Teaching Statistics with Fictional Anecdotes

by

Hershey H. Friedman
Department of Business and Economics
Brooklyn College of the City University of New York

Noemi Halpern
Department of Mathematics
Brooklyn College of the City University of New York

David Salb
Department of Mathematics and Computer Science
Kingsborough Community College of the City University of New York


ABSTRACT


       The authors demonstrate that anecdotes are a simple, yet effective,
       way of communicating some major principles of statistics.  Anecdotes
       are used to demonstrate various important statistical concepts
       including:  the importance of obtaining a representative sample and
       achieving a high rate of response, the consequence of improper
       wording of questions, the usefulness of measures of dispersion, and
       why correlation does not prove causality. 

Teaching Statistics with Fictional Anecdotes

       Historian Andrew Lang said that some people use statistics "as a
drunken man uses lampposts - for support rather than illumination." There
has been a large increase of students enrolled in statistics classes in
recent years. With the advent of calculators and computers, statistical
analysis is occurring at an unprecedented rate and its study has become
an essential component of high school and college education. The ability,
interest, and motivation vary greatly among elementary statistics
students. The purpose of this article is to demonstrate how fictitious
anecdotes, especially humorous ones, can be used to effectively teach
statistical concepts and to dispel some common misconceptions in
statistics.  More importantly, anecdotes can be used to demonstrate the
relevance of statistics to everyday situations. This anecdotal approach is
an especially helpful tool in motivating a diverse student body with a
wide variety of mathematical backgrounds.

Representative Samples

(1)  Millions of Alien Abductions?
       Professor Ima Klutz is doing research on individuals who claim to
have been abducted by aliens from another planet.  Professor Klutz sent
out 2 million questionnaires and 100,000 were returned by respondents.
The questionnaire included such questions as: "In the last 30 days, how
many aliens from other planets have you encountered?"  Professor Klutz
has cited his research on various television talk shows.  Based on his
research, he claims that thirty percent of Americans have encountered
aliens.  Indeed, he insists that twenty percent of Americans claim to
have been forcibly abducted, and ten percent of respondents claim to have
been molested by these aliens from other planets.  Professor Klutz's
research was challenged by other panelists on the television show.
Professor Klutz's response was that the Gallup poll and other well-known
polls are based on samples of 2,000 individuals, whereas the Klutz study
was based on 100,000 respondents.
       
       Students are asked to think about this fictional anecdote and
decide whether Professor Klutz's research is valid.  This anecdote can be
used to illustrate that it is not the number of responses that matter but
the rate of response.  Students can also be taught the important idea of
representative samples.  Respondents to surveys, in general, are overly
interested in the subject of the questionnaire and those with little
interest in the subject will often choose not to respond. 

(2) Sample of 1,000?
       The Hercules Juice Company purchases apples from various orchards.
They have been offered apples from Aphrodite Apple Orchards at an
excellent price.  Professor Klutz is hired as a quality control
consultant to determine whether the apples are sufficiently sweet for
Hercules apple juice.  Professor Klutz has determined that a sample size
of 1,000 apples should be adequate.  Professor Klutz arrives at the
orchard and is told that the orchard consists of 25,000 trees and the
typical tree has about 1,000 apples.  He tells the Aphrodite people to
pick apples from a tree that he has selected randomly and bring him the
apples. The apples are picked from one tree and brought to Professor
Klutz.  There are slightly more than 1,000 apples in the sample.
Professor Klutz brings the apples to the Hercules Company for testing and
the apples are perfect for juice.  The Hercules Juice Company buys all of
Aphrodite's apples.  Subsequently, the juice produced by the company is
found to be too tart and cannot even be given away, much less sold for
profit.  Professor Klutz is blamed.  He is adamant in his claim that a
sample of 1,000 apples is more than adequate for a representative sample.  

       A week later, Professor Klutz goes to the Hercules company doctor
for a checkup. The doctor informs Professor Klutz that he needs a blood
sample.  The doctor takes a blood sample from the Professor's fingers,
nose, ears, toes, knees, hips, lips, cheeks, buttocks, and elbows.
Professor Klutz asks the doctor why he needs so much blood.  The doctor,
Dr.  Mal Practice, informs the Professor that he wants a representative
sample and he will not be satisfied with anything less than a sample of
1000. Professor Klutz is duly impressed. 

       This example illustrates that a sample of 1,000 from a perfectly
homogeneous cluster may actually be the same as a sample of one.  The
blood supply of the human body is perfectly homogeneous and, thus, a
sample from any part of the body is adequate. Similarly, the 1,000 apples
coming from a single tree was actually an effective sample of only one.

       Since apples from the same tree are homogeneous, one apple should
have been randomly sampled from 1,000 different randomly selected trees
in order to obtain a representative sample of 1,000.

Response Errors

(3) How Many Tax Cheaters?
       Two researchers were interviewed on the Honey Money talk show.
One researcher, Dr. Lane Payne, claims that his research shows that only
10% of Americans cheat on their Federal income taxes.  The other
researcher, Dr. Jack Asch, claims that his research shows that 90% of
Americans cheat on their taxes.  Dr. Klutz, a panelist on the show, is
asked to determine whose research is more valid.  He finds out that Dr.
Asch used an anonymous mail survey with a 60% response rate; Dr. Payne 
used a telephone interview and achieved an 85% rate of response.  Dr.
Klutz's conclusion was that Dr. Payne's research is more likely to be
valid given the higher rate of response.  
       
       This example demonstrates that the interviewing method can affect
responses.  Very few people are likely to admit cheating on their taxes
to an anonymous telephone interviewer who claims to be working for a
research company and may very well be an I.R.S. agent.

(4) Expert Interviewers?
       Two different research companies are hired to determine whether
illegal drug usage is a problem on campus.  The ABC Research Company
finds that 52% of students say they have used an illegal drug within the
last thirty days.  The XYZ Research Company, on the other hand, finds that
only 13% of students claim to have used an illegal drug within the last
thirty days.  Dr. Klutz, consultant to the college, determines that both
companies used personal interviewers.  However, ABC interviewers were
students at the college and XYZ interviewers were mainly retired women who
have worked for the company for the last several years.  Dr. Klutz
concludes that the XYZ company's research is more valid since they used
professional interviewers, not novices.

       This example illustrates the problem of interviewer effects.  The
age, sex, race, clothing, social class, vocal intonations, and mannerisms
of the interviewer can affect the responses. Students are probably more
likely to admit to drug usage to a fellow student wearing torn jeans and
one earring than to an elderly woman in a conservative dress. 

Wording the Questionnaire

(5) Roast Cardboard.
       The Pecunious Research Company is hired by the Brown Shack hamburger
chain to determine whether Brown Shack hamburgers are preferred to the
hamburgers made by other fast food outlets.  The Brown Shack hamburgers
taste like dried cardboard but are baked (overbaked?) in an oven, not
grilled.  The question asked of a random sample of respondents is the
following:  How do you prefer your hamburgers: fried in greasy fat or
roasted over a fire?  The results show that 80% of respondents prefer their
hamburgers roasted rather than fried. Brown Shack is citing this research
in their advertising. 

       This example illustrates that the connotations of words can
influence the responses. Words such as "fried," "greasy," and "fat" have
negative connotations.  This is similar to what actually happened when
Burger King did a survey which asked individuals whether they preferred
their hamburgers "flame broiled or fried?"

Measures of Dispersion

(6) Buying good chips.
       Professor Ima Klutz has been hired as a consultant by the Atlas
Computer Company. They manufacture a computer with a three year warranty,
i.e., they will replace or repair any computer that fails within the three
year warranty period.  The current supplier of the chip which is the main
component of their computer is Supplier X, and only about 3% of Atlas
computers do not make it through the warranty period.  Currently, the
Atlas Company is being offered computer chips from two suppliers, Supplier
X and Supplier Y. Professor Klutz tells the buyer to take a random sample
of ten chips from each supplier's chips and calculate the mean life.  The
buyer informs Professor Klutz that the chips manufactured by Supplier X
have an average life of 10 years and the average life of chips
manufactured by Supplier Y is 20 years.  Professor Klutz orders the buyer
to stop purchasing chips from Supplier X and switch to Supplier Y.  After
switching to Supplier Y, the Atlas Company computers start failing at an
alarmingly high rate.  More than 50% of the computers break down in the
warranty period and the company is about to declare bankruptcy.  Professor
Klutz is blamed for switching to a new supplier.  He claims that the new
supplier's chips have an average life that is twice that of the old
supplier.  He offers to do additional consulting and discover the problem.  
       
       This example can be used to demonstrate the importance of measures
of dispersion. Students can be shown the original data.

Supplier X chips, n = 10 (life in years):  
10, 10, 10, 10, 9, 11, 10, 11, 9, 10 
Mean = 10 years, range = 2 years, and standard deviation = 0.67 years

Supplier Y chips, n=10 (life in years):  
40, 2, 37, 1, 0, 31, 49, 1, 38, 1 
Mean = 20 years, range = 48 years, and standard deviation = 20.50 years

       Students should be able to appreciate the necessity of utilizing
measures of dispersion after thinking about this example. Students will
note that 50% of Supplier Y's chips have an average life of less than
three years.  This example also illustrates the importance of sampling in
quality control.

(7) Blackout.
        The New York City Transit Authority has just purchased a brand
new type of fluorescent light bulb.  These bulbs have an average life of
5.10 years with a standard deviation of 0.00.  Dr. Klutz, consultant for
the Transit Authority, recommends that on one Saturday evening between
midnight and 5 a.m., when the number of riders is small, all the bulbs
in the subway system should be replaced.  A worker for the system suggest
replacing 5% of the bulbs each Saturday evening for the next twenty weeks
until all the bulbs are replaced.  Dr. Klutz insists that it is cheaper
and more efficient to replace all the bulbs at once.  The Transit
Authority listens to Dr. Klutz.
       
       Question:  Is Dr. Klutz causing a problem for the Transit
Authority?  Answer:  This example illustrates that with a standard
deviation of 0.00, all the bulbs will burn out at about the same time and
the whole subway system will suddenly be plunged into darkness.

Levels of Measurement

(8) Twice as hot?
       Mr. Hale Snowball, meteorologist for a prominent network, is
reporting on the weather.  The temperature in New York City is 60 degrees
Fahrenheit and 120 degrees Fahrenheit in Las Vegas.  Mr. Snowball remarks
that it is twice as hot in Las Vegas as in New York. After the broadcast,
Dr. Statlove informs Mr. Snowball that the fact that 120 degrees is
double 60 degrees does not allow one to say that it is "twice as hot."
Dr.  Snowball's colleague, Dr. Klutz, disagrees and says that it is twice
as hot in Las Vegas than New York City.  
       
       This example could be used to demonstrate the levels of
measurement:  nominal, ordinal, interval, and ratio.  Only with a ratio
scale, i.e., where there is a "true" zero, may one use ratios.

Probability

(9) Too many components? 
       The Prometheus Printer Company manufactures a laser printer made
of 298 individual components.  The warranty on their printers is one year,
and they desire to reduce the failure rate (within the warranty period) of
their printers to about 1%.  Professor Klutz, their new consultant, orders
buyers to purchase components from suppliers that have defect rates of no
more than 1%. Subsequently, the Prometheus Printer Company discovers that
95% of their printers do not work.  Professor Klutz claims that the
suppliers have been dishonest about their defect rates and orders that all
suppliers be fired immediately.
       
       This example can be used to illustrate the idea of joint
probabilities and might be useful as a way of introducing the concept of
the binomial distribution.  The probability of 0 defects with 298 parts
and the probability of a defective is .01, is a binomial and is equal to
5%.  There is a 95% chance that 1 or more of the components are defective.
This example can be used to demonstrate that another way to improve
quality is to reduce the number of components of a product. 

Reliability and Validity

(10) Buying a good scale. 
       Jane Seeless goes to a department store to purchase a scale to keep
tabs on her weight.  The store carries four different brands of scales,
and Jane does not know which scale is more accurate.  A friend suggest
stepping on five or six different scales made by each of the four
companies and recording the weight.  Any brand whose readings are
inconsistent should not be purchased.  Only a brand that provides
consistent readings should be considered.  Dr. Klutz, renowned consultant,
overhears the conversation and disagrees with the friend.  Dr. Klutz
asserts that even a brand that provides consistent readings might be
providing consistently wrong readings.  His suggestion is to simply buy
the cheapest scale.
       
       This example can be used to illustrate the concept of reliability
and validity.  If a measurement is not reliable, it cannot be valid.
However, reliability does not guarantee validity.  The scales that are
not reliable should not be considered under any circumstance. Dr. Klutz
got it almost right this time.

Statistical Inference

(11) Significant difference? 

       A hospital corporation is interested in knowing whether the average
stay for their two maternity units is the same.  They look at the records
of every single maternity patient who stayed at the two hospitals in the
last ten years (both hospitals were opened 10 years previously) and find
that the average stay at Hospital X is 2.48 days (1506 patients) and
2.61 days at Hospital Y (1950 patients).  The director of Hospital Y
claims that this difference is not statistically significant.  Management
is unsure and hires Dr. Klutz.  He performs a t-test on the data and
concludes that the difference is statistically significant.
       
       This example can be used to illustrate that statistical tests
should not be used on a census.  There is no sampling error when a census
is taken and any difference, no matter how small, will be "significant,"
i.e., not due to sampling error.  It is the job of management to determine
whether the observed difference is of sufficient magnitude to warrant
taking action or not.

(12) Cooling off.
        Professor Ima Klutz has performed research demonstrating that
there is a significant inverse correlation between the amount of clothing
people wear and the temperature.  There has been a terrible heatwave for
several weeks.  Professor Klutz has come up with a solution.  He asserts
that since there is a significant and strong negative correlation between
the amount of clothing and temperature, then everyone should go out
wearing their heaviest winter clothing and it will get cooler.
       
       Students are asked to think about this fictional anecdote and
decide why Professor Klutz's research is wrong.  It should not take them
long to realize that a correlation between A and B does not necessarily
prove that A causes B.  It is possible that B causes A, or that C affects
both A and B.  Sometimes correlations can be spurious.  In this case, it
is clear that the weather affects the amount of clothing worn, and not
vice versa.  In the past, Prof.  Klutz has also recommended that farmers
go out with open umbrellas in order to cause it to rain (sigh!).

       In a similar vein, the following story has been circulating. 

              There was once a cholera epidemic in Russia. The
              government, in an effort to stem the disease, sent doctors to
              the worst-affected areas. The peasants of the province of
              S_____ discussed the situation and observed a very high
              correlation between the number of doctors in a given area and
              the incidence of cholera in that area (i.e. more doctors were
              observed in cholera areas than elsewhere). Relying on this
              hard fact, they rose and murdered their doctors. 


       Clearly, the misapplication of statistics can be dangerous to one's
health and life.