Example 4 An economics text gives the following graph and notes that from 1966
to 1982 the prices of stocks were generally going down.
The text then presents the following graphs for two other time periods, noting
in particular that from 1993 to 1998 stock prices were generally going up.
272 CHAPTER 13 Numbers?
A much longer and less-biased choice of period (1925-1998) gives a less distorted
picture. It indicates that investments in stocks are sometimes profitable and sometimes
unprofitable. W. Baumol and A. S. Blinder, Economics: Principles and Policy
SECTION C Averages 273
Analysis Why is the longer period apt for comparison to the present day? If we
looked at 1890 onwards, we'd have a different picture still ("Full History" is a bad
label). Maybe the best comparison for an analogy about investing in stocks is with
the later periods because of new regulations on buying and selling stocks. These
graphs, however, do compensate for inflation by stating the values in 1983 dollars—
if they didn't, the comparisons would be apples and oranges.
C. Averages
"It ought to be safe to cross here. I heard that the average depth is only two feet."
Beware: The average is not the maximum or most likely depth.
The average or mean of a collection of numbers is obtained by adding the numbers and then dividing by the number of items. For example,
The average of 7, 9, 37, 22, 109 is calculated:
Add 7 + 9 + 37 + 2 2 + 1 0 9 = 1 8 4
Divide 184 by 5 = 36.8, the average
An average is a useful figure to know if there isn't too much variation in the
figures. For example, suppose the marks Dr. E gave for his course were:
score
# students
95
3 students
94
7 students
92
1 student
90
4 students
75
1 student
62
4 students
57
5 students
55
4 students
52
2 students
The grading scale was 90-100 = A, 80-89 = B, 70-79 = C, 60-69 = D, 59 and
below = F. When Dr. E's department head asked him how the teaching went, he told
her, "Great, just like you wanted, the average mark was 75%, a C."
274 CHAPTER 13 Numbers?
But she knows Dr. E too well to be satisfied. She asks him, "What was the
median score?" The median is the midway mark: the same number of items above
as below. Again Dr. E can reply, "75." As many got above 75 as below 75.
But knowing how clever Dr. E is with numbers, she asks him what the mode
score was. The mode is the number most often obtained. Dr. E flushes, "Well,
94." Now she knows something is fishy. When she said that she wanted the average
score to be about 75, she was thinking of a graph that looked like:
students
The distribution of the marks should be in a bell-shape, clustered around the median.
Unless you have good reason to believe that the average is pretty close
to the median and that the distribution is more or less bell-shaped, the
average doesn't tell you anything important.
Sometimes people misuse the word "average" by confusing it with the mode or
most, as in "The average American enjoys action movies."
Gel your class to stand up. Look around. Do you think the average height is
the same as the median height? How can you tell? Come up with a physical
way to determine the median height and the mode of the heights.
Suppose your class had just eight players from the men's basketball team
and five women gymnasts. Do you think the median and the average would
be the same?
Summary Numbers are our way of measuring. They are important in our reasoning.
But it's easy to be misled or use them wrong. A vague claim doesn't get any better
by using numbers. Both sides of a comparison must be made clear. The numbers
must represent quantities someone could actually know. And often it's not the
average that's significant, but the median or the mode.
We also have to be careful in reading graphs, for they can mislead or conceal
claims by not taking the base of the comparison to be zero, or by using bars, or by
spacing the numbers on the axes in certain ways.
Key Words apples and oranges mean
two times zero is still zero median
average mode
EXERCISES for Chapter 13 275
Exercises for Chapter 13
1. Find an advertisement that uses a claim with percentages that is misleading or vague.
2. Find an advertisement that uses a claim with numbers other than percentages that is
misleading or vague.
3. Compare a sundial on a sunny day with a digital watch that is set wrong.
a. Which is more accurate at telling the time?
b. Which is more precise?
4. Dick is contemplating getting a new printer. It's faster than his old one. He prints out a
cartoon and finds that it takes 7 minutes. On his old printer it took 10 and a half minutes.
Tom tells him he'll save 1/3 of his time. Dick says no, he'll save about 50% of his time.
Who is right?
5. "The birth control pill is 99% effective." What does this mean?
6. Find the average, mean, median, and mode of the scores of Dr. E's students who took
his critical thinking final exam: 92, 54, 60, 86, 62, 76, 88, 88, 62, 68, 81.
7. Estimate the average age of students in your class. Do you think it's the same as the
median? As the mode?
8. The experts say that over the long term the stock market is the best place to invest. So
you invested most of your retirement in stocks. You've just turned 70 and need cash to
retire. But the market went down 15% last week. Evaluate those experts' advice now.
For Exercises 9-27 point out any use of numbers that is vague, misleading, or wrong.
9. [Advertisement] Our employees have a combined 52 years of experience!
10. [On a box of Texmati® rice] Amount per serving
Serving size 1/4 cup (45g) Calories 150 %DV*
Servings Per Package about 22 Total Fat 0.5g 1 %
Sodium 0mg 0%
Total Carb. 34g 11%
Protein 3g
* Percent Daily Values are based on a 2,000 calorie diet.
11. [From a glossy brochure "Why do I need a water softener?" by Pentair Water Treatment]
The Bureau of Statistics found that between 17 and 20.8 cents of every dollar are
spent on cleaning products. . . . The bottom line? Soft water can save you thousands
of dollars.
12. [Advertisement for 3 Musketeers® candy bars]
The sweetest part is finding out how little fat it has.
(45% less fat than the average of the 25 leading chocolate brands, to be exact.)*
*Not a low-fat food. 8 fat grams per serving for single bar vs. 15 gram average for
leading chocolate brands.
276 CHAPTER 13 Numbers?
13. [Advertisement] Studies have shown that three cups of Cheerios® a day with a low-fat
diet can help lower cholesterol.
14. Dick: Gee, cars are really expensive now. My uncle said he bought a new Ford
Mustang in 1968 for only $2,000.
15. [On the box of a fan made by Lasco™ that Dr. E bought]
NEW WIND RING™ 30% MORE Air Velocity
16.
[Concerning the way the U.S. Census Bureau operates] In 1990, 65% of the question-
naires that were mailed were filled out and returned. Census counters went back to
every household that didn't mail back a form. Even then, the bureau was able to count
only 98.4% of the U.S. population. USA Today, April 15,1998
17. Less than 10% of women who get breast cancer have the gene for breast cancer.
Therefore, if you have the gene, there's only a 10% chance you'll get breast cancer.
18. In America 7 out of 10 people believe that they are one of the 3 out of 10, whereas in
Japan 7 out of 10 people believe they are one of the 7.
Advanced Reasoning Forum, May, 2000
19. Roadway Congestion
Cities with highest and lowest roadway congestion index. A value
greater than 1.0 indicates significant congestion.
Highest Index Lowest Index
Los Angeles 1.57 Bakersfield, Calif. 0.68
Washington 1.43 Laredo, Texas 0.73
Miami-Hialeah 1.34 Colorado Springs 0.74
Chicago 1.34 Beaumont, Texas 0.76
San Francisco 1.33 Corpus Christi, Tex. 0.78 USA Today, 4/13/99
20. New Mexico Lodging Report: May 2001
Available
Occupied
Occupancy
Average
Room-Nights
Room-Nights
Rate
Albuquerque
190,373
125,780
66.1%
$67.84
Santa Fe
100,752
72,512
72.0%
$120.72
Farmington
18,197
12,667
69.6%
$57.66
Carlsbad
17,647
10,753
61.6%
$56.20
Las Cruces
29,884
19,218
64.3%
$57.32
Taos
25,345
12,250
48.3%
$68.04
Other
53,040
31,247
59.3%
$57.83
STATE
435,058
284,607
65.4%
$78.61
Albuquerque Tribune, June 28, 2001
21. Dick: I read that drinking a shot of whiskey a day is good for your health. I didn't
drink much last year, so I better make up for it tonight.
EXERCISES for Chapter 13 277
22. Dick: I read that on average, women think of sex about every 12 minutes.
Zoe: Really? I guess some woman out there is thinking about sex only once a year.
23. [Advertisement] Mitsubishi is the fastest growing Japanese car company in America.
24. [Advertisement]
Official Royal Flush Results! Fiesta 2,115 Texas 1,735
It's not even close
Fiesta backs up its claim:
"We Pay More Royal Flushes per Machine Than Any Other Casino Hotel in the World!"
For the month of September, Texas [Casino] claimed that it paid out a total of 1,735
Royals, with approximately 2,000 machines, but for that same period, Fiesta Casino paid
out 2,115 Royal Flushes, with just 1,200 machines. Here's proof, once again, that
Fiesta's Slots and Video Poker Machines are the loosest on Earth!
25. The Vacancy Rate of Albuquerque Apartment Complexes
98%
96%
94%
92%
90%
88%
86%
84%
82%
80%
SOURCE: Apartment Association of New Mexico
Albuquerque Journal, March 3, 2002
26. S. Korea declares war on leftovers
Because of the feeling of bounty and plenty that it gives, Koreans routinely cook more at
home than they can eat, and restaurants serve more than any customer could reasonably
consume. . . .
"Koreans are used to thinking 'the more the better,'" said Koh, the restaurant
manager.
It's a philosophy the government is battlingto change. In the latest round, the govern-
men
a
t announce
Man
toughe
Th
metric e y
restaurant
r
tons of d
waste Dec
Korean
,
problems. .
government .
garbag 6
say
which
e tha
the
th
. .
say
as e t
y
th i
e
day. t wil
are l mak
carefu
country'
P l
governmens e
t a
at
45
auline majo
hom
says e r
millio pus
to
n
h
ea
accountt i
s
Jelinekn
peopl; 199
for
e 42 7
leftover
thro t
s
w o cu
the
percent
awa foo
nex
o
y tf
Associatedd wast
day
foo
nearl .
d
y
Press e
, b
Bu y
t
48,00 half
garbage
0 , i .
s
11/23/96
278 CHAPTER 13 Numbers?
27. Artery narrowing can be reversed
A new study has shown what many researchers have thought all along—cardiovascular
disease (i.e., narrowing of the arteries) can be moderately reversed.
The well-known secret: lifestyle changes.
In the study, heart patients who had coronary artery (heart) disease—diagnosed
through angiograms (X-rays of the arteries)—were: 1) put on a vegetarian diet, 2) told
to stop smoking, 3) started on a mild to moderate aerobic exercise program (three hours
per week), and 4) told to practice stress management techniques (e.g., meditation) one
hour a day.
Five-year findings: In a control group of heart patients who had not made the
above lifestyle changes, 45% had coronary narrowing that became worse; 50% showed
no change; and 5% showed improvement.
By comparison, 99% of the group who made significant lifestyle changes (see
above) had healthier arteries (i.e., improved blood flow) or their condition remained
stable.
From the heart, Washoe Health System, Fall, 1996
Which of the following should be trusted to give you a good idea of the population as a
whole? For which would you prefer to know the median or mode? Explain.
28. The average wage in the U.S. is $28,912.
29. The average wage in one rural county of Utah was $14,117.
30. The average wage of concert pianists in the U.S. is less than the average wage of
university professors.
31. The average number of people in a household in Las Vegas is 2.1.
32. The average GPA of a graduating senior at this college in 2000 was 2.86, while in
1972 it was 2.41.
33. Dick: Which section of English Lit should I take, Zoe, Professor Zzzyzzx's or
/> Professor 011eb0d's?
Zoe: It doesn't really matter. You can't understand either, and the department info
on the sections said the average mark in both their classes was a C.
34. The average income of a woman in the U.S. was only 82% that of a man.
14 Generalizing
A. Generalizing 279
• Exercises for Section A 281
B. What is a Good Sample?
1. How you can go wrong 283
2. Representative samples 284
• Exercises for Section B 286
C. When Is a Generalization Good?
1. Sample size 287
2. Is the sample studied well? 288
3. Three premises needed for a good generalization 288
4. The margin of error and confidence level 289
5. Variation in the population 290
6. Risk 290
7. Analogies and generalizations 291
Summary 291
• Exercises for Chapter 14 292
A. Generalizing
I think I'll get a border collie. Every one I've met has been friendly and loyal.
I'd better not visit your home. You've got a cat, and every time I've been
around a cat I get a terrific sneezing fit and asthma.
We generalize every day, arguing from a claim about some to a claim about more.
It's how we make sense of our world: What's happened before is likely to happen
again. My experience is typical, until I learn otherwise. As we experience more, we
generalize better because we have more examples from which to generalize.
But it's not only our own experience. Poll takers and scientists generalize, too,
as when they say that the President's approval rating is 54%, or they report that 28%
of all people who smoke get cancer. Those are generalizations from the groups of
people that were interviewed or studied.
279
280 CHAPTER 14 Generalizing
Generalizing We are generalizing if we conclude a claim about a
group, the population, from a claim about some part of it, the sample.
To generalize is to make an argument.
Sometimes the general claim that is the conclusion is called the
generalization; sometimes we use that word for the whole argument.
Plausible premises about the sample are called the inductive evidence
for the generalization.
To evaluate whether a generalization is good, we need to see it as an argument.
Strong arguments with plausible premises will be the best we're likely to get, since
there will always be the possibility that there's an exception to a generalization.
E x a m p l e s Are the following generalizations? If so, what is the sample? What is
the population?
Example 1 In a study of 816 people who owned sport utility vehicles in Cincinnati,
Richard L Epstein Page 35