Starnes_3e_CH01_002-093_v2_LR_clean watermark

Goodluz/Shutterstocka

ns of quantitative data, it’s not enough just to list
bility of each distribution. You have to explicitly
ressions like “greater than,” “less than,” or “about

. What percent of the nitrate concentration
measurements for each stream exceeded
10 milligrams per liter (mg/l)?

. Compare the centers of these two distributions.
. Is the variability in nitrate concentrations for

the two streams similar or different? Justify your
answer.

Applet

2. Select 1 as the number of groups and Raw data
as the input method.

3. Enter the data. Be sure to separate the data
values with commas or spaces as you type them.

4. Click Begin analysis. A dotplot of the data
should appear.

The applet can also be used to make parallel dotplots
or comparing the distribution of a variable in two or
more groups, like the graph shown in the Household
Size example on page 25.

29/03/16 9:09 pm

LESSON

Lesson 1.3

W h a t D i d Yo u L e a r n ?
Learning Target

Make and interpret dotplots of quantitative data.
Describe the shape of a distribution.
Compare distributions of quantitative data with dotplo

Exercises Lesson 1.3

Mastering Concepts and Skills

1. Magic words Here are data on the lengths of th
pg 22 first 25 words on a randomly selected page from

Harry Potter and the Prisoner of Azkaban.

2 3 4 10 2 11 2 8 4 3 7 2 7
536 44 2 582344

(a) Make a dotplot of these data.
(b) Explain what the dot above 6 represents.
(c) What percent of the words have more than 4 letters

2. Frozen pizza Here are the number of calories pe
serving for 16 brands of frozen cheese pizza.18

340 340 310 320 310 360 350 330

260 380 340 320 310 360 350 330

(a) Make a dotplot of these data.
(b) Explain what the dot above 260 represents.
(c) What percent of the frozen pizzas have fewer tha

330 calories?

3. How fuel efficient? Here are the EPA estimate
of city gas mileage in miles per gallon (mpg) fo
the sample of 21 model year 2014 midsize cars.1
Make a dotplot of these data.

Model mpg Model mpg
Acura RLX 20 Kia Optima 20
Audi A8 18 Lexus ES 350 21
BMW 550i 17 Lincoln MKZ 22
Buick Lacrosse 18 Mazda 6 28
Cadillac CTS 18 Mercedes-Benz E350 21
Chevrolet Malibu 21 Nissan Maxima 19
Chrysler 200 21 Subaru Legacy 24
Dodge Avenger 21 Toyota Prius 51
Ford Fusion 22 Volkswagen Passat 24
Hyundai Elantra 28 Volvo S80 18
Jaguar XF 19

Starnes_3e_CH01_002-093_v2.indd 27

N 1.3  •  Displaying Quantitative Data: Dotplots 27

Examples Exercises
p. 22 1–6
p. 24 7–10

ots. p. 25 11–14

4. Gooooaaal! How good was the 2012 U.S. wom-
en’s soccer team? With players like Abby Wam-
he bach, Megan R­ apinoe, and Carli Lloyd, the team
m put on an impressive showing en route to win-

ning the gold medal at the 2012 Summer Olym-
pics in London. Here are data on the number of
goals scored by the team in games played in the
12 months prior to the 2012 Olympics.20 Make a
dotplot of these data.

s? 1 3 1 14 13 4 3 4 2 5 2 0 4
134 3 42431242

er
5. Visualizing fuel efficiency The dotplot shows the
difference (Highway – City) in EPA mileage ratings
for each of the 21 model year 2014 midsize cars
from Exercise 3.

an d
d
d d
dd
dddd
dd d d d
dd d d dd

es –4 –2 0 2 4 6 8 10 12 14
or
Difference (Highway – City)

19 (a) The dot above –3 is for the Toyota Prius. Explain

what this dot represents.

(b) What percent of these car models get fuel effi-
ciency of at least 10 mpg more on the highway
than in the city?

6. Look at that gooooaaal! The dotplot shows
the  difference in the number of goals scored in
each game (U.S. women’s team – Opponent) in
Exercise 4.

dd dd
dd
dd
ddddd
ddddd
dd d d d d d

–2 0 2 4 6 8 10 12 14

Difference in goals (U.S. team – Opponent)

29/03/16 9:09 pm

28 C H A P T E R 1   •  Analyzing One -Variable Data

(a) Explain what the dot above –1 represents.
(b) What percent of its games did the 2012 U.S.

women’s team win?

7. Looking at the old How old is the oldest person
you know? Prudential Insurance Company asked
pg 24 400 people to place a blue sticker on a huge wall
next to the age of the oldest person they have ever
known. An image of the graph is shown here. De-
scribe the shape of the distribution.

11

pg 25

8. Off to school The dotplot displays data on the 12
travel time to school (in minutes) reported by 13
50 ­Canadian students. Describe the shape of the
d­ istribution.

dd

ddd

ddd

dddd

dddd

ddd d d

ddd d d d

ddd d d d d d

ddd d d d d d d d d d d dd

0 20 40 60 80 100
Travel time (min)

9. Pair-a-dice The dotplot shows the results of roll-
ing  a pair of 6-sided dice and finding the sum of
the up faces 100 times. Describe the shape of the
distribution.

d

d

d

d
dd
dd
dd
dd
dd
dd
dd
dd
dd d

dd d

dd d

ddd d d
dddddd d
dddddddd
ddddddddd
dddddddddd
dddddddddd
ddddddddddd
ddddddddddd

2 4 6 8 10 12

Dice rolls

10. Phone numbers The dotplot displays the last digit
of 100 phone numbers chosen at random from a
phone book. Describe the shape of the distribution.

Starnes_3e_CH01_002-093_v2.indd 28

a

d

dd

dd

dd
dd
ddd
dd
dd d
dd
dd dd
dd d ddd
dd
dd d d d d ddd

dd d d d d d d d d

dd d d d d d d d d

dd d d d d d d d d

dd d d d d d d d d

dd d d d d d d d d

dd d d d d d d d d

02 4 6 8

Last digit

1. Making money The following parallel dotplots
5 show the total family income of randomly chosen

individuals from Indiana (38 individuals) and New
Jersey (44 individuals). Compare the distributions
of total family incomes in these two samples.

d dd d d
dd d d dd
d dddd ddd dd

Indiana ddd dd dd dd ddd dd dd d

d d
d d
dd d dd d
dd dd d dd d d d d
New Jersey dd dd dd d dddd dd dd d dddd d d d

0 25 50 75 100 125 150 175

Total family income ($1000)

2. Movie lengths The following parallel dotplots
display the lengths of the best 15 movies in each
of three decades according to the Internet Movie
Database (www.imdb.com). Compare the distribu-
tions of movie lengths for these three time periods.

1960–1969 d d dd d d
d d ddddd dd d

Decade 1980–1989 dd d 220
dd
d d dddd d d d

2000–2009 dd dd d
dd d d ddd d dd

100 120 140 160 180 200
Movie length (min)

3. Sugar high(er)? Researchers collected data on 76
brands of cereal at a local supermarket.21 For each
brand, the values of several variables were recorded,
including sugar (grams per serving), calories per serv-
ing, and the shelf in the store on which the cereal was
located (1 = bottom, 2 = middle, 3 = top). Here are
parallel dotplots of the data on sugar content by shelf.

d dd
d
d dd ddd d
dd
3 d dd ddd dd d
d d dd d
d dd d dd d d d

Shelf d

2d d

d
d dd
d d dd
d d d d d d d d dd

d

d

dd

1 d d d d dd d
dd d d d d dd
14 16
0 24 6 8 10 12
Sugars

29/03/16 9:09 pm

LESSON

(a) Critics claim that supermarkets tend to put sugar
kids’ cereals on lower shelves, where the kids ca
see them. Do the data from this study support thi
claim? Justify your answer.

(b) Is the variability in sugar content of the cereals on th
three shelves similar or different? Justify your answe

14. Enhancing creativity Do external rewards—thing
like money, praise, fame, and grades—promote cre
ativity? Researcher Teresa Amabile recruited 47 ex
perienced creative writers who were college student
and divided them at random into two groups. Th
students in one group were given a list of statement
about external reasons (E) for writing, such as publi
recognition, making money, or pleasing their parent
Students in the other group were given a list of state
ments about internal reasons (I) for writing, such a
expressing yourself and enjoying playing with word
Both groups were then instructed to write a poem
about laughter. Each student’s poem was rated sepa
rately by 12 different poets using a creativity scale.2
These ratings were averaged to obtain an overall cre
ativity score for each poem. Parallel dotplots of th
two groups’ creativity scores are shown here.

Reward d d d d
Internal (I) External (E) dd d dd dd
d dd
dddd dd ddd

d d d d
ddd d d ddd d
dd d dddddddd

0 5 10 15 20 25 30
Average rating

(a) What do you conclude about whether external re
wards promote creativity? Justify your answer.

(b) Is the variability in creativity scores similar o
d­ ifferent for the two groups? Justify your answer.

Applying the Concepts

15. Bad dotplot Janie asked 10 friends how many pho
tos they posted on Instagram yesterday. Then sh
made the following dotplot to display the data
What’s wrong with Janie’s dotplot?

0 1 2 3 5 6 9 12
Number of photographs

Starnes_3e_CH01_002-093_v2.indd 29

N 1.3  •  Displaying Quantitative Data: Dotplots 29

ry 16. Another bad dotplot Herschel asked the students
an in his English class how many siblings they have.
is Then he made the following dotplot to display the

data. What’s wrong with Herschel’s dotplot?
he
er.

gs
e- 0 1 2 3 4 5 6
x- Number of siblings
ts
he Recycle and Review

ts
ic 17. Brands that sell (1.1) The brands of the last 45 digi-
ts. tal single-lens reflex (SLR) cameras sold on a popu-
e- lar Internet auction site are listed here. Summarize
as the distribution of camera brands with a frequency
ds. table and a relative frequency table.
m
a- Canon Sony Canon Nikon Fujifilm
22 Nikon Canon Sony Canon Canon
e-
he Nikon Canon Nikon Canon Canon

Canon Nikon Fujifilm Canon Nikon

Nikon Canon Canon Canon Canon

Olympus Canon Canon Canon Nikon

Olympus Sony Canon Canon Sony

Canon Nikon Sony Canon Fujifilm

Nikon Canon Nikon Canon Sony

18. Divorce American Style (1.2) The bar chart com-
pares the marital status of U.S. adult residents (18
years old or older) in 1980 and 2010.23 Write a
e- few sentences comparing the distributions of mar-
ital status for these two years.

or 70
.
60

50

o- 40 Percent
he 30
a.

20

10

0 1980 2010 1980 2010 1980 2010 1980 2010
Never married Married Widowed Divorced
Marital status

29/03/16 9:09 pm

Lesson 1.4

Displaying Q
Data: Stemp

Learning Ta

dd Make stemplots of quantitat
dd Interpret stemplots.
dd Compare distributions of qu

Another simple type of graph for d
a stem-and-leaf plot).

D E F I N I T I O N   Stemplot
A stemplot shows each data value
but the final digit, and a leaf, the fin
and arranged in a vertical column. T
the appropriate stems.

Figure 1.4 shows a stemplot of
s­ everal popular soft drinks. You’ll le

1 556
2 033344
2 55667778
3 113
3 55567778
4 33
4 77

FIGURE 1.4 Stemplot showing the caffei
soft drinks.

Making and Interpretin

It is fairly easy to make a stem
Stemplots give us a quick picture
values in the graph.

How to Make a Stemplo

1. Make stems. Separate each ob
and a leaf, the final digit. Write
Draw a vertical line at the right
data value for a particular stem

2. Add leaves. Write each leaf in
3. Order leaves. Arrange the leav
4. Add a key. Provide a key that e

30

Starnes_3e_CH01_002-093_v2.indd 30

Quantitative
plots

argets

tive data.

uantitative data with stemplots.
displaying quantitative data is a stemplot (also called

separated into two parts: a stem, which consists of all
nal digit. The stems are ordered from least to greatest
The leaves are arranged in increasing order out from

f data on the caffeine content per 8-ounce serving for
earn how to make and interpret stemplots in this lesson.

8888899 KEY: 2 | 8 means the
8 soft drink contains
28 mg of caffeine per
8-ounce serving.

ine content (in milligrams or mg per 8-ounce serving) of various

ng Stemplots

mplot by hand for small sets of quantitative data.
of a distribution that includes the actual numerical

ot

bservation into a stem, consisting of all but the final digit,
the stems in a vertical column with the smallest at the top.
t of this column. Do not skip any stems, even if there is no
m.

the row to the right of its stem.

ves in increasing order out from the stem.

explains in context what the stems and leaves represent.

29/03/16 9:09 pm

LESSON

What does the Electoral College do?
Making a stemplot

PROBLEM:  To become president of the United States
popular vote. The candidate does, however, have to w
Electoral College. The table shows the number of elec
District of Columbia. Make a stemplot of these data.

State EV State EV
20
Alabama  9 Illinois 11
 6
Alaska  3 Indiana  6
 8
Arizona 11 Iowa  8
 4
Arkansas  6 Kansas 10
11
California 55 Kentucky 16
10
Colorado  9 Louisiana  6
10
Connecticut  7 Maine

Delaware  3 Maryland

District of Columbia  3 Massachusetts

Florida 29 Michigan

Georgia 16 Minnesota

Hawaii  4 Mississippi

Idaho  4 Missouri

SOLUTION:

0
1
2
3
4
5

0 9 3 6 9 733 44 6 6 8 84 6 3 5 64 5 37 74 9 3 6 353
1 1 61016 00 45 8 3210
2 9 09 0
38
4 Be sure to include this stem
55 even though it contains no data.

0 3333333344444 55 566 666 67 7788999
1 000011112345668
2 0099
38
4 KEY: 1|5 is a state with 15
55 electoral votes.

Starnes_3e_CH01_002-093_v2.indd 31

1.4  •  Displaying Quantitative Data: Stemplots 31

EXAMPLE

s, a candidate does not have to receive a m­ ajority of the
win a majority of the 538 e­ lectoral votes that are cast in the

toral votes (EV) in 2016 for each of the 50 states and the

V State EV State EV
0 Montana  3 Rhode Island  4
1 Nebraska  5 South Carolina  9
6 Nevada  6 South Dakota  3
6 New Hampshire  4 Tennessee 11
8 New Jersey 14 Texas 38
8 New Mexico  5 Utah  6
4 New York 29 Vermont  3
0 North Carolina 15 Virginia 13
1 North Dakota  3 Washington 12
6 Ohio 18 West Virginia  5
0 Oklahoma  7 Wisconsin 10
6 Oregon  7 Wyoming  3
0 Pennsylvania 20

1.  Make stems. The smallest number of electoral votes for
any state is 3 and the largest is 55. Thinking of single-digit
numbers like 3 as 03, we use the first digit of each value as
the stem. So we need stems from 0 to 5 (even though there
is no value with a stem of 4).

2.  Add leaves. For Alabama’s 9 electoral votes, we place
a 9 on the 0 stem. Then we place a 3 on the 0 stem for
Alaska’s 3 electoral votes. Next we place a 1 on the 1 stem
for Arizona’s 11 electoral votes. And so on.

3.  Order leaves.

4.  Add a key. Explain in context what the stems and
leaves represent.

FOR PRACTICE  TRY EXERCISE 1.

29/03/16 9:09 pm

32 C H A P T E R 1   •  Analyzing One -Variable Data

We can get a better picture of
1.5(a), the values from 0 to 9 are p
stemplot of the same data. This ti
one stem, while values ending in 5
see the shape of the distribution m

FIGURE 1.5 Two (a) 0 333333334444455566
stemplots showing the
electoral vote data. The 1 000011112345668
graph in (b) improves on 2 0099
the graph in (a) by split- 3 8
ting stems. 4
5 KEY: 1|5 is
5 15 elector

EXAMPLE Here are a few tips to consider be

•• There is no magic number of
it difficult to see the distributi

•• If you split stems, be sure tha
leaf digits.

•• When the data have too many
truncating the data. See Exerc

a Who has the most votes?

Interpreting a stemplot S
(a
PROBLEM:  Here once again is the stemplot with split
stems for the electoral vote data. Use the stemplot and
the data table from the previous example to answer
these questions.

0 3333333344444

0 55566666677788999

1 00001111234

1 5668

2 00

2 99

3

38

4
4
5 KEY: 1|5 is a state with
55 15 electoral votes.

(a) What percent of states get 10 or more electoral votes? (b

(b) Describe the shape of the distribution.

(c) Which two values appear to be outliers? What
states are these?

FOR PRACTICE  TRY EXERCISE 5. (c

Starnes_3e_CH01_002-093_v2.indd 32

a

the electoral vote data by splitting stems. In Figure
placed on the “0” stem. Figure 1.5(b) shows another
ime, values having leaves 0 through 4 are placed on
5 through 9 are placed on another stem. Now we can
more clearly.

6666677788999 (b) 0 3333333344444
s a state with 0 55566666677788999
ral votes. 00001111234
1 5668
1 00
99
2
2 8
3
3 5

4
4
5
5

efore making a stemplot:

stems to use. Too few or too many stems will make
ion’s shape. Five stems is a good minimum.

at each stem is assigned an equal number of possible

y digits, you can get more flexibility by rounding or
cises 13 and 14.

OLUTION:
a) 21/51 ≈ 0.412. About 41.2% of states get 10 or more

electoral votes.

0 3 3 33 33 3 34 4444
0 5556 666 667 7788999
1 00001111234
1 5668
2 00
2 99
3
38
4
4
5
55

b) The stemplot is skewed to the right with a single peak on
the 05–09 stem and two clear gaps—one from 29 to 38
and the other from 38 to 55.

Note that skewed right always means skewed toward the
larger values, regardless of the orientation of the graph.

c) Apparent outliers: 38 (Texas) and 55 (California).

29/03/16 9:09 pm

LESSON

Comparing Distributions with Stem

You can use a back-to-back stemplot with common s
of a quantitative variable in two groups. The leaves
leading from the common stem.

How can we distinguish oaks?

Comparing distributions with stemplots

PROBLEM: Of the many species of oak trees in the Un
California. How does the distribution of acorn sizes co
the average volumes of acorns (in cubic centimeters) f

Atlantic 0.3 0.4 0.6 0.8 0.9
1.8 1.8 1.8 2.0 2.5

California 0.4 1.0 1.6 2.0 2.6

(a) Make a back-to-back stemplot for these data.
(b) Which of the two regions’ oak tree species has the
(c) Are the shapes of the acorn size distributions simi

SOLUTION: California
(a)  Atlantic Coast

998643 0 4 KEY: | 5 | 9 is a California
88864211111 1 06 oak species with acorn
volume 5.9 cm3.
50 2 06

6640 3
84 1
5 59
86 0
71
18
19
5 10
11
12
13
14
15
16
17 1

(b) California oak tree species tend to have larger acorn
volumes (center ≈ 4.1 cm3) than Atlantic Coast oak tree
species (center ≈ 1.7 cm3).

(c) The distributions of acorn size in the two regions are
quite different. For Atlantic Coast oak tree species, th
distribution is skewed to the right and single-peaked.
For California oak tree species, the distribution has tw
distinct clusters (0.4 to 2.6 cm3 and 4.1 to 7.1 cm3)
a large gap from 7.1 cm3 to 17.1 cm3, and no clear pea

Starnes_3e_CH01_002-093_v2.indd 33

1.4  •  Displaying Quantitative Data: Stemplots 33

mplots

stems to compare the distribution
s on each side are placed in order

EXAMPLE

nited States, 28 grow on the Atlantic Coast and 11 grow in
ompare for oak trees in these two regions? Here are data on
for these 39 oak species:24

0.9 1.1 1.1 1.1 1.1  1.1 1.2 1.4  1.6
3.0 3.4 3.6 3.6 4.8  6.8 8.1 9.1 10.5
4.1 5.5 5.9 6.0 7.1 17.1

e larger acorns?
ilar or different in the two regions? Justify your answer.

1.  Make stems. The smallest acorn volume for any oak
species is 0.3 cubic centimeter (cm3) and the largest
volume is 17.1 cm3. We use the whole number part of
each value as the stem and the decimal part as the leaf.
So we need stems from 0 to 17.

2.  Add leaves.

3.  Order leaves. Put the Atlantic Coast oak species on
the left side of the stem and the California oak species on
the right side of the stem. The data values for both groups
have been sorted in increasing order, so you can add the
leaves on each stem from lowest to highest as you go. For
the first Atlantic oak acorn, place a 3 to the left of the 0
stem. For the first California oak acorn volume, place a 4
on the right side of the 0 stem, and so on.

4.  Add a key. Explain in context what the stems and
leaves represent.

he

wo
),
ak. FOR PRACTICE  TRY EXERCISE 9.

29/03/16 9:09 pm

34 C H A P T E R 1   •  Analyzing One -Variable Data

l e sso n A pp 1.  4

How many shoes are too many shoes?

How many pairs of shoes does a typical teenager own?
To find out, a group of statistics students surveyed
separate random samples of 20 female students and
20 male students from their large high school. Then
they recorded the number of pairs of shoes that each
person owned. Here are the data.

50 26 26 31 57 19 24 22 23 38
Females

13 50 13 34 23 30 49 13 15 51

Males 14 7 6 5 12 38 8 7 10 10
10 11 4 5 22 7 5 10 35 7

1. Make a stemplot of the female data. Do not split stems.

2. Describe the shape of the distribution.

3. Explain why we should split stems for the male data.

4. The back-to-back stemplot with split stems displays
the data for both genders. Write a few sentences
comparing the male and female distributions.

TCOERCNEHR Using an Applet To Make a S

You can use the One Quantitative Variable applet at Th
highschool.bfwpub.com/spa3e to make a stemplot. th
For the electoral vote data on page 3: Le

1. Enter Electoral votes as the Variable name.

2. Select 1 as the number of groups and Raw data
as the input method.

3. Enter the data. Be sure to separate the data
values with commas or spaces as you type
them.

4. Click Begin analysis.

5. Change the Graph type to a stemplot.

6. Split stems to get a better picture of the
distribution.

Starnes_3e_CH01_002-093_v2.indd 34

a

MissKadri/Getty Images

Females Males

04

0 555677778

333 1 0000124

95 1

4332 2 2

66 2

410 3

8 3 58

4 KEY: 2|2 represents
94 a male student with
100 5 22 pairs of shoes.
75

Stemplot

he applet can also make a back-to-back stemplot of
he male and female shoe data like the one shown in
esson App 1.4.

29/03/16 9:09 pm

LESSON

Lesson 1.4

W h a t D i d Yo u L e a r n ?
Learning Target

Make stemplots of quantitative data.
Interpret stemplots.
Compare distributions of quantitative data with stemp

Exercises Lesson 1.4

Mastering Concepts and Skills

1. Science gets your heart beating! Here are th
resting heart rates of 26 ninth-grade biolog
pg 31 students. Make a stemplot of these data. Do no
split stems.

61 78 77 81 48 75 70 77 70 76 86 55 65
60 63 79 62 71 72 74 74 64 66 71 66 68

2. Hot enough for you? Here are the high tempera
ture readings in degrees Fahrenheit for Phoenix
Arizona, for each day in July 2013. Make a stem
plot of these data. Do not split stems.

111 107 115 108 106 109 111 113 104 103 97
99 104 110 109 100 105 107 102 101 84 93
101 105 99 102 104 108 106 106 109

3. Something fishy As part of a study on salmo
health, researchers measured the pH of 25 salmo
fillets. Here are the data. Make a stemplot of thes
data using split stems.

6.34 6.39 6.53 6.36 6.39 6.25 6.45 6.38 6.33 6.26 6.24 6.37 6.32
6.31 6.48 6.26 6.42 6.43 6.36 6.44 6.22 6.52 6.32 6.32 6.48

4. Eat your beans! Beans and other legumes are
great source of protein. The following data give th
protein content of 31 different varieties of bean
in grams per 100 grams of cooked beans.25 Make
stemplot of these data using split stems.

7.5  8.2  8.9 9.3 7.1 8.3 8.7 9.5 8.2 9.1  9.0 9.0
9.7  9.2  8.9 8.1 9.0 7.8 8.0 7.8 7.0 7.5 13.5 8.3
6.8 16.6 10.6 8.3 7.6 7.7 8.1

Starnes_3e_CH01_002-093_v2.indd 35

1.4  •  Displaying Quantitative Data: Stemplots 35

plots. Examples Exercises
p. 31 1–4
p. 32 5–8
p. 33 9–12

5. Science gets your heart beating! Here is a stemplot
he pg 32 using split stems for the heart-rate data from Exercise 1.
gy 4 8
ot 5
55

6 01234
6 5668

7 0011244
7 567789

81
a- 8 6
x, (a) What percent of these ninth-grade biology students
m- have resting heart rates below 70 beats per minute?

(b) Describe the shape of the distribution.

(c) Which value appears to be an outlier? Give the
stemplot a key using this value.

6. Hot enough for you? Here is a stemplot using split
stems for the daily high temperature in Phoenix
on data from Exercise 2.
on 8 4
se 8
93

9 799

2 10 011223444
10 556667788999
11 0113
11 5

a (a) What percent of days in this month were hotter
he than 100 degrees Fahrenheit (°F)?
ns,
a (b) Describe the shape of the distribution.

(c) Which value appears to be an outlier? Give the
stemplot a key using this value.

7. Where are the older folks? Following is a stemplot
of the percents of residents aged 65 and older in the
50 states and the District of Columbia.26

29/03/16 9:09 pm

36 C H A P T E R 1   •  Analyzing One -Variable Data

77 KEY: 12|2 represents (b
8 a state in which (c
12.2% of residents 10
90 are 65 and older.
10 379 F
M
11 44 (a
12 02333445899 (b
13 02234555557788889 (c
14 012334445689 11
15 49
12
16 0
17 3

(a) What percent of states have more than 15% of resi-
dents aged 65 and older?

(b) Describe the shape of the distribution.
(c) Which value appears to be an outlier? Can you

guess what state this is?

8. South Carolina counties Here is a stemplot of the
areas of the 46 counties in South Carolina. Note
that the data have been rounded to the nearest 10
square miles (mi2).

3 9999 KEY: 6 | 4 represents a
4 0116689 county with an area
5 01115566778 of 640 square miles
6 47899 (rounded to the
nearest 10 mi2)
7 01245579

8 0011
9 13
10 8
11 233
12 2

(a) What percent of South Carolina counties have ar-
eas of less than 500 mi2?

(b) Describe the shape of the distribution.
(c) What is the area of the largest South Carolina county?

9. Basketball scores Here are the numbers of points
scored by teams in the California Division I
pg 33 high school basketball playoffs in a single day’s
games.27

71 38 52 47 55 53 76 65 77 63 65 63 68
54 64 62 87 47 64 56 78 64 58 51 91 74
71 41 67 62 106 46

On the same day, the final scores of games in Division V
were as follows:

98 45 67 44 74 60 96 54 92 72 93 46
98 67 62 37 37 36 69 44 86 66 66 58

(a) Make a back-to-back stemplot to compare the
points scored by the 32 teams in the Division
I playoffs and the 24 teams in the Division V
playoffs.

Starnes_3e_CH01_002-093_v2.indd 36

a

b) In which of the two divisions did teams score more
points in their playoff games? Justify your answer.

c) Are the shapes of the distribution of points scored
similar or different in the two divisions? Justify your
answer.

0. Who’s taller? Who is taller, males or females? A
sample of 14-year-olds from the United Kingdom
was randomly selected. Here are the heights of the
students (in centimeters).

160 169 152 167 164 163 160 163 169 157 158
Female

153 161 165 165 159 168 153 166 158 158 166

Male 154 157 187 163 167 159 169 162 176 177 151
175 174 165 165 183 180

a) Make a back-to-back stemplot for these data.
b) Who tends to be taller in the United Kingdom:

14-year-old females or 14-year-old males? Justify
your answer.
c) Are the shapes of the male and female height distri-
butions similar or different? Justify your answer.

1. Who hits the books more? Researchers asked the
students in a large first-year college class how many
minutes they studied on a typical weeknight. The
back-to-back stemplot displays the responses from
random samples of 30 women and 30 men from
the class, rounded to the nearest 10 minutes. Write
a few sentences comparing the male and female dis-
tributions.

Women Men

0 03333

96 0 56668999

22222222 1 02222222
888888888875555 1 558

4440 2 00344
2

30 KEY: 2 | 3 = 230 min
63

2. Fill ’er up The back-to-back stemplot displays the
prices for regular gasoline at stations in Reading,
Pennsylvania, and Yakima, Washington, in spring
2014. Write a few sentences comparing the two dis-
tributions.

Reading Yakima

4 36

996 36 7999
422 37 333

999999776655 37 5589999
1 38 33
38 55799

39 KEY: 36 | 7 = $3.67
39 5

29/03/16 9:09 pm

LESSON

Applying the Concepts

13. Chasing food dollars A marketing consultant ob
served 50 consecutive shoppers at a supermarket t
find out how much each shopper spent in the store
Here are the data (in dollars), arranged in increas
ing order:

3.11 8.88 9.26 10.81 12.69 13.78 15.23 15.62 17.00 17.39
18.36 18.43 19.27 19.50 19.54 20.16 20.59 22.22 23.04 24.47
24.58 25.13 26.24 26.26 27.65 28.06 28.08 28.38 32.03 34.98
36.37 38.64 39.16 41.02 42.97 44.08 44.67 45.40 46.69 48.65
50.39 52.75 54.80 59.07 61.22 70.32 82.70 85.76 86.37 93.34
(a) Round each amount to the nearest dollar. The

make a stemplot using tens of dollars as the stem
and dollars as the leaves.
(b) Make another stemplot of the data by splittin
stems. Which graph shows the shape of the distr
bution better?
(c) Write a few sentences describing the amount o
money spent by shoppers at this supermarket.
14. What does ERA mean? One way to measure the ef
fectiveness of baseball pitchers is to use their earne
run average, which measures how many earne
runs opposing teams score, on average, every nin
innings pitched. The overall earned run average fo
all pitchers in the major leagues in 2013 was 3.86
Here are the earned run averages for all 25 player
who pitched for the Boston Red Sox in 2013.
3.75 3.52 4.57 4.32 1.74 1.09 3.16 1.81 2.64 3.77 4.04 4.97 4.86
5.34 4.88 4.62 3.86 3.52 5.56 3.60 8.60 5.40 6.35 9.82 9.00
(a) Truncate the hundredths place of each data valu
and make a stemplot, using the ones digit as th
stem and the tenths digit as the leaves.
(b) Make another stemplot of the data by splittin
stems. Which graph shows the shape of the distr
bution better?
(c) Write a few sentences describing the distribution o
earned run averages of Boston’s pitchers in 2013.

Extending the Concepts

Sometimes, the variability in a data set is so sma
that splitting stems in two doesn’t produce a stemplo
that shows the shape of the distribution well. We ca
often solve this problem by splitting the stem into fiv
parts, each consisting of two leaf values: 0 and 1,
and 3, 4 and 5, and so on.

Starnes_3e_CH01_002-093_v2.indd 37

1.4  •  Displaying Quantitative Data: Stemplots 37

Exercises 15 and 16 refer to the following setting.

b- Here are the weights, in ounces, of 36 navel oranges
to selected from a large shipment to a grocery store.

e.
s- 5.7 5.4 5.8 5.3 4.6 4.9 5.6 5.3 5.5 5.5 5.4 5.8

5.3 5.5 5.5 5.4 5.8 5.9 5.4 5.1 5.0 5.5 5.7 4.9

9 5.0 5.3 5.1 5.2 5.7 5.6 5.8 4.5 5.2 5.4 5.7 5.6

7 15. Weights of oranges Make a stemplot of the data
8 by splitting stems into two parts. Explain why this
5 graph does not display the distribution of orange
4 weights effectively.

en 16. Splitting the oranges Make a stemplot of the data
ms by splitting stems into five parts. Describe the shape
of the distribution.

ng Recycle and Review

ri-
17. More gas guzzlers (1.2) The EPA-estimated high-
way fuel efficiency for four different sedans is given
of in the bar chart.28 Explain how this graph is mis-

f- leading.

ed 32
ed 30
ne Miles per gallon
or 28
6. 26

rs 24

22

6 20

Toyota Buick Hyundai Ford
Camry LaCrosse Genesis Taurus

Vehicle make and model

ue
he 18. Comparing tuition (1.3) The dotplot shows the
2014 out-of-state tuition for the 40 largest colleges
and universities in North Carolina.29 Describe the
ng overall pattern of the distribution and identify any
ri- clear departures from the pattern.

of d
. d
d

d

d

d

d

d

all d
d

ot d
dd

an d d
dd d

ve ddd ddd d dd
d dd d dddddd d dd

2 5 10 15 20 25 30 35 40 45 50

Annual tuition ($1000)

29/03/16 9:09 pm

Lesson 1.5

Displaying Q
Data: Histog

Learning Ta

dd Make histograms of quantita
dd Interpret histograms.
dd Compare distributions of qu

You can use a dotplot or stemplot
individual data value. For large d
pattern in the graph. We often get a
values together. Doing so allows us

D E F I N I T I O N   Histogram
A histogram shows each interval a
or relative frequencies of values in e

Figure 1.6 shows a dotplot an
eruptions of the Old Faithful geys
together.

d
d
d
d
d
d
d
d
d
d
d
dd
d ddd
d ddd
d ddd
d ddd
dd ddd
dd ddd
dd ddd
ddd ddd
ddd ddd
ddd dd dddd
ddd dd dddd
dddd dd dddd
dddd dd dddd
dddd dd dddd
dddd ddd d ddd
dddd dd ddd d ddd
dddd dd ddd d ddd
dddd dd ddd d ddd
dddd dd ddd d ddd
ddddd dd ddd d ddd
ddddd dd d ddd d ddd d
ddddd dd d d dddddddddd
dddddddd dd d d dddddddddddd
dddddddddddd d d dddddddddddd
dddddddddddd d d ddddd dddddddddddddd

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

(a) Duration (min)

FIGURE 1.6 (a) Dotplot and (b) histogra
Faithful geyser.

Making and Interpretin

It is fairly easy to make a histogra

38

Starnes_3e_CH01_002-093_v2.indd 38

FrequencyQuantitative
grams

argets

ative data.

uantitative data with histograms.

to display quantitative data. Both graphs show every
data sets, this can make it difficult to see the overall
a cleaner picture of the distribution by grouping nearby
s to make a new type of graph: a histogram.

as a bar. The heights of the bars show the frequencies
each interval.

nd a histogram of the durations (in minutes) of 220
ser. Notice how the histogram groups nearby values

70
60
50
40
30
20
10

0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

(b) Duration (min)
am of the duration (in minutes) of 220 eruptions of the Old

ng Histograms

am by hand. Here’s how you do it.

29/03/16 9:09 pm

LESSON 1

How to Make a Histogram

1. Choose equal-width intervals that span the data. F
2. Make a table that shows the frequency (count) or re

proportion) of data values in each interval.
3. Draw and label the axes. Put the name of the quan

axis. To the left of the vertical axis, indicate if the grap
relative frequency (percent or proportion) of data va
4. Scale the axes. Place equally spaced tick marks at th
along the horizontal axis. On the vertical axis, start a
marks until you exceed the largest frequency or relat
5. Draw bars above the intervals. Make the bars equal
them. Be sure that the height of each bar correspond
frequency of data values in that interval. An interval
bar of height 0 on the graph.

It is possible to choose intervals of different wi
Such graphs are beyond the scope of this book.

Which states do immigrants choose?
Making a histogram

PROBLEM:  How does the percent of foreign-born res
the country? The table presents the data for all 50 stat
display the data.

State Percent State P

Alabama 3.4 Louisiana
Alaska 6.2 Maine
Arizona 13.4 Maryland
Arkansas 4.3 Massachusetts
California 27.1 Michigan
Colorado 9.7 Minnesota
Connecticut 13.3 Mississippi
Delaware 8.6 Missouri
Florida 19.4 Montana
Georgia 9.6 Nebraska
Hawaii 18.2 Nevada
Idaho 5.9 New Hampshire
Illinois 13.9 New Jersey
Indiana 4.6 New Mexico
Iowa 4.3 New York
Kansas 6.7 North Carolina
Kentucky 3.3 North Dakota

Starnes_3e_CH01_002-093_v2.indd 39

1.5  •  Displaying Quantitative Data: Histograms 39

Five intervals is a good minimum.
elative frequency (percent or

ntitative variable under the horizontal
ph shows the frequency (count) or
alues in each interval.
he smallest value in each interval
at 0 and place equally spaced tick
tive frequency in any interval.
in width and leave no gaps between
ds to the frequency or relative
with no data values will appear as a

idths when making a histogram.

EXAMPLE

sidents in your state compare to the rest of
tes in 2011.30 Make a frequency histogram to

Percent State Percent

3.8 Ohio 3.9
3.3 Oklahoma 5.5
13.7 Oregon 9.5
14.9 Pennsylvania 5.9
6.1 Rhode Island 13.5
7.4 South Carolina 4.7
2.3 South Dakota 2.9
4.1 Tennessee 4.7
2.0 Texas 16.5
6.2 Utah 8.4
19.2 Vermont 3.9
5.3 Virginia 11.1
21.3 Washington 13.4
10.2 West Virginia 1.3
22.2 Wisconsin 4.8
7.3 Wyoming 2.9
2.4

29/03/16 9:10 pm

40 C H A P T E R 1   •  Analyzing One -Variable Data

SOLUTION:
The data vary from 1.3% to 27.1%. We choose intervals of
width 5, beginning at 0:

0 to <5 5 to <10 10 to <15 15 to <20 20 to <25 25 to <30

Interval Frequency
0 to <5 19
5 to <10 15
10 to <15  9
15 to < 20  4
20 to <25  2
25 to <30  1

Number of states

Number of states Percent foreign-born residents
20
Number of states 15
10
5
0

0 5 10 15 20 25 30
Percent foreign-born residents

20
15
10

5
0

0 5 10 15 20 25 30
Percent foreign-born residents

caution Figure 1.7 shows two different
the left (a) uses the intervals of w
! right (b) uses intervals half as wid
tervals in a histogram can affect the
intervals show more detail but ma

Starnes_3e_CH01_002-093_v2.indd 40

a

1.  C hoose equal-width intervals that span the
data. Note that this choice results in more than the
minimum of five intervals. Also notice that we follow
the convention of including the left endpoint of an
interval and excluding the right endpoint when making
histograms.

2.  Make a table. The table shows the count of data
values in each interval. This type of table is sometimes
called a grouped frequency table.

3.  Draw and label the axes. We label the horizontal axis
with the name of the quantitative variable, "Percent foreign-
born residents." On the vertical axis, we put "Number of
states" for a frequency histogram.

4.  Scale the axes. The scale on the horizontal axis runs
from 0 to 30 with tick marks every 5 units to match the
intervals we chose earlier. Because the highest frequency
in an interval is 19, we scale the vertical axis from 0 to 20
with tick marks every 5 units.

5.  Draw bars. The completed graph is shown at left.

FOR PRACTICE  TRY EXERCISE 1.

t histograms of the foreign-resident data. The one on
width 5 from the previous example. The one on the
de: 0 to <2.5, 2.5 to <5, and so on. The choice of in-
e appearance of a distribution. Histograms with more
ay have a less clear overall pattern.

29/03/16 9:10 pm

LESSON 1

20 20
15
Number of states15
Number of states 10
10 5

5 0
0
0 30
0 5 10 15 20 25 (b)

(a) Percent foreign-born residents

Where do immigrants live?

Interpreting a histogram

PROBLEM:  Use the data table from the previous
example and the histograms in the figure above to
answer these questions.
(a) What percent of states have less than 10% foreign

born residents?
(b) Describe the shapes of the two graphs.
(c) Which state is the possible outlier in the right-

hand graph?
FOR PRACTICE  TRY EXERCISE 5

Think About It What are we actually doing whe
plot in part (a) shows the foreign-born resident data.
intervals of width 5, beginning with 0 to <5, as indica
counted the number of values in each interval. The do
of that process. Finally, we drew bars of the appropri
the completed histogram shown.

dd d d
dd d d d
ddd d d d d
ddddd dd d d d d d
ddddddddddd d d dd d dd d d d dd dd d d
d
0 5 10 15 20 25 30 d
d
(a) Percent foreign-born residents d
d
d
d
d
d
d
d
d
d
d

0

(b) P

Frequency 20
15
10

5
0

0 5 10 15 20

(c) Percent foreign-born re

Starnes_3e_CH01_002-093_v2.indd 41

1.5  •  Displaying Quantitative Data: Histograms 41

0 5 10 15 20 25 30 FIGURE 1.7 (a) F­ requency
Percent foreign-born residents histogram of the percent
of foreign-born residents in
the 50 states with intervals
of width 5, from the previ-
ous example. (b) Frequency
histogram of the data with
intervals of width 2.5.

EXAMPLE

SOLUTION:
(a) (19 + 15)/50 = 34/50 or 68% of states have less than

10% foreign-born residents.
n- (b) The histogram with intervals of width 5 is skewed to the right

with a single peakin the 0% to <5%interval. Thehistogram
with intervals ofwidth 2.5is skewedto the rightwithtwo
clear peaks—one in the 2.5% to <5%intervaland theother in
the 12.5% to <15%interval—and agapfrom 22.5% to 25%.
5. (c) California, with 27.1% foreign-born residents.

en we make a histogram? The dot-
. We grouped the data values into
cated by the dashed lines. Then we
otplot in part (b) shows the results
iate height for each interval to get

d
d
d
d
d
d
d d
d d d
d d d
d d
d d
d d
d d
dddd
ddddd

5 10 15 20 25 30
Percent foreign-born residents

25 30
esidents

29/03/16 9:10 pm

42 C H A P T E R 1   •  Analyzing One -Variable Data

EXAMPLE Comparing Distribution

Histograms can also be used to com
or more groups. It’s a good idea to u
comparing, especially if the groups
when making comparative histogr
horizontal axis scale.

a Why should you get your diploma?

Comparing distributions with histograms

PROBLEM:  Is it true that students who graduate from high school
earn more money than students who do not graduate from high
school?To find out, we took a random sample of 371 U.S. residents
aged 18 and older from a recent census.The educational level and
total personal income of each person were recorded. Following are
relative frequency histograms of the data for the 57 non-graduates
(No) and the 314 graduates (Yes). Compare the distributions.

SOLUTION:

Shape:  Both distributions are skewed to the right and single-
peaked. The graph for the graduates is more strongly skewed.
Center:  The center of the distribution is higher for graduates,
indicating that graduates typically have higher incomes than
non-graduates in this sample.
Variability:  The incomes for graduates vary a lot more than
the incomes for non-graduates.
Outliers:  There are some possible high outliers in the gradu-
ate distribution.

l e sso n A pp 1.  5
How old are U.S. presidents?

The table gives the ages of the first 44 U.S. presidents when they

President Age President Age President Age Pres
64 B. Harrison 55 Eise
Washington 57 Taylor 50 Cleveland 55 Kenn
48 McKinley 54 L. B.
J. Adams 61 Fillmore 65 T. Roosevelt 42 Nixo
52 Taft 51 Ford
Jefferson 57 Pierce 56 Wilson 56 Cart
46 Harding 55 Reag
Madison 57 Buchanan 54 Coolidge 51 G. H
49 Hoover 54 Clint
Monroe 58 Lincoln 51 F. D. Roosevelt 51 G. W
47 Truman 60 Oba
J. Q. Adams 57 A. Johnson

Jackson 61 Grant

Van Buren 54 Hayes

W. H. Harrison 68 Garfield

Tyler 51 Arthur

Polk 49 Cleveland

Starnes_3e_CH01_002-093_v2.indd 42

a

ns with Histograms

mpare the distribution of a quantitative variable in two
use relative frequencies (percents or proportions) when
s have different sizes. Be sure to use the same intervals
rams, so the graphs can be drawn using a common

60%
50%
40%
30%
20%
10%

0%
60%
50%
40%
30%
20%
10%

0%
0 20 40 60 80 100 120 140 160
Total personal income (thousands)

FOR PRACTICE  TRY EXERCISE 9.
High school graduate?
y took office.Yes No

sident National Park ServiceAge
enhower
nedy 61
. Johnson
on 43
d
ter 55
gan
H. W. Bush 56 1. M ake a frequency histogram of
ton 61 the data using intervals of width
W. Bush 52 4 starting at age 40.
ama 69

64 2. D escribe the shape of the distri-

46 bution.

54 3. W hat percent of presidents took
47 office before the age of 60?

29/03/16 9:10 pm

LESSON 1

TCOERCNEHR Making a Histogram w
Calculator

You can use an applet or a graphing calculator to
make a histogram. The technology’s default choice
of intervals is a good starting point, but you should
adjust the intervals to fit with common sense. For the
foreign-born resident data (page 39), use either of the
following.

Applet

1. Launch the One Quantitative Variable applet at
highschool.bfwpub.com/spa3e.

2. E nter Percent foreign-born as the Variable
name.

3. Select 1 as the number of groups and Raw data
as the input method.

4. E nter the data from the example on page 39.
Be sure to separate the data values with
commas or spaces as you type them.

5. Click Begin analysis.

6. Change the Graph type to a histogram. You can
adjust the intervals if desired.

The applet can also be used to make parallel
histograms for comparing the distribution of a var
able in two or more groups, like the graph shown
in the value of a high school diploma example on
page 42.

TI-83/84

1. Enter the percent of foreign-born residents for
each state in your statistics list editor.

dd Press STAT and choose Edit….
dd Type the values into list L1.

Starnes_3e_CH01_002-093_v2.indd 43

1.5  •  Displaying Quantitative Data: Histograms 43

with an Applet or a Graphing

e
e

2. Set up a histogram in the statistics plots menu.
dd Press 2nd Y= (STAT PLOT).
a dd Press ENTER or 1 to go into Plot1.
dd Adjust the settings as shown.

n

3. Use ZoomStat to let the calculator choose inter-
vals and make a histogram.

dd Press ZOOM and choose 9: ZoomStat.
dd Press TRACE and   to examine the intervals.

ri-

Note the calculator’s
unusual choice of
intervals.

29/03/16 9:10 pm

44 C H A P T E R 1   •  Analyzing One -Variable Data

4. Adjust the intervals to match those from the d
example and then graph the histogram. d

dd Press WINDOW . Enter the values shown.

Lesson 1.5

W h a t D i d Yo u L e a r n ?
Learning Target

Make histograms of quantitative data.
Interpret histograms.
Compare distributions of quantitative data with h

Exercises Lesson 1.5

Mastering Concepts and Skills

1. Measuring carbon dioxide Burning fuels in power C
plants and motor vehicles emits carbon dioxide N
pg 39 (CO2), which may contribute to global warming. N
The table displays CO2 emissions in metric tons per
person from 48 countries with populations of at least P
20 million.31 Make a histogram of the data using
intervals of width 2, starting at 0. P

Country CO2 Country CO2 Country CO2 P
Algeria 3.3 Egypt 2.6 Italy 6.7 P
R
Argentina 4.5 Ethiopia 0.1 Japan 9.2
2.
Australia 16.9 France 5.6 Kenya 0.3

Bangladesh 0.4 Germany 9.1 Malaysia 7.7

Brazil 2.2 Ghana 0.4 Mexico 3.8

Canada 14.7 India 1.7 Morocco 1.6

China 6.2 Indonesia 1.8 Myanmar 0.2

Colombia 1.6 Iran 7.7 North Korea 11.5

Congo 0.5 Iraq 3.7 South Korea 2.9

Starnes_3e_CH01_002-093_v2.indd 44

a

dd Press GRAPH .
dd Press TRACE and   to examine the intervals.

histograms. Examples Exercises
p. 39 1–4
p. 41 5–8
p. 42 9–12

Country CO2 Country CO2 Country CO2
Nepal 0.1 Russia 12.2 Turkey 4.1
Nigeria 0.5 Saudi 17.0 Ukraine 6.6

Pakistan Arabia 9.0 United 7.9
0.9 South Kingdom
Peru 17.6
Africa 5.8 United
2.0 Spain States 3.7
6.9
Philippines 0.9 Sudan 0.3 Uzbekistan 1.7
Poland 8.3 Tanzania 0.2 Venezuela
Romania 3.9 Thailand 4.4 Vietnam

. Off to work I go How long do people travel each
day to get to work? The following table gives
the average travel times to work (in minutes) for
workers in each state and the District of Colum-
bia who are at least 16 years old and don’t work
at home.32 Make a histogram of the travel times
using intervals of width 2 minutes, starting at 14
minutes.

29/03/16 9:10 pm

LESSON 1

State Travel State Travel State Travel
AL time to time to time to
AK work work work
AZ (min) (min) (min)
AR
CA 23.6 LA 25.1 OK 20.0
CO 21.8
CT 17.7 ME 22.3 OR 25.0
DE 22.3
DC 25.0 MD 30.6 PA 22.9
FL 15.9
GA 20.7 MA 26.6 RI 23.5
HI 24.6
ID 26.8 MI 23.4 SC 20.8
IL 21.2
IN 23.9 MN 22.0 SD 26.9
IA 25.2
KS 24.1 MS 24.0 TN 25.6
KY 20.8
23.6 MO 22.9 TX 17.9

29.2 MT 17.6 UT

25.9 NE 17.7 VT

27.3 NV 24.2 VA

25.5 NH 24.6 WA

20.1 NJ 29.1 WV

27.9 NM 20.9 WI

22.3 NY 30.9 WY

18.2 NC 23.4

18.5 ND 15.5

22.4 OH 22.1

3. A bell curve? The IQ scores of 60 randoml
selected fifth-grade students from one school ar
shown here.33

145 139 126 122 125 130 96 110 118 118
101 142 134 124 112 109 134 113 81 113
123 94 100 136 109 131 117 110 127 124
106 124 115 133 116 102 127 117 109 137
117 90 103 114 139 101 122 105 97 89
102 108 110 128 114 112 114 102 82 101

(a) Make a histogram that displays the distribution o
IQ scores effectively.

(b) Many people believe that the distribution of IQ
scores follows a “bell curve,” like the one shown a
top right. Does the graph you drew in part (a) sup
port this belief? Explain.

Starnes_3e_CH01_002-093_v2.indd 45

1.5  •  Displaying Quantitative Data: Histograms 45

o

4. Slow country tunes Here are the lengths, in min-
utes, of the 50 most popular mp3 downloads of
songs by country artist Dierks Bentley.

4.2 4.0 3.9 3.8 3.7
4.7 3.4 4.0 4.4 5.0
4.6 3.7 4.6 4.4 4.1
3.0 3.2 4.7 3.5 3.7
4.3 3.7 4.8 4.4 4.2
4.7 6.2 4.0 7.0 3.9
3.4 3.4 2.9 3.3 4.0
4.2 3.2 3.4 3.7 3.5
3.4 3.7 3.9 3.7 3.8
3.1 3.7 3.6 4.5 3.7
(a) Make a histogram that displays the distribution of
song lengths effectively.
(b) Describe what you see.
5. Carbon dioxide emissions Refer to Exercise 1.
pg 41 The histogram displays the data using intervals of
width 1.
ly
re 12

Number of countries 10

88
3

6
4
74

92

10
0 3 6 9 12 15 18

of CO2 emissions (metric tons per person)

Q (a) In what percent of countries did CO2 emissions ex-
at ceed 10 metric tons per person?

p- (b) Describe the shape of the distribution.
(c) Which countries are possible outliers?

29/03/16 9:10 pm

46 C H A P T E R 1   •  Analyzing One -Variable Data

6. Traveling to work Refer to Exercise 2. The
­histogram displays the data using intervals of
width 1.

Number of states 9
8
7 18 21 24 27 30 (a
6 Average travel time (min)
5 (b
4 (c
3
2 9.
1
0 pg 42

15

(a) In what percent of states is the average travel time
at least 20 min?

(b) Describe the shape of the distribution.
(c) Which two states are possible outliers with average

travel times of less than 16 min?

7. Looking at returns on stocks The return on a
stock is the change in its market price plus any
dividend payments made. Total return is usually
expressed as a percent of the beginning price.
The figure shows a histogram of the distribu-
tion of the monthly returns for all common
stocks listed on U.S. markets over a 273-month
­period.34

Number of months 80
60
40
20

0
−25 −20 −15 −10 −5 0 5 10 15
Monthly percent return on common stocks

(a) A return less than zero means that stocks lost value
in that month. About what percent of all months
had returns less than zero?

(b) Describe the shape of the distribution.
(c) Identify the interval(s) that include(s) any possible

outliers.
8. Healthy cereal? Researchers collected data on

calories per serving for 77 brands of breakfast
cereal. The following histogram displays the
data.35

Starnes_3e_CH01_002-093_v2.indd 46

Frequencya

Percent 30
25
Percent 20
15
10

5
0

50 60 70 80 90 100 110 120 130 140 150 160
Calories

a) About what percent of the cereal brands have 130
or more calories per serving?

b) Describe the shape of the distribution.
c) Identify the interval(s) that include(s) any possible

outliers.
. Households and income Rich and poor households
2 differ in ways that go beyond income. Here are his-

tograms that display the distributions of household
size (number of people) for low-income and high-
income households.36 Low-income households
had annual incomes less than $15,000, and high-
income households had annual incomes of at least
$100,000. Compare the distributions.

60
50
40
30
20
10

0
1234567
Household size, low income

60
50
40
30
20
10

0
1234 567
Household size, high income

29/03/16 9:10 pm

LESSON 1Percent of words

Percent of words 10. The statistics of writing style Numerical data ca
distinguish different types of writing and, some
Percent in age group times, even individual authors. Here are histo
grams that display the distribution of word lengt
Percent in age group in Shakespeare’s plays and in articles from Popula
Science magazine. Compare the distributions.
25 Shakespeare

20

15

10

5

0
1 2 3 4 5 6 7 8 9 10 11 12
Number of letters in word

20 Popular Science

15

10

5

0
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Number of letters in word

11. The shape of populations The following histogram
show the distribution of age for 2015 in Vietnam
and Australia from the U.S. Census Bureau’s inter
national database.
9 Australia

8
7
6
5
4
3
2
1
0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Age

9 Vietnam
8
7
6
5
4
3
2
1
0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Age

Starnes_3e_CH01_002-093_v2.indd 47

1.5  •  Displaying Quantitative Data: Histograms 47

an (a) T he total population of Australia at this time was
e- 22,751,014. Vietnam’s population was 94,348,835.
o- Why did we use percents rather than counts on the
th vertical axis of these graphs?
ar (b) What important differences do you see between the

age distributions?

12. Who makes more: men or women? A manufactur-
ing company is reviewing the salaries of its full-
time employees below the executive level at a large
plant. The following histograms display the distri-
bution of salary for male and female e­ mployees:

35 Salary distribution for men

30

Percent of men 25

20

15

10

5

0 40 50 60 70
30 Salary ($1000)

35 Salary distribution for womenPercent of women
30

25

20

15

ms 10
m5
r- 0

30 40 50 60 70
Salary ($1000)

(a) There were 756 female employees and 2451 male
employees at the plant. Why did we use percents
rather than counts on the vertical axis of these
graphs?

(b) Do men or women tend to earn higher salaries at
this plant? Justify your answer.

Applying the Concepts

13. Off to school A random sample of 50 Canadian
students was selected to complete an online survey
in a recent year. The dotplot displays data on the
travel time to school (in minutes) reported by each
student. Make a histogram of the data. Describe
what you see.

dd
ddd
ddd
dddd
dddd
ddd d d
ddd d d d
ddd d d d d d
ddd d d d d d d d d d d dd

0 20 40 60 80 100
Travel time (min)

29/03/16 9:10 pm

48 C H A P T E R 1   •  Analyzing One -Variable Data

14. Unprovoked gator! The dotplot shows the total Ex
number of unprovoked attacks by wild alligators
on people in Florida in each year from 1971 to 17
2013. Make a histogram of these data. Describe
what you see. C
A
d dd St
(a
dd d dd (b
18
dddd d d ddd
ddddddddd ddd R
ddddddddddddd
d 19
2 4 6 8 10 12 14
16 2
(a
Attacks (b
20
15. Communicate, eh? We chose a random sample of
50 Canadian students who completed an online W
survey that included the question “Which of these
methods do you most often use to communicate
with your friends?” The graph displays data on
students’ responses. Jerry says that he would de-
scribe this graph as skewed to the right. Explain
why Jerry is wrong.

30

Frequency of method 25
of communication
20

15

10

5

0 In Internet Facebook Telephone Text

Cell

phone person chat (landline)message

Method of communication

16. The Brooklyn Half The histogram shows the dis-
tribution of age for runners in the Brooklyn, New
York, half-marathon in 2013.37 Explain what is
wrong with this histogram.

Percent 25
20
15
10

5
0

12 20 25 30 35 40 45 50 55 60 70 80
Age

Starnes_3e_CH01_002-093_v2.indd 48

a

xtending the Concepts

7. Do the math The table gives the distribution of
grades earned by students taking the AP® Calculus
AB and AP® Statistics exams in 2014.38

Grade

5 4 3 2 1 Total no.
of exams

Calculus 72,332 48,873 51,950 31,340 89,577 294,072
AB

tatistics 26,265 38,512 45,052 32,748 41,596 184,173

a) M ake an appropriate graphical display to compare
the grade distributions for AP® Calculus AB and
AP® S­ tatistics.

b) Write a few sentences comparing the two distribu-
tions of exam grades.

8. Rolling the die Imagine rolling a fair, six-sided die
60 times. Draw a plausible graph of the distribu-
tion of die rolls. Should you use a bar chart or his-
togram to display the data?

Recycle and Review

9. Runs Scored (1.4) Listed here are the number of
runs scored by players on the Chicago White Sox
who played regularly during the 2013 season.39

24 41 46 68 46 43 84 57 60 38 14 15 19 8 7 4 6 7 2

a) Make a stemplot of these data.
b) Describe the shape of the distribution of runs

scored.

0. Hot enough for you? (1.3) St. Louis, Missouri, and
Washington, D.C., are at the same latitude, but are
their summer temperatures similar? Here are dot-
plots for each city of the high temperature on July
4 for the years from 1980 through 2013.40 Write a
few sentences comparing the distributions of July 4
high temperature in these two cities.

St. Louis d d d d dd
d dd d
d dd d d dd ddd d
dddd ddd d dd
d dd

Washington, D.C. d dd d d d 105
70 d d d dd ddd d d dd
ddd dddd dddddd ddd
100
75 80 85 90 95

High temperature on July 4, 1980 to 2013

29/03/16 9:10 pm

Lesson 1.6

Measuring Center

Learning Targets

dd Find and interpret the median of a distribution o
dd Calculate the mean of a distribution of quantita
dd Compare the mean and median of a distribution

appropriate measure of center in a given setting

How long do people typically spend traveling to wo
where they live. Here are the travel times in minutes
in New York state, along with a dotplot of the data:

10 30 5 25 40 20 10 15 30 20 15 20 8

d

dd

d dd d d d
dd dd
dd dddd

0 10 20 30 40 50 60

Travel time to work (min

The distribution is right-skewed and single-peaked.
appears to be an outlier.

How should we describe where this (or some o
data is centered? The two most common ways to m
the mean.

The Median

We could report the value in the “middle” of a dis
idea of the median.

D E F I N I T I O N   The Median

The median is the midpoint of a distribution, the numb
vations are smaller and about half are larger. To find the
from smallest to largest.

•• If the number n of data values is odd, the median is th
•• If the number n of data values is even, the median is

values in the ordered list.

The median is easy to find by hand for small sets

Starnes_3e_CH01_002-093_v2.indd 49

of quantitative data.
ative data.
n, and choose the more
g.

ork? The answer may depend on
s of 20 randomly chosen workers
:41
85 15 65 15 60 60 40 45

d

70 80 90
n)

The longest travel time (85 min)
other) distribution of quantitative
measure center are the median and

stribution as its center. That’s the

ber such that about half the obser-
e median, arrange the data values

he middle value in the ordered list.
the average of the two middle

s of data.

49

29/03/16 9:10 pm

50 C H A P T E R 1   •  Analyzing One -Variable Data

EXAMPLE
a How unhealthy are fast-food sandwiches?

Finding and interpreting the median when n is odd

PROBLEM:  Here are data on the amount of fat (in grams) in 9
es on McDonald’s menu in 2014.42 Find the median. Interpret th

Sandwich Fat (g)

Filet-O-Fish® 19
McChicken® 16
Premium Crispy Chicken Classic Sandwich 22
Premium Crispy Chicken Club Sandwich 33
Premium Crispy Chicken Ranch Sandwich 27
Premium Grilled Chicken Classic Sandwich
Premium Grilled Chicken Club Sandwich 9
Premium Grilled Chicken Ranch Sandwich 20
Southern Style Crispy Chicken Sandwich 14
19

SOLUTION:
9 14 16 19 19 20 22 27 33

The median is 19. About half of McDonald’s fish and chicken
sandwiches have more than 19 g of fat and about half have less.

A dotplot of the fat content
confirm that the median is 19 by
mum values.

dd

5 10 1

EXAMPLE
a Where’s the beef?

Finding and interpreting the median when n is even

PROBLEM: Here are data on the amount of fat (in grams) in 12
McDonald’s 2014 menu, along with a dotplot. Find and interpre

Starnes_3e_CH01_002-093_v2.indd 50

a
different fish and chicken sandwich-
his value in context.

Sort the data values from smallest to largest. Because
there are n = 9 data values (an odd number), the median
is the middle value in the ordered list.

FOR PRACTICE  TRY EXERCISE 1.

t data from the example is shown here. You can
y “counting inward” from the minimum and maxi-

d d d d d
dd
15 25 35
20 30

Fat (g)

2 different beef sandwiches from
et the median.

29/03/16 9:10 pm

Sandwich Fat (g)
Big Mac® 27
Cheeseburger 11
Daily Double 22
Double Cheeseburger 21
Bacon Clubhouse Burger 40
Hamburger  8
McDouble® 17
BBQ Ranch Burger 15
Quarter Pounder® Bacon and Cheese 29
Quarter Pounder® Bacon Habanero Ranch 31
Quarter Pounder® Deluxe 27
Quarter Pounder® with Cheese 26

SOLUTION:
8 11 15 17 21 22 26 27 27 29 31

The median is 22 + 26 = 24. About half of ­McDonald’s be
2

sandwiches have more than 24 g of fat and about half have l

The Mean

The most common measure of center is the mean.

D E F I N I T I O N   The mean x

The mean x (pronounced “x-bar”) of a quantitative data
values. To find the mean, add all the values and divide b

x = sum of data value
number of data valu

If the values in a data set are given by x1, x2, …, x
calculating the mean as

x = x1 + x2 + c+ xn =
n

The Σ (capital Greek letter sigma) in the formula
The subscripts on the observations xi are just a w
distinct. They do not necessarily indicate order o
the data.

Starnes_3e_CH01_002-093_v2.indd 51