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.
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
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.
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
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.
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.
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
Key Words: Null hypothesis; Statistical testing; Type I error rate;
Type II error rate; P value; Argument; Framework; Teaching statistics.
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
Key Words: Measures of skewness; Estimating power; Simulation.
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
Key Words: Confidence intervals; Hypothesis testing; Informal reasoning;
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.
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.
In this column, we would like to highlight some recent new additions to CAUSEweb and MERLOT.
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.