This article illustrates that not all statistical software packages are correctly calculating a p-value for the classical F test comparison of two independent Normal variances. This is illustrated with a simple example, and the reasons why are discussed. Eight different software packages are considered.
Two recent developments in statistical education provide the opportunity for significant advances in helping non-statisician judge statistical claims.<br>(1) Research in statistical thinking has begun to yield models of people's conceptions that are detailed enough to have practical, pedagogical implications.<br>(2) Powerful new software tools designed explicitly for statistical education provide new visualizations with enormous potential for making statistical thinking accessible, for the first time, to the wide range of people who need to use it.<br>While these two developments are exciting in and of themselves, a collaboration between researchers and software designers would accelerate the development of both research and software in important ways. Currently both researchers and developers benefit in limited ways from each others' expertise and ongoing work, but taking full advantage of their respective contributions requires an explicit and organized effort with careful planning to achieve the benefits to each. We propose to create a paradignmatic illustration of such a collaboration, one which will have an impact on both software developers and researchers and, ultimately, on the productivity of the statistical education field.
The development of a Revised National Curriculum Statement is seen as a key project in the transformation of South African Society. The thrust of the project is towards achieving "a prosperous, truly united, democratic and internationally competitive country with literate, creative and critical citizens leading productive, self-fulfilled lives in a country free of violence, discrimination and prejudice." (Curriculum 2005, Learning for the 21st Century 1997, Department of Education, Pretoria.)<br><br>Curriculum reform in South Africa thus faces a two-fold challenge. The first is the post-apartheid challenge which requires developing the knowledge, values and skills base for South Africa's citizens necessary for greater social justice and development. Secondly, there is the challenge of participating in a global economy. This raises questions about the knowledge, values, skills and competencies for innovation and economic growth for the 21st Century.<br><br>The view taken by the curriculum designers is that the best route to greater social justice and development is through a high-knowledge and high skills curriculum.<br><br>This paper will explore the meaning and importance of numeracy and in particular of statistical literacy, within this context. The paper will focus largely on the relationship between values and mathematical/statistical literacy within the South African context.
After 4 decades of severe criticism, the ritual of null hypothesis significance testing - mechanical dichotomous decisions around a sacred .05 criterion - still persists. This article reviews the problems with this practice, including its near-universal misinterpretation of p as the probability that H 0 is false, the misinterpretation that its complement is the probability of successful replication, and the mistaken assumption that if one rejects H 0 one thereby affirms the theory that led to the test. Exploratory data analysis and the use of graphic methods, a steady improvement in and a movement toward standardization in measurement, an emphasis on estimating effect sizes using confidence intervals, and the informed use of available statistical methods is suggested. For generalization, psychologists must finally rely, as has been done in all the older sciences, on replication.
Book description<br>How can anyone be rational in a world where knowledge is limited, time is pressing, and deep thought is often an unattainable luxury? In our book, "Simple Heuristics That Make Us Smart," we invite readers to embark on a new journey into a land of rationality that differs from the familiar territory of cognitive science and economics. Traditional models of rationality in these fields have tended to view decision-makers as possessing supernatural powers of reason, limitless knowledge, and an eternity in which to make choices. But to understand decisions in the real world, we need a different, more psychologically plausible notion of rationality. This book provides such a view. It is about fast and frugal heuristics-simple rules for making decisions with realistic mental resources. These heuristics can enable both living organisms and artificial systems to make smart choices, judgments, and predictions by employing bounded rationality.<br><br>But when and how can such fast and frugal heuristics work? What heuristics are in the mind's "adaptive toolbox," and what building blocks compose them? Can judgments based simply on a single reason be as accurate as those based on many reasons? Could having less knowledge even lead to systematically better predictions than having more knowledge? We explore these questions by developing computational models of heuristics and testing them through theoretical analysis and practical experiments with people. We show how fast and frugal heuristics can yield adaptive decisions in situations as varied as choosing a mate, dividing resources among offspring, predicting high-school drop-out rates, and profiting from the stock market.<br><br>We have worked to create an interdisciplinary book that is both useful and engaging and will appeal to a wide audience. It is intended for readers interested in cognitive psychology, evolutionary psychology, and cognitive science, as well as in economics and artificial intelligence. We hope that it will also inspire anyone who simply wants to make good decisions.
Statistical reasoning is an art and so demands both mathematical knowledge and informed judgment. When it is mechanized, as with the institutionalized hybrid logic, it becomes ritual, not reasoning. Many colleagues have argued that it is not going to be easy to get researchers in psychology and other sociobiomedical sciences to drop this comforting crutch unless one offers an easy-to-use substitute. But this is exactly what I want to avoid - the substitution of one mechanistic dogma for another. It is our duty to inform our students of the many good roads to statistical inference that exist and to teach them how to use informed judgment to decide which one to follow for a particular problem. At the very least, this chapter can serve as a tool in arguments with people who think they have to defend a ritualistic dogma instead of good statistical reasoning. Making and winning such arguments is indispensable to good science.
It is claimed here that the confidence mathematics education researchers have in statistical significance testing (SST) as an inference tool par excellence for experimental research is misplaced. Five common myths about SST are discussed, namely that SST: (a) is a controversy-free, recipe-like method to allow decision making; (b) answers the question whether there is a low probability that the research results were due to chance; (c) logic parallels the logic of mathematical proof by contradiction; (d) addresses the reliability /replicability question; and (e) is a necessary but not sufficient condition for the credibility of results. It is argued that SST's contribution to educational research in genera, and mathematics education research in particular, as not beneficial, and that SST should be discontinued as a tool for such research. Some alternatives to SST are suggested, and a call is made for mathematics education researchers to take the lead in using these alternatives.
I have found writing this response to be a difficult task, as evidenced by my inability to resist the combination of clichés in the title. As I read Menon's article I found myself agreeing with much of what he had written, although sometimes I wondered why it was considered to be worth stating. Then Menon would take a more extreme line which had not really been justified by what had preceded it, and I found myself frustrated by the lack of continuity as much as by the extreme view itself. I will give some examples of what I found to be problems with Menon's position, based around the themes of (a) was it worth saying anyway; (b) the function of over-statement; (c) methodology and the role of theory in educational research; and (d) the proposed ideal world of educational research. IN this response I have taken research in mathematics education to be entirely subsumed in educational research generally.
I recall, in the mid-1970s, a research student of mine who, on carrying out an analysis of her data using statistical significance testing (SST), found that the p value, for what she regarded as her most important hypothesis, was .07, which was not significant at the .05 level. The student asked whether it would be legitimate for her to change the 2-tailed test she had use to a 1-tailed test, and on receiving a negative answer from me, went away disappointed. A couple of days later she returned saying that she had decided to remove some of the "outliers" from the data set, and that when these were removed she had got a p value of .04. In her thesis she honestly reported the sequence of events, but still claimed that she had obtained a "statistically significant" result. The external examiners for her thesis accepted this as legitimate tactic.
Many people are familiar with the calculus reform movement that has been sweeping the country for the last five years, heavily supported by the National Science Foundation. Less well-known is a similar movement within the statistics community that recommends major changes in the content and teaching of introductory statistics courses. NSF has funded numerous projects designed to implement aspects of this reform This proposal outlines a three-stage evaluation to determine the impact of statistics reform efforts on the current teaching of college-level statistics courses.