Journal of Statistics Education v.5, n.2 (1997)
Copyright (c) 1997 by Karla Ballman, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.
Key Words: Characteristics of random variation; Probability; Statistics education.
Many introductory courses teach traditional probability concepts. The objectives of these courses may be better met by emphasizing characteristics of random variation rather than formal probability. To illustrate a different approach, some alternative concepts and related activities are described and discussed.
1 Statistics educators recommend that introductory statistics courses place greater emphasis on data collection, understanding and modeling variation, graphical display of data, design of experiments and surveys, problem solving, and process improvements, and less emphasis on mathematical and probabilistic concepts (Easton, Roberts, and Tiao 1988; Hogg 1991, 1992; Moore 1992; Snee 1990). As a positive consequence, many textbooks and courses now begin with data analysis rather than formal inference based on probability theory. However, these texts and courses still teach formal probability: probabilistic computation, conditional probability, random variables, probability distributions, and expectation. This approach usually fails to help the student develop an understanding of variation and its relevance in statistics.
2 Moore (1992) suggests that too much probability is taught in introductory statistics courses and that only the necessary probabilistic concepts required to further statistical thinking should be taught. He maintains that for statistical reasoning, the student must recognize the omnipresence of variation and learn how this variation is quantified and explained (Moore 1990). A study by Konold (1995) indicates that instruction on traditional probability topics fails to provide students with the intuition and concepts they need to master statistical reasoning.
3 If the primary goal of an introductory course is to promote an understanding of statistical concepts and reasoning, I suggest that a way to achieve this is to replace traditional probability with topics and activities that develop a sound intuition for the characteristics of random variation and its role in statistics.
4 Because most textbooks teach formal probability, I outline relevant topics and activities that I teach to provide an example of a course that emphasizes characteristics of random variation. I do not propose that this is the optimal set of topics and activities, but rather wish to prompt others to create their own. Although some of the activities I describe take place in a computer classroom, they can easily be modified for use without computers.
Concept of Probability
5 (Two class periods) Students are introduced to the relative frequency definition of probability because it is intuitive and reappears later in discussions of sampling distributions and statistical inference. In addition, students learn notation such as P(heads) and a few basic conventions of probability: probabilities are greater than or equal to zero and less than or equal to one, the probabilities of all possible outcomes sum to one, and the combined probability of two distinct outcomes on a single random trial is equal to the sum of their individual probabilities.
6 Activity 1: Students are asked to define probability as part of a class discussion. Since this is done before the students have had any assigned reading, the various definitions that surface are representative of the different interpretations of probability. However, the most common definition is usually some form of relative frequency. Next, relative frequency is demonstrated via physical and computer simulations. Class ends with a discussion of two issues: (1) whether it is reasonable to apply a relative frequency interpretation of probability to a single trial, and (2) characteristics of situations best-suited for the relative frequency interpretation of probability.
7 Activity 2: (Computer classroom) Class begins with a five-minute lecture on probability notation. Next, working in pairs at a computer, students complete simulation exercises that lead them to discover the basic facts of probability along with some of their immediate implications. The period ends with a summary and discussion of their discoveries.
Variation of a Single Random Outcome
8 (One class period) Two aspects of the variation in a single random outcome are emphasized. First, not all random outcomes are equally likely. Although this seems obvious, there are a surprisingly large number of students who believe otherwise. Second, the outcome of a random trial cannot be predicted with certainty, even when an outcome has a relatively high probability of occurring. This generates a discussion of many related issues. Why is it reasonable to predict the most likely outcome for the next trial? What does it mean if the most likely outcome does not occur on the next trial? Is the outcome of one trial adequate for assessing the accuracy of a probability assignment? If not, what are some reasonable methods for obtaining sufficient evidence in support of or against a particular probability assessment?
9 Activity: The class begins with the investigation of two random experiments: a fair coin flip and a random draw from a box containing six white and two black balls. Students are prompted to describe the outcomes, assign probabilities to the outcomes, and make a prediction for the outcome of the next trial for each experiment. Students are also asked to describe similarities and differences between the two experiments. Their answers are discussed. Next, each student flips a fair coin and randomly draws a ball and is asked to evaluate the quality of her/his original probability assignments in light of the actual outcomes. After some discussion, the class concludes that one outcome is not adequate to judge the quality of a probability estimate. To reinforce the concepts, I produce a brown paper bag and claim that the proportion of white balls is the same as that in the box. As a class, the students estimate the probability of randomly drawing a white ball and predict the next outcome using the available information. A ball is randomly drawn and the students are asked to provide two plausible scenarios consistent with the outcome. For example, if a white ball is drawn there are two possibilities: the hypothesis P(W) = .75 is true and the most likely outcome occurred or the hypothesis P(W) = .75 is not true and a white ball was drawn. Based on a single trial, it is not possible to decide which is correct. Students are asked how they might test my claim using only sampling with replacement. This generates a lively discussion of when there is enough evidence one way or the other. As a final lesson relevant to statistics, I do not reveal the contents of the bag. The class ends with small group reactions to the accuracy of weather forecasts and an article on earthquake predictions.
Long-Term Variation / Large Sample Variation
10 (One class period) After learning it is not possible to predict the outcome of a random trial with certainty, students learn that random phenomena display regularity in the long-run. In addition, their innate belief that larger (random) samples produce better estimates is reinforced. Unfortunately, most students have serious misconceptions regarding the more subtle implications of long-term variation. Two misconceptions I try to correct are (1) larger samples (more trials) always produce a better estimate, and (2) a long streak of one outcome must be "balanced" with a similar streak of the other.
11 Activity: (Computer classroom) Students work in groups at a computer. They are asked to predict the appearance of two plots: a plot of the proportion of heads versus the number of flips in a large number of flips of a fair coin, and a plot of the proportion of sixes versus the number of rolls in a large number of rolls of a fair die. They then perform simulations and compare the actual plots with their predictions. Next, groups report their proportions of heads in 500 and 1000 trials. These are recorded on the board and discussed. The class learns that, generally, the larger the sample size, the better the estimate. However, this is not guaranteed to be true. During the last segment of the class, groups investigate and prepare a written report on the validity of the claim that "A run of heads must be balanced by a run of tails of equal length." As part of their investigation, I suggest that they analyze two specific items: (1) the proportion of heads in 1000 flips of a fair coin when the first ten flips are tails and the next 990 flips are alternating heads and tails, and (2) the runs present in a large number of simulated flips of a fair coin.
Short-Term Variation / Small Sample Variation
12 (One class period) Commonly, students presume that small samples are more representative of the population than they really are. This is the result of applying characteristics of large-sample variation to small samples. The goal is to make students aware that often observed patterns in small samples are simply the product of random variation. In particular, students should understand that there are two plausible explanations for an observed pattern: either it is the result of chance variation, or it is indicative of some other influence. Understanding small sample variation plays a crucial role in both data analysis and inference.
13 Activity: In the first segment of class, students again examine the behavior of flips of a fair coin. Each student is asked to (1) generate a hypothetical sequence of typical outcomes for ten flips, and (2) record the outcomes of ten actual flips. Both the hypothetical and actual sequences are recorded on the board. Groups then compare the characteristics of the two types of sequences, paying particular attention to the lengths of the longest runs and the proportion of heads, and report their conclusions. Finally, the groups react to media articles that attribute patterns in small samples to some cause other than chance. The groups are asked to propose two plausible explanations for a pattern and to indicate which of the two is more believable and why. For one article, groups are required to support their observations with the results of an appropriate simulation.
Independent Events and Mutually Exclusive Events
14 (One class period) The concepts of independent and mutually exclusive events play important roles in statistical reasoning. To understand why random sampling is so important, students need to understand the characteristics of independent events. An understanding of mutually exclusive events is needed for probability statements accompanying two-tailed tests. As a secondary objective, students learn to calculate probabilities of conjunctive, independent events, and disjunctive, mutually exclusive events.
15 Activity: The class begins with a 15-minute lecture on the definitions of mutually exclusive events and independent events. Groups then work on exercises that provide practice at identifying events that are mutually exclusive, independent, or neither. Some of these exercises are typical textbook problems, some are drawn from the media, and some require groups to create their own events.
Binomial Distribution and Normal Distribution
16 (One class period) Many inference techniques assume either a binomial or a normal distribution. The students learn about some general properties of these distributions: their parameters, location (mean), spread (standard deviation), and shape. Next, they discover the effects of changing the parameters. The emphasis for the binomial distribution is on recognizing a binomial setting. Students learn to calculate normal probabilities using both the computer and a table. Although students are not taught about random variables and probability distributions, they seem quite comfortable with the concepts outlined here. In particular, they do not have difficulty understanding that a probability associated with a normal distribution corresponds to an area under its curve.
17 Activity: (Computer classroom) Students explore the different distributions, in particular, how their shapes change by varying their parameters. In groups, they practice identifying a binomial setting in real applications, and they compute probabilities for a normal distribution. A concluding exercise leads students to discover the 68-95-99.7 rule.
Later Segments of the Course
18 There are several major probability topics missing from this course that may trouble some instructors. One is conditional probability. Conditional probability may be used to explain a p-value. However, in the standard non-Bayesian interpretation of a p-value, conditional probability is basically irrelevant. In fact, casting the p-value as a conditional probability tends to confuse students. Overall, I am more successful when I stress that the p-value is dependent on initial assumptions. Specifically, I state that the p-value is the probability of observing the statistic value obtained from the data assuming that the null hypothesis is true. With this emphasis and the previous discussion of plausible explanations for patterns in data, students understand that a small p-value could mean one of two things: the initial assumptions are true and an unlikely event occurred by chance, or the initial assumptions are incorrect.
19 Two other notable omissions are random variables and expectations. These concepts play a major role in sampling distributions, a difficult concept to teach. However, I achieve greater success now using simulation to demonstrate and explain sampling distributions than I did when I explained the concept in terms of random variables, expectation, and the central limit theorem.
20 Based on previous research (Fischbein and Gazit 1982; Garfield and Ahlgren 1988; Green 1983; Konold 1995; Pfannkuch and Brown 1996; Shaughnessy 1993; Tversky and Kahneman 1982), I teach topics in a manner that challenges the student's conception of randomness directly. Typically, a purely random setting is presented to the student, who is asked to predict some aspect of its behavior in writing, to simulate many trials, and to compare the simulation results to the original prediction. The simulation phase generally starts with physical simulation, e.g., the student actually flips a coin, and then moves onto the computer. In addition, most activities present the student with a realistic situation containing random characteristics similar to those under study. This step is essential because even when students understand characteristics of randomness in a purely random setting, they often cannot identify them in an authentic setting. Furthermore, using real studies illustrates how statisticians use probabilistic reasoning in their investigations.
21 I am not aware of an introductory textbook that covers the topics discussed here. Most popular introductory texts teach traditional probability concepts. Some resources I have found useful when generating class activities and accompanying notes include the Chance Database (http://www.geom.umn.edu/docs/snell/chance/welcome.html), items accessible from the UCLA home page -- especially the case studies and Xlisp-Stat archive (http://www.stat.ucla.edu/), the Data and Story Library (http://lib.stat.cms.edu/DASL/), and material found in the Journal of Statistics Education (http://www.stat.ncsu.edu/info/jse/homepage.html). In addition, there are two new books that instructors may find useful: Workshop Statistics: Discovery with Data by Rossman (1996) (for more information see http://stats.dickinson.edu/math/rossman/wshome.html) and Activity-Based Statistics: Instructor Resources by Scheaffer, Gnanadesikan, Watkins, and Witmer (1996). Finally, I have developed some activities that I would be happy to share with interested individuals.
22 My goal is to encourage instructors of introductory statistics courses to evaluate the role of probability in light of their course objectives. There are several reasons that instructors should consider placing more emphasis on characteristics of random variation. First, most formal probability concepts are not explicitly needed or used later in many courses. Second, probability is a difficult subject. To do it justice and to ensure that students learn it, much more time is required than is typically spent in most courses. A better place for probability is in a subsequent course. Most importantly, students taught traditional probability topics fail to develop a sound intuition for variation. Arguably, understanding variation is more important than learning the rules of probability in light of the objectives of most introductory courses and textbooks.
23 Although I have created a list of topics that serve the objectives of most introductory courses better than traditional probability, I am not suggesting that it is the optimal list. The ideal set of topics depends on the specific course and instructor. I offer my own thoughts on course content and activities as a means to motivate others to develop their own.
I would like to thank the individuals of the Statistics Department at The University of Auckland for providing me with administrative support and a congenial and intellectually stimulating atmosphere during my sabbatical. In addition, I would like to thank the anonymous referees for their useful and insightful comments.
Easton, G., Roberts, H. V., and Tiao, G. C. (1988), "Making Statistics More Effective in Schools of Business," Journal of Business and Economic Statistics, 6, 247-260.
Fischbein, E., and Gazit, A. (1982), "Does the Teaching of Probability Improve Probabilistic Intuitions?" Proceedings of the First International Conference on Teaching Statistics, eds. D. R. Grey, P. Holmes, V. Barnett, and G. M. Constable, pp. 738-752.
Garfield, J., and Ahlgren, A. (1988), "Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research," Journal for Research in Mathematics Education, 19(1), 44-63.
Green, D. R. (1983), "School Pupils' Probability Concepts," Teaching Statistics, 5, 34-42.
Hogg, R. V. (1991), "Statistical Education: Improvements Are Badly Needed," The American Statistician, 45, 342-343.
----- (1992), "Report of Workshop on Statistical Education," in Heeding the Call for Change, ed. Lynn Steen, MAA Notes No. 22, Washington: Mathematical Association of American, pp. 34-43.
Konold, C. (1995), "Issues in Assessing Conceptual Understanding in Probability and Statistics," Journal of Statistics Education [Online], 3(1). (http://jse.amstat.org/v3n1/konold.html)
Moore, D. S. (1990), "Uncertainty," in On the Shoulders of Giants: New Approaches to Numeracy, ed. L. A. Steen, Washington: National Academy Press, pp. 95-137.
----- (1992), "Teaching Statistics as a Respectable Subject," in Statistics for the Twenty-First Century, eds. Florence Gordon and Sheldon Gordon, MAA Notes No. 26, Washington: Mathematical Association of America, pp. 14-25.
Pfannkuch, M. and Brown, C. M. (1996), "Building on and Challenging Students' Intuitions About Probability: Can We Improve Undergraduate Learning?" Journal of Statistics Education [Online], 4(1). (http://jse.amstat.org/v4n1/pfannkuch.html)
Rossman, A. J. (1996), Workshop Statistics: Discovery with Data, New York: Springer-Verlag.
Scheaffer, R. L., Gnanadesikan, M., Watkins, A., and Witmer, J. A. (1996), Activity-Based Statistics: Instructor Resources, New York: Springer-Verlag.
Shaughnessy, J. M. (1993), "Probability and Statistics," The Mathematics Teacher, 86(3), 244-248.
Snee, R. D. (1990), "Statistical Thinking and Its Contribution to Total Quality," The American Statistician, 44, 116-121.
Tversky, A., and Kahneman, D. (1982), "Judgment Under Uncertainty: Heuristics and Biases," in Judgment Under Uncertainty: Heuristics and Biases, eds. D. Kahneman, P. Solvic, and A. Tversky, New York: Cambridge University Press, pp. 3-20 (originally published in Science (1974), 185, 1124-1131).
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