INTRODUCTORY STATISTICS: A CONTEMPORARY APPROACHAuthor(s): JOHN D. EMERSONSource: The Mathematics Teacher, Vol. 70, No. 3 (MARCH 1977), pp. 258-261Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960795 .

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INTRODUCTORY STATISTICS: A CONTEMPORARY APPROACH

Three years of high school mathematics can enable students to explore an interesting collection of statistical concepts.

By JOHN D. EMERSON Middlebury College

Middlebury, VT 05753

THE National Council of Teachers of Mathematics has provided leadership in

promoting the effective teaching of statis tics and probability at increasingly early stages of the mathematics curriculum. To this end, the Joint Committee with the

American Statistical Association was estab lished in 1967 to deal with questions on the statistics and probability curriculum. (For a progress report on the work of the Joint

Committee, we refer to an article by Fred erick Mosteller [1970] that appeared in this

journal.) This marriage of efforts of two national mathematics organizations has borne fruit. Among its notable successes is the paperback Statistics: A Guide to the Unknown, edited by Judith M. Tanur, Frederick Mosteller, and others (1972). The Guide consists of a collection of de

scriptive essays by experts in various spe cializations that describe in non mathematical language diverse applications of statistics and probability. The short and

stimulating essays in this book effectively construct a bridge from this important area of mathematics to the outside world.

In recent decades a distinct trend has

developed within the statistical world to ward increased understanding and use of the so-called nonparametric statistical

methods. A nonparametric procedure is a

procedure whose applicability does not re

quire the assumption of a particular para metric population model like the normal model. Such procedures are characterized

by their general applicability (they are "dis

tribution-free") and by the ease with which

they can be applied. In statistical jargon, they are "robust" methods. The need to make nonparametric meth

ods accessible at an elementary level has become increasingly apparent. Both high school and college students who have a rea

sonably good preparation (perhaps three

years) in high school mathematics can learn to use nonparametric methods. In

troductory precalculus texts have been de

signed to fulfill the need for sources that

emphasize nonparametric statistics. Gott fried Noether, the author of one such text

(1971), has written in this journal about the

many inherent advantages of the non

parametric approach in elementary statis tics (1974).

In designing a course for the experimen tal and intensive four-and-one-half week

curriculum of Middlebury College's winter term, I had several objectives in mind:

1. To bridge the gulf, perceived by many students, between mathematics and the "real world."

2. To emphasize nonparametric statistics and to study its methods before beginning a detailed study of the normal-theory meth ods.

3. To stimulate the students, including those whose formal experience in math ematics may have ended after three years of

high school mathematics.

4. To provide a successful experience with real mathematical content for non science majors.

The course was entitled Applicable Sta tistics for Ordinary People. We adopted two

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texts, those of Tanur and Noether. Al

though one-third of the students who en

rolled in the course had had some exposure to calculus, fully two-thirds were majoring in areas outside the physical sciences and

mathematics. In response to a question on an initial survey, many of the students said that they were taking the course in order to

"get into an area where I might never be otherwise exposed," or "to find out what statistics is all about without taking several

college mathematics courses." The mathematical content of the course

was designed around the chapters of Noe ther's text. Since we met together for two hours each day and since the students were

taking only this course in the winter term, we were able to cover most sections in all

chapters of the Noether text. The progres sion of topics included a brief introduction to probability; binomial and normal distri

butions; an introduction to hypothesis test

ing and estimation; the chi-square statistic and its uses; nonparametric methods in

volving one sample, paired comparisons, two independent samples, k independent samples, and Kendall rank correlation; and normal theory methods paralleling those of the nonparametric development. The price extracted by this unconventional selection of topics was the loss of time necessary for more than a very modest treatment of two

large topics?analysis of variance and re

gression. The time available permitted only an exposure to, and qualitative treatment

of, these topics. Throughout the course,

daily problem sets were assigned, and de tailed mimeographed solutions were

distributed in an effort to minimize class time spent going over the homework as

signments.

Perhaps the most novel feature of the course was the way in which it attempted to meet the first of the stated objectives. Along with daily assignments, the students were

asked to read fifteen of what I considered to

be the more interesting essays in the Guide. I selected these assigned readings partly on

the basis of their relevance to the current

topics for class discussion. In the lectures

the statistical methods being introduced were then illustrated with examples de

signed to reflect some aspect of an experi ment the students had read about in the Guide. With this approach, contemporary uses of statistics in our society could be

discussed as an integral and nondisjoint part of the course.

To illustrate the implementation of this

approach, let us consider the use of the chi

square test for homogeneity. We began this

topic after the students had read an article

by Lincoln Moses and Frederick Mosteller on a national study of the relative safety of

four anesthetics. With this as a back

ground, I described a simplified hypothet ical experiment for evaluating the extent of

adverse side effects for each of the four anesthetics and for testing for significant differences in these effects among the anes

thetics. An example such as this one pro vides ample opportunity to discuss the very real difficulties that such an experimental design presents. The adage "Correlation is

not causation" can be readily illustrated in

this context: Is one anesthetic more com

monly administered to patients who are

critically ill or who are older and thus per

haps more prone to side effects? Were there differences among hospitals that were not

explainable by random data fluctuation? Are occasional individuals sensitive to

some anesthetics? How is the severity of side effects to be determined? For precisely what sort of experiment and with what as

sumptions is a chi-square test for

homogeneity valid? Indeed, such questions must be carefully considered, and the ar

ticle itself focuses well on some of these issues.

As a second illustration, consider the

Wilcoxon-Mann-Whitney method for test

ing hypotheses about differences between

independent, randomly selected groups. At

this point, the students had read an essay

by Frank A. Haight about controlled ex

periments to test the effects of speed limits on traffic safety. It was then natural to con

struct a hypothetical experiment involving many large cities and to choose some cities

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in which to conduct a media campaign for traffic safety. Other cities without such a

campaign comprised a control group. The data consist of the numbers of reported accidents following the advertising campaign, and they are used to test the effectiveness of the safety program.

Are such examples too contrived? To be sure, they are contrived, but this can ac

tually work to the instructor's advantage. The instructor should carefully point out the assumptions that are required to vali date the choice of a particular statistical

procedure. The real-world difficulties that are obstacles to these idealized assumptions must be emphasized. However, one of the

strengths of many essays in the Guide is the care with which such difficulties are dis cussed. Such matters are all too frequently overlooked in traditional introductory courses. The primarily nonparametric ap proach does aid in making such difficulties

surmountable, since it needs no assump tions concerning the types of distributions that underlie the variables of a given experi ment. For this reason the first two goals stated earlier were quite compatible and even complementary.

Were the objectives of the course achieved? I think that they were, for most students. They found it satisfying to see the relevance of the mathematics they were

learning to such familiar aspects of their world as TV election reports, Gallup polls, jury selection biases, product quality con

trol, smoking and health, and the consumer

price index. Still, the primary focus of the course was on statistics, and I believe that this approach to the subject was more pala table for some of the less quantitatively oriented students and more interesting for the mathematically inclined students.

I do wish to recommend the adoption of a substantial project toward the end of the course to provide a culminating experience. There is no adequate substitute for formu

lating pertinent questions, designing an ex

periment, collecting and sorting one's own

data, and finally answering the questions using appropriate statistical techniques. If

time and opportunity permit, the collection of real data in the local community can

provide the student with the ideal learning experience. For a discussion of some rather novel approaches to the design of experi ments and the collection and analysis of

data, see the article by Joiner and Campbell that appeared in the May 1975 issue of this

journal. Alternatively, source materials are available that give students an opportunity to collect data in a controlled environment within the classroom. One source that has

recently been made available, called STAT LAB (1975), enables the student to select data randomly from an actual census popu lation using a pair of dice. The use of

Monte Carlo techniques provides still an other approach to the problem of data gen eration. See articles on the subject by Tanis

(1973) and by Simon and Holmes (1969), both of which appeared in the Mathematics Teacher.

Teachers designing a course along the lines of the one I have described can find a

variety of texts around which to structure the course. Precalculus texts that use the

nonparametric approach to statistics are

becoming increasingly available. In addi tion to Noether's fine text, there is a text by Kraft and van Eeden (1968) that is slightly more advanced. The forthcoming text by Nemenyi, Dixon, and White may be partic ularly appropriate for the high school teacher. The text by Carlson (1973) pre supposes only high school algebra and is

more conventional in its approach to statis tics. It, too, includes an introduction to

nonparametric statistics.

Finally, there are sources that, like the

Guide, convey to the reader an appreciation of statistics without presenting the techni cal aspects of the subject. For two decades, Darrell Huff's classic How to Lie with Sta tistics (1954) has informed and entertained even the most mathematically na?ve reader.

Walter Federer's Statistics and Society (1973) is written in a more technical vein but is designed as the basis for a "liberal arts" course in statistics. Each of these texts, in its individual way, is designed to

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convey an understanding of the use and relevance of statistics in modern society.

In writing this paper, I have suggested an

approach for a statistics course at the in

troductory college level. Since it uses pri marily nonparametric procedures that have no calculus prerequisite and that are ana

lytically straightforward, the approach could be just as appropriate for high school students who are well prepared in algebra. In my view, it would then become necessary to devote some class time to the consid eration of homework problems instead of

relying on prepared solutions. At either

level, the course provides one resolution of the conflict between proponents and oppo nents of the "math must be relevant" phi losophy. One can teach courses that are

primarily mathematical in content and yet demonstrate convincingly the relevance of that mathematics to our complex society.

REFERENCES

Bedford, Crayton W. "Ski Judge Bias." Mathematics Teacher 65 (May 1972):397-400.

Booth, Ada. "Two-thirds of the Most Successful. . . ." Mathematics Teacher 66 (November 1973):593-97.

Carlson, Roger. Statistics. San Francisco: H olden

Day, 1973.

F?d?rer, Walter T. Statistics and Society. New York: Marcel Dekker, 1973.

Hodges, J. L., David Krech, and Richard S. Crutchfield. STA TLAB: An Empirical Introduction to Statistics. New York: McGraw-Hill Book Co., 1975.

Joiner, Brian L., and Cathy Campbell. "Some Inter

esting Examples for Teaching Statistics." Math ematics Teacher 68 (May 1975):364-69.

Kraft, Charles H., and Constance van Eeden. A Non

parametric Introduction to Statistics. New York: Macmillan Co., 1968.

Mosteller, Frederick. "Progress Report of the Joint Committee of the American Statistical Association and the National Council of Teachers of Mathemat ics." Mathematics Teacher 63 (March 1970): 199-208.

Mosteller, Frederick, Judith M. Tanur, William H.

Kruskal, Richard F. Link, Richard S. Pieters, and Gerald R. Rising, eds. Statistics by Example: Ex

ploring Data, Weighing Chances, Detecting Patterns, and Finding Models. 4 vols. Reading, Mass.: Ad

dison-Wesley Publishing Co., 1973.

Nemenyi, Peter, Sylvia K. Dixon, and Nathaniel B.

White, Jr., Statistics from Scratch. San Francisco:

Holden-Day, forthcoming.

Noether, Gottfried E. Introduction to Statistics, a

Fresh Approach. Boston: Houghton Mifflin Co., 1971.

-. "The Nonparametric Approach in Elementary Statistics." Mathematics Teacher 67 (February 1974): 123-25.

Simon, Julia L., and Allen Holmes. "A New Way to Teach Probability Statistics." Mathematics Teacher 62 (April 1969):283-88.

Tanis, Elliot A. "A Statistical Hypothesis Test for the Classroom." Mathematics Teacher 66 (November" 1973): 657-58.

Tanur, Judith M., Frederick Mosteller, William H.

Kruskal, Richard F. Link, Richard S. Pieters, and Gerald R. Rising, eds. Statistics: A Guide to the Unknown. San Francisco: Holden-Day, 1972.

St. John's MEETING

ome and join us in another world next door. The site of the 1977 Canadian regional NCTM

name-of-site meeting is St. John's, Newfoundland, Canada's youngest province. Eric

MacPherson will be the opening speaker on Wednesday evening. A wide variety of eminent

mathematics educators will highlight the three-day program. Speakers and workshop leaders include

John C. Egsgard (president of NCTM), Bob Eicholz, Frank Ebos, John Del Grande, E. Glenadine

Gibb, Carole Greenes, Jack Lesage, Evan Maletsky, Doyal Nelson, Henry Pollack, Fernand Pr?vost,

Gerald Rising, Tom Romberg, Joan Routledge, Harry Ruderman, George Immerzeel, Peter

Weygang, Dora Whittaker and many others. Collectively, these people represent a tremendous variety

in background, expertise, and teaching experience. The Newfoundland Mathematics Council invites you to attend this conference. We promise a

varied program for your professional interests and a worthwhile personal experience in visiting a

unique and friendly province. Come and enjoy a less hectic pace of life.

March 1977

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