Carine A. Bellera, Marilyse Julien, and James A. Hanley
Normal Approximations to the Distributions of the Wilcoxon Statistics: Accurate to What N? Graphical Insights

The Wilcoxon statistics are usually taught as nonparametric alternatives for the 1- and 2- sample Student-t statistics in situations where the data appear to arise from non-normal distributions, or where sample sizes are so small that we cannot check whether they do. In the past, critical values, based on exact tail areas, were presented in tables, often laid out in a way that saves space but makes them confusing to look up. Recently, a number of textbooks have bypassed the tables altogether, and suggested using normal approximations to these distributions, but these texts are inconsistent as to the sample size n at which the standard normal distribution becomes more accurate as an approximation. In the context of non-normal data, students can find the use of this approximation confusing. This is unfortunate given that the reasoning behind—and even the derivation of—the exact distributions can be so easy to teach but also help students understand the logic behind rank tests. This note describes a heuristic approach to the Wilcoxon statistics. Going back to first principles, we represent graphically their exact distributions. To our knowledge (and surprise) these pictorial representations have not been shown earlier. These plots illustrate very well the approximate normality of the statistics with increasing sample sizes, and importantly, their remarkably fast convergence.

Key Words: Graphics; Mann-Whitney U statistic; Nonparametrics; Normal approximation; Ranking methods; Sampling distribution; Wilcoxon rank-sum statistic; Wilcoxon signed-rank statistic.

Linda L. Cooper and Felice S. Shore
The Effects of Data and Graph Type on Concepts and Visualizations of Variability

Recognizing and interpreting variability in data lies at the heart of statistical reasoning. Since graphical displays should facilitate communication about data, statistical literacy should include an understanding of how variability in data can be gleaned from a graph. This paper identifies several types of graphs that students typically encounter-histograms, distribution bar graphs, and value bar charts. These graphs all share the superficial similarity of employing bars, and yet the methods to perceive variability in the data differ dramatically. We provide comparisons within each graph type for the purpose of developing insight into what variability means and how it is evident within the data's associated graph. We introduce graphical aids to visualize variability for histograms and value bar charts, which could easily be tied to numerical estimates of variability.

Key Words: Descriptive statistics; Histograms; Bar graphs; Value bar charts; Time plots; Misconceptions.

Sean Duffy
Random Numbers Demonstrate the Frequency of Type I Errors: Three Spreadsheets for Class Instruction

This paper describes three spreadsheet exercises demonstrating the nature and frequency of type I errors using random number generation. The exercises are designed specifically to address issues related to testing multiple relations using correlation (Demonstration I), t tests varying in sample size (Demonstration II) and multiple comparisons using analysis of variance (Demonstration III). These demonstrations highlight the purpose and application of hypothesis testing and teach students the dangers of data dredging and a posteriori hypothesis generation.

Key Words: Power, Simulation, Correlation, P-value, Hypothesis testing.

Felicity B. Enders, Sarah Jenkins, and Verna Hoverman
Calibrated Peer Review for Interpreting Linear Regression Parameters: Results from a Graduate Course

Biostatistics is traditionally a difficult subject for students to learn. While the mathematical aspects are challenging, it can also be demanding for students to learn the exact language to use to correctly interpret statistical results. In particular, correctly interpreting the parameters from linear regression is both a vital tool and a potentially taxing topic. We have developed a Calibrated Peer Review (CPR) module to aid in learning the intricacies of correct interpretation for continuous, binary, and categorical predictors. Student results in interpreting regression parameters for a continuous predictor on midterm exams were compared between students who had used CPR and historical controls from the prior course offering. The risk of mistakenly interpreting a regression parameter was 6.2 times greater before the introduction of the CPR module (p=0.04). We also assessed when learning took place for a specific item for three students of differing capabilities at the start of the assignment. All three demonstrated achievement of the goal of this assignment; that they learn to correctly evaluate their written work to identify mistakes, though one did so without understanding the concept. For each student, we were able to qualitatively identify a time during their CPR assignment in which they demonstrated this understanding.

Key Words: Statistics education; Writing assignment; Interpreting regression coefficients.

Martin Griffiths
Maximizing a probability: A student workshop on an application of continuous distributions

For many students meeting, say, the gamma distribution for the first time, it may well turn out to be a rather fruitless encounter unless they are immediately able to see an application of this probability model to some real-life situation. With this in mind, we pose here an appealing problem that can be used as the basis for a workshop activity introducing, and subsequently encouraging the exploration of, many of the well-known continuous distributions in a meaningful way. We provide suggestions as to how the session might be run, discuss any pedagogical issues that arise and highlight particularly interesting features of the distributions.

Key Words: Continuous distribution; Continuous random variable; Probability density function (pdf); Probability model.

Jennifer Kaplan, Diane G. Fisher, and Neal T. Rogness
Lexical Ambiguity in Statistics: How students use and define the words: association, average, confidence, random and spread

Language plays a crucial role in the classroom. The use of specialized language in a domain can cause a subject to seem more difficult to students than it actually is. When words that are part of everyday English are used differently in a domain, these words are said to have lexical ambiguity. Studies in other fields, such as mathematics and chemistry education, suggest that in order to help students learn vocabulary instructors should exploit the lexical ambiguity of the words. The study presented here is the second in a sequence of studies designed to understand the effects of and develop techniques for exploiting lexical ambiguities in statistics classrooms. In particular, this paper looks at five statistical terms and the meanings of these terms most commonly expressed by students at the end of an undergraduate statistics course.

Key Words: Statistics education, Lexical ambiguity, Language, Word usage.

Steven D. LeMire
An Argument Framework for the Application of Null Hypothesis Statistical Testing in Support of Research

This paper proposes an argument framework for the teaching of null hypothesis statistical testing and its application in support of research. Elements of the Toulmin (1958) model of argument are used to illustrate the use of p values and Type I and Type II error rates in support of claims about statistical parameters and subject matter research constructs. By viewing the application of null hypothesis statistical testing within this framework, the language and intent of statistical support for research can be more precisely understood and taught.

Key Words: Null hypothesis; Statistical testing; Type I error rate; Type II error rate; P value; Argument; Framework; Teaching statistics.

Josh Tabor
Investigating the Investigative Task: Testing for Skewness. An Investigation of Different Test Statistics and their Power to Detect Skewness

On the 2009 AP© Statistics Exam, students were asked to create a statistic to measure skewness in a distribution. This paper explores several of the most popular student responses and evaluates which statistic performs best when sampling from various skewed populations.

Key Words: Measures of skewness; Estimating power; Simulation.

Aaron Weinberg, Emilie Wiesner, and Thomas J. Pfaff
Using Informal Inferential Reasoning to Develop Formal Concepts: Analyzing an Activity

Inferential reasoning is a central component of statistics. Researchers have suggested that students should develop an informal understanding of the ideas that underlie inference before learning the concepts formally. This paper presents a hands-on activity that is designed to help students in an introductory statistics course draw informal inferences about a bag of bingo chips and connect these ideas to the formal T-test and confidence interval. This activity is analyzed using a framework and recommendations drawn from the research literature.

Key Words: Confidence intervals; Hypothesis testing; Informal reasoning; Statistical inference.

Amanda S. Williams
Statistics Anxiety and Instructor Immediacy

The purpose of this study was to investigate the relationship between instructor immediacy and statistics anxiety. It was predicted that students receiving immediacy would report lower levels of statistics anxiety. Using a pretest-posttest-control group design, immediacy was measured using the Instructor Immediacy scale. Statistics anxiety was measured using the Statistics Anxiety Rating Scale (STARS).

Results indicated that instructor immediacy is significantly related to six factors of statistics anxiety, with immediacy explaining between 6% and 20% of the variance in students' anxiety levels. Instructors should attempt to increase their use of immediacy behaviors in order to decrease anxiety.

Key Words: Anxiety; Psychological support; Graduate student anxiety.

Audbjorg Bjornsdottir and Joan Garfield
Teaching Bits: Statistics Education Articles from 2009 and 2010

We located 34 articles that have been published from November 2009 through June 2010 that pertained to statistics education. In this column, we highlight a few of these articles that represent a variety of different journals that include statistics education in their focus. We also provide information about the journal and a link to their website so that abstracts of additional articles may be accessed and viewed.

Michelle Everson and Ellen Gundlach
Teaching Bits: What's New with CAUSEweb and MERLOT?

In this column, we would like to highlight some recent new additions to CAUSEweb and MERLOT.

Deborah J. Rumsey
Teaching Bits: Random Thoughts on Teaching. Rejoinder to "Let's Just Eliminate the Variance"

In the letter "Keep Teaching the Variance" the author makes some excellent counterpoints regarding my "Random Thoughts on Teaching" column in the November 2009 issue. The title of that column was "Let's Just Eliminate the Variance." I very much appreciate the author's letter, and I'm happy to have the opportunity to respond.