
GraphingFriel and O'Connor (1999) used a previously published data set that examined 37 brands of peanut butter to introduce students to graphing within a realworld context. Following the discussion of the data set, students discussed what characteristics make good peanut butter (e.g., should not be too salty), conducted their own taste test between two brands, and graphed the resultant data using several different methods (e.g., stemandleaf plots, boxplot). The Normal CurveThere are limited published demonstrations devoted to illustrating the normal curve. Existing examples tended to focus on providing students with real life illustrations of the normal distribution. For example, Brooks (2000) had students take a field trip to a nearby beach to examine the dimension of pebbles on the beach. Students recorded measurements for over 800 pebbles, calculated descriptive statistics, and plotted the data using a histogram and boxplot. Brooks noted that the exercise provided students the opportunity to explore a naturally occurring normal distribution and the importance of random sampling, proper graphing, and working together to achieve a common goal. Measures of Central TendencyThere are numerous additional published central tendency exercises designed to capture student's attention and get them involved in the learning process. For example, Tyrrell (2003) described a novel way of illustrating income data—he linked an individual's height to their income level. Tyrrell represented someone making the average income as being almost six feet tall. Someone making the minimum wage was only 9 inches tall as opposed to the richest individuals who stood at over 230 feet tall. Tyrrell used a public United Kingdom income database and images available through Microsoft Excel to create this illustration. He reported that the technique proved useful when he discussed the impact of outliers on income data. Educators have also used sample word counts from novels (Salzinger, 1990), music (Lesser, 2001), opinion questionnaires (Connor, 2003), simulated union strikes (Shatz, 1985), and climate data (Lindquist & Hammel, 1998) to teach the mean, median, and mode. VariabilityThere has been a growing body of research devoted to the role of variation in several different areas such as the graphical representation of data (Meletiou & Lee, 2002), correlation and regression (Nicholson, 1999), teaching chance and data analysis (Watson & Kelly, 2002), repeated samples (Reading & Shaughnessy, 2004), and in the comparison of data sets (BenZvi, 2002; Makar & Confrey, 2004). However, Reading and Shaughnessy lamented that statistics courses overemphasize central tendency and minimize the importance of variability. Indeed, Gould (2004) remarked, "The conceptualization of data as 'signal versus noise'…teaches students that the central tendency, however it's measured, is of primary importance and variability is simply a nuisance. A noisy one at that" (p. 7). Trice, Trice, and Ogden (1990), Gelman and Glickman (2000), and Melton (2004) provided several additional exercises to illustrate variation in the classroom. CorrelationAdditional demonstrations and techniques are available on the topics such as the relationship among Pearson's r and variability, reliability, and validity (Huck, Wright, & Park, 1992; Odom & Morrow, 2006); understanding the coefficient of determination (Barrett, 2000); teaching Pearson's r without reliance on crossproducts (Huck, Ren, & Yang, 2007); and quantifying students' qualitative understanding of the scatterplot (Holmes, 2001). RegressionA similar exercise proposed by Stern (1999) examined the relationship between the number of shoes owned and gender. Yoshiwara and Yoshiwara (2000) invited instructors to use examples in class that incorporated bicycles, birds, bats, and balloons into examples designed to illustrate linear regression and other algebra problems. Chan (2001) proposed that instructors could use the concept of the breakdown point in linear regression to illustrate the influence that extreme data points have on ordinary least square (OLS) regression results. Dietz (1989) encouraged instructors to add a nonparametric alternative (e.g., Theil's estimator of slope and intercept) to simple linear regression when teaching a section or course on nonparametric statistics. She felt that these nonparametric estimators of slope and intercept were "robust, efficient, and easytocalculate alternatives to least squares" (p. 35). Additional materials exist for the teaching of the linear regression with unbalanced (Little, 1982) and clustered (Hedeker, Gibbons, & Flay, 1994) data. Regression to the MeanStatistical educators have also provided examples to illustrate regression to the mean. For example, Cutter (1976) provided a means to illustrate regression to the mean using two sets of dice tosses. Levin (1982) modified Cutter's original demonstration by using two decks of playing cards instead of dice sums to reduce the artificiality of the exercise. Karylowski (1985) offered an additional modification by introducing psychological content to the demonstration by having students test individuals on a personality measure. After determining the cutoffs for the high and low groups (e.g., upper and lower 25%), the instructor had students retest the subjects on the same personality measure. Karylowski noted that the constituency of both extreme groups would change (i.e., regress to the mean) after the second testing session. Watkins (1986) offered a series of trivia questions that instructors could use to illustrate the regression to the mean among college students. Becker and Greene (2005) provided information on how various Nobel Laureates (e.g., Daniel Kahneman, Milton Friedman) have addressed regression to the mean throughout their careers. Computer ApplicationsStockburger (1982) created three computer simulation exercises to illustrate the mean, normal curve, and correlation coefficients. He reported that students who used these exercises were significantly more successful on subsequent assignments than those students that did not utilize the simulations. Educators have also created programs to illustrate nonnormal data sets (Walsh, 1992), correlation (Goldstein & Strube, 1995; Mitchell & Jolley, 1999; Strube, 1991), regression and heteroscedasticity (Bradley, Hemstreet, & Ziegenhagen, 1992), and scatterplots (Goldstein & Strube, 1995; Hassebrock & Snyder, 1997). ReferencesBecker, W. E., & Greene, W. H. (2005). Using the Nobel Laureates in economics to teach quantitative methods. Journal of Economic Education, 36, 261277. BenZvi, D. (2002). Seventh grade students' sense making of data and data representations. Paper presented at the Sixth International Conference on the Teaching of Statistics, South Africa. Retrieved July 31, 2007, from http://www.stat.auckland.ac.nz/~iase/publications/1/2c5_benz.pdf Bowen, R. W. (1992). Graph it!: How to make, read, and interpret graphs. Englewood Cliffs, NJ: PrenticeHall. Brooks, A. (2000). Do the normal lengths of pebbles on Hastings Beach follow a normal distribution? Teaching Statistics, 22, 7376. Chan, W. (2001). Teaching the concept of breakdown point in simple linear regression. International Journal of Mathematical Education in Science and Technology, 32, 745794. Cutter, G. R. (1976). Some examples for teaching regression toward the mean from a sampling viewpoint. The American Statistician, 30, 194197. Dietz, E. J. (1989). Teaching regression in a nonparametric statistics course. The American Statistician, 43, 3540. Friel, S. N., & O'Connor, W. T. (1999). Sticks to the roof of your mouth? Mathematics Teaching in the Middle School, 4, 404411. Gelman, A., & Glickman, M. E. (2000). Some classparticipation demonstrations for introductory probability and statistics. Journal of Educational and Behavioral Statistics, 25, 84100. Gnanadesikan, M., Scheaffer, R. L., Watkins, A. E., & Witmer, J. A. (1997). An activitybased statistics course. Journal of Statistics Education, 5(2). Retrieved July 31, 2007 from, http://www.amstat.org/publications/jse/v5n2/gnanadesikan.html Hawkins, A. (1997). Mythconceptions! In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 114). Voorburg, The Netherlands: International Statistical Institute. Hedeker, D., Gibbons, R. D., & Flay, B. R. (1994). Randomeffects regression models for clustered data with an example from smoking prevention research. Journal of Counseling and Clinical Psychology, 62, 757765. Holmes, P. (2001). Correlation: From picture to formula. Teaching Statistics, 23, 6771. Huck, S. W., Ren, B., & Yang, H. (2007). A new way to teach (or compute) Pearson's r without reliance on crossproducts. Teaching Statistics, 29, 1316. Huck, S. W., Wright, S. P., & Park, S. (1992). Pearson's r and spread: A classroom demonstration. Teaching of Psychology, 19, 4547. Karylowski, J. (1985). Regression to the mean effect: No statistical background required. Teaching of Psychology, 12, 229230. Lesser, L. M. (2001). Musical means: Using songs in teaching statistics. Teaching Statistics, 23, 8185. Levin, J. R. (1982). Modifications of a regressiontothemean demonstration. Teaching of Psychology, 9, 237238. Lindquist, P. S., & Hammel, D. J. (1998). Applying descriptive statistics to teaching the regional classification of climate. Journal of Geography, 97, 7282. Little, R. J. A. (1982). Direct standardization: A tool for teaching linear models for unbalanced data. The American Statistician, 36, 3843. Makar, K., & Confrey, J. (2004). Secondary teachers’ reasoning about comparing two groups. In D. BenZvi & J. Garfield (Eds.), The challenges of developing statistical literacy, reasoning, and thinking (pp. 353373). Dordrecht, The Netherlands: Kluwer Academic Publishers. Marasinghe, M. G., Meeker, W. Q., Cook, D., & Shin, T. (1996). Using graphics and simulation to teach statistical concepts. The American Statistician, 50, 342351. McClain, K. (1999). Reflecting on students' understanding of data. Mathematics Teaching in the Middle School, 4, 374380. Meletiou, M., & Lee, C. (2002). Student understanding of histograms: A stumbling stone to the development of intuitions about variation. Paper presented at the Sixth International Conference on the Teaching of Statistics, South Africa. Retrieved July 31, 2007, from http://www.stat.auckland.ac.nz/~iase/publications/1/10_19_me.pdf Melton, K. I. (2004). Statistical thinking activities: Some simple exercises with powerful lessons. Journal of Statistics Education, 12(2), Retrieved July 31, 2007, from http://www.amstat.org/publications/jse/v12n2/melton.html Moore, D. S. (1990). Uncertainty. In L. A. Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 95137). Washington, DC: National Academy Press. Morita, J. G. (1999). Capture and recapture your students' interest in statistics. Mathematics Teaching in the Middle School, 4, 412418. Nicholson, J. (1999). Understanding the role of variation in correlation and regression. Paper presented at the First International Research Forum on Statistical Reasoning, Thinking, and Literacy, Be'eri, Israel. Odom, L. R., & Morrow, J. R., Jr. (2006). What's this r? A correctional approach to explaining validity, reliability and objectivity coefficients. Measurement in Physical Education and Exercise Science, 10, 137145. Salzinger, K. (1990). On the average… In V. P. Makosky, C. C. Sileo, L. G. Whittemore, C. P. Landry, & M. L. Skutley (Eds.), Activities handbook for the teaching of psychology (Vol. 3, pp. 185186). Washington, DC: American Psychological Association. Singer, J. D., & Willett, J. B. (1990). Improving the teaching of applied statistics: Putting the data back into data analysis. The American Statistician, 44, 223230. Stern, S. E. (1999). The effect of gender on the number of shoes owned: Gathering data for statistical and methodological demonstrations. In L. T. Benjamin, B. F. Nodine, R. M. Ernst, & C. B. Broeker (Eds.), Activities handbook for the teaching of psychology (Vol. 4, pp. 7476). Washington, DC: American Psychological Association. Stockburger, D. W. (1982). Evaluation of three simulation exercises in an introductory statistics course. Contemporary Educational Psychology, 7, 365370. Strube, M. J. (1991). Demonstrating the influence of sample size and reliability on study outcome. Teaching of Psychology, 18, 113115. Trice, A. D., Trice, O. A., & Ogden, E. P. (1990). Teaching the concept of statistical variability. In V. P. Makosky, C. C. Sileo, L. G. Whittemore, C. P. Landry, & M. L. Skutley (Eds.), Activities handbook for the teaching of psychology (Vol. 3, pp. 189191). Washington, DC: American Psychological Association. Walsh, J. F. (1992). A simple procedure for generating nonnormal data sets: A FORTRAN program. Teaching of Psychology, 19, 243244. Watkins, A. E. (1986). The regression effect; or, I always thought that the rich get richer… Mathematics Teacher, 79, 644647. Yoshiwara, B., & Yoshiwara, K. (2000, November). Bicycles, birds, bats and balloons: New applications for algebra classes. Paper presented at the Annual Meeting of the American Mathematical Association of TwoYear Colleges, Chicago, IL.
