Visualizing and Understanding Probability and Statistics: Graphical Simulations Using Excel

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Jul/Aug 2009 by Gordon, Sheldon P, Gordon, Florence S

Abstract:

The authors describe a collection of dynamic interactive simulations for teaching and learning most of the important ideas and techniques of introductory statistics and probability. The modules cover such topics as randomness, simulations of probability experiments such as coin flipping, dice rolling and general binomial experiments, a simulation of the Law of Large Numbers, simulations of the distributions of various sampling distributions (the mean, median, mode, and sample proportions), a simulation of confidence intervals, a simulation of hypothesis testing, and simulations for regression analysis.

Keywords: Statistics, statistics education, simulations, Excel, probability, Central Limit Theorem, sampling distributions, randomness, variability among samples, Law of Large Numbers, regression.

1. INTRODUCTION

Students in introductory statistics courses are often asked to accept many statements and procedures on faith since the mathematical justification may be far too sophisticated for them to comprehend. In large measure, this can be attributed to the nature of statistics, which is quite unlike most other undergraduate mathematics offerings. At most large universities, introductory statistics courses are given by a statistics department, though often by individual departments (such as economics, business, biology, psychology, etc.) that use statistics heavily. At some small colleges, the introductory statistics courses are given by trained statisticians who are often in the mathematics department.

However, at many schools, particularly two-year colleges, introductory statistics is offered by the mathematics department with instructors (often parttime faculty) having little, or no, formal statistical training. According to the 2000 CBMS study [1], over 400,000 students took introductory statistics in a mathematics department; this number represents about four times the number taking such a course through a statistics department. Moreover, the AP Statistics program [2] in the high schools has been growing extremely rapidly; in 2002, about 50,000 students took the AP exam at the end of the course, and this number had been growing at an annual rate of about 25%. As with the college mathematicians who teach the introductory statistics course, most of the high school teachers have had little formal training (though some teacher development workshops) in statistics. It is diese two groups to whom this article is primarily directed.

In matiiematics courses, the underlying theory applies directly to the objective in question, whether it is a function being optimized, a system of linear equations being solved, or a differential equation being solved. In statistics, however, the dieory applies to some unseen underlying population, whereas usually the student has just one sample in hand. In fact, the sample is only used to make a statistical inference about the unknown population, either to estimate a population parameter such as the mean µ or the proportion π or to test a hypothesis about one of these (or some other) parameters.

Unfortunately, students see only the sample, but have no direct way to perceive the population or to develop any deep understanding of its properties. It is therefore not surprising that many students come out of an introductory statistics course having mastered, at best, a series of computational procedures, but with relatively little statistical understanding. Many others have been through an introductory course that focuses heavily on the use of technology, either calculator or software, to perform statistical computations, but that experience alone does not necessarily translate into statistical understanding.

Two key notions that underlie virtually every concept and method in statistics are randomness and variation among samples. Has the sample been collected in a truly random fashion that reflects the underlying population? How representative of that population is this one sample? How does this single sample compare to other possible samples drawn from the same population?

Fortunately, most of the critical topics in probability and statistical inference can be dramatically presented using computer graphics simulations to allow students to visualize the underlying statistical populations and so enhance their understanding of the statistical concepts and methods. Many years ago, the current authors addressed the challenge of making these notions evident to students by developing a comprehensive package of computer graphics simulations using BASIC that addressed virtually every topic in introductory probability and statistics.

Unfortunately, BASIC routines have become outmoded over the years, especially in terms of being able to provide students (or even colleagues) with copies of the files to explore the statistical concepts on their own computers. We have lately returned to this challenge and are developing a comparable package of Dynamic Interactive Graphical simulations of the mathematics (DIGMath) using Excel because it is available on almost all computers today. Moreover, Excel has the further advantages that it does not require any plug-ins or internet connections, as for example, using Flash, Java, or JavaScript, and it is the standard technology tool for virtually every discipline outside of mathematics. (The complete package can be downloaded from the author's website, as discussed at the end of this article.)