Ann E. Watkins

Is Human Height Bimodal

The combined distribution of heights of men and women has become the canonical illustration of bimodality when teaching introductory statistics. But is this example appropriate? This article investigates the conditions under which a mixture of two normal distributions is bimodal. A simple justification is presented that a mixture of equally weighted normal distributions with common standard deviation σ is bimodal if and only if the difference between the means of the distributions is greater than 2σ. More generally, a mixture of two normal distributions with similar variability cannot be bimodal unless their means differ by more than approximately the sum oftheirstandard deviations. Examination of national survey data on young adults shows that the separation between the distributions of mens and womens heights is not wide enough to produce bimodality. We suggest reasons why histograms of height nevertheless often appear bimodal.

An Activity-Based Statistics Course

So that students can acquire a conceptual understanding of basic statistical concepts, the orientation of the introductory statistics course must change from a lecture-and-listen format to one that...

Statistics Teaching in Colleges and Universities: Courses, Instructors, and Degrees in Fall 1995

Abstract The Conference Board of the Mathematical Sciences (CBMS) is made up of 14 professional organizations in the mathematical sciences. The American Statistical Association is one of these organizations. Every five years since 1965 CBMS has conducted an extensive survey of undergraduate programs in the mathematical sciences in the United States. These studies give a detailed picture of enrollment, faculty, and instructional methods in two-year colleges, four-year colleges, and universities. This article presents selected results from the fall 1995 CBMS survey (Loftsgaarden, Rung, and Watkins 1997) about statistics courses, faculty, and degrees in departments of statistics, departments of mathematics or mathematical sciences, and in mathematics programs at two-year colleges.

Advanced Placement Statistics—Past, Present, and Future

Abstract The Advanced Placement Statistics course, first offered in 1997 after more than 12 years of planning, allows high school students who pass an examination to receive college credit for a statistics course taken at their high school. In its third year 25,240 students took this examination, and 57% “passed.” The three-hour exam consists of both multiple choice and free response questions, which cover the four sections of the AP Statistics syllabus: Data exploration, study design, probability distributions through simulation, and inference. Although some problems remain, both students and teachers find the course to be challenging and valuable.

Simulation of the Sampling Distribution of the Mean Can Mislead.

Although the use of simulation to teach the sampling distribution of the mean is meant to provide students with sound conceptual understanding, it may lead them astray. We discuss a misunderstanding that can be introduced or reinforced when students who intuitively understand that “bigger samples are better” conduct a simulation to explore the effect of sample size on the properties of the sampling distribution of the mean. From observing the patterns in a typical series of simulated sampling distributions constructed with increasing sample sizes, students reasonably—but incorrectly—conclude that, as the sample size, n, increases, the mean of the (exact) sampling distribution tends to get closer to the population mean and its variance tends to get closer to ś2/n, where ś2 is the population variance. We show that the patterns students observe are a consequence of the fact that both the variability in the mean and the variability in the variance of simulated sampling distributions constructed from the means of N random samples are inversely related, not only to N, but also to the size of each sample, n. Further, asking students to increase the number of repetitions, N, in the simulation does not change the patterns.

A Suggestion for Sunflower Plots

Abstract Although sunflower plots are highly effective for displaying bivariate data with coincident observations, they possess certain disadvantages involving graphical perception of data. Moreover, sunflower plots for data that arrive “on line” can be updated only by completely redrawing the affected sunflowers. We propose a variation of the traditional sunflower plot that addresses these issues.