CHAPTER 1
REPRESENTING DATA GRAPHICALLY
Competitive swimmers use statistics to track and
improve their performance.
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Great athletes use many tools to gain an edge. For instance,
swimmers can track the speed of their performances to ensure
they are peaking at just the right time. Why would swimmers use
a graph to represent their performance data? A graph paints a
picture that can help them see trends and patterns.
In This Chapter
You will use tables, line plots, histograms, stem-and-leaf plots, and time series plots to
interpret data. You’ll also learn how to select and create statistical graphs for specific
data and purposes.
Topic List
► Chapter 1 Introduction
► Data and Variables
► Graphs of Categorical Data
► Two-Way Tables
► Line Plots
► Frequency Tables
► Histograms
► Stem-and-Leaf Plots
► Time Series Plots
► Chapter 1 Wrap-Up
REPRESENTING DATA GRAPHICALLY
3
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Chapter 1 Introduction
Every sport has lots of data. Athletes and coaches
can use data to track their performance.
Reporters and broadcasters use data to help fans understand each
performance as part of a trend. Fans can then compare the data from
that performance to other athletes past and present. Presenting the data
graphically can help whoever is using the data to better understand and
make sense of them. An appropriate graph can make it easier to see trends
and relationships.
Swimming Gold Medals
The table shows how many gold medals different countries have earned in
Olympic swimming.
Country
Swimming Gold Medals
214
Sweden
Australia
56
Soviet Union
East Germany
38
Germany
Hungary
23
Japan
20
Netherlands
17
Hungary
Great Britain
15
East Germany
Germany
13
Australia
Soviet Union
12
United States
Sweden
8
Great Britain
Country
United States
Gold Medals
Netherlands
Japan
0
50
100
150
200
250
Medals
Which countries have dominated the Olympic swimming competitions?
The table has all the data, but a graph shows a more vivid story.
From the graph, we can easily see that the United States has historically
dominated Olympic swimming.
4
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_IN_RG.indd 4
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Chapter 1 Introduction
Butterfly Records
When you examine numbers in a table, you can see some patterns.
For instance, these data show the world-record times for the 200-meter
butterfly at the beginning of each decade since 1960.
Butterfly World-Record Times
Time (min)
02:35.5
1960
02:16.4
02:18.2
1970
02:05.7
02:01.0
1980
01:59.2
01:43.7
1990
01:56.2
2000
01:55.2
2010
01:51.5
Time (min)
Year
01:26.4
01:09.1
00:51.8
00:34.6
How has the world-record time changed over the
years? When did the world-record time decrease
most quickly? When was it most stable?
00:17.3
00:00.0
1950
1960
It’s easier to see a trend when you look at the same
information in a graph.
1970
1980
1990
Year
2000
2010
2020
From the graph, you can see that the world-record time has decreased
since 1960. It went down more quickly in the 1960s and was most stable
in the 1990s.
Applying It
In this chapter, you will see how to create and
interpret statistical graphs.
CHAPTER 1 INTRODUCTION
5
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Preparing for the Chapter
Review the following skills to prepare for the
concepts in Chapter 1.
►
Write a ratio in simplest form.
►
Convert between fractions and decimals.
►
Convert between decimals and percents.
►
Convert between fractions and percents.
►
Round numbers to a certain place value.
►
Solve problems involving percents.
Problem Set
Write each ratio in simplest form.
1.
2.
10 : 25
2
___
80
3.
4.
252 : 36
44
___
82
5.
6.
4 to 38
64
____
352
Write each fraction as a decimal.
7.
8.
1
___
10
1
____
100
9.
10.
3
__
4
9
___
12
11.
55
____
12.
440
1
3__
8
0.86
6.7
17.
18.
1.07
4.44
0.5%
75%
23.
24.
120%
0.07%
0.09
0.35
29.
30.
3.5
0.02
Write each decimal as a fraction.
13.
14.
0.4
0.04
15.
16.
Write each percent as a decimal.
19.
20.
50%
5%
21.
22.
Write each decimal as a percent.
25.
26.
6
0.6
0.45
CHAPTER 1
27.
28.
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_IN_PS.indd 6
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Chapter 1 Introduction
Write each fraction as a percent.
31.
32.
1
__
4
3
__
4
33.
34.
5
__
8
5
___
80
35.
1
___
36.
40
5
__
2
41.
42.
52%
5.2%
Write each percent as a fraction.
37.
38.
8%
80%
39.
40.
0.8%
120%
Round each number to the indicated place value.
43.
6.9 to the nearest whole number
44.
34.49 to the nearest tenths place
45.
234.5 to the nearest tens place
46.
9.909 to the nearest hundredths place
47.
54,092 to the nearest thousands
48.
0.911 to the nearest tenths
49.
1203.3 to the nearest whole number
50.
45,067 to the nearest hundreds
Solve.
51.
What is 20% of 180?
59.
52.
What is 5% of 7.2?
60.
53.
50 is 40% of what number?
61.
0.5 is what percent of 100?
1?
What is 350% of __
4
Find the percent increase from 25 to 30.
54.
1.8 is 2% percent of what number?
62
Find the percent decrease from 40 to 25.
55.
12 is what percent of 120?
63.
56.
12 is what percent of 1200?
Find the percent increase from $450.50
to $540.60.
57.
What is 250% of 4.8?
64.
58.
What is 0.05% of 2.5 million?
Find the percent decrease from $12,068.40
to $9654.72.
CHAPTER 1 INTRODUCTION
7
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HS_PS_S1_01_IN_PS.indd 7
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Data and Variables
Statistical data can be categorical or quantitative.
And quantitative data can be discrete or continuous.
Identifying Types of Data
The World Cup is the most popular sporting event in the world. More
people watch the soccer tournament, which occurs every four years, than
watch any other sports tournament. The teams that compete in the World
Cup represent the six confederations of the Federation Internationale
de Football Association (FIFA). These confederations are geographical
groupings of countries and soccer organizations.
FIFA Confederations
Year
Goals
Asia
1950
88
Africa
1954
140
North America, Central America,
and the Caribbean
1958
126
1962
89
1966
89
1970
95
1974
97
1978
102
1982
146
1986
132
1990
115
1994
141
1998
171
2002
161
2006
147
2010
145
South America
Oceania
Europe
The list of confederations represents categorical data. Categorical data
refer to descriptions (called categories) that can be observed, but not
measured. Categorical data are ordinal if the categories can be ordered
(such as small, medium, and large categories for shirts), otherwise they
are called nominal. Since there is no particular order to confederations,
the confederations can more specifically be called nominal. You could put
the confederations in alphabetical order, but that ordering is not inherent
in the confederations and could be different in different languages (for
instance, South America is known as Amerique du Sud in French).
Look at the data for the total number of goals scored in the tournament
since 1950. Both the year and the number of goals are measured numerically,
so they are each quantitative variables. Quantitative data can be either
discrete or continuous.
8
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T01_RG.indd 8
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BY THE WAY
Discrete Data vs. Continuous Data
Discrete
Quantitative data with gaps
between values
Continuous
Quantitative data with no gaps
between values
Examples
Number of victories, goals, team
members
Examples
Temperature, height, weight
The word discrete is derived from
a Latin word meaning separate.
Understanding Statistical Variables
In algebra, a variable is a symbol (usually a letter) that stands for an
unknown value. In statistics, we often think of variables as any quantity for
which we have or want information. In algebra, we often solve an equation
to find the value of a variable. In statistics, we often perform a measurement
or use a survey or research to find values of variables.
In this table, there are three
variables: confederation,
teams, and championships.
The first is categorical,
while the other two are
quantitative. Both the
quantitative variables are
discrete—part of a team
can’t play in a tournament,
and a confederation can’t
win part of a championship.
Confederation
Teams
Championships
Asia
28
0
Africa
34
0
North, Central
America, and
Caribbean
35
0
South America
74
9
Oceania
4
0
Europe
218
10
Look at the data about the
positions and ages for the
men on Argentina’s World Cup
championship team in 1986.
There are two variables: position
and age. Position is categorical.
It can’t be measured; it can only
be observed. However, age can
be measured.
Though the age data shown here
are rounded to the nearest whole
year, there is no logical reason for
the gaps between ages. Ages are
continuous.
Position
Age
Position
Age
Forward
27
Midfield
24
Midfield
23
Defense
29
Midfield
32
Midfield
29
Midfield
21
Goalkeeper
20
Defense
29
Defense
27
Defense
33
Forward
26
Midfield
23
Goalkeeper
28
Defense
23
Defense
24
Defense
25
Midfield
23
Midfield
25
Midfield
31
Forward
30
Goalkeeper
29
BY THE WAY
The youngest player to participate
in the World Cup was Mario
Mendez, who was 16 years,
1 month, and 5 days old when he
played for Uruguay in 1954.
The oldest player was Roger Milla,
who was 42 years, 1 month, and
8 days old when he played for
Cameroon in 1994.
DATA AND VARIABLES
9
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HS_PS_S1_01_T01_RG.indd 9
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Problem Set
Describe the variable as either categorical or quantitative.
1.
favorite movie
5.
type of bird
2.
type of fruit
6.
car speed
3.
movies watched
7.
number of students
4.
house prices
8.
days of rain
Describe the quantitative variable as either discrete or continuous.
number of vegetables
13.
phone calls received
10.
income
14.
systolic blood pressure
11.
books read
15.
tree height
12.
amount of water
16.
website visits
9.
Describe the categorical variable as either nominal or ordinal.
17.
breed of dog
20.
police officer rank
18.
type of food
21.
place in a race
19.
letter grades
22.
movie type
The bar graph shows the amount of sales for three products for each
quarter of the fiscal year.
How many variables are represented
in the graph?
24.
What name could be given to each
variable?
25.
Which variable(s) are categorical?
26.
Which variable(s) are quantitative?
27.
Describe the quantitative variable(s)
as either discrete or continuous.
28.
Describe the categorical variable(s)
as either ordinal or nominal.
Sales by Quarter
Products
Product Z
Product Y
Product X
Quarter
23.
$0
$3000
$6000
$9000
$12,000
$15,000
Sales
10
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T01_PS.indd 10
8/23/11 8:08 PM
A 4-year college student completed a survey as shown.
29.
How many variables are being measured?
30.
Which variables are categorical?
31.
Which categorical variable is nominal? How
many categories does the nominal variable have?
32.
Which categorical variable is ordinal? What are
the levels of the ordinal variable, from lowest to
highest?
33.
SURVEY FORM
Which variables are quantitative? Describe each
quantitative variable as discrete or continuous.
Age (in years)
19
Academic year
Sophomore
Height
5 feet, 8 inches
Gender
Female
Give the number of variables displayed in the graph. Describe each
variable as either categorical or quantitative.
Blood Type
Type
A
Type B
Type AB
Type O
35.
Spending by Presidential Candidates in the 2008 Election
$60
Amount spent (millions)
34.
$50
$40
$30
$20
$10
$0
Mitt
Romney
Barack
Obama
Hillary Rudolph
Clinton Giuliani
Candidate
John
McCain
John
Edwards
DATA AND VARIABLES
11
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HS_PS_S1_01_T01_PS.indd 11
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Graphs of
Categorical Data
A graph can help you compare data and put them
in context.
Creating a Circle Graph
In 2011, the total amount in
the U.S. federal budget was
$3.8 trillion.
How would you present these
data? Because the data are
categorical, you could use a circle
or bar graph to show them visually.
Senator Polly Tick is trying to
make a point about the federal
budget.
Category
Spending
(billions of dollars)
Pension Plans
788
Health Care
898
Education
141
Defense
929
Welfare
465
Interest
251
Other
362
Here is her circle graph for the
2011 budget.
With her circle graph, Senator Tick
shows that 21% of the total budget
is spent on pension plans, including
Social Security.
REMEMBER
Categorical data can be broken
down into separate categories or
classes.
Spending
Pension Plans
9%
7%
21%
12%
Health Care
Education
23%
Defense
Welfare
24%
TIP
A circle graph is a good way
to show data that are part of a
whole.
Interest
4%
Other
Q&A
HOW TO
To create a circle graph from a set of data
Step 1 Write a ratio that represents each category.
Step 2 Convert each ratio to a percent.
Step 3 Multiply the ratio for each category by 360° to find each sector’s
degree measure.
Q
Why is a circle graph a good
choice for Senator Tick?
A
A circle graph is a good
choice because she is trying
to show how the pieces
relate to the whole.
Step 4 Use the degree measures to draw and label each sector.
12
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T02_RG.indd 12
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Creating a Bar Graph
Senator Phil Buster wants to look at portions of the budget. These data do
not represent parts of a whole, so it makes sense to use a bar graph.
Military Spending
(billions of dollars)
Year
1950
294
1960
526
1970
1038
1980
2788
1990
5800
2000
9951
2010
14,624
Here is Senator Buster’s bar graph.
Military Spending
Q&A
Amount (billions of dollars)
16,000
14,000
12,000
Q
What is the percent increase
in military spending from
1950 to 2010?
A
4874%
10,000
8000
6000
4000
2000
0
1950
1960
1970
1980
Year
1990
2000
2010
With his bar graph, Senator Buster shows that spending on the military has
increased very significantly since 1950.
HOW TO
To create a bar graph from a set of data
Step 1 Find the greatest value.
Step 2 Use the greatest value to create an appropriate scale for the
vertical axis.
Step 3 Draw each bar according to its height. Use a grid to make sure all
the heights are correct. Also make sure each bar is the same width.
GRAPHS OF CATEGORICAL DATA
13
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Problem Set
Determine whether a bar graph or a circle graph is appropriate for
the situation.
1.
A toy store wants to compare sales of some of its
top-selling toys.
4.
A legislative body wants to show its breakdown
by political party.
2.
A sales manager wants to compare sales for all
five of her sales associates.
5.
An investment company wants to show its net
earnings in three of its newest regions.
3.
A newspaper reporter wants to display the final
results of an election.
6.
A university wants to show the home states of its
freshman class.
Use the circle graph to solve.
7.
A 2010 estimate of Australia’s population is
about 21,515,754. Estimate the number of
Australians with type AB blood in 2010.
8.
Estimate the number of Australians with type A
or type O blood.
9.
The annual population growth rate of Australia
is estimated to be about 1.17%. According to this
estimate, how many Australians had type B blood
in 2009?
Blood Types in Australia
Type B
10%
Type A
38%
Type O
49%
Type AB
3%
The Natural Grocery Store chain wants to display and compare the
number of full- and part-time employees at its three largest stores.
Store
Number of Employees
Part-Time Employees (%)
Cherry Grove
30
50%
Downtown
35
20%
Lakeshore
44
25%
10.
Create a circle graph that shows the percent of
full-time and part-time employees at the Cherry
Grove store.
11.
Create a circle graph that shows the percent
of full-time and part-time employees at the
Downtown store.
14
CHAPTER 1
12.
Create a bar graph that shows the number of fulltime employees at each store.
13.
Create a bar graph that shows the number of parttime employees at each store.
14.
Create a double bar graph that displays the number
of full- and part-time employees at each store.
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T02_PS.indd 14
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Match the circle graph with the data set to which it most closely corresponds.
A. {20, 30, 20, 20}
C. {10, 20, 30, 60}
B. {20, 40, 20, 40}
D. {60, 10, 5, 10}
15.
17.
8%
12% 12%
6%
17%
50%
25%
70%
18.
16.
17%
22%
33%
34%
17%
22%
22%
33%
The double bar graph shows the number of imports and exports for a
company in a certain week.
On how many days did the number of imports
exceed the number of exports?
20.
Which day had the highest ratio of imports
to exports?
21.
What was the ratio of exports to imports for
the week?
22.
What was the percent decrease in the number
of exports from Wednesday to Thursday?
Company Imports and Exports
20
18
16
14
Quantity
19.
12
10
8
6
4
2
0
Mon.
Tues.
Wed.
Thurs.
Day of the week
Imports
Fri.
Exports
An eastbound passenger plane is carrying 130 passengers, and a
westbound plane is carrying 106 passengers. The table shows the ratio
of men to women riding on each plane.
Plane
23.
Ratio of Men to Women
Eastbound
15 : 11
Westbound
21: 32
Create a circle graph that displays the percent
of people riding on the eastbound plane and the
percent of people riding on the westbound plane.
24.
Create a circle graph that displays the percent
of men riding on the eastbound plane and the
percent of women riding on the eastbound plane.
GRAPHS OF CATEGORICAL DATA
15
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HS_PS_S1_01_T02_PS.indd 15
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Two-Way Tables
When data have several categories, you need more
complicated tables and graphs to organize and
present the data.
Comparing Categorical Data
How many servings of milk and soft drinks children and teenagers drink
can change over time. Because there are two dimensions (type of drink and
age range), a two-way table can help present the data.
Milk and Soft Drink Consumption
(number of 8 oz servings per day) by Age (years)
1–3
4–8
9–13
14–18
Milk
1.4
1.0
0.9
0.7
Soft Drinks
0.6
1.0
1.5
2.7
You can make quantitative comparisons (differences and ratios) based on
the two-way table data.
What is the ratio of milk to soft drinks for the average 1- to 3-year-old?
7
14 __
___
0.6 = 6 = 3
1.4
___
What about for the average 9- to 13-year-old?
0.9
___
9 __
3
___
1.5 = 15 = 5
What percent of a 14- to 18-year-old’s milk and soda consumption is milk?
0.7
________
0.7
___
0.7 + 2.7 = 3.4 ≈ 0.21 = 21%
Creating Stacked Bar Graphs
To see trends with the milk and soft drink data, it can help to make a graph
of the data. Because the data have two dimensions (or categories), a stacked
bar graph can be used to show the data.
16
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T03_RG.indd 16
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HOW TO
To create a stacked bar graph
Step 1 Create a bar graph for one dimension.
Step 2 Choose a different color to create bars for the second dimension.
Step 3 Place bars for the same horizontal category right on top of the
bars for the first dimension so they touch.
Step 4 Include a legend so people viewing your graph know what the
colors represent.
Here is the stacked bar graph for the data.
Amount consumed
(8 oz serving per day)
4
3.5
3
2.5
2
Soft Drinks
Milk
1.5
1
0.5
0
1–3
4–8
9–13
Age (years)
14–18
You can tell a lot from the stacked bars. For instance,
• The total amount of milk and soft drinks (the total height of each stacked
bar) remains steady until age 8, then increases a bit from age 9 to age 13,
but increases steeply between the ages of 14 and 18.
REMEMBER
Stacked bar graphs and multiple
bar graphs are used to compare
categorical variables.
• The ratio of soft drinks to milk consumed rises considerably during the
teenage years.
You could also represent a two-way table with a double bar graph.
Here is the double bar graph for the data.
Amount consumed
(8 oz serving per day)
3
2.5
2
1.5
Milk
Soft Drinks
1
0.5
0
1–3
4–8
9–13
Age (years)
14–18
Two-way tables compare two categorical variables. Stacked bar graphs and
double bar graphs are the most common ways to represent two-way tables.
TWO-WAY TABLES
17
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Problem Set
Solve.
1.
From 1998 to 2010, Swiss tennis player Roger
Federer won 43 out of 54 matches at the French
Open and won 56 out of 62 matches at the U.S.
Open. Create a two-way table that represents the
number of wins and losses at each tournament.
2.
In a music class there are 30 students, and each
student knows how to play at least 1 instrument.
Five out of the 12 females in the class play more
than 1 instrument. Four of the males in the class
play more than 1 instrument. Create a two-way
table that represents this situation.
In an election, Greenwell and Wong were the only candidates running
for a public office.
Votes for Candidates by District
District A
District B
District C
District D
Greenwell
600
710
355
558
Wong
820
682
400
702
3.
How many residents in all four districts voted for
Greenwell?
6.
The total population of District D is 2133. How
many residents in this district did not vote?
4.
What percent of votes in District B did Wong
receive? Round your answer to the nearest percent.
7.
5.
In District A, what is the ratio of the number of
votes Greenwell received to the number of votes
Wong received?
Create a stacked bar graph to represent the
number of votes the candidates received in
each of the four districts.
A marble manufacturer makes large and small marbles in five different
colors. The stacked bar graph shows the daily production output for
each type of marble.
9.
10.
11.
How many small red marbles does the company
make each day?
How many large marbles does the company
make each day?
What percent of black marbles produced each
day are small?
Create a two-way table that represents the
stacked bar graph.
80
70
60
Quantity
8.
50
40
30
20
10
0
Blue
Red Yellow White
Color of marble
14 mm diameter
18
CHAPTER 1
Black
20 mm diameter
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T03_PS.indd 18
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Match the two-way table with the correct stacked bar graph.
12.
Group 1
Group 2
Category A
4
4
Category B
3
5
Category C
6
4
A.
12
10
8
6
4
Group 1
2
Group 2
0
Category A
13.
Group 1
Group 2
Category A
2
6
Category B
4
4
Category C
7
3
B.
Category B
Category C
12
10
8
6
4
Group 1
2
Group 2
0
Category A
14.
Group 1
Group 2
Category A
5
2
Category B
3
5
Category C
6
4
C.
Category B
Category C
12
10
8
6
4
Group 1
2
Group 2
0
Category A
15.
Group 1
Group 2
Category A
7
2
Category B
3
7
Category C
6
4
D.
Category B
Category C
12
10
8
6
4
Group 1
2
Group 2
0
Category A
Solve.
16.
Category B
Category C
Challenge On a commuter flight from Washington, D.C., to New York,
one-third of the passengers are male and the ratio of passengers under
the age of 18 to passengers age 18 or over is 5 : 4. Also the number
of male passengers age 18 or over is equal to the number of female
passengers age 18 or over. Of the 54 passengers on the flight, what
percent are females age 18 or over?
TWO-WAY TABLES
19
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HS_PS_S1_01_T03_PS.indd 19
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Line Plots
When you understand the shape of a data set,
you can better see how each data value relates to
the whole.
Creating a Line Plot
In the NCAA Women’s Soccer tournament, the final game determines
the champion. The table shows the number of goals scored in each
championship game since the tournament began in 1982.
A line plot can help you see the shape of the data.
HOW TO
To create a line plot
Step 1 Find the minimum and maximum values in the data set.
Step 2 Draw a number line that includes the minimum and maximum
values and has an appropriate tick interval.
Step 3 Draw a dot for every data point. Move up by uniform increments.
Step 4 Count the total number of data items and the total number of
dots. They must be the same.
For the soccer final data, the line plot looks like this.
0
20
CHAPTER 1
1
2
3
4
5
6
7
Number of Goals
8
9
10
11
Year
Goals
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
1
4
2
2
2
1
5
2
6
4
10
6
5
1
1
2
1
2
3
1
3
6
2
4
3
2
3
1
1
REPRESENTING DATA GRAPHICALLY
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Interpreting a Line Plot
You can tell quite a bit from a line plot. For instance,
• Would a total of only 2 goals be unusual? Not at all. In most games, either
1 or 2 goals were scored.
• In how many games were more than 4 goals scored? There were 6 games
in which more than 4 goals were scored (that’s about 21% of the time).
• Would a total of 7 goals be a lot for a final? Yes, it would. Only once was
more than 6 goals scored. So any value more than 6 would be pretty rare.
Distribution Shapes
When looking at a line plot, the shape of the data is called the distribution
of the data. There are many different possible shapes for a distribution of
data, but there are three characteristics of shapes that we often look for.
Symmetrical: A data set has a symmetrical
distribution if the values below the median
are approximately a mirror image of those
above the median. In a symmetrical data
set, the mean and median are generally
pretty close to each other.
Skewed left: A data set is skewed left if
the frequency of data is clustered at the
higher end of the distribution. In a graph,
the left tail of a left-skewed data set is
longer than the right tail.
Skewed right:A data set is skewed right
if the frequency of data is clustered at the
lower end of the distribution. In a graph,
the right tail of a right-skewed data set is
longer than the left tail.
Symmetrical Distribution
Skewed Left
Lower frequencies on left
Skewed Right
Lower frequencies on right
Gaps and Outliers
A line plot has a gap when there are no data values for a certain number and
when there are data values for numbers on either side. It is easy to spot a
gap in a line plot because it looks like an empty spot between data values.
An outlier is a data value that is either much greater or much less than most
of the other data values. There is always a gap between an outlier and the rest
of the data set. Outliers can be identified easily on a line plot because they
are either far to the right or far to the left of the other data values. There are
methods for calculating outliers, but for now, we’ll identify outliers visually.
LINE PLOTS
21
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HS_PS_S1_01_T04_RG.indd 21
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Problem Set
Use the U.S. State Temperatures data set on pp. A-7 to A-8.
1.
Create a line plot that displays the record high
temperatures for states in the South.
4.
Create a line plot that displays the record high
temperatures for states in the West.
2.
Create a line plot that displays the record high
temperatures for states in the Midwest.
5.
Create a line plot that displays record high
temperatures of 120°F or higher for all states.
3.
Create a line plot that displays the record high
temperatures for states in the Northeast.
6.
Create a line plot that displays record low
temperatures of −50°F or lower for all states.
Blood pulse rate is a measure of the number of heart beats per minute
(BPM). For a school project, a student took his blood pulse rate every
30 minutes throughout the day, starting at 10:00 a.m. The student
recorded each measure.
BPM
67
68
71
79
80
81
84
90
92
Frequency
1
2
1
3
5
4
1
2
1
7.
Create a line plot that displays the data.
8.
What was the most frequent blood pulse rate for
the student throughout the day?
9.
10.
What was the highest blood pulse rate taken
during the day?
How many times did the student take his blood
pulse rate during the day? What time of day was
the last blood pulse rate measured?
A group of high school students applied for a research grant. The
students’ grade point averages (GPAs) are displayed in this line plot.
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
Grade Point Average
11.
How many students applied for the research
grant?
12.
How many students had a GPA of less than 3.5?
13.
What percent of students had a GPA of less than
3.5? Round your answer to the nearest percent.
22
CHAPTER 1
14.
3.7
3.8
3.9
4.0
4.1
Only those students whose GPA was in the
highest 20% of the group have priority for the
grant. How many students in the group fit this
description? What are their GPAs?
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T04_PS.indd 22
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Each data set represents data from a line plot. Match the data set with
the description.
A. skewed right
D. symmetric
E. asymmetric
B. uniform
C. skewed left
15.
{2, 4, 2, 1, 4, 1, 2, 3, 5, 1, 5, 4}
18.
{5, 3, 1, 2, 4, 1, 2, 3, 4, 2, 2, 1}
16.
{2, 3, 4, 5, 1, 3, 2, 4, 4, 2, 3, 3}
19.
{4, 5, 1, 2, 3, 1, 1, 3, 3, 4, 5, 5}
17.
{1, 3, 5, 4, 2, 3, 2, 4, 5, 5, 4, 5}
Adult coyotes that live in the lower desert regions of the southwestern
United States usually weigh approximately 20 pounds, while coyotes in
the nearby mountain regions can weigh up to twice as much. Weights,
in pounds, of 32 randomly selected coyotes from these regions are
shown in the data set. Note: Data values with an asterisk (*) represent
weights of coyotes from the mountain region.
26*
21
32*
24*
24
23
29*
31*
25
22
31*
28*
24
29*
30*
23
31*
22
22
21
30*
23
30*
27*
31*
28*
23
29*
30*
31*
22
30
20.
Create a line plot that displays the weights of all
32 coyotes. Describe the shape of the line plot,
and identify any outliers.
22.
21.
Create a line plot that displays the weights of
the coyotes in the mountain region. Describe the
shape of the resulting line plot, and identify any
outliers.
Create a line plot that displays the weights of the
coyotes in the lower desert region. Describe the
shape of the line plot, and identify any gaps or
outliers.
Create a line plot for the data set, and describe the shape of the
distribution.
23.
{41, 43, 44, 47, 46, 45, 42, 42, 43, 44, 46, 45, 44}
24.
{905, 908, 910, 909, 907, 906, 906, 906, 908, 907, 907, 906, 905, 905}
25.
{27, 30, 29, 32, 31, 28, 28, 31, 30, 32, 32, 28, 29}
26.
{1040, 1010, 1020, 1040, 1030, 1030, 1000, 1040, 1030, 1040}
LINE PLOTS
23
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HS_PS_S1_01_T04_PS.indd 23
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Frequency Tables
Tables and graphs can show how often data fall into
various class intervals.
Interpreting a Frequency Table
Augusta National Golf Club, which is about 3 hours from Atlanta, Georgia,
hosts the Masters golf tournament. In golf, the goal is to take as few strokes
as possible, so the lower the score, the better the result.
In 2010, Phil Mickelson won the Masters. How impressive was Mickelson’s
tournament? To determine the answer to this question, take a look at the
frequency table that shows the results of all the players who made it to the
final rounds of the 2010 Masters.
Each class interval contains a range of 5 scores.
What can you tell from the table?
• A total of 48 players played in the final rounds.
1 + 5 + 5 + 17 + 13 + 6 + 1 = 48
• Only one player took fewer than 275 strokes, and one unlucky player took
at least 300 strokes as well.
• Twenty-two players scored in the 280s (frequencies 5 + 17).
REMEMBER
A class interval is a way of
organizing data. Each interval
defines specific ranges for the data.
Total
Strokes
Frequency
(f )
270–274
275–279
280–284
285–289
290–294
295–299
300–304
1
5
5
17
13
6
1
Creating Relative Frequency Tables
How can you compare a set of data with 1000 values to another set with
20 values? A relative frequency table is a good way to compare data sets.
HOW TO
To create a relative frequency table
Step 1 Add the frequencies to find the total number of data values.
Step 2 Divide each frequency by the total number of data values.
Step 3 Convert the fraction to a decimal or percent.
When you add the frequencies, you see that there are a total of 48 scores
in the table. Use 48 as the denominator of a fraction to describe how each
class’s frequency compares to the whole.
24
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T05_RG.indd 24
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Compare these two relative frequency tables. The first is for the 2010
Masters. The second is for the 2010 AT&T National golf tournament.
2010 Masters
Total
Strokes
f
270–274
1
275–279
5
280–284
5
285–289
17
290–294
13
295–299
6
300–304
1
2010 AT&T National
Relative
Frequency
Total
Strokes
f
1
__
48 = 0.02
5
__
48 = 0.10
5
__
48 = 0.10
17
__
48 = 0.35
13
__
48 = 0.27
6
__
48 = 0.13
1
__
48 = 0.02
270–274
5
275–279
19
280–284
28
285–289
15
290–294
4
295–299
0
300–304
0
Relative
Frequency
5
__
71 = 0.07
19
__
71 = 0.27
28
__
71 = 0.39
15
__
71 = 0.21
4
__
71 = 0.06
0
__
71 = 0.0
0
__
71 = 0.0
What percent of players in each tournament scored better than 280?
For each tournament, add the frequencies for the first two class intervals.
BY THE WAY
The two most common categories
cover the range of strokes from
285 to 294. As a matter of fact,
those two categories contain
more golfers than the other five
categories combined.
United States
Ryder Cup
Scores
13.5
16.5
9.5
9.5
12.5
14.5
For the Masters:
0.02 + 0.10 = 0.12 = 12%
13.5
For the AT&T National:
0.07 + 0.27 = 0.34 = 34%
15
13.5
14.5
The Ryder Cup is a biannual golf competition in which a team representing
the United States competes against a team representing Europe. Points are
earned by winning matches in various formats. The United States team
scores from 1961 to 2010 are shown in the table. The data in the United
States Ryder Cup Scores table can be used to create a frequency table.
14
13
11.5
14.5
Score
Tally
f
8–9.5
2
10–11.5
1
12–13.5
6
14–15.5
6
16–17.5
3
18–19.5
4
20–21.5
1
22–23.5
2
Relative
Frequency
2
__
25 = 0.08
1
__
25 = 0.04
6
__
25 = 0.24
6
__
25 = 0.24
3
__
25 = 0.12
4
__
25 = 0.16
1
__
25 = 0.04
2
__
25 = 0.08
18.5
17
12.5
21
19
18.5
16
23.5
19.5
23
14.5
FREQUENCY TABLES
25
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HS_PS_S1_01_T05_RG.indd 25
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Problem Set
Use the data in the table to solve.
15
10
12
9
8
3
4
11
10
10
10
1
6
2
3
7
5
12
9
7
6
9
10
12
11
7
11
9
11
4
1.
Create a frequency table without grouping
the data.
2.
Create a relative frequency table without
grouping the data. Use proportions rounded
to the nearest hundredth.
3.
Create a frequency table using class widths of 2.
4.
Create a relative frequency table using class
widths of 2. Use proportions rounded to the
nearest hundredth.
5.
Create a frequency table using class widths of 3.
6.
Create a relative frequency table using class
widths of 3. Use proportions rounded to the
nearest hundredth.
7.
Explain why it would not make sense to create a
frequency table for the data using class widths of
4 or more.
The frequency table shows the results from a test that was worth
80 points.
8.
How many students took the test?
9.
What is the width of each class?
10.
How many students earned less than 50 points on the test?
11.
Use the frequency table to create a relative frequency table.
Round to the nearest tenth of a percent.
12.
What percent of students earned 60 points or more on the test?
Scores
80–84
75–79
70–74
65–69
60–64
55–59
50–54
45–49
40–44
35–39
30–34
f
1
6
7
8
3
0
3
3
0
0
1
Ages
64–71
56–63
48–55
40–47
32–39
24–31
16–23
8–15
0–7
Rel f (%)
?
8
9
12
30
24
10
0
2
The relative frequency table shows the percentages of age groups of
200 passengers riding on a commuter train.
13.
What percent of passengers are ages 0 to 15?
14.
How many passengers are older than 55?
15.
What percent of passengers are ages 64 to 71? How many
passengers are in this age group?
16.
What percent of passengers are younger than 40? How many
passengers are in this age group?
26
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T05_PS.indd 26
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Use the Nutrition data set on p. A-11.
17.
Create a frequency table for the number of
calories for all foods in the data set. Use classes
of width 8.
18.
Create a relative frequency table that displays
the number of calories for all foods in the data
set. Use classes of width 8. Round to the nearest
tenth of a percent.
A bookstore is open 6 days a week. At the end of each business day, the
store owner writes down on a calendar the number of books sold.
19.
Why is it necessary to group the data
when creating a frequency table for the
number of books sold in April?
20.
Create a frequency table for the April
2010 data using classes of width 4.
21.
Create a frequency table for the April
2010 data using classes of width 5.
22.
Why wouldn’t class widths of 10 be
used for creating a frequency table for
these data?
April 2010
Sun
Mon
Tue
Wed
Thu
Fri
Sat
1
2
3
23 books 28 books 40 books
sold
sold
sold
4
5
6
7
8
9
10
28 books 20 books 28 books 25 books 27 books 42 books
sold
sold
sold
sold
sold
sold
11
12
13
14
15
16
17
28 books 16 books 24 books 16 books 20 books 48 books
sold
sold
sold
sold
sold
sold
18
19
20
21
22
23
24
29 books 19 books 26 books 23 books 27 books 32 books
sold
sold
sold
sold
sold
sold
25
26
27
28
29
30
35 books 29 books 24 books 22 books 25 books
sold
sold
sold
sold
sold
A group of adults and a group of teenagers gave estimates of the
number of text messages they usually send each day.
23.
Use the frequency table to create a relative
frequency table. Round to the nearest percent.
24.
What percent of adults said they send fewer than
3 text messages each day?
25.
What percent of teenagers said they send
between 2 and 6 text messages each day?
26.
What percent of adults said they send between
20 and 30 text messages each day?
27.
What percent of teenagers said they send 15 or
more text messages each day?
Text Messages
30–32
27–29
24–26
21–23
18–20
15–17
12–14
9–11
6–8
3–5
0–2
Adults
0
2
0
8
12
9
15
14
10
6
5
Teenagers
3
6
10
17
12
10
8
5
0
3
1
FREQUENCY TABLES
27
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HS_PS_S1_01_T05_PS.indd 27
8/23/11 10:36 PM
Histograms
When you want to see the shape of a distribution, a
histogram is a specific type of bar graph that can help.
Creating a Histogram
In the 1983–84 season, the University of Southern California’s women’s
basketball team won the National Collegiate Athletic Association
championship. If you look at the scores from the games, you can see that
the team won several games, but analyzing the data can help you see how
good the team was.
How would you present these data? There are several options, but one way
to do it is to start with a frequency table.
USC Points
f
Opponents Points
f
60–69
5
40–49
1
70–79
7
50–59
8
80–89
13
60–69
12
90–99
5
70–79
9
100–109
3
80–89
2
90–99
0
100–109
1
A histogram is a graph showing the frequency distribution of data.
HOW TO
To create a histogram from a set of data
Step 1 Create a frequency table.
Step 2 Draw axes. The horizontal axis should have numbers that correspond
to the frequency table. The vertical axis should be either frequencies
or relative frequencies.
Step 3 Be sure to draw the bars of the histogram so they touch each
other. There should be no gap between adjacent bars.
28
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
USC
Opponent
86
78
67
81
86
81
82
102
79
91
68
66
90
95
74
101
101
82
86
79
78
88
86
85
82
81
85
67
97
76
90
62
72
68
64
65
79
58
66
74
73
66
57
77
75
102
80
53
67
84
56
44
69
60
76
67
58
61
58
63
71
72
51
74
57
61
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HS_PS_S1_01_T06_RG.indd 28
8/23/11 10:50 PM
Here are histograms for USC’s scores and their opponents’ scores.
USC Women’s Basketball Scores 1983–84
16
THINK ABOUT IT
Frequency
12
8
4
0
60
70
80
Points
90
100
110
USC Women’s Basketball Opponents’ Scores 1983–84
It might seem as though a
histogram is the same as a bar
graph, but they are different.
First, a bar graph is based
on categorical data, while a
histogram is based on numerical
data. Second, the bars of a
histogram should touch, whereas
a bar graph usually has spaces
between each pair of bars.
16
Frequency
12
8
4
0
40
50
60
70
Points
80
90
100
110
Interpreting a Histogram
So, if you have a histogram, how do you interpret it? What can you tell from
the graph? Looking at the USC scores histogram, you can see several things.
Bin widths: All the bins are 10 points wide. This is the same as the
frequency table. You can determine the bin width on the histogram from the
labels on the horizontal axis. For instance, the graph has a bin that is from
50 to 59, which yields a width of 10 points.
TIP
All bins for a single graph must
have the same width.
Total number of data values: If you want to know how many games the
team played, add all the bin heights.
5 + 7 + 13 + 5 + 3 = 33
The team played 33 games that season.
Data in a range: To find the number of times USC scored at least 80 points,
just add all the bin heights starting with the 80 to 90 bin.
13 + 5 + 3 = 21
Divide to find the percent of times the team scored at least 80 points.
21
___
33 ≈ 0.636 = 63.6%
Histograms provide a visual way to see how data are distributed. Aside from
all the specific values you can find, you can also see at a glance that USC
was much better than its competition. That visual is very valuable.
HISTOGRAMS
29
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HS_PS_S1_01_T06_RG.indd 29
8/23/11 10:50 PM
Problem Set
Use the data in the table to solve.
1.
Create a frequency histogram without grouping
the data. Describe the shape of the distribution.
2.
Create a relative frequency histogram without
grouping the data. Use proportions rounded to
the nearest tenth of a percent.
3
7
8
8
5
6
8
7
5
4
8
7
8
5
1
4
7
6
7
3
The histogram shows the heights in meters of trees in a certain
section of a park.
3.
Describe the shape of the histogram.
4.
What are the bin widths of the histogram?
5.
How many trees are represented in the
histogram?
6.
What bin contains the greatest number of trees?
How many tree heights does it contain?
7.
How many trees are less than 20 meters tall?
2
8.
How many trees are at least 16 meters tall?
1
Heights of Trees
6
Frequency
5
4
3
0
4
8
12
16 20
Height (m)
24
28
32
9.
{0.5, 1.6, 1.6, 3.1, 1.2, 3.9, 2.9, 4.3, 4.6, 7.3}
10.
{1.4, 0.3, 0.9, 1.9, 2.0, 2.9, 4.6, 3.1, 3.1, 4.3}
11.
{0.3, 1.5, 3.1, 1.4, 4.2, 1.9, 2.0, 4.4, 7.8, 6.3}
12.
{6.0, 4.6, 3.2, 3.0, 3.0, 1.5, 1.9, 2.8, 1.3, 1.3}
Frequency
Determine whether the frequency histogram represents the data set.
3
2
1
0
1.5
3
4.5
6
7.5
Use the U.S. State Population data set on pp. A-5 to A-6.
13.
30
Create a frequency histogram for the 2010 Census
of all states in the Midwest. Use 0 to 4,000,000
as the lowest bin interval. Describe the shape
of the distribution and determine whether there
are outliers.
CHAPTER 1
14.
Create a frequency histogram for the 2010 Census
of all states in the West. Use 0 to 4,000,000 as
the lowest bin interval. Describe the shape of
the distribution and determine whether there
are outliers.
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T06_PS.indd 30
8/23/11 10:56 PM
A total of 200 fish were caught during a fishing expedition. The weights
of the fish are shown in the relative frequency histogram.
Describe the shape of the histogram.
16.
What are the bin widths of the histogram?
17.
What percent of the fish in the sample weighed
less than 4 pounds?
18.
How many fish in the sample weighed less than
4 pounds?
19.
What percent of the fish weighed 4 pounds
or more?
20.
Weights of Fish
30
Relative frequency (%)
15.
25
20
15
10
5
How many fish weighed 4 pounds or more?
0
0.8
1.6
2.4
3.2
4
4.8
Weight (lb)
5.6
6.4
7.2
21.
{0.5, 1.6, 1.6, 3.1, 1.2, 3.9, 2.9, 4.3, 4.6, 7.3}
22.
{1.4, 0.3, 0.9, 1.9, 2.0, 2.9, 4.6, 3.1, 3.1, 4.3}
23.
{0.3, 1.5, 3.1, 1.4, 4.2, 1.9, 2.0, 4.4, 7.8, 6.3}
24.
{6.0, 4.6, 3.2, 3.0, 3.0, 1.5, 1.9, 2.8, 1.3, 1.3}
Relative frequency (%)
Determine whether the relative frequency histogram represents the
data set.
30
20
10
0
1.5
3
4.5
6
7.5
These data are the top 40 batting averages for the 2010 Major League
Baseball season.
25.
26.
27.
Create a frequency histogram using .280 to .290
as the lowest bin interval. Describe the shape
of the histogram and determine whether there
are outliers.
Create a relative frequency histogram using .280
to .290 as the lowest bin interval. What percent
of these averages is below .330?
Challenge Suppose a histogram with bin widths
of 200 is created to represent the data. Find the
smallest possible value of x so that the shape of
the distribution is uniform.
.315
.294
.359
.300
.293
.296
.336
.302
.292
.297
.328
.304
.307
.297
.327
.290
.290
.298
.324
.307
.288
.298
.321
.307
.288
.298
.321
.312
.287
.300
.319
.312
.286
.300
.318
.312
.285
.300
.315
.293
400
250
309
691
201
60
980
451
582
756
800
x
850
87
80
HISTOGRAMS
31
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HS_PS_S1_01_T06_PS.indd 31
8/23/11 10:56 PM
Stem-and-Leaf Plots
You don’t need fancy bar graphs or histograms to see
the shape of a data set. Even a simple, but elegant,
listing of data can show you the shape.
Interpreting a Stem-and-Leaf Plot
There are earthquakes every day, all over the world. Here is a plot showing
magnitudes on the Richter scale of earthquakes detected in various parts of
the world on February 14, 2011.
From the stem-and-leaf plot, you can see
many things. For instance,
• There were four earthquakes that registered
5.0 or higher.
• The strongest earthquake registered 6.6.
2
3
4
5
6
1
1
4
0
6
8
3 3 8 9 9
5 5 5 8 9 9
2 2
BY THE WAY
On February 14, 2011, the
earthquake with the greatest
magnitude occurred in the
Pacific Ocean about 300 km
southwest of Santiago, Chile.
If you look at the stem-and-leaf plot sideways, you can see it as a simple
histogram, where the categories are defined by the stems. This is a quick
way to see the shape of a data set. In this case, the distribution is rather
symmetrical.
1
1
4
0
6
2.1, 2.8, 3.1, 3.3, 3.3, 3.8, 3.9, 3.9, 4.4, 4.5, 4.5, 4.5, 4.8, 4.9, 4.9, 5.0, 5.2,
5.2, 6.6
2
3
4
5
6
To count the number of data values in the data set, just count the number
of leaves. In this case, there are 19 data values. It’s also easy to list the data
values in order.
8
3 3 8 9 9
5 5 5 8 9 9
2 2
Key: 2 | 1 = 2.1
Creating a Stem-and-Leaf Plot
On February 7, 2011, earthquakes with the following magnitudes
were detected:
6.4, 4.7, 4.5, 4.5, 4.4, 4.9, 4.7, 2.7, 5.3, 4.2, 4.7, 2.5, 4.3
32
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T07_RG.indd 32
8/23/11 11:01 PM
HOW TO
To create a stem-and-leaf plot
Step 1 Arrange the data in increasing order.
Step 2 Draw a vertical line to separate stems from leaves.
Step 3 Write your stems down one side of the line.
Step 4 Write each leaf in increasing order on the right side of the line,
and include a key.
Step 5 Count your leaves to ensure you have the right number.
Here are the steps for creating a stem-and-leaf plot for the February 7 data.
Step 1 Arrange the data in increasing order.
2.5, 2.7, 4.2, 4.3, 4.4, 4.5, 4.5, 4.7, 4.7, 4.7, 4.9, 5.3, 6.4
Steps 2–4 Draw a vertical line,
and then write your stems in order.
Finally write each leaf to the right in
increasing order.
2
3
4
5
6
5 7
2 3 4 5 5 7 7 7 9
3
4
Key: 4 | 5 = 4.5
Step 5 Count the leaves. There are 13 leaves in the plot, and there are
13 data values in the original data set. No data values were skipped.
Double Stem-and-Leaf Plots
You can also combine the data into a single plot known as a double
stem-and-leaf plot. When presented this way, you can see the stems
straight down the middle with one set of leaves going to the left and the
other going to the right.
February 7
7 5
9 7 7 7 5 5 4 3 2
3
4
2
3
4
5
6
February 14
1 8
1 3 3 8 9 9
4 5 5 5 8 9 9
0 2 2
6
Key: 6 | 2 = 6.2
How would you describe the differences between these two sets of data?
• Both sets of data are rather symmetrical.
• The data for February 7 has a gap when the stem is 3 making 2.5 and
2.7 possible outliers.
REMEMBER
An outlier is an extreme value
that is far from the rest of the
values in a data set.
STEM-AND-LEAF PLOTS
33
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HS_PS_S1_01_T07_RG.indd 33
8/23/11 11:01 PM
Problem Set
Write the data set represented by the stem-and-leaf plot.
1.
4
5
6
7
8
9
10
7
1
5
0
2
3.
2
9
0 4
3 9
9
10
11
12
13
14
0
1
1
2
0
2
4 4
9
3 6 8 8
5 5
Key: 12 6 = 12.6
1
Key: 8 5 = 85
2.
18
19
20
21
22
0
0
0
0
5
0
0
0
5
5
5
0
5
5
5
Key: 18 5 = 185
4.
−3
−2
−1
0
1
2
3
1 3
0 1 3 6
5
7 8
4 9
1
Key: 1 5 = 1.5
The stem-and-leaf plot displays the birth years of a group of people.
5.
How many people are in the group?
6.
Describe the shape of the distribution. Identify any outliers.
7.
How many people in the group were born before 1970?
8.
During what year were the most number of people in the
group born?
9.
What is the difference in years between the oldest person and
the youngest person in the group?
194
195
196
197
198
199
200
0
5
0
2
4
1
9
7
3
6
1
7
8
7 9
1 3 5
Key: 198 5 = 1985
Create a stem-and-leaf plot for the data set. Describe the shape of the
distribution and identify any outliers.
10.
{2, 3, 10, 11, 9, 33, 21, 22, 22, 10, 55, 15}
11.
{0.1, 1.1, 2.4, 3.6, 2.4, 4.5, 4.7, 3.7, 3.2, 2.3, 0.1, 1.9, 2.4}
12.
{3.3, 4.5, 5.0, 4.4, 5.0, 2.2, 4.9, 3.3, 3.4, 2.0, 2.1}
13.
{88, 90, 102, 110, 123, 142, 136, 131, 80, 109, 108, 124, 143, 149}
14.
{90, 82, 75, 55, 77, 83, 95, 84, 79, 88, 89, 61, 99}
15.
{8.0, 7.1, 6.0, 5.1, 4.0, 4.6, 5.6, 6.0, 7.5, 8.5, 7.6, 6.0, 6.8, 6.9}
16.
{1954, 1961, 1977, 1986, 1995, 1986, 1974, 1966, 1951, 1960, 1960}
17.
{90, 101, 112, 129, 133, 134, 128, 112, 109, 99, 98, 126}
34
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T07_PS.indd 34
8/23/11 11:07 PM
The double stem-and-leaf plot shows the number of gold and silver
medals won by countries that earned more than 10 medals overall in
the 2008 Summer Olympics in Beijing.
18.
Describe the shape of the distribution of gold and
silver medals. Identify any outliers.
19.
What was the greatest number of gold medals
earned by a country?
20.
What was the greatest number of silver medals
earned by a country?
21.
How many countries in this group earned more
than 10 silver medals?
Gold
9 8 7 7 7 6 5 4 3 2
9 6 4 3
3
6
0
1
2
3
4
5
1
Silver
4 4 5 5 5 5 5 9
0 0 0 0 1 3 5 6
1 1
8
Key: 4 5 = 45
Use the Nutrition data set on p. A-11.
22.
Create a stem-and-leaf plot for the number of
calories for all nuts in the data set. Describe the
shape of the distribution.
23.
Create a double stem-and-leaf plot that compares
the number of calories for the cereal and nut data
in the data set. What conclusion can you draw
from the double stem-and-leaf plot?
Use the U.S. State Temperatures data set on pp. A-7 to A-8.
24.
Create a stem-and-leaf plot that displays the
record low temperatures for states in the South.
25.
Create a double stem-and-leaf plot that compares
the record low temperatures for states in the
South and the Northeast. What conclusion can
you draw from the double stem-and-leaf plot?
The data represent ages of people at a social gathering. Note: Data
values with an asterisk (*) represent females.
22
51
18*
30
24*
31*
31
31*
29*
22
22*
44
30
49
26.
Create a stem-and-leaf plot for the data.
27.
Describe the shape of the distribution.
28.
Create an appropriate side-by-side stem-and-leaf
plot. What differences in the groups does the
double stem-and-leaf plot reveal?
29.
How many more men are over the age of 25,
compared to women?
STEM-AND-LEAF PLOTS
35
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HS_PS_S1_01_T07_PS.indd 35
8/23/11 11:07 PM
Time Series Plots
When data change over time, a time series plot can
help you see patterns and trends.
Creating a Time Series Plot
Malaria has been largely eliminated from some places (for example, the
United States, Western Europe, Taiwan, and Morocco), but it’s still a big
problem in other places.
The table shows how many cases of malaria were diagnosed in Africa each
year of the first decade of the twenty-first century. The data change with
time, so a good graph would show the trends or patterns in the data.
A bar graph could work for this, but when the independent variable (on the
horizontal axis) is time, it is often a good idea to use a line graph. Simply
treat a pair of corresponding data values as an ordered pair and graph the
ordered pair.
From the time series plot, you can see that malaria cases increased
steeply at the beginning of the decade and have remained consistently
high since then.
Year
Cases
(millions)
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
32
43
45
64
69
68
71
72
60
69
Africa Malaria Cases
80
70
Cases (millions)
60
50
40
30
20
10
0
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Year
36
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T08_RG.indd 36
8/23/11 11:16 PM
THINK ABOUT IT
HOW TO
To create a time series plot
Step 1 Identify the independent variable and dependent variable.
Step 2 Create a coordinate grid. Set the horizontal axis to have the
appropriate scale for the independent variable, and the vertical
axis to have the appropriate scale for the dependent variable.
Though the points are connected
with segments, don’t imagine
that you can interpolate between
them. Each value is discrete, but
the segments can help show
trends and patterns.
Step 3 Plot data values on the coordinate grid.
Step 4 Draw line segments to connect consecutive points.
Comparing Time Series Data
The malaria stories in Southeast Asia and the Western Pacific are completely
different from the story in Africa.
Southeast Asia and Western Pacific Malaria Cases
9
8
7
Cases (millions)
6
5
Southeast Asia
4
Western Pacific
3
2
1
0
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Year
If you use different colors or shapes, you can include more than one set
of data on a single plot. How many diagnoses were there in Southeast Asia
between 2000 and 2004?
5.1 + 7.7 + 7.4 + 7.2 + 7.6 = 35
So there were 35 million diagnoses.
How could you use the graph to describe what happened to the incidence of
malaria in the two regions?
Southeast Asia has seen a sharp decrease since its high between 2001 and
2004. From 2004 to 2009, the incidence dropped.
7.6 – 3.1 = 4.5
4.5 ÷ 7.6 = 0.592 = 59.2%
A 59% drop is pretty good. The Western Pacific region also saw a drop in
that time frame.
2.6 – 1.7 = 0.9
0.9 ÷ 2.6 = 0.346 = 34.6%
Time series plots help you see trends in data over time.
TIME SERIES PLOTS
37
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HS_PS_S1_01_T08_RG.indd 37
8/23/11 11:16 PM
Problem Set
A company is testing an ultrasound insect repellent on a colony of insects.
The time series plot shows the number of insects present in an area during
the first 40 seconds after continuous exposure to the sound.
How many insects were present at the
start of the test?
2.
How many insects fled during the first
20 seconds after exposure to the sound?
3.
How many seconds after exposure to the
sound had about 50% of the insects fled?
Ultrasound Repellent Test
60
50
Insects present
1.
40
30
20
10
0
0
5
10
15
20
25
Time (s)
30
35
40
45
Use the Mobile Phone data set on p. A-9.
4.
Create a time series plot for the mobile phone
data for Argentina.
5.
Between what 2 consecutive years was the
increase in the number of mobile phones for
Argentina the greatest?
6.
Between what 2 consecutive years was there
no change in the number of mobile phones for
Argentina?
7.
Between what 2 consecutive years was there a
decrease in the number of mobile phones for
Argentina?
Determine whether the data set can be represented by the type of graph
or table indicated.
8.
{(0 h, 5 ft), (2 h, 4 ft), (1.5 h, 4 ft), (1 h, 5 ft),
(0.5 h, 1.5 ft)}; time series plot
12.
{1.2, 2.3, 3.5, 2.7, 1.8, 5.5, 4.9, 9.1, 4.2,
7.6, 4.9}; stem-and-leaf plot
9.
{2, 3, 4, 1, 6, 7, 8, 1, 9, 4, 6, 7, 8, 2, 8, 6, 4};
stem-and-leaf plot
13.
{3 min., 4 min., 3 min., 7 min., 5 min., 2 min.,
6 min., 4 min.}; time series plot
10.
{40, 41, 65, 71, 89, 44, 51, 67, 66, 65, 80, 91,
61, 58, 44, 72}; histogram
14.
(25, 102, 58, 98, 88, 45, 34, 48, 65, 110, 55,
51, 29, 47, 62}; frequency table
11.
{males: 45, females: 60}; circle graph
15.
{−2, −4, −3, −6, −5, −1, −8, −6, −4, −5,
−1, −9}; line plot
38
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_T08_PS.indd 38
8/23/11 11:22 PM
A second grade student taking a test on a computer had 2 minutes to
complete the 50 questions on single-digit addition.
Time (s)
15
30
45
60
75
90
105
120
Total Correct
8
17
25
33
39
44
46
46
16.
Create a time series plot for these data.
17.
How many questions did the student miss
overall?
18.
During what time interval did the student answer
the greatest number of questions correctly? How
many questions were answered correctly during
this interval?
19.
Describe what happened during the last
15 seconds of the test. What reason could be
given for what occurred?
The time series plot displays the amount of time in minutes that it
took a college student to drive to and from school each day during a
single week.
What was the student’s approximate
morning commute time on Wednesday?
21.
During which days of the week was the
student’s afternoon commute time less than
30 minutes?
22.
23.
On which days of the week was the student’s
total commute time more than an hour?
To the nearest hour, how long did the student
spend commuting to school during the week?
Commute Time
45
40
35
Time (minutes)
20.
30
25
20
15
10
5
0
Mon.
Tues.
Wed.
Day of the week
Morning
Thurs.
Fri.
Afternoon
Use the Sugar Consumption data set on p. A-10.
24.
Create a multiple time series plot for the sugar
consumption data for Belgium and Ireland.
25.
Which country had the greatest overall change in
sugar consumption from 1968 to 2004?
26.
In approximately what year was the sugar
consumption for the 2 countries about the same?
TIME SERIES PLOTS
39
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Chapter 1 Wrap-Up
Swimming Medals and Records
When you look at a line plot or histogram that represents the data for
Olympic swimming medals, you can see that the United States has
dominated this sport.
Swimming Gold Medals
Sweden
Soviet Union
Germany
Country
Great Britain
Netherlands
Japan
Hungary
East Germany
Australia
United States
0
50
100
150
200
250
Medals
What about the record time data? When you look at a time series plot of
the record times, you can see that the times have steadily decreased over
time, but the record saw its greatest improvement between 1960 and 1970.
The record changed very little between 1990 and 2000, then decreased a bit
more quickly again between 2000 and 2010.
Butterfly World-Record Times
02:35.5
02:18.2
02:01.0
Time (min)
01:43.7
01:26.4
01:09.1
00:51.8
00:34.6
00:17.3
00:00.0
1950
1960
1970
1980
1990
Year
2000
2010
2020
Whatever you are trying to show or understand in a set of data, choosing the
right type of statistical graph can help tremendously.
40
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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In Summary
There are many types of statistical graphs. Each has different purposes and
can be used to communicate aspects of data.
Type of Graph
Use It to . . .
Bar Graph
Show how categories compare
Circle Graph
Show how parts of a whole compare
Line Plot
Show frequencies of discrete data
Histogram
Show frequencies of discrete or continuous data
Stem-and-Leaf Plot
Show ordered discrete data
Time Series Plot
Show trends for quantities that change over time
CHAPTER 1 WRAP-UP
41
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Practice Problems
Representing Data Graphically
Describe the variable as either categorical or quantitative. If the
variable is categorical, describe it as either nominal or ordinal. If the
variable is quantitative, describe it as either discrete or continuous.
1.
income
3.
density
2.
blood type
4.
firefighter rank
The table shows the number of hours that a group of college students
spent working and studying during a particular week.
Student
5.
Working
Studying
Toby
12
20
Millie
21
14
Allison
15
14
Rishab
10
18
Create a circle graph that displays how Millie
spent all of her time during the week.
6.
Create a double bar graph that compares the number
of hours each student spent working and studying.
The U.S. Census Bureau divides the states (including the District of
Columbia) into four regions. The circle graph shows the percent of the
U.S. population living in each region, according to the 2000 Census count.
7.
8.
What percent of the population lived in the West,
according to the 2000 Census count?
U.S. Regions
Northeast
19%
According to the 2000 Census count, the total
U.S. population was 281,421,906. Find the
population of the Midwest region in 2000.
West
South
36%
Midwest
23%
Determine whether the data set can be represented by the type of graph
or table indicated.
9.
10.
42
{(0 s, 0.5 m), (2 s, 1.5 m), (1.5 s, 5.2 m),
(1 s, 2.25 m), (0.5 s, 2.5 m)}; stem-and-leaf plot
11.
{30, 44, 65, 71, 89, 44, 51, 67, 66, 65, 106, 91,
61, 68, 44, 82}; frequency table
{3, 4, 7, 9, 1, 0, 2, 5, 4, 4, 6}; bar graph
12.
{1881, 1899, 1900, 1905, 1876, 1746, 1911, 1890,
1921, 1905, 1887, 1891}; stem-and-leaf plot
CHAPTER 1
REPRESENTING DATA GRAPHICALLY
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HS_PS_S1_01_WU_PS.indd 42
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Chapter 1 Wrap-Up
The two-way table shows the blood types for a sample of men and women.
Type O
Type A
Type B
Type AB
Men
45
32
12
4
Women
50
38
6
3
13.
How many people in the sample have type A blood?
14.
How many women in the sample do not have
type B blood?
15.
What percent of people in the sample are men?
Round your answer the nearest percent.
16.
What percent of people in the sample are women
with type O blood? Round your answer the
nearest percent.
17.
What percent of women in the sample do not
have type O blood? Round your answer the
nearest percent.
A sample of 24 mice were observed. The observers measured how many
milliliters of water each mouse consumed during a 24-hour period. The
results are shown in the data set.
18.
Create a line plot that displays the data.
19.
Describe the shape of the resulting line plot and
identify any outliers.
20.
What percent of the mice drank more than
10 milliliters of water?
8
10
11
16
14
9
10
10
12
13
12
8
9
14
12
14
9
13
14
10
9
19
11
12
Create a stem-and-leaf plot for the data set. Describe the shape of the
distribution and identify any outliers.
21.
{5.4, 4.4, 3.0, 7.7, 6.0, 5.1, 4.5, 3.1, 5.2, 7.9, 6.0,
4.5, 6.3}
22.
{44, 66, 101, 102, 90, 88, 77, 79, 68, 79, 103,
105, 90, 91, 109, 92, 89, 97}
23.
{1, 12, 23, 31, 34, 3, 37, 20, 28, 16, 19, 3}
24.
{1998, 1994, 2001, 1999, 2002, 1985, 1978,
1991, 2004, 1988, 1977}
As of 2010, Robert Horry had played in a record 244 play-off games
during his career in the National Basketball Association (NBA). These
data show the number of play-off games played by 20 different NBA
players during their career (as of 2010).
25.
Create a frequency table for these data using
class widths of 10.
26.
Create a relative frequency table for these data
using class widths of 10.
27.
Create a histogram for these data using class
widths of 15.
28.
Create a relative frequency histogram for these
data using class widths of 15. Use the histogram
to give the approximate percentile rank of a
player who played in 210 play-off games.
168
169
175
180
190
193
208
237
244
203
193
184
183
182
179
175
172
170
169
167
CHAPTER 1 WRAP-UP
43
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