8-9 misleading statistics - glencoe/mcgraw- · pdf filelesson 8-9 misleading statistics 445...

MAIN IDEARecognize when statistics and graphs are misleading.

Math Online

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8-9

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.

2. Do you think

this graph is

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

tickets increased.

Spring Dance Tickets

4

8

12

16

Pric

e ($

)

20

24

Year

0’02 ’03 ’04 ’05 ’06

Graph A

Spring Dance Tickets

2

4

6

8

Pric

e ($

)

10

12

Year

0’02 ’03 ’04 ’05 ’06

Graph B

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.

Misleading Statistics

Source: ESPN Sports Almanac

All-Time Stanley Cup Playoff Leaders

Poin

tsWayne Gretzky

MarkMessler

JariKurri

GlennAnderson

PaulCoffey

Players

430

380

330

280

230

180

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.

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.

Prof

it ($

)

15,00014,500

15,100

14,000

0

15,20015,30015,400

Month

Oct. Nov.Dec

.Jan

.Fe

b.Mar.

Graph AMonthly Profits

Prof

it ($

)

15,000

14,500

15,500

14,000

0

16,000

16,500

17,000

Month

Oct.Nov.Dec

.Jan

.Fe

b.Mar.

Graph BMonthly Profits

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

2 FITNESS The president of a large

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.

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.

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.

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.

Do You Use the Fitness Room?

Num

ber o

f Res

pons

es

500450400350300250200150100500

Yes

338

No

116

Changing ScalesChanging ScalesTo emphasize a change over time, reduce the scale interval on the vertical axis.

Lesson 8-9 Misleading Statistics 445

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b. MOVIES The manager of a movie

What Do You Like to Doon a Saturday Night?

Num

ber o

f Res

pons

es

Mov

ie

Danc

ing

Stay

at H

ome

Spor

ting

Even

t

Oth

er

100

80

60

40

20

0

52

178 7

16

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.

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 Height (ft)

Viper 109

Monster 135

Red Zip 115

Tornado 365

Riptide 126

mean: 109 + 135 + 115 + 365 + 126

___ 5 = 850

_ 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. 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

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.

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.

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.

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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Example 1(pp. 444–445)

1. BASEBALL Refer to the graphs below. Which graph suggests that Cy Young

had three times as many wins as Jim Galvin? Is this a valid conclusion?

Explain.

Example 2(pp. 445–446)

2. PHONES The manager of a telephone

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.

Example 3(p. 446)

3. TUNNELS The table lists the five largest 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.

For Exercises

4, 85, 96, 7

See Examples

123

HOMEWORK HELPHELP 4. GAS The bar graph

shows monthly gas

prices for 2006–2007.

Why is the graph

misleading?

See margin.

U.S. Vehicle Tunnels on Land

Name Length (ft)Anton Anderson Memorial 13,300

E. Johnson Memorial 8,959

Eisenhower Memorial 8,941

Allegheny 6,072

Liberty Tubes 5,920

Most Career Wins by a Pitcher

Pitc

hers

0 200Wins

400 600

CyYoung

WalterJohnson

GroverAlexander

ChristyMathewson

JimGavin

Graph A

Most Career Wins by a Pitcher

Pitc

hers

350 400 450Wins

500 550

CyYoung

WalterJohnson

GroverAlexander

ChristyMathewson

JimGavin

Graph B

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.

Gas Prices

Pric

e ($

)

Month

3.5

3

2.5

2

1.5Jun July Aug Sep Oct Nov Dec Jan Feb Mar Apr May

How Do You Prefer to Pay Your Telephone Bill?

Num

ber o

f Res

pons

es

Pay by M

ail

Pay Onlin

e

Direct

Withdraw

al fro

m

Checkin

g Acco

unt Other

2001501251007550250


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See pages 690, 711.EXTRA PRACTICEPRACTICE

5. SCHOOL To determine how often his students

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.

TRAVEL For Exercises 6 and 7, use the table. Annual Sight-Seeing Visitors

Sight 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

6. Find the mean, median, and mode of the

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.

8. STOCK The graphs below show the increases

and decreases in the monthly closing prices of Skateboard Depot’s stock.

Monthly Stock Prices

20100

4050

30

Pric

e pe

r Sha

re ($

)

Month

Graph A

Jan.

Feb.

Mar

.

May

Apr.

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,

Perc

ent o

f Bat

terie

s

30

20

10

0

Battery Production

40

50

60

70

80

90

100

Product

Exce

llent

Cond

ition

Does

Not

Hol

da

Char

geBr

oken

or

Crac

ked

Casin

g

95

3 2

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.

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.

10. Rent has remained fairly stable.

11. Rent has increased dramatically.

Tardy to Class

Num

ber o

f Res

pons

es

0–56–10

11–1516–20

14121086420

Number of Times Tardy


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14. The bar graph shows the average

number of hours each week that a

group of students attend an

extracurricular activity after school.

Tim

e (h

)

0

Time Spent onExtracurricular Activities

9111316182426

Hect

or

Chip

Pear

lJa

mes

Min

dy

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?

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.

15. A department store mailed 100

surveys to teenagers about their

preferred style of jeans. The graph

shows the results.

Preferred Style of Jeans

Num

ber o

f Tee

nage

rs

Types of Jeans

Low Rise

Bootcut

Slouch

Relaxe

d

50454035302520151050

19

42

23

4

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)

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. MATHWRITING IN Describe at least two ways in which the display of

data can influence the conclusions reached.


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8-9 misleading statistics - glencoe/mcgraw- · pdf filelesson 8-9 misleading statistics 445...

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