8-9
Misleading Statistics
MAIN IDEA
Math Online
glencoe.com
• Extra Examples
• Personal Tutor
• Self-Check Quiz
HOCKEY The graph shows the all-time Stanley Cup playoff leaders.
1. According to
the size of the
hockey players,
how many
times more
points does
Mark Messier
appear to have
than Jari Kurri?
Explain.
All-Time Stanley Cup Playoff Leaders
430
380
Points
Recognize when
statistics and graphs
are misleading.
330
280
230
180
2. Do you think
Wayne
Gretzky
Mark
Messler
Jari
Kurri
Glenn
Anderson
Paul
Coffey
Players
this graph is
Source: ESPN Sports Almanac
representative
of the players’
number of points? 1, 2. See margin.
Graphs let readers analyze and interpret data easily, but are sometimes
drawn to influence conclusions by misrepresenting the data. The use of
different scales can influence conclusions drawn from graphs.
Changing the Interval of Graphs
1 SCHOOL DANCES The graphs show how the price of spring dance
Graph A
Graph B
Spring Dance Tickets
Spring Dance Tickets
24
12
20
10
16
8
Price ($)
Price ($)
tickets increased.
12
8
4
0
’02 ’03 ’04 ’05 ’06
Year
6
4
2
0
’02 ’03 ’04 ’05 ’06
Year
Do the graphs show the same data? If so, explain how they differ.
The graphs show the same data. However, the graphs differ in that
Graph A uses an interval of 4, and Graph B uses an interval of 2.
444
Chapter 8 Statistics: Analyzing Data
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Which graph makes it appear that the prices increased more rapidly?
Graph B makes it appear that the prices increased more rapidly even
though the price increase is the same.
Which graph might Student Council use to show that while ticket
prices have risen, the increase is not significant? Explain. They
might use Graph A. The scale used on the vertical axis of this graph
makes the increase appear less significant.
Changing Scales
To emphasize a change
over time, reduce the
scale interval on the
vertical axis.
a. BUSINESS The line graphs show monthly profits of a company
from October to March. Which graph suggests that the business
is extremely profitable? Is this a valid conclusion? Explain.
Graph B
Monthly Profits
Graph A
Monthly Profits
15,400
15,300
17,000
16,500
Profit ($)
Profit ($)
a. Sample answer:
Although both graphs
show a profit, in Graph A,
the increase in profits is
exaggerated due to the
intervals of both $500
and $100.
15,200
15,100
15,000
16,000
15,500
15,000
14,500
14,500
14,000
14,000
0
t. v. c. n. b. r.
Oc No De Ja Fe Ma
0
t. v. c. n. b. r.
Oc No De Ja Fe Ma
Month
Month
Sometimes the data used to create the display comes from a biased
sample. In these cases, the data and the display are both biased and
should be considered invalid.
Identify Biased Displays
Do You Use the
Fitness Room?
company mailed a survey to 500
of his employees in order to
determine if they use the fitness
room at work. The results are
shown in the graph. Identify any
sampling errors and explain why
the sample and the display might
be biased.
Number of Responses
2 FITNESS The president of a large
500
450
400
350
300
250
200
150
100
50
0
338
116
s
Ye
No
Not all of the surveys were returned since 338 + 116 < 500. This is a
biased, voluntary response sample. The sample is not representative
of the entire population since only those who wanted to participate in
the survey are involved in the sampling.
The display is biased because the data used to create the display
came from a biased sample.
Lesson 8-9 Misleading Statistics
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What Do You Like to Do
on a Saturday Night?
80
52
60
17
40
20
8
7
16
e
he
r
Ot
om
at
H
Sta
y
Mo
ng
rti
Sp
o
Da
nc
ing
0
vie
theater asked 100 of his customers
what they like to do on a Saturday
night. The results are shown in
the graph. Identify any sampling
errors and explain why the sample
and the display might be biased.
100
Ev
en
t
b. MOVIES The manager of a movie
Number of Responses
b. The sample is a
biased convenience
sample. Customers
at a movie theater
probably like to
watch movies
more than other
activities. The
display is biased
because the data
used to create the
display came from
a biased sample.
Statistics can also be used to influence conclusions.
Misleading Statistics
3 MARKETING Refer to the table that gives the height of roller coasters
at an amusement park. The park boasts that the average height of
their roller coasters is 170 feet. Explain how this is misleading.
Park Rollercoaster Heights
Coaster
mean:
Height (ft)
Viper
109
Monster
135
Red Zip
115
Tornado
365
Riptide
126
109 + 135 + 115 + 365 + 126
850
___
=_
5
5
= 170
median: 109, 115, 126, 135, 365
mode:
none
The average used by the park was the mean. This measure is much
greater than most of the heights listed because of the outlier, 365 feet.
So, it is misleading to use this measure to attract visitors.
A more appropriate measure to describe the data would be the
median, 126 feet, which is closer to the height of most of the coasters.
c. Sample answer:
The mean price of
the items listed is
$3.50; however,
they are calculating
the average price
by including coffee,
a low priced item.
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c. FOOD A restaurant claims its
average menu price is $3.50.
Use the table to explain how
this is misleading.
Menu
Burger
$4.00
Fish Sandwich
$4.45
Chicken Sandwich
$4.35
Garden Salad
$3.90
Coffee
$0.80
Chapter 8 Statistics: Analyzing Data
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Example 1
1. BASEBALL Refer to the graphs below. Which graph suggests that Cy Young
(pp. 444–445)
had three times as many wins as Jim Galvin? Is this a valid conclusion?
Explain.
Graph A
Graph B
Most Career Wins by a Pitcher
Most Career Wins by a Pitcher
Cy
Young
Cy
Young
Walter
Johnson
Walter
Johnson
Pitchers
Pitchers
1. Graph B; From
the length of the
bars, it appears
that Cy Young had
about 3 times as
many wins as Jim
Galvin. However,
Jim Galvin had 365
wins and Cy Young
had 511 wins. So,
the conclusion is
not valid.
Grover
Alexander
Grover
Alexander
Christy
Mathewson
Christy
Mathewson
Jim
Gavin
Jim
Gavin
0
200
400
600
350
400 450
Wins
Wins
2. PHONES The manager of a telephone
550
How Do You Prefer to
Pay Your Telephone Bill?
company mailed a survey to 400
households asking each household how
they prefer to pay their monthly bill.
The results are shown in the graph at the
right. Identify any sampling errors and
explain why the sample and the display
might be biased. See margin.
Number of Responses
Example 2
(pp. 445–446)
500
200
150
125
100
75
50
25
0
ail
r
e
e
m
th
ro t
l f un O
wa cco
a
P
dr g A
ith kin
t W hec
c
ire C
y
yb
Pa
M
in
nl
O
ay
D
(p. 446)
HOMEWORK
For
Exercises
4, 8
5, 9
6, 7
HELP
See
Examples
1
2
3
3. TUNNELS The table lists the five largest land
U.S. Vehicle Tunnels on Land
vehicle tunnels in the U.S. Write a convincing
argument for which measure of central
tendency you would use to emphasize the
average length of the tunnels. See margin.
4. GAS The bar graph
shows monthly gas
prices for 2006–2007.
Why is the graph
misleading?
See margin.
Name
Length (ft)
Anton Anderson Memorial
13,300
E. Johnson Memorial
8,959
Eisenhower Memorial
8,941
Allegheny
6,072
Liberty Tubes
5,920
Gas Prices
3.5
3
Price ($)
Example 3
2.5
2
1.5
Jun
July
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Month
Lesson 8-9 Misleading Statistics
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5. SCHOOL To determine how often his students
Tardy to Class
Number of Responses
are tardy, Mr. Kessler considered his first period
class. The results are shown in the graph at the
right. Identify any sampling errors and explain
why the sample and the display might be biased.
14
12
10
8
6
4
2
0
5
0–
10
6–
0
5
–1
11
–2
16
Number of Times Tardy
TRAVEL For Exercises 6 and 7, use the table.
Annual Sight-Seeing Visitors
6. Find the mean, median, and mode of the
Sight
data. Which measure might be misleading
in describing the average annual number
of visitors who visit these sights? Explain.
7. Which measure would be best if you
wanted a value close to the most number
of visitors? Explain.
Visitors*
Cape Cod
4,600,000
Grand Canyon
4,500,000
Lincoln Memorial
4,000,000
Castle Clinton
4,600,000
Smoky Mountains
10,200,000
Source: The World Almamac
*Approximation
8. STOCK The graphs below show the increases
and decreases in the monthly closing prices of Skateboard Depot’s stock.
'RAPH"
-ONTHLY3TOCK0RICES
50
40
30
20
10
0
Fe
Jan
*AN
&E
B
-A
R
!P
R
-A
Y
0RICEPER3HARE
Monthly Stock Prices
.
b.
Ma
r.
Ap
r.
Ma
y
Price per Share ($)
Graph A
Month
-ONTH
Suppose you are a stockbroker and want to show a customer that the
price of the stock has been fairly stable since January. Write a convincing
argument as to which graph you should show the customer.
9. MANUFACTURING To evaluate their product,
EXTRA
PRACTICE
See pages 690, 711.
448
10. Rent has remained fairly stable.
11. Rent has increased dramatically.
100
95
90
80
70
60
50
40
30
20
10
3
2
0
E
Co xcel
nd len
itio t
n
Do
es
No
aC tH
ha old
rge
Cr
ac Bro
ke ke
dC n
as or
ing
APARTMENTS For Exercises 10 and 11, create a
display that would support each argument given
the monthly costs to rent an apartment for the
last five years are $500, $525, $560, $585, and $605.
Battery Production
Percent of Batteries
the manager of an assembly line inspects the
first 100 batteries that are produced out of
30,000 total batteries produced that day. He
displays the results in the graph at the right
and then releases it to the local newspaper.
Identify any sampling errors and explain why
the sample and the display might be biased.
Product
Chapter 8 Statistics: Analyzing Data
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H.O.T. Problems
12. CHALLENGE Does adding values that are much greater or much less than
the other values in a set of data affect the median of the set? Give an
example to support your answer.
13.
WR ITING IN MATH Describe at least two ways in which the display of
data can influence the conclusions reached.
14. The bar graph shows the average
number of hours each week that a
group of students attend an
extracurricular activity after school.
15. A department store mailed 100
surveys to teenagers about their
preferred style of jeans. The graph
shows the results.
Number of Teenagers
Preferred Style of Jeans
y
nd
es
42
50
45
40
35
30
25
20
15
10
5
0
4
A The vertical scale should show
days instead of hours.
B The graph does not show which
activity each person attended.
C The intervals on the vertical scale
are inconsistent.
D The graph’s title is misleading.
t
se
cu
ot
Bo
Which statement best tells why the
graph may be misleading if you want
to use the graph to compare the
number of hours the students attend
an extracurricular activity?
23
19
Mi
ip
arl
Jam
Pe
He
cto
r
26
24
18
16
13
11
9
0
Ch
Time (h)
Time Spent on
Extracurricular Activities
w
Lo
Ri
h
uc
Slo
ed
lax
Re
Types of Jeans
Which of the following is true
concerning the sample and the display?
F Both the display and the sample
are unbiased.
G The display is biased because the
sample is a biased, voluntary
response sample.
H The display is biased because the
sample is a biased, convenience
sample.
J
The sample is biased but the
display is unbiased.
16. CARS To determine what kind of automobile is preferred by most
customers, the owner of an auto dealership surveys every 10th person
who enters the dealership. Of these, 54% state that they prefer 4-door
sedans. Based on these results, if the dealership stocks 150 cars, about
how many of them should be 4-door sedans? (Lesson 8-8)
17. MP3 PLAYERS In a survey, 46% of randomly selected teens said they own
an MP3 player. Predict how many of the 850 teens at Harvey Middle
School own an MP3 player. (Lesson 8-7)
Lesson 8-9 Misleading Statistics
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