Teaching Bits: A Resource for Teachers of Statistics

Journal of Statistics Education v.6, n.3 (1998)

Robert C. delMas
General College
University of Minnesota
333 Appleby Hall
Minneapolis, MN 55455
612-625-2076

William P. Peterson
Department of Mathematics and Computer Science
Middlebury College
Middlebury, VT 05753-6145
802-443-5417

This column features "bits" of information sampled from a variety of sources that may be of interest to teachers of statistics. Bob abstracts information from the literature on teaching and learning statistics, while Bill summarizes articles from the news and other media that may be used with students to provoke discussions or serve as a basis for classroom activities or student projects. We realize that due to limitations in the literature we have access to and time to review, we may overlook some potential articles for this column, and therefore encourage you to send us your reviews and suggestions for abstracts.

From the Literature on Teaching and Learning Statistics

Statistical Education -- Expanding the Network

eds. Lionel Pereira-Mendoza (Chief Editor), Lua Seu Kea, Tang Wee Kee, and Wing-Keung Wong (1998). Proceedings of the Fifth International Conference on Teaching of Statistics, Singapore, June 21-26, 1998.

This three-volume set contains over 200 papers presented by statistics educators from around the world. Each paper falls into one of the following broad categories:

VOLUME 1

Statistical Education at the School Level

Statistical Education at the Post-Secondary Level

Statistical Education for People in the Workplace

Statistical Education and the Wider Society

VOLUME 2

An International Perspective of Statistical Education

Research in Teaching Statistics

The Role of Technology in the Teaching of Statistics

VOLUME 3

Other Determinants and Developments in Statistical Education

Contributed Papers

Posters

The three volume set can be purchased from:

CTMA Ltd., 425 Race Course Road, Singapore 218671
Telephone: (65) 299 8992
FAX: (65) 299 8983

The cost is:
IASE/ISI Member $65 plus shipping/handling
Non-member $80 plus shipping/handling

"Dice and Disease in the Classroom"

by Marilyn Stor and William L. Briggs (1998). The Mathematics Teacher, 91(6), 464-468.

This article presents an interesting mathematical modeling project. The goal of the activity is to model the exponential growth of communicable diseases by adding a risk factor. The required equipment is fairly simple: each student needs a die and a data sheet that is illustrated in the article. To simulate disease transmission, a student walks around the classroom and meets another student. The two students then roll their dice and sum the two outcomes. If the sum is below some predetermined cutoff, the encounter is designated as "risky," meaning that if either student was a carrier, the disease has been passed on to the other student. At the end of the activity, one student is randomly chosen to be the initial carrier of the disease, and the spread of the disease through the classroom is tracked. The article describes how the data are collected, graphed, and analyzed, and also presents suggestions for discussion and variations on the activity.

"Roll the Dice: An Introduction to Probability"

by Andrew Freda (1998). Mathematics: Teaching in the Middle School, 4(2), 8-12.

The author describes a simple dice game that helps students understand why it is important to collect data to test beliefs. The game can also be used to help students develop an understanding of why large samples are more beneficial than smaller samples. To play the game, two students each roll a die. The absolute difference of the two outcomes is computed. Player A wins if the difference is 0, 1, or 2. Player B wins if the difference is 3, 4, or 5. Most students' first impressions are that this is a fair game. Over several rounds of the game the students come to realize that the stated outcomes are not equiprobable. The activity promotes skills in data collection and hypothesis testing, as well as mathematical modeling. The author presents examples of a program written in BASIC and a TI-82 calculator simulation, both of which can be used to help students explore the effects of sample size in modeling the dice game.

The American Statistician: Teacher's Corner

"A One-Semester, Laboratory-Based, Quality-Oriented Statistics Curriculum for Engineering Students" (with discussion)

by Russell R. Barton and Craig A. Nowack (1998). The American Statistician, 52(3), 233-243.

This article describes a new laboratory-based undergraduate engineering statistics course being offered by the Department of Industrial and Manufacturing Engineering at Penn State. The course is intended as a service course for engineering students outside of industrial engineering. We describe the topics covered in each of the eight modules of the course and how the laboratories are linked to the lecture material. We describe how the course is implemented, including facilities used, the course text, grading, student enrollment, and the implementation of continuous quality improvement in the classroom. We conclude with some remarks on the benefits of the laboratories, the effects of CQI in the classroom, and dissemination of course materials.

"The Blind Paper Cutter: Teaching About Variation, Bias, Stability, and Process Control"

by Richard A. Stone (1998). The American Statistician, 52(3), 244-247.

The intention of this article is to provide teachers with a student activity to help reinforce learning about variation, bias, stability, and other statistical quality control concepts. Blind paper cutting is an effective way of generating tangible sequences of a product, which students can use to address many levels of questions. No special apparatus is required.

"Expect the Unexpected from Conditional Expectation"

by Michael A. Proschan and Brett Presnell (1998). The American Statistician, 52(3), 248-252.

Conditioning arguments are often used in statistics. Unfortunately, many of them are incorrect. We show that seemingly logical reasoning can lead to erroneous conclusions because of lack of rigor in dealing with conditional distributions and expectations.

"Some Uses for Distribution-Fitting Software in Teaching Statistics"

by Alan Madgett (1998). The American Statistician, 52(3), 253-256.

Statistics courses now make extensive use of menu-driven, interactive computer software. This article presents some insight as to how a new class of PC-based statistical software, called "distribution fitting" software, can be used in teaching various courses in statistics.

Teaching Statistics

A regular component of the Teaching Bits Department is a list of articles from Teaching Statistics, an international journal based in England. Brief summaries of the articles are included. In addition to these articles, Teaching Statistics features several regular departments that may be of interest, including Computing Corner, Curriculum Matters, Data Bank, Historical Perspective, Practical Activities, Problem Page, Project Parade, Research Report, Book Reviews, and News and Notes.

The Circulation Manager of Teaching Statistics is Peter Holmes, ph@maths.nott.ac.uk, RSS Centre for Statistical Education, University of Nottingham, Nottingham NG7 2RD, England. Teaching Statistics has a website at http://www.maths.nott.ac.uk/rsscse/TS/.

Teaching Statistics, Autumn 1998
Volume 20, Number 3

"Lawn Toss: Producing Data On-the-Fly" by Eric Nordmoe, 66-67.

The author describes an activity based on common lawn tossing games such as horseshoes, flying disc golf, and lawn darts. The activity involves helping students to design and carry out a two-factor experiment. The two factors are Distance to the Target (short or long) and Hand Used for Throwing (left or right). An example of a tabulation sheet for collecting the data is provided, and a simple paired t-test is described for testing whether people tend to have a dominant hand. Suggestions are also provided for how to use the data to illustrate the effects of outliers, conduct a two-sample t-test, explore issues of experimental design, and conduct nonparametric tests.

"Why Stratify?" by Ted Hodgson and John Borkowski, 68-71.

The article describes an activity to help students understand the benefits of using stratified random sampling. The materials consist of equal numbers of red cards and black cards. Each card has a number written on one side. The values on the red cards are consistently smaller than those on the black cards, which creates two strata in the population of cards that differ with respect to the measure of interest. Students draw simple random samples and stratified random samples for samples of the same size from the population of cards and create distributions for the sample means. Students can determine that the average sample mean from both types of samples provide good estimates of the population mean. Comparison of the two distributions illustrates that the sample means from stratified random samples show much less variability.

"Introducing Dot Plots and Stem-and-Leaf Diagrams" by Chris du Feu, 71-73.

The author describes an activity that introduces dot plots and stem-and-leaf diagrams to elementary school students by capitalizing on a basic skill that is well rehearsed: measuring the length of things with a ruler. The list of equipment includes a worksheet, a ruler, a protractor, and a piece of string. An example of the worksheet, which contains lines and angles that can be measured, is provided for photocopying. One line is curved in a complex way so that the piece of string is used to match the curved line, and then a ruler is used to measure the length of the string as an estimate. The measurements produced by the children for a particular line typically show some variation. A dot plot or stem-and-leaf diagram of the measurements can visually demonstrate that there is variability, yet one measurement occurs more often than other measurements. The author has found this activity to spontaneously generate discussion of many statistical ideas.

"A Constructivist Approach to Teaching Permutations and Combinations" by Robert J. Quinn and Lynda R. Wiest, 75-77.

The authors present an approach to teaching about permutations and combinations where students are given an opportunity to explore these concepts within the context of a problem situation. Students are not instructed in the formal mathematics of permutations and combinations. Instead, they are asked to solve a problem regarding how many different ways someone can wallpaper three rooms given there are four different patterns to choose from. Students work in groups to come up with answers to the question. Each group reports back to the class with their answer and provides arguments for why their answer is reasonable. The instructor then identifies which groups have produced answers based on permutations and which have produced answers based on combinations. The formal terminology, symbolic notation, and methods are then introduced and students are shown how the approach taken by a group is modeled by the formal approach. The authors find that this approach empowers students by reinforcing the validity of their own intuitions.

"BUSTLE -- A Bus Simulation" by John Appleby, 77-80.

The author describes a computer simulation that demonstrates to students how statistical modeling can be used to account for the perception that buses always arrive in bunches. The program assumes that the rate at which passengers arrive at a bus stop follows a Poisson process, and that the delay caused as passengers board can account for buses eventually bunching up along a route. The program allows various parameters to be changed such as the initial delay between bus departures from the terminal, the number of buses, and the number of stops along the route. The author provides a web address from which the program can be downloaded and used for educational purposes.

"Testing for Differences Between Two Brands of Cookies" by Rhonda C. Magel, 81-83.

The author describes two activities that involve measurements of two different brands of chocolate chip cookies. In the first activity, students count the number of chips in cookies from samples of each brand. The students use the data to conduct a two-sample t-test, first testing for assumptions of normality. In the second activity, each student provides a rating for the taste of each brand. This provides an opportunity for students to conduct a matched-pairs t-test, as well as an opportunity to see the distinction between the two-sample and matched-pairs situations. The author has found these to be very motivating activities that provide students with a concrete and personal understanding of hypothesis testing.

"A Note on Illustrating the Central Limit Theorem" by Thomas W. Woolley, 89-90.

The article describes the use of the phone book to generate data for illustrating the Central Limit Theorem. In class, students discuss the expected shape of the distribution of the last four digits of a telephone number. It can be argued that if the digits are produced randomly, they should form a uniform distribution of digits between 0 and 9 with an expected value of 4.5 and a standard deviation of approximately 2.872. Outside of class, students randomly select a page of the phone book, randomly select a telephone number from the page, record the last four digits of the telephone number, and compute the average of the four digits. Each student repeats this process until 20 sample means are generated. The sample means from all students are collected in class and entered into statistical software to produce a distribution and obtain summary statistics. The distribution is typically unimodal, normal in its shape, centered around 4.5, with a standard deviation of approximately 1.436. This concrete example allows students to empirically test the Central Limit Theorem.

Topics for Discussion from Current Newspapers and Journals

"Following Benford's Law, or Looking Out for No. 1"

by Malcolm W. Browne. The New York Times, 4 August 1998, F4.

This article is based on "The First Digit Phenomenon" by Theodore P. Hill (American Scientist, July-August 1998, Vol. 86, pp. 358-363). It describes how Hill and other investigators have successfully applied a statistical phenomenon known as Benford's Law to detect problems ranging from fraud in accounting to bugs in computer output. The law is named for Dr. Frank Benford, a physicist at General Electric who identified it using a variety of datasets some sixty years ago. But in fact, as Hill's original article notes, the law had already been discovered in 1881 by the astronomer Simon Newcomb. Newcomb observed that tables of logarithms showed greater wear on the pages for lower leading digits. Most people intuitively expect leading digits to be distributed uniformly, but the log table evidence suggests that everyday calculations involve relatively more numbers with lower leading digits. Newcomb theorized that the chance of leading digit d is given by the base 10 logarithm of (1 + 1/d) for d = 1,2,...,9. Thus the chance of a leading 1 is not one in nine, but rather log(2) = .301 -- nearly one in three!

Benford's law has been observed to hold in a wide range of datasets, including the numbers on the front page of The New York Times, tables of molecular weights of compounds, and random samples from a day's stock quotations. Dr. Mark Nigrini, an accounting consultant now at Southern Methodist University, wrote his Ph.D. dissertation on using Benford's Law to detect tax fraud. He recommended auditing those returns for which the distribution of digits failed to conform to Benford's distribution. In a test on data from Brooklyn, his method correctly flagged all cases in which fraud had previously been admitted. However, Nigrini points out that Benford's Law is not universally applicable. For example, analyses of corporate accounting data often turn up too many 24's, apparently because business travelers have to produce receipts for expenses of $25 or more. He also notes that the law won't help you pick lottery numbers. For example, in a "Pick Four" numbers game, lottery officials take great pains to ensure that the four digits are independently selected and are uniformly distributed on d = 1,2,...,9.

The Times article describes a classroom experiment conducted by Dr. Hill in his classes at Georgia Tech. For homework, Hill asks those students whose mother's maiden name begins with A through L to flip a coin 200 times and record the results and the rest of the students to imagine the outcomes of 200 flips and write them down. As the article points out, the odds are overwhelming that there will be a run of at least six consecutive heads or tails somewhere in a sequence of 200 real tosses. In his class experiment, Hill reports a high rate of success at detecting the imaginary sequences by simply flagging those that fail to contain a run of length six or greater. While this is not an example of Benford's Law, it is another situation in which people are surprised by the true probability distribution. Readers of JSE may be familiar with this coin tossing experiment from the activity "Streaky Behavior: Runs in Binomial Trials" in Activity-Based Statistics by Schaeffer, Gnanadesikan, Watkins, and Witmer (1996, Springer).

"Fate ... or Blind Chance"

by Bruce Martin. The Washington Post, 9 September 1998, H1.

This article on coincidences is excerpted from Martin's article in the September-October issue of The Skeptical Inquirer. The article starts with the famous (to statisticians!) birthday problem, illustrated with birthdays and death days of US presidents. The problem is extended to treat the chance that at least two people in a random sample will have birthdays within one day (i.e., on the same day or on two adjacent days). In this formulation, only 14 people are required to have a better than even chance. The article then mentions some popularly reported examples of coincidences, such as the similarities between the Lincoln and Kennedy assassinations.

Martin observes that, as far as we know, the decimal digits of the number are "random." It follows that we can model coin-tossing by using the expansion for as a random digit generator, interpreting an even digit as "heads" and an odd digit as "tails." Examining the results for the first 100 digits reveals a streak of eight heads in a row, from the 62nd to 69th digits. As noted in the previous article on Benford's Law, this does not show that the digits of are non-random; rather, it is nothing more than the behavior one should expect in a random sequence. Martin's point is that such "coincidences" occur every day in people's lives. It is only our search for patterns that leads us to select some of these events after the fact and treat them as amazing.

In the original Skeptical Inquirer article, Martin discusses the above examples, and also investigates the randomness of prices in the stock market. Again using the digits from the expansion of , he simulates stock prices, and proceeds to find in these random data the kinds of behavior that market analysts consider peculiar to the stock market. Martin's position on all of these examples is that the principle of Occam's Razor should apply: among competing explanations for a phenomenon, the simplest should be preferred. So if chance alone would produce the patterns we perceive, why do we search for more complicated causes?

Deadly Disparities; Americans' Widening Gap in Incomes May be Narrowing our Lifespans"

by James Lardner. The Washington Post, 16 August 1998, C1.

Since the 1970s, virtually all income gains in the US have gone to households in the top 20% of the income distribution -- the greatest inequality observed in any of the world's wealthy nations. Beyond the fairness issues, a growing body of research indicates that countries with more pronounced differences in incomes tend to experience shorter life expectancies and greater risks of chronic illness in all income groups. Moreover, the magnitude of these risks appears to be larger than the more widely publicized health risks associated with cigarettes or high-fat foods.

Richard Wilkinson, an economic historian at Sussex University, found that, among nations with gross domestic products of at least $5000 per capita, one nation could have twice the per capita income of another, yet still have a lower life expectancy. On the other hand, income equality emerged as a reliable predictor of health. This finding ties together a variety of international comparisons. For example, the greatest gains in British civilian life expectancy came during World War I and World War II, periods characterized by compression of incomes. In contrast, over the last ten years in Eastern Europe and the former Soviet Union, small segments of the population have had tremendous income gains, while living conditions for most people have deteriorated. These countries have actually experienced decreases in life expectancy. Among developed nations, the US and Britain today have the largest income disparities and the lowest life expectancies. Japan has a 3.6 year edge over the US in life expectancy (79.8 years vs. 76.2 years) even though it has a lower rate of spending on health care. The difference is roughly equal to the gain the US would experience if heart disease were eliminated as a cause of death!

The July 1998 issue of the American Journal of Public Health presents analogous data comparing US states, cities, and counties. Research directed by John Lynch and George Kaplan of the University of Michigan found that mortality rates are more closely associated with measures of relative, rather than absolute, income. Thus the cities Bixoli, Mississippi; Las Cruces, New Mexico; and Steubenville, Ohio have both high inequality and high mortality. By contrast, Allentown, Pennsylvania; Pittsfield, Massachusetts; and Milwaukee, Wisconsin share low inequality and low mortality.

"Driving While Black; A Statistician Proves that Prejudice Still Rules the Road"

by John Lamberth. The Washington Post, 16 August 1998, C1.

Lamberth is a member of the psychology department of Temple University. In 1993, he was contacted by attorneys whose African-American clients had been arrested on the New Jersey Turnpike for possession of drugs. It turned out that 25 blacks had been arrested over a three-year period on the same portion of the turnpike, but not a single white. The attorneys wanted a statistician's opinion of the trend. Lamberth was a good choice. Over 25 years his research on decision-making had led him to consider issues including jury selection and composition, and application of the death penalty. He was aware that blacks were underrepresented on juries and sentenced to death at greater rates than whites.

In this article, Lamberth describes the process of designing a study to investigate the highway arrest issue. He focused on four sites between Exits 1 and 3 of the Turnpike, covering one of the busiest segments of highway in the country. His first challenge was to define the "population" of the highway, so he could determine how many people traveling the turnpike in a given time period were black. He devised two surveys, one stationary and one "rolling." For the first, observers were located on the side of the road. Their job was to count the number of cars and the race of their occupants during randomly selected three-hour blocks of time over a two-week period. From June 11 to June 24, 1993, his team carried out over 20 recording sessions, counting some 43,000 cars, 13.5% of which had one or more black occupants. For the "rolling survey," a public defender drove at a constant 60 miles per hour (5 miles per hour over the speed limit), counting cars that passed him as violators and cars that he passed as non-violators, noting the race of the drivers. In all, 2096 cars were counted, 98% of which were speeding and therefore subject to being stopped by police. Black drivers made up 15% of these violators.

Lamberth then obtained data from the New Jersey State Police and learned that 35% of drivers stopped on this part of the turnpike were black. He says, "In stark numbers, blacks were 4.85 times as likely to be stopped as were others." He did not obtain data on race of drivers searched after being stopped. However, over a three-year period, 73.2% of those arrested along the turnpike by troopers from the area's Moorestown barracks were black, "making them 16.5 times more likely to be arrested than others." These findings led to a March 1996 ruling by New Jersey Superior Court Judge Robert E. Francis, who ruled that state police were effectively targeting blacks, violating their constitutional rights. Francis suppressed the use of any evidence gathered in the stops.

Lamberth speculates that department drug policy explains police behavior in these situations. Testimony in the Superior Court case revealed that troopers' performance is considered deficient if they do not make enough arrests. Since police training targets minorities as likely drug dealers, the officers had an incentive to stop black drivers. But when Lamberth obtained data from Maryland (similar data has not been available from other states), he found that about 28% of drivers searched in that state have contraband, regardless of their race. Why, then, is there a continued perception that blacks are more likely to carry drugs? It turns out that, of 1000 searches in Maryland, 200 blacks were arrested, compared to only 80 non-blacks. But the problem is that the sample was biased: of those searched, 713 were black, and 287 were non-black.

"Excerpts from Ruling on Planned Use of Statistical Sampling in 2000 Census."