A Mathematical View of Our World

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A Mathematical View of Our World

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Title: A Mathematical View of Our World

1
A Mathematical View of Our World

1st ed.
Parks, Musser, Trimpe, Maurer, and Maurer

2
Chapter 8

Descriptive Statistics
Data and Patterns

3
Section 8.1Organizing and Picturing Data

Goals
Study visual displays of data
Dot plots
Stem-and-leaf plots
Histograms
Bar graphs
Line graphs
Pie charts

4
8.1 Initial Problem

You need to give a sales report showing that
District A had 135,000 in sales.
District B had 85,000 in sales.
District C had 115,000 in sales.
How can you present this data clearly to compare
the 3 districts?
The solution will be given at the end of the
section.

5
Obtaining Data

Data sets are sets of numbers collected from the
real world.
Data can be obtained from
Previously published research
A designed experiment
An observational study
A survey

6
Obtaining Data, contd

Once data has been collected, exploratory data
analysis takes an initial look at data to see
what patterns might emerge or what further
questions need to be asked.
One way to carry out exploratory data analysis is
to represent data pictorially.

7
Dot Plots

A dot plot is a graph in which
The horizontal axis represents the data values.
The vertical axis represents the frequency of the
data values.
One dot is placed for each occurrence of each
data value.

8
Example 1

Create a dot plot for the test scores 26, 32,
54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87,
87, 87, 89, 93, 95, 96.
Solution Notice the scores have been arranged
in order.

a. 50
b. 51
c. 52
d. 53

9
Stem-and-Leaf Plot

A stem-and-leaf plot is a graph in which
The digit furthest to the right is called the
leaf.
The other digits are called the stem.
The stems and leaves are placed in vertical
columns, with the leaves arranged in numerical
order.

10
Example 2

Create a stem-and-leaf plot for the test scores
26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81,
84, 87, 87, 87, 89, 93, 95, 96.
Solution The tens digits will be the stems and
the ones digits will be the leaves.

11
Example 2, contd

Solution, contd The plot at right shows
A cluster of values between 54 and 96.
A gap between 54 and 32.
The values 32 and 26 are outliers, separated from
the other scores by a large gap.

12
Example 3

Create and interpret a stem-and-leaf plot for the
pizza prices9.20, 10.50, 10.70, 10.80,
12.00, 12.10, 12.20, 12.20, 12.30.
Solution The dollar amounts will be the stems
and the tens of cents will be the leaves.

13
Example 3, contd

Solution, contd The plot at right shows
Two clusters of prices separated by a gap.
The price 9.20 may be considered an outlier.

14
Histograms

A histogram is a graph in which
The data is separated into intervals called
measurement classes or bins.
Various interval sizes can be chosen, depending
on the situation.
A frequency table, showing the number of data
values in each bin, can be created to aid in
drawing a histogram.

15
Example 4

Create a histogram for the test scores 26, 32,
54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87,
87, 87, 89, 93, 95, 96.
Solution Make a frequency table first, using
bins of width 10.

16
Example 4, contd

Solution, contd Create the histogram.
The height of each bar is equal to the frequency
of the bin.

17
Example 4, contd

Note The choice of bin size affects the
appearance of the graph.
A histogram of the same data set with a bin size
of 5 is shown next.

18
Question

Why is the histogram with bin size 5 not the
best choice to represent the data set from the
previous example? Choose the best answer.

19
Question contd

a. There are outliers.
b. There are a wide range of values.
c. It is hard to see the overall pattern of the
scores.
d. The bars are too narrow.

20
Example 4, contd

A histogram of the same data set with a bin size
of 20 is shown next.

21
Question

Why is the histogram with bin size 20 not the
best choice to represent the data set from the
previous example? Choose the best answer.
a. The bars are too tall.
b. A lot of information
about the data is lost.
c. There are a wide
range of values.
d. The frequencies of the
bins are not the same.

22
Relative Frequency Histograms

A relative frequency histogram is a graph in
which
The data is separated into bins.
The relative frequency (percent of the whole data
set) of each bin is calculated.
The height of each bar is equal to the relative
frequency of the bin.
A relative frequency table can be created to aid
in drawing a relative frequency histogram.

23
Example 5

Create a relative frequency histogram for the
test scores 26, 32, 54, 62, 67, 70, 71, 71, 74,
76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96, using
a bin size of 10.
Solution Find the relative frequency of each
bin.

24
Example 5, contd

Solution, contd The graph is shown below.

25
Bar Graphs

A bar graph is any graph in which the height or
length of bars is used to represent quantities.
A histogram is a special type of bar graph.

26
Example 6

Create a bar graph to display the data in the
table.

27
Example 6, contd
28
Line Graphs

A line graph is used to graph data values that
occur over time.
The horizontal axis represents the time.
The vertical axis represents the data value.
Each data value is plotted and the dots are
connected by a line.

29
Example 7

Create a line graph for the data shown in the bar
graph below.

30
Example 7, contd

Solution

31
Example 8

Interpret the line graph shown here.

32
Example 8, contd

Solution The general trend in the graph is an
increase in the number completing college,
although there were a few years with decreases.
In 1980, about 17.5 completed.
In 2002, about 26.7 completed.
In order to emphasize the most recent statistic,
the percentage for 2002 was highlighted in the
graph.

33
Pie Charts

A pie chart is used to graph relative proportions
of quantities.
Pie charts are also called circle graphs.
Each quantity is graphed as a wedge-shaped
portion of the circle.

34
Example 9

The pie chart shows the average number of hours
of sleep for a certain group of adults.
Interpret the chart.

35
Example 9, contd

Solution Most of the people sleep 7 or 8 hours
per night.
Also, 6 of the people get 5 hours of sleep or
less per night.

36
Choosing a Graph

The different types of graphs and their uses are
summarized below.

37
Example 10

The table shows the average number of hours
worked in different countries.
What type of graph would be most effective?

38
Example 10, contd

Solution
We do not need to show a trend over time or
percentages, so rule out line graphs and pie
charts.
A bar graph would make comparison between
countries easy.
The categories are the countries.
The height of each bar will represent the number
of hours worked per year.

39
Example 10, contd

Solution, contd A bar graph for the data is
shown below.

40
8.1 Initial Problem Solution

You need to give a sales report showing that
District A had 135,000 in sales.
District B had 85,000 in sales.
District C had 115,000 in sales.
How can you present this data clearly to compare
the 3 districts?
Either a bar graph or a pie chart allows for easy
comparison between categories.

41
Initial Problem Solution, contd

A pie chart will clearly show the difference in
proportions of sales from the different
districts.
Calculate the total sales.
Find what portion of a circle represents each
districts sales.
The results are shown at right.

42
Section 8.2Comparisons

Goals
Study comparison graphs
Double-stem-and-leaf plots
Comparison histograms
Multiple bar graphs
Multiple line graphs
Multiple pie charts
Proportional bar graphs

43
8.2 Initial Problem

How can the monthly sales of the 3 items be
presented to show and compare the sales trends?
The solution will be given at the end of the
section.

44
Double-Stem-and-Leaf Plots

A double-stem-and-leaf plot compares two data
sets.
The stems are placed in the middle column.
The leaves of one data set are placed on the
left, and the leaves of the other set on the
right.

45
Example 1

Create a double-stem-and-leaf plot to compare
scores from the two classes.
Class 1 26, 32, 54, 62, 67, 70, 71, 71, 74, 76,
80, 81, 84, 87, 87, 87, 93, 95, 96
Class 2 34, 45, 52, 57, 63, 65, 68, 70, 71, 72,
74, 76, 76, 78, 83, 85, 85, 87, 92, 99

46
Example 1, contd

Solution Since more leaves are at the top on the
left than on the right, it appears that Class 1
did somewhat better on the test than Class 2.

47
Question

Choose the statement that is not true.
a. Class 1 had more low scores than Class 2.
b. Class 2 has a larger
gap than Class 1.
c. Class 2 has fewer
scores in the 80s and
80s than Class 1.
d. Class 2 has a higher
score than Class 1.

48
Comparison Histogram

A comparison histogram compares two data sets.
The same bin size is chosen for both sets.
Bars for both sets are placed side-by-side in
each interval, where necessary.

49
Example 2

Create a comparison histogram to compare the
scores from the two classes.
Class 1 26, 32, 54, 62, 67, 70, 71, 71, 74, 76,
80, 81, 84, 87, 87, 87, 93, 95, 96
Class 2 34, 45, 52, 57, 63, 65, 68, 70, 71, 72,
74, 76, 76, 78, 83, 85, 85, 87, 92, 99

50
Example 2, contd

Solution A bin size of 10 was used.

51
Comparison Bar Graphs

A comparison bar graph compares two data sets.
As before, bar graphs can be used to represent
frequencies, relative frequencies, and trends
over time.
This type of graph is also called a double bar
graph.

52
Example 3

Create a comparison bar graph for the two data
sets.

53
Example 3, contd
54
Question

Choose the statement that is true.

55
Question contd
a. The ratio of female to male doctors is largest
in the field of pediatrics. b. The field with the
fewest female doctors is family practice. c. The
field with the most male doctors is family
practice. d. There are more females in the field
of obstetrics and gynecology than there are men.
56
Example 4

This comparison bar graph shows that the majority
of kids in all age groups have access to
computers, and that older children use the
Internet more than younger children.

57
Multiple Line Graphs

A multiple line graph compares two data sets.
As before, line graphs are usually used to
represent trends over time.

58
Example 5

The double line graph shows that the gap between
mens and womens earnings has decreased over the
years.

59
Example 6

Create a double line graph to compare the scores
from the two classes.
Class 1 26, 32, 54, 62, 67, 70, 71, 71, 74, 76,
80, 81, 84, 87, 87, 87, 93, 95, 96
Class 2 34, 45, 52, 57, 63, 65, 68, 70, 71, 72,
74, 76, 76, 78, 83, 85, 85, 87, 92, 99

60
Example 6, contd
61
Multiple Pie Charts

A multiple pie chart compares two data sets.
As before, pie charts are used to show portions
of a whole.

62
Example 7

Use multiple pie charts to compare the
composition of the population over time.

63
Example 7, contd

Solution A pie chart is created for each year.

64
Question

Which year had the smallest percentage of
children under the age of 15?

65
Proportional Bar Graphs

Proportional bar graphs show relative amounts and
trends simultaneously.
All the bars are the same height.
Each bar corresponds to 100 of a whole.
Each bar is divided into pieces to represent the
portions of the different categories.

66
Example 8

The proportional bar graph illustrates how the
U.S. population has been distributed among 4
regions over time.

67
Choosing a Graph

The type of comparison graph selected depends on
The type of data.
The features of the data that will be emphasized.

68
Example 9

What type of graph could be used to make the
comparison between the two years in the following
table striking?

69
Example 9, contd
70
Example 9, contd

Solution A double bar graph

71
8.2 Initial Problem Solution

How can the monthly sales of the 3 items be
presented to show and compare the sales trends?

72
Initial Problem Solution, contd

Use a multiple line graph in order to
Show trends in sales over time.
Allow for comparison between items.

73
Initial Problem Solution, contd
74
Section 8.3Enhancement, Distraction,and
Distortion

Goals
Study misleading graphs
Study scales and axis manipulation
Study line graphs and cropping
Study three-dimensional effects
Study pictographs
Study graphical maps

75
8.3 Initial Problem
76
8.3 Initial Problem, contd

Use the data to make one graph that is
pessimistic about the debt and one that is
optimistic.
The solution will be given at the end of the
section.

77
Scaling and Axis Manipulation

To emphasize differences among the bars of a
histogram or bar graph, you can leave off part of
the vertical axis.
Reversing the axes or the orientation of one of
the axes is another way to create a misleading
graph.

78
Example 1

The graph appears to show that Beary Sticks has
far less sugar than the other cereals.

79
Example 1, contd

The first graph was misleading because the scale
is not shown and the axis actually begins at 8,
not at 0.
A better graph is shown here.

80
Example 2

The price of 3 brands of baked beans are as
follows.
Brand X 0.79
Brand Y 0.89
Brand Z 0.99
Create a bar graph that emphasizes the
differences in the prices.

81
Example 2, contd

Solution Exaggerate the differences by starting
the vertical scale at 75 cents instead of at 0.

82
Example 3

This bar graph shows that a companys profits
decline over time.

83
Example 3, contd

When the axes are switched and the years are
placed in reverse order, the graph has a more
positive feel and may be misleading.

84
Question

Estimate the total decrease in the company
profits from 1999 to 2003.
a. -140
b. -240
c. -140,000,000
d. -240,000,000

85
Example 4

Create a graph for the data that might give the
impression that things are getting better rather
than worse.

86
Example 4, contd

Solution
The years are placed in reverse order.
The vertical scale is started at 20.
The graph is drawn tall and narrow.

87
Line Graphs and Cropping

A type of scale manipulation used to make line
graphs misleading is called cropping.
A viewing window is chosen in order to make a
trend look more or less impressive.
Examples are shown on the following slide.

88
Cropping, contd
89
Example 5

Draw two line graphs of the data that give
different impressions.

90
Example 5, contd

Solution Begin one vertical axis at 0 and the
other at 24.

91
Example 6

A graph of the price of a stock from April 25
through May 5 seems to show a large increase.
Notice that the vertical axis begins at 98.

92
Example 6, contd

Another graph of the same stock over a longer
time period seems to show a gradual decline
overall.
Notice that the vertical axis begins at 0 in this
graph.

93
Example 6, contd

The final graph shows the same information as in
the last graph, but with a different choice of
vertical scale.
The decrease is more dramatic because the
vertical axis begins at 100.

94
Three-Dimensional Effects

Three-dimensional effects
Are often used in newspapers and magazines.
Can make a graph more attractive.
Can obscure a true picture of the data.

95
Example 7

It can be difficult to read exact values from a
3-D graph.
For example, the profits were almost 100,000 in
2003 but it might appear much lower from glancing
at this graph.

96
Example 8

It is also hard to read exact values from this
3-D line graph.

97
Example 9

In a 3-D pie chart, the exploded sector has more
emphasis, making it appear larger than it really
is.

98
Pictographs

A pictograph is a type of graph in which
pictures, symbols, or icons represent quantities.
Pictographs can represent data in interesting
ways, but they can also be misleading.

99
Example 10

This pictograph predicts the world population.
Each person icon represents 1 billion people.

100
Example 11

Each hotdog represents 10 of campers.

101
Example 12

This pictograph compares amounts spent on
different types of holiday gifts.

102
Example 13

Why is this pictograph misleading?

103
Example 13, contd

Solution The bars are not proportional in height
to the amounts they represent.
For example, there were nearly 4 times as many
students in grades 1-8 as there were in
pre-elementary.
The bar representing grades 1-8 is only about 3
times as tall as the one representing
pre-elementary students.
This causes the difference to look smaller than
it really is.

104
Example 14

Why is this pictograph misleading?

105
Example 14, contd

Solution The bars are not proportional to the
amounts they represent.
The bars are also angled, emphasizing the length
of the top bar and making the bottom bar look
shorter.

106
Question

If the graph were accurate, what should be the
ratio of the bar representing under 25 to the
one representing 35-44? Round to the nearest
hundredth.
a. 0.43
b. 1.29
c. 1.81
d. 2.33

107
Example 15

Why is this pictograph misleading?

108
Example 15, contd

Solution It must be clear whether a
2-dimensional or 3-dimensional object is being
used to represent a quantity.
The amount of milk sold in 2003 was about twice
the amount sold in 1997.
It is not clear whether the volume or the height
of the carton represents the milk quantity.
The volume of the second carton is 8 times as
large as the volume of the first carton. This is
misleading.

109
Example 16

Why is this pictograph misleading?

110
Example 16, contd

Solution The number of bottles in each stack are
not proportional to the actual dollar amounts.
The heights of the stacks are proportional.
It is not clear from the presentation whether it
should be viewed as a bar graph or as a
pictograph.

111
Customized Pie Charts

Pie charts can be customized by embedding the
chart in another picture or by adding other
content.
An example of a customized pie chart is shown on
the next slide.

112
Example 17
113
Example 18

This pie chart distorts the data.

114
Graphical Maps

A graphical map summarizes information about
geographical areas.

115
Example 19

The map below shows population increases.

116
Example 19, contd

The map is misleading because the areas of the
shaded regions are not proportional to the
population increases.
Some small states experienced large increases,
but that may not show up well on this map.

117
8.3 Initial Problem Solution