Starnes_3e_CH01_002-093_v2_LR_clean watermark

a

b) Do these data support the belief that men and
women differ in their study habits and attitudes
toward learning? Give appropriate evidence to sup-
port your answer.

6. Well connected Who has more contacts—males or
females? The data show the number of contacts
that a sample of high school students had in their
cell phones.

Male 124  41  29  27  44  87  85 260 290  31
168 169 167 214 135 114 105 103  96 144

 30  83 116  22 173 155 134 180 124  33
Female

213 218 183 110

a) Make parallel boxplots to compare the distribu-
tions.

b) Based on your graphs in part (a), which gender
tends to have more contacts in their cell phones?
Give appropriate evidence to support your answer.

xtending the Concepts

7. Measuring skewness Here is a boxplot of the num-
ber of electoral votes in 2016 for each of the 50
states and the District of Columbia, along with
summary statistics. You can see that the distribu-
tion is skewed to the right with 3 high outliers.
How might we compute a numerical measure of
skewness?

** *

0 10 20 30 40 50 60
Electoral votes

Variable n Mean Minimum Q1 Median Q3 Maximum

Electoral 51 10.55 3.00 4.00 8.00 12.00 55.00
votes

a) One simple formula for calculating skewness
m ax im u m − m e d ian
is  m e d ian − m in im u m . Compute this value for

the electoral vote data. Explain why this formula

should yield a value greater than 1 for a right-skewed

distribution.

b) Choosing only from the summary statistics provid-
ed below, define a formula for a different statistic
that measures skewness. Compute the value of this
statistic for the electoral vote data. What values of
the statistic might indicate that a distribution is
skewed to the right? Explain.

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LESS

Recycle and Review

18. Best in Show (1.1, 1.2) Here is a frequency table
showing the breed group of the dog that won Bes
in Show at the Westminster Kennel Club dog show
for 96 years.60 Make a relative frequency bar char
for these data. Describe what you see.

Breed group Frequency Breed group Frequency
Terrier Toy
Sporting 38 Non-sporting 11
Working 18 Hound  9
15  5

Lesson 1.9

Describing Location
Distribution

Learning Targets

dd Find and interpret a percentile in a distribution o
dd Estimate percentiles and individual values using

frequency graph.
dd Find and interpret a standardized score (z-score)

quantitative data.

Here are the scores of all 25 students in Mr. Pryor’s

79 81 80 77 73 83 74 93 78

77 83 86 90 79 85 83 89 84

The bold score is Jenny’s 86. How did she perform on
The following dotplot displays the class’s test sc

in red. The distribution is roughly symmetric with
dotplot, we can see that Jenny’s score is above the m
bution. We can also see that Jenny did better on the
the class.

dd

d d dd d
dddd
d dddddddddd

65 70 75 80 85
Score

Starnes_3e_CH01_002-093_v2.indd 77

O N 1.9  •  Describing Location in a Distribution 77

e 19. Heartbeats (1.7) Here are the resting heart rates of
st 26 ninth-grade biology students. Decide on an ap-
w propriate measure of variability for these data and
rt calculate it. Justify your choice.

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

n in a

of quantitative data.
g a cumulative relative
) in a distribution of

statistics class on their first test:

80 75 67 73
82 77 72

this test relative to her classmates?
cores, with Jenny’s score marked
h no apparent outliers. From the
mean (balance point) of the distri-
e test than most other students in

d dd d

90 95

29/03/16 9:11 pm

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

Finding and Interpretin

One way to describe Jenny’s loca
her percentile.

D E F I N I T I O N   Percentile
An individual’s percentile is the pe
individual’s data value.

caution Using the dotplot, we see tha
class. Because 21 of the 25 obser
! 84th percentile in the class’s test s

Be careful with your language
locations in a distribution, so an
is “at” the 84th percentile.

EXAMPLE

a What are the results of the first test?

Finding and interpreting percentiles

PROBLEM:  Refer to the dotplot of scores on Mr. Pryor’s
first statistics test.
(a) Find the percentile for Norman, who scored 72.
(b) Maria’s test score is at the 48th percentile of the

distribution. Interpret this value in context. What
score did Maria earn?

SOLUTION:
(a) 1/25 = 0.04, so Norman scored at the 4th percentile

on this test.
(b) (0.48)(25) = 12, so Maria’s score was higher than 12 of

the 25 students in the class. Maria earned an 80 on the test.

caution The median of a distribution i
median score on Mr. Pryor’s first
! score of 80 put her at the 48th p
is roughly the 25th percentile of
of values from the upper 75%. L
percentile.

A high percentile is not alway
pressure is at the 90th percentile
blood pressure!

Starnes_3e_CH01_002-093_v2.indd 78

a

ng Percentiles

ation in the distribution of test scores is to calculate

ercent of values in a distribution that are less than the

at Jenny’s 86 places her fourth from the top of the
rvations (84%) are below her score, Jenny is at the
score distribution.

when describing percentiles. Percentiles are specific
observation isn’t “in” the 84th percentile. Rather, it

dd

d d dd d
dddd
d d d d dd d d d d d dd d

65 70 75 80 85 90 95
Score

Only 1 of the 25 scores in the class is below Norman’s 72.

One other student in the class scored an 80 on the test. This
student’s score is also at the 48th percentile because 12 of
the 25 students in the class earned lower scores.

FOR PRACTICE  TRY EXERCISE 1.

is roughly the 50th percentile. For instance, 80 is the
test. As you saw in part (b) of the example, Maria’s
percentile of the distribution. The first quartile Q1
a distribution because it separates the lowest 25%
Likewise, the third quartile Q3 is roughly the 75th

ys a good thing. For example, a man whose blood
for his age group may need treatment for his high

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LESS

Cumulative Relative Frequency Grap

There are some interesting graphs that can be made
common starts with a frequency table for a quantitat
quency table summarizes the ages of the first 44 U.S.

Age Frequency
40–44  2
45–49  7
50–54 13
55–59 12
60–64  7
65–69  3

Let’s expand this table to include columns fo
­frequency, and cumulative relative frequency.

• To get the values in the relative frequency column
by 44, the total number of presidents. Multiply by

•• To fill in the cumulative frequency column, add th
for the current interval and all intervals with sm

•• For the cumulative relative frequency column, di
frequency column by 44, the total number of
convert to a percent.

Here is the original frequency table with the relative
and cumulative relative frequency columns added.

Age Frequency Relative frequency Cumu
40–44  2 2/44 = 0.045, or 4.5% frequ
45–49  7 7/44 = 0.159, or 15.9%
50–54 13 13/44 = 0.295, or 29.5%  
55–59 12 12/44 = 0.273, or 27.3%  
60–64  7 7/44 = 0.159, or 15.9% 2
65–69  3 3/44 = 0.068, or 6.8% 3
4
4

Now we can make a cumulative relative frequenc

D E F I N I T I O N   Cumulative relative frequency grap

A cumulative relative frequency graph plots a point c
relative frequency in each interval at the smallest value
a point at a height of 0% at the smallest value of the firs
then connected with a line segment to form the graph.

Figure 1.9 on the next page shows the complet
graph for the presidential age at inauguration data.

• The leftmost point is plotted at a height of 0% at
the first interval. This point tells us that none of th
office before age 40.

Starnes_3e_CH01_002-093_v2.indd 79

O N 1.9  •  Describing Location in a Distribution 79

phs

with percentiles. One of the most
tive variable. For instance, the fre-
. presidents when they took office.

y

or relative frequency, cumulative

n, divide the count in each interval
y 100 to convert to a percent.
he counts in the frequency column
maller values of the variable.
ivide the entries in the cumulative
individuals. Multiply by 100 to

e frequency, cumulative frequency,

ulative Cumulative relative
uency frequency
 2
 9 2/44 = 0.045, or 4.5%
22 9/44 = 0.205, or 20.5%
34 22/44 = 0.500, or 50.0%
41 34/44 = 0.773, or 77.3%
44 41/44 = 0.932, or 93.2%
44/44 = 1.000, or 100%

cy graph.

ph

corresponding to the cumulative
of the next interval, starting with
st interval. Consecutive points are

ted cumulative relative frequency
Notice the following:

Age = 40, the smallest value in
he first 44 U.S. presidents took

29/03/16 9:11 pm

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

FIGURE 1.9 Cumula- • The next point to the right is p Cumulative relative frequency (%)
tive relative frequency tells us that 4.5% of presidents
graph for the ages of U.S. 45 years old.
presidents when they
took office. • The graph grows very gradual
when they were in their 40s. T
cause most U.S. presidents wer
id growth in the graph slows a

• The rightmost point on the gra
frequency 100%. That’s becaus

100

80

60

40

20

0d
40

EXAMPLE A cumulative relative frequenc
individual within a distribution o

a Is the president old?

Interpreting cumulative relative frequency graphs

PROBLEM:  Use the graph in Figure 1.9 to help you answer eac
(a) Was Barack Obama, who was first inaugurated at age 47, u
(b) Estimate and interpret the 65th percentile of the distributi

SOLUTION:
(a)  100
Cumulative relative frequency (%)
80

60

40

20
11
0
40 45 47 50 55 60 65 70
Age at inauguration

Barack Obama’s inauguration age places him at about the 11th percentile
took office at a younger age than Obama did. So Obama was fairly young, bu

Starnes_3e_CH01_002-093_v2.indd 80

a

plotted at a height of 4.5% at Age = 45. This point
s (i.e., two of them) were inaugurated before they were

lly at first because few presidents were inaugurated
Then the graph gets very steep beginning at age 50 be-
re in their 50s when they were inaugurated. The rap-
at age 60.
aph is plotted above age 70 and has cumulative relative
se 100% of U.S. presidents took office before age 70.

d
d
d

d

d
d

45 50 55 60 65 70
Age at inauguration

cy graph can be used to describe the position of an
or to locate a specified percentile of the distribution.

ch question:
unusually young?
ion.

To find President Obama’s location in the distribution,
draw a vertical line up from his age (47) on the
horizontal axis until it meets the graph. Then draw a
horizontal line from this point to the vertical axis.

e. About 11% of all U.S. presidents first
ut not unusually young, when he took office.

29/03/16 9:11 pm

LESSCumulative relative frequency (%)

(b)  100

80
65

60
40
20

0
40 45 50 55 58 60 65 70
Age at inauguration

The 65th percentile is about 58. About 65% of all U.S. preside

Finding and Interpreting Standardiz

A percentile is one way to describe the location of an ind
tative data. Another way is to give the standardized sco

D E F I N I T I O N   Standardized score (z-score)

The standardized score (z-score) for an individual valu

many standard deviations from the mean the value falls

standardized score (z-score), compute

z = value − mean
standard deviatio

Values larger than the mean have positive z-scor
have negative z-scores.

Let’s return to the data from Mr. Pryor’s first sta
displays the data, with Jenny’s score marked in red. T
maries for these data.

dd

d d dd d
dddd
d ddddddddd

65 70 75 80 85
Score

Summary Statistics

n mean SD min Q1 Med Q3 max
25 80 6.07 67 76 80 83.5 93

Where does Jenny’s 86 fall within the distribution?

z = value − mean = 86 −
standard deviation 6.0

That is, Jenny’s test score is 0.99 standard deviations a

Starnes_3e_CH01_002-093_v2.indd 81

O N 1.9  •  Describing Location in a Distribution 81

The 65th percentile of the distribution is the age with
cumulative relative frequency 65%. To find this value,
draw a horizontal line across from the vertical axis at
a height of 65% until it meets the graph. Then draw a
vertical line from this point down to the horizontal axis.

0

ents were younger than 58 when they took office.
FOR PRACTICE  TRY EXERCISE 5.

zed Scores (z-Scores)

dividual in a distribution of quanti-
ore (z-score) for the observed value.

ue in a distribution tells us how
s, and in what direction. To find the

on

res. Values smaller than the mean
atistics test. The following dotplot
The table provides numerical sum-

dd dd d

5 90 95

Her standardized score (z-score) is

− 80 = 0.99
07

above the mean score of the class.

29/03/16 9:11 pm

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

EXAMPLE

a How well did Lionel do?

Finding and interpreting z-scores S
z
PROBLEM:  Find the standardized score (z-score) for
Lionel, who earned a 67 on Mr. Pryor’s first test. Inter-
pret this value in context.

FOR PRACTICE  TRY EXERCISE 9. Li
m

We often standardize observe
example, we might compare the h
by calculating their z-scores.

• At age 7, Jordan is 51 in. tall. H
is 1 standard deviation above t

• Zayne’s height at age 9 is 54 in
Zayne is one-half standard dev

So Jordan is taller relative to girls
standardized heights tell us where
for his or her age group.

l e sso n A pp 1.  9

Which states are rich?

Cumulative percentThe following cumulative relative frequency graph and the
numerical summaries describe the distribution of ­median
household incomes in the 50 states in a recent year.61 1
2
100 d d d 3
80 d
60 d
40 d

20 d
0 dd
35 40 45 50 55 60 65 70 75
Median household income

Use the information provided above and in this table
to help you answer the following ­questions.

Median household income

n 50

Mean $51,742.44

SD $8,210.642

Starnes_3e_CH01_002-093_v2.indd 82

a

OLUTION:

= 67 − 80 = − 2.14
6.07

ionel’s score is 2.14 standard deviations below the class
mean of 80.

ed values to express them on a common scale. For
heights of two children of different ages or genders

Her height puts her at a z-score of 1. That is, Jordan
the mean height of 7-year-old girls.
n. His corresponding z-score is 0.5. In other words,
viation above the mean height of 9-year-old boys.

s her age than Zayne is relative to boys his age. The
e each child stands (pun intended!) in the distribution

© Andre Jenny/Alamy Stock Photo

1. At what percentile is North Dakota, with a median
household income of $55,766?

2. Estimate and interpret the first quartile Q1 of the
distribution.

3. Find and interpret the standardized score (z-score)
for New Jersey, with a median household income
of $66,692.

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LESS

Lesson 1.9

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

Find and interpret a percentile in a distribution of quan
Estimate percentiles and individual values using a cum
frequency graph.
Find and interpret a standardized score (z-score) in a di
quantitative data.

Exercises Lesson 1.9

Mastering Concepts and Skills

1. Play ball! The dotplot shows the number of win
pg 78 for each of the 30 Major League Baseball teams i

the 2014 season:

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

60 65 70 75 80 85 90 95 100
Wins

(a) Find the percentile for the Boston Red Sox, wh
won 71 games.

(b) The New York Yankees’ number of wins is at th
60th percentile of the distribution. Interpret this valu
in context. How many games did New York win?

2. Stand up tall The dotplot shows the heights of the 2
students in Mrs. Navard’s statistics class.

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

60 62 64 66 68 70 72 74
Height (in.)

(a) Find the percentile for Lynette, the student who i
65 in. tall.

(b) Asher’s height is at the 88th percentile of the distribu
tion. Interpret this value in context. How tall is Asher

3. A boy and his shoes How many pairs of shoes doe
a typical teenage boy own? To find out, a group o
statistics students surveyed a random sample of 2
male students from their large high school. The
they recorded the number of pairs of shoes tha
each boy owned. Here are the data.

14 7 6 5 12 38 8 7 10 10
10 11 4 5 22 7 5 10 35 7
(a) Martin is the student who reported owning 2

pairs of shoes. Find Martin’s percentile.
(b) Luis is at the first quartile Q1 of the distribution

How many pairs of shoes does Luis own?

Starnes_3e_CH01_002-093_v2.indd 83

O N 1.9  •  Describing Location in a Distribution 83

ntitative data. Examples Exercises
mulative relative p. 78 1–4
p. 80 5–8

istribution of p. 82 9–12

ns 4. Unlocked for sale The “sold” listings on a popu-
in lar auction website included 21 sales of used

­“unlocked” phones of one popular model. Here
are the sales prices.

450 415 495 300 325 430 370

400 325 400 235 330 304 415

ho 355 405 449 355 425 299 345

(a) Find the percentile of the phone that sold for $325.
he (b) What was the sales price of the phone that was at
ue the third quartile Q3?

25 5. Supermarket sweep The following figure is a cu-
pg 80 mulative relative frequency graph of the amount
spent by a sample of 50 grocery shoppers at
a store.

Cumulative relative frequency (%) 100 d d

90 d dd
d
is 80
70 d
d
60

u- 50
r? 40
30 d
es 20

of 10 d
20 0
en 0 10 20 30 40 50 60 70 80 90 100
at Amount spent ($)

(a) What is the percentile for the shopper who spent
$19.50?

(b) Estimate and interpret the 80th percentile of the
22 distribution.

6. Light life The following graph is a cumulative
n. relative frequency graph showing the lifetimes (in

hours) of 200 lamps.62

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

100Cumulative relative 10
frequency (%)
80 11
Cumulative relative (a
60frequency (%) (b
12
40
(a
20 (b

0 A
500 700 900 1100 1300 1500
13
Lifetimes (hr)
14
(a) What is the percentile for a lamp that lasted 900 hr?
(b) Estimate and interpret the 60th percentile of this

distribution.
7. Call me maybe? The graph displays the cumula-

tive relative frequency of the lengths of phone calls
made from the math department office at Gabalot
High last month.

100 d d d
80 d

d

60 d d d

d

40 d

20 d

d

0d d d

0 5 10 15 20 25 30 35 40 45
Call length (min)

(a) About what percent of calls lasted 30 min or
more?

(b) Estimate the interquartile range (IQR) of the
­distribution.

8. That tall? The graph displays the cumulative relative
frequency of the heights (in inches) of college basket-
ball players in a recent season.

Cumulative relative 100 65 70 75 80 85 15
frequency (%) 75 Height (in.) 16
50
25
0

60

(a) About what percent of players were at least 75 in. tall?
(b) Estimate the interquartile range (IQR) of the

­distribution.

9. The Nationals play During the 2014 season, the
mean number of wins for Major League Baseball
pg 82 teams was 81 with a standard deviation of 9.6
wins. Find the standardized score (z-score) for the
Washington Nationals, who won 96 games. Inter-
pret this value in context.

Starnes_3e_CH01_002-093_v2.indd 84

a

0. Stand tall The heights of the 25 students in
Mrs. Navard’s statistics class have a mean of 67 in.
and a standard deviation of 4.29 in. Find the stan-
dardized score (z-score) for Boris, a member of the
class who is 76 in. tall. Interpret this value in context.

1. Where are the old folks? Based on data from the
2010 U.S. Census, the percent of residents aged 65
or older in the 50 states and the District of Columbia
has mean 13.26% and standard deviation 1.67%.

a) Find and interpret the standardized score (z-score)
for the state of Colorado, which had 9.7% of its
residents age 65 or older.

b) The standardized score for Florida is z = 2.60. Find
the percent of the state’s residents that were 65 or
older.

2. Meaning of the Dow The Dow Jones Industrial
Average (DJIA) is a commonly used index of the
overall strength of the U.S. stock market. In 2013
the mean daily change in the DJIA for the 252 days
that the stock markets were open was 13.59 points
with a standard deviation of 94.05 points.

a) Find and interpret the standardized score (z-score)
for the change in the DJIA on May 7, 2013, which
was 87.31 points.

b) The standardized score for May 1, 2013, was
z = –1.62. Find the change in the DJIA for that date.

Applying the Concepts

3. Setting speed limits According to the Los Angeles
Times, speed limits on California highways are set
at the 85th percentile of vehicle speeds on those
stretches of road. Explain to someone who knows
little statistics what that means.

4. Percentile pressure Larry came home very excited
after a visit to his doctor. He announced proudly to
his wife, “My doctor says my blood pressure is at
the 90th percentile among men like me. That means
I’m better off than about 90% of similar men.”
How should his wife, who is a statistician, respond
to Larry’s statement?

5. Big or little? Mrs. Munson is interested to know how
her son’s height and weight compare with those of
other boys his age. She uses an online calculator to
determine that her son is at the 48th percentile for
weight and the 76th percentile for height. Explain to
Mrs. Munson what these values mean.

6. Run faster Peter is a star runner on the track team.
In the league championship meet, Peter records a
time that would fall at the 80th percentile of all his
race times that season. But his performance places
him at the 50th percentile in the league champion-
ship meet. Explain how this is possible. (Remember
that lower times are better in this case!)

29/03/16 9:12 pm

© Imagerymajestic/Alamy Stock Photo LESS

17. SAT versus ACT During her senior year, Courtney
took both the SAT and ACT. She scored 680 on
the SAT math test and 27 on the ACT math test.
Scores on the math section of the SAT vary from
200 to 800, with a mean of 514 and standard de-
viation of 117. Scores on the math section of the
ACT vary from 1 to 36, with a mean of 21.0 and
a standard deviation of 5.3. Calculate Courtney’s
standardized score on each test. Which of her two
test scores was better? Explain.

18. Generational GPA Rebecca and her father both
graduated from the same high school. When her fa-
ther looked at Rebecca’s transcript, he noticed that
her high school GPA (4.2) was higher than his high
school GPA (3.9). After letting Rebecca gloat for a
minute, he pointed out that there were no weight-
ed grades when he went to school. To settle their
argument, they called the registrar at the school
and got information about the distribution of GPA
in each of their graduation years. When the father
graduated, the mean GPA was 2.8 with a standard
deviation of 0.6. When Rebecca graduated, the
mean GPA was 3.2 with a standard deviation of
0.7. Calculate Rebecca’s and her father’s standard-
ized GPA. Who had the better GPA? Explain.

Extending the concepts

19. Medical exam results People with low bone den-
sity have a high risk of broken bones. Currently,
the most common method for testing bone density
is dual-energy X-ray absorptiometry (DEXA). A
patient who undergoes a DEXA test usually gets
bone density results in grams per square centime-
ter (g/cm2) and in standardized units.
Judy, who is 25 years old, has her bone density
measured using DEXA. Her results indicate a bone
density in the hip of 948 g/cm2 and a standardized

Chap

Sta

Does

Recall
domly
ing wit
hand sanitizer, and 10 stu
three days of incubation
counted. Here are the dat

Starnes_3e_CH01_002-093_v2.indd 85

O N 1.9  •  Describing Location in a Distribution 85

y score of z = −1.45. In the population of 25-year-old
n women like Judy, the mean bone density in the hip
. is 956 g/cm2.63
m
- (a) Judy has not taken a statistics class in a few years.
e Explain in simple language what the standardized
d score tells her about her bone density.

s (b) Use the information provided to calculate the
o standard deviation of bone density in the popula-
tion of 25-year-old women.

h Recycle and Review
-
t 20. Birthrates in Africa (1.6, 1.7, 1.8) One of the impor-
h tant factors in determining population growth rates
a is the birthrate per 1000 individuals in a population.
- Here are a dotplot and five-number summary for the
r birthrates per 1000 individuals in 54 African nations.
ld
A dd
dd d
r dd d dd d d
dd d d dd d d d d d d d

d d dd d dd d d d dd d d d d dd d d d d d d d d d d d
e
f 18 24 30 36 42 48 54
Birthrate (per 1000 population)

- Minimum Q1 Median Q3 Maximum

14 29 37.5 41 53

(a) Construct a boxplot for these data.

- (b) Suppose the maximum value of 53 was in error and
y, should have been 45. For each statistic, indicate
y whether this correction would result in an increase,
A a decrease, or no change. Justify your answer in
s each case.
- • Mean

y • Median
e • Standard deviation

d • Interquartile range

pter 1

tats applied

hand sanitizer work?

all:  Using 30 identical petri dishes, Daniel and Kate ran-
assigned 10 students to press one hand in a dish after wash-
th soap, 10 students to press one hand in a dish after using
udents to press one hand in a dish after using nothing. After
n, the number of bacteria colonies on each petri dish was
ta from Daniel and Kate’s experiment.

29/03/16 9:12 pm