 65
57 51 55 54 52 64 67 59 56 63
51, 52, 53, … ,79, 80
58 58 57 64 60 57 64 75 60 61
 Choose 6 classes with a
class width of 5.
51-55
56-60
etc.
Sorted data
51 51 52 52 54 54 55 55 56 57
57 57 58 58 58 59 60 60 60 61
62 63 64 64 64 64 65 67 75 80
MAT 141 (Sullivan 4e) - 2.2
Slide 13
GHK 01/2014
Histogram
Quantitative continuous variable
Class
51 - 55
56 - 60
61 - 65
66 - 70
71 - 75
76 - 80
Freq.
8
11
8
1
1
1
Rel. Freq.
0.27
0.37
0.27
0.03
0.03
0.03
Relative Frequency
Frequency and
relative frequency
distribution
 Bars have equal widths.
 No gaps between bars unless there
are values with frequency = 0.
Lower class limits
51
shown on left side of
each
bar. “right number”
The
56
61
66
71
76
81
of classes is the number of
classes that best conveys your message
MAT 141 (Sullivan 4e) - 2.2
Slide 14
5 classes
51-80
Width = 6
GHK 01/2014
6 classes
51-80
Width = 5
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 38
geoffrey.krader@morton.edu
8 classes
51-82
Width = 4
8 classes
50-81
Width=4
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 9
EXAMPLE: Age of Fortune 30 CEOs (cont’d)

How does each of the four histograms describe the data?
Summary: Frequency Distributions and Histograms for Quantitative Variables
Type of Quantitative Variable
Discrete variable with a relatively small
number of different values.
Discrete variable with a large number of
different values.
Continuous variable.
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Type of Frequency Distribution and
Histogram
Ungrouped (i.e., each bar is associated with a
specific value).
Grouped (i.e., values are grouped into classes of
equal width and each bar is associated with a
specific class).
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 10
Stem-and-Leaf Plots
Stem-and-leaf plots are similar to histograms but they preserve the raw data values.
Because the “leaf” is usually the last digit in the observed value, the class width is often 10.
EXAMPLE: Players’ Weights – 2005 Chicago Bears
A histogram tells us
how many players are
in each weight class,
but we do not know
the specific values
within each class.
Histogram
Quantitative Continuous Variable
2005 Chicago Bears
12
Frequency
10
8
6
4
2
0
175
200
225
250
275
300
325
350
375
Weight (lbs.)
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 41
GHK 01/2012
Stem-andleaf plot
Stems
Multiple occurrences
of the same value are
repeated
Histogram
Class width = 10
18
0 0 1 5 6 7
19
2 2 5 5 6
20
0 5 7
21
0 2 7 8
22
0 0 3 3 8
23
5 5 8 8
24
0 2
25
3 4 8
26
0 0 2 5
27
2 4 5
28
2005 Chicago Bears
Player’s Weights (lbs.)
MAT 141 (Sullivan 4e) - 2.2
Slide 19
A stem-and-leaf plot
preserves the data
values. This stemand-leaf plot is similar
to a histogram with a
class with of 10.
Leaves
29
2
30
0 0 0 0 2
31
5 8 8 8
32
0
33
0
Data values:
272, 274, 275
34
35
5
GHK 01/2014
How to Construct a Stem-and-Leaf Plot for a Quantitative Variable:



Sort the observed values from highest to lowest.
Choose the stem (generally all digits except the right-most digit).
List the remaining digit(s) in ascending order as leaves.
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 11
EXAMPLE: All-in-One Printers – Text Cost Per Page
When the data are rounded to one decimal place (i.e., tenths), the leaves represent the tenths
digit. Thus, the stem and leaf plot is similar to a histogram with a class with of 1.0.
Stem and leaf plot
Text cost per page (cents)
All-in-one inkjet printers rated by Consumer Reports, December 2010
0 9
1 1 1
2 1 6 7 7
Multiple occurrences
of the same value are
repeated
3 0 1 1 2 3 4 5 5 6 9
4 2 4 4 5 5 5 5 5 6 7 7 8 8 8 8 8 9 9
5 0 1 2 3 3 4 7 7
6 0 0 1 2 3 4 5 6
7 1 8
Data values:
7.1, 7.8
8 3
Stems
Leaves
9
10
11
MAT 141 (Sullivan 4e) - 2.2
Slide 20

12 6
GHK 01/2014
What observations can you make about the cost of printing a page of text?
Stem and leaf plot
Sorted data
51
51
52
52
54
54
55
55
56
57
57
57
58
58
58
59
60
60
60
61
62
63
64
64
64
64
65EXAMPLE:
67 75 80
Age of Fortune 30 CEOs
Each stem may be split into an upper and a
lower stem to avoid bunching the data into
just a few categories.
Age of Fortune 30 CEOs
We can split each
stem into two stems
to avoid bunching
the data into just a
few categories.
5 1 1 2 2 4 4
5 5 5 6 7 7 7 8 8 8 9
60-64
6 0 0 0 1 2 3 4 4 4 4
65-69
6 5 7
7
7 5
Data values:
65, 67
8 0
AT 141 (Sullivan 3e) - 2.1-2.2
de 45
GHK 01/2012
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 12
Back-to-Back Stem-and-Leaf Plots
A back-to-back stem-and-leaf plot allows us to compare two different data sets.
The number of observations should be roughly the same in the two data sets so any difference in
the shape of the plots is due to different data distributions, not a different number of
Back-to-back stem and leaf plot can be
observations.
used to compare two data sets
EXAMPLE: Ages at Death of US Presidents and Vice Presidents
Who live longer – US Presidents or Vice Presidents?
Vice Presidents of the US (N=41)
Presidents of the US (N=38)
4
69
77541
5
36678
887666643200
6
003344567778
98877643211000
7
0112347889
853110
8
01358
8630
9
0033
People who served in both positions (e.g., Nixon, Ford) are shown twice.
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 45

GHK 01/2012
What observations can you make about the life expectancy of US presidents and vice
presidents? Is there a difference in life expectancy between the two offices?
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 13
Shape of a Data Distribution
The distribution of a variable describes how the values of the variable (i.e., the data) vary, e.g.:
 What values does the variable assume?
 Which values occur more frequently? Less frequently?
Different variables have different distributions, e.g.:
 Height of students in this class.
 Height (length) of newborn children.
 Odometer readings of cars in the parking lot.
The shape of a histogram can tell us a lot about a variable’s data distribution.
 What does each
shape tell
us about the data distribution?
Common
data
distributions
Common data distributions
Uniform or rectangular
Normal or bell-shaped
Uniform or rectangular
Normal or bell-shaped
(Right) skewed
Bi-modal
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 47
(Right) skewed
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 47
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GHK 01/2012
Bi-modal
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 14
Shape of a Data Distribution (cont’d)

Which distributions are more or less symmetrical? What does that tell you about the
data distribution?
NOTE: These shapes are not used to describe bar charts of qualitative variables, because the
bars can often be arranged in any order.
EXAMPLE: Data Distributions
Identify the shape of the following data distributions and describe what the shape tells you
about the data distribution.

Five-Year Compensation of Selected Fortune 30 CEOs (page 6).

All-in-One Printers – Text Cost Per Page (page 11).
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 15
Time Series Graphs
A time series graph shows the value of a quantitative variable over a period of time. The time
intervals depend on the data (e.g., years, months, days, etc.).
Time Series Graph
EXAMPLE: Average Life Span in US
Average Life Span in US
Year
90.0
Life Span (yrs)
80.0
70.0
60.0
Men
50.0
40.0
30.0
Male
Life
Span
Female
Life
Span
1907
45.6
49.9
1927
59.0
62.1
1947
64.4
69.7
1967
67.0
74.3
1987
71.4
78.3
2007
75.1
79.7
20.0
10.0
0.0
1907
1927
1947
1967
1987
2007
Year
Source: AARP (March/April 2007)

What observations can you make about the average life span of men in the US during the
20th century?
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 52
GHK 01/2012
How to Construct a Time Series Graph




Collect the data over a period of time. List the data in chronological order.
Label the time intervals on the horizontal axis.
o Equal time intervals should have equal widths, despite the number of
observations. For example, if there were three observations in the 1980s and
eight observations in the 1990s, the intervals for each decade would still have
the same width on the horizontal axis.
Plot a point for each observation. The x-coordinate represents the time of the
observation and the y-coordinate represents the value of the variable at that time.
Use line segments to connect the points. Do not attempt to smooth the graph.
EXAMPLE: Temperatures
Draw a rough time series graph to show
average daily high temperatures in Chicago
over a one-year period.
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 16
A time-series
graph
is
EXAMPLE:
Temperatures
(cont’d)
not a histogram and
it does not describe a data distribution!
A time-series graph
shows how a variable
changes over time.
This is not a bell-shaped distribution …
in fact it’s not even a distribution!
Source: chicagonow.com, January 27, 2013.
MAT 141 (Sullivan 4e) - 2.2
Slide 28
GHK 01/2014
NOTE: Do not confuse a time series graph with a histogram. A time series graph describes a
This
histogram
describes
the
distribution
variable
over
time – not a data
distribution
– and,
therefore, it is not classified as uniform, bellshaped,
etc.
of temperatures on January 1
A histogram shows the
data distribution, i.e.,
how often each value of
the variable occurs.
0
10
20
30
40
50
60
70
Source: www.crh.noaa.gov
MAT 141 (Sullivan 4e) - 2.2
Slide 29
GHK 01/2014
Distinguishing Between Time-Series Graphs and Histograms
Function
x-axis
(Horizontal)
y-axis
(Vertical)
Time-Series Graphs
Shows how the value of a
variable changes over
time.
Time of observation
Values of the variable
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Histograms
Shows the data
distribution, i.e., how
often each value of the
variable occurs.
Values of the variable
Frequency or relative
frequency of each value.
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MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 17
Compound Time-Series Graphs
A compound time
series graph
shows
the value of more than one variable over a period of
Compound
Time
Series
Graph
time.
Life Span (yrs)
Average Life Span in US
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
1907
Year
Women
Men
1927
1947
1967
1987
2007
Male
Life
Span
Female
Life
Span
1907
45.6
49.9
1927
59.0
62.1
1947
64.4
69.7
1967
67.0
74.3
1987
71.4
78.3
2007
75.1
79.7
Year
Source: AARP (March/April 2007)

What observations can you make about the average life span of men and women in the
US during the 20th century?
MAT 141 (Sullivan 3e) - 2.1-2.2
Slide 53
GHK 01/2012
EXAMPLE: College Tuition
Time series graphs are sometimes drawn with vertical bars. There is one bar for each
observation, and the height of the bar represents the value of the variable.
geoffrey.krader@morton.edu
kradermath.jimdo.com
01/2014
MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 18
EXAMPLE: College Tuition (cont’d)
Use the compound time-series graph below to answer the following:

Which has increased more: public or private college tuition?

Which has increased by a larger percentage: public or private college tuition?
Compound Time-Series Graph
Private
Public
Source: trends.collegeboard.org/college_pricing
MAT 141 (Sullivan 4e) - 2.2
Slide 33
Academic
Year
1981-82
1982-83
1983-84
1984-85
1985-86
1986-87
1987-88
1988-89
1989-90
1990-91
1991-92
1992-93
1993-94
1994-95
1995-96
1996-97
1997-98
1998-99
1999-00
2000-01
2001-02
2002-03
2003-04
2004-05
2005-06
2006-07
2007-08
2008-09
2009-10
2010-11
2011-12
Private
Public
Nonprofit
Four-Year Four-Year
$2,242
$10,144
$2,389
$10,749
$2,596
$11,518
$2,665
$12,058
$2,762
$12,828
$2,917
$13,737
$2,948
$13,992
$3,008
$15,260
$3,080
$15,733
$3,306
$16,182
$3,495
$16,276
$3,753
$16,800
$3,966
$17,221
$4,118
$17,841
$4,164
$18,097
$4,281
$18,698
$4,379
$19,404
$4,495
$20,362
$4,556
$21,031
$4,586
$21,013
$4,793
$22,117
$5,141
$22,655
$5,706
$23,280
$6,114
$23,910
$6,350
$24,257
$6,443
$24,766
$6,715
$25,754
$6,770
$25,859
$7,396
$27,412
$7,889
$28,254
$8,244
$28,500
GHK 01/2014
01/2012
Normalized Time-Series Graph
In a normalized time-series graph, the y-coordinate represents the following ratio:
Thus the initial y-coordinate is 1. A y-coordinate of 3 would indicate an observed value that is 3
times the initial value.
geoffrey.krader@morton.edu
kradermath.jimdo.com
01/2014
MAT 141 – Statistics
Section 2.2 (Sullivan 4e)
Page 19
EXAMPLE: College Tuition
Normalized Compound
Time-Series Graph
Academic
Year
1981-82
1982-83
1983-84
1984-85
1985-86
1986-87
1987-88
Public
Private
To normalize, divide each data point by the
starting value (i.e., $2,242 or $10,144)
Source: trends.collegeboard.org/college_pricing
MAT 141 (Sullivan 4e) - 2.2
Slide 34

1988-89
1989-90
1990-91
1991-92
1992-93
1993-94
1994-95
1995-96
1996-97
1997-98
1998-99
1999-00
2000-01
2001-02
2002-03
2003-04
2004-05
2005-06
2006-07
2007-08
2008-09
2009-10
2010-11
2011-12
Private
Public
Nonprofit
Four-Year Four-Year
1.00
1.00
1.07
1.06
1.16
1.14
1.19
1.19
1.23
1.26
1.30
1.35
1.31
1.38
1.34
1.37
1.47
1.56
1.67
1.77
1.84
1.86
1.91
1.95
2.00
2.03
2.05
2.14
2.29
2.55
2.73
2.83
2.87
3.00
3.02
3.30
3.52
3.68
1.50
1.55
1.60
1.60
1.66
1.70
1.76
1.78
1.84
1.91
2.01
2.07
2.07
2.18
2.23
2.29
2.36
2.39
2.44
2.54
2.55
2.70
2.79
2.81
GHK 01/2014
Which has increased by a larger percentage: public or private college tuition?
geoffrey.krader@morton.edu
kradermath.jimdo.com
01/2014