Statistics: Psychological

Statistics for Psychology (3rd Edition) by Arthur Aron, and Elaine N. Aron (Prentice Hall) In the 1950s and 1960s statistics texts were dry, daunting, mathematical tomes that quickly left most students behind. In the 1970s, there was a revolution‑in swept the intuitive approach, with much less emphasis on derivations, proofs, and mathematical foundations. The approach worked. Students became less afraid of statistics courses and found the material more accessible, even if not quite clear.

The intuitive trend continued in the 1980s, adding in the 1990s some nicely straightforward writing. A few texts have now also begun to encourage students to use the computer to do statistical analyses. However, discussions of intuitive understandings are becoming briefer and briefer. The standard is a cursory overview of the key idea and sometimes the associated definitional formula for each technique. Then come the procedures and examples for actually doing the computation, using another "computational" formula.

Even with all this streamlining, or perhaps because of it, at the end of the course most students cannot give a clear explanation of the logic behind the techniques they have learned. A few months later they can rarely carry out the procedures either. Most important, the three main purposes of the introductory statistics course are not accomplished: Students are not able to make sense of the results of psychology research articles, they are poorly prepared for further courses in statistics (where instructors must inevitably spend half the semester reteaching the introductory course), and the exposure to deep thinking that is supposed to justify the course's meeting general education requirements in the quantitative area has not occurred.

We continue to do what the best of the newer books are already doing well: emphasizing the intuitive, de-­emphasizing the mathematical, and explaining everything in direct, simple language. But what we have done differs from these other books in 11 key respects.

1. The definitional formulas are brought to center stage because they provide a concise symbolic summary of the logic of each particular procedure. All our explanations, examples, practice problems, and test bank items are based on these definitional formulas. (The amount of data to be processed in practice problems and test bank items are reduced appropriately to keep computations manageable.)
Why this approach? To date, statistics texts have failed to adjust to technological reality. What is important is not that the students learn to calculate a test with a large data set‑computers can do that for them. What is important is that students work problems in a way that they remain constantly aware of the underlying logic of what they are doing. Consider the population variance‑the average of the squared deviations from the mean. This concept is directly displayed in the definitional formula (once the student is used to the symbols): Variance. Repeatedly working problems using this formula engrains the meaning in the student's mind. In contrast, the usual computational version of this formula only obscures this meaning: Teaching the old computational formulas today is an anachronism. Researchers do their statistics on computers now. At the same time, the use of statistical soft­ware makes the understanding of the basic principles, as they are symbolically ex­pressed in the definitional formulas, more important than ever. Students still need to work lots of problems by hand to learn the material. But they need to work them using the definitional formulas that reinforce the concepts, not using the computa­tional formulas that obscure them. Those formulas once made some sense as time­savers for researchers who had to work with large data sets by hand, but they were always poor teaching tools. (Because some instructors may feel naked without them, we still provide the computational formulas, usually in a brief footnote, at the point in the chapter where they would traditionally have been introduced.)
2. Each procedure is taught both verbally and numerically‑and usually vi­sually as well. In fact, when we introduce every formula, it has attached to it a con­cise statement of the formula in words. Typically, each example lays out the procedures in worked‑out formulas, in words (often with a list of steps), and illus­trated with an easy‑to‑grasp figure. Practice problems and test bank items, in turn, require the student to calculate results, write a short explanation in layperson's lan­guage of what they have done, and make a sketch (for example of the distributions involved in a t test). The chapter material completely prepares the student for these kinds of practice problems and test questions.
It is our repeated experience that these different ways of expressing an idea are crucial for permanently establishing a concept in a student's mind. Many psychol­ogy students are more at ease with words than with numbers. In fact, some have a positive fear of all mathematics. Writing the formula in words and providing the lay‑language explanation gives them an opportunity to do what they do best.
3. A main goal of any introductory statistics course in psychology is to prepare students to read research articles. The way a procedure such as a t test or an analy­sis of variance is described in a research article is often quite different from what the student expects from the standard textbook discussions. Therefore, as this book teaches a statistical method, it also gives examples of how that method is reported in the journals (excerpts from current articles). And we don't just leave it there. The practice problems and test bank items also include excerpts from articles for the stu­dent to explain.
4. The book is unusually up to date. For some reason, most introductory statis­tics textbooks read as if they were written in the 1950s. The basics are still the ba­sics, but statisticians and researchers think far more subtly about those basics now. Today, the basics are undergirded by a new appreciation of effect size, power, the accumulation of results through meta‑analysis, the critical role of models, the un­derlying unity of difference and association statistics, the growing prominence of regression and associated methods, and a whole host of new orientations arising from the central role of the computer. We are much engaged in the latest develop­ments in statistical theory and application, and this book reflects that engagement. For example, we devote an entire early chapter to effect size and power and then re­turn to these topics as we teach each technique.
5. We capitalize on the students' motivations. We do this in two ways. First, our examples emphasize topics or populations that students seem to find most inter­esting. The very first example is from a real study in which 151 students in their first week of an introductory statistics class rate how much stress they feel they are under. Other examples emphasize clinical, organizational, social, and educational psychology while being sure to include sufficient interesting examples from cogni­tive, developmental, behavioral and cognitive neuroscience, and other areas to inspire students with the value of those specialties. (Also, our examples continually emphasize the usefulness of statistical methods and ideas as tools in the research process, never allowing students to feel that what they are learning is theory for the sake of theory.)
Second, we have worked to make the book extremely straightforward and systematic in its explanation of basic concepts so that students can have frequent "aha" experiences. Such experiences bolster self‑confidence and motivate further learning. It is quite inspiring to us to see even fairly modest students glow from having mastered some concept like negative correlation or the distinction between failing to reject the null hypothesis and supporting the null hypothesis. At the same time, we do not constantly remind them how greatly oversimplified we have made things, as some books do. Instead, we show students, in the controversy sections in particular, how much there is for them to consider deeply, even in an introductory course.
6. We emphasize statistical methods as a living, growing field of research. We take the time to describe the issues, such as the recent upheaval about the value of significance testing. In addition, each chapter includes one or more "boxes" about famous statisticians or interesting side‑lights. The goal is for students to see statistical methods as human efforts to make sense out of the jumble of numbers generated by a research study; to see that statistics are not "given" by nature, not infallible, not perfect descriptions of the events they try to describe but rather constitute a language that is constantly improving through the careful thought of those who use it. We hope that this orientation will help them maintain a questioning, alert attitude as students and later as professionals.
7. Chapter 16 integrates the major techniques that have been taught, explaining that the t test is a special case of the analysis of variance and that both the t test and the analysis of variance are special cases of correlation and regression. (In short, we introduce the general linear model.) In the past, when this point has been made at all, it has usually been only in advanced texts. But many students find it valuable for digesting and retaining what they have learned, as well as for sensing that they have penetrated deeply into the foundations of statistical methods.
8. The final chapter looks at advanced procedures without actually teaching them in detail. It explains in simple terms how to make sense out of these statistics when they are encountered in research articles. Most psychology research articles today use methods such as analysis of covariance, multivariate analysis of variance, hierarchical multiple regression, factor analysis, or structural equation modeling. Students completing the ordinary introductory statistics course are ill‑equipped to comprehend most of the articles they must read to prepare a paper or study a course topic in further depth. This chapter makes use of the basics that students have just learned (along with extensive excerpts from current research articles) to give a rudimentary understanding of these advanced procedures. This chapter also serves as a reference guide that students can keep and use in the future when reading such articles.
9. The accompanying Student's Study Guide and Computer Workbook focuses on mastering concepts and also includes instructions and examples for working problems on the computer. Most study guides concentrate on plugging numbers into formulas and memorizing rules (which is consistent with the emphasis of the textbooks they accompany). For each chapter, our Student's Study Guide and Computer Workbook provides learning objectives, a detailed chapter outline, the chapter's formulas (with all symbols defined), and summaries of steps of conducting each procedure covered in the chapter, plus a set of self tests, including multiple-choice, fill‑in, and problem/essay questions. In addition, for each procedure covered in the chapter, the study guide furnishes a thorough outline for writing an essay explaining the procedure to a person who has never had a course in statistics (a task they are frequently given in the practice problems and test bank items.).
Also, our Student's Study Guide and Computer Workbook provides the needed support for teaching students to carry out analyses on the computer. First, there is a special appendix on getting started with SPSS. Then, in each chapter corresponding to the text chapters, there is a section showing in detail how to carry out the chapter's procedures with SPSS. (These sections include step‑by‑step instructions, examples, and illustrations of how each menu and each output appears on the screen.) There are also special activities for using the computer to strengthen understanding. As far as we know, no other statistics textbook package provides this much depth of explanation.
10. We have written an Instructor's Resource Manual that really helps teach the course. The manual begins with a chapter summarizing what we have gleaned from our own teaching experience and the research literature on effectiveness in college teaching. The next chapter discusses alternative organizations of the course, including tables of possible schedules and a sample syllabus. Then each chapter, corresponding to the text chapters, provides full lecture outlines and additional worked‑out examples not found in the text (in a form suitable for copying onto transparencies or for student handouts). These worked‑out examples are especially useful to new instructors or those using our book for the first time, since creating good examples is one of the most difficult parts of preparing statistics lectures.
11. Our Test Bank makes preparing exams easy. We supply approximately 40 multiple‑choice, 25 fill‑in, and 10 to 12 problem/essay questions for each chapter. Considering that the emphasis of the course is so conceptual, the multiple­choice questions will be particularly useful for those of you who do not have the resources to grade essays.

We did the revision for the third edition over a summer in Tiburon, a small town overlooking the San Francisco Bay. We hope that this has not resulted in a loss of whatever romance the first edition gained from being written in Paris. On the other hand, this edition has been leavened by some beautiful Bay views.

More important, this revision is enriched by what we learned teaching with the first and second editions and by what we learned from the many instructors and students who have written to us about their experiences using the book. This revision is also informed by our own use of statistical methods. The last several years have been quite productive for the two of us in our own research programs in personality and social psychology. Our most recent adventure has been in social neuroscience, learning brain‑imaging techniques, which it turns out are almost as fascinating for the statistical analysis challenges they pose as for the opportunities they provide for deepening knowledge of the issues we were previously studying with more conventional methods. Perhaps particularly useful has been that one of us (A. A.) has been serving as an associate editor for the Journal of Personality and Social Psychology. This has kept us in touch with how the best researchers are using statistics (as well as how reviewers assess their colleagues' use of statistics). In addition to reworking the book to keep it up to date in obvious and subtle ways, we have made a special effort in this edition to bring in to the text significant new pedagogical features.

1. New pedagogic features. The most obvious changes to those familiar with the book will be the following additions we made to ease the learning process:
• "How Are You Doing?" sections. These are brief self‑tests focusing on concepts, inserted at three or four appropriate points in each chapter. These give students a chance to check that they have learned what they have just read, help them identify the central material in what they have just read, reinforce this material before going on to the next section, and divide the chapter into more accessible "chunks."
• Doubling the number of practice problems. Each chapter now has at least 20. This provides the instructor with greater flexibility in the kinds and numbers of problems to assign.
• Examples of Worked‑Out Computational Problems. These are included just before the practice problems at the end of each chapter. These give the student the chance to check their knowledge before starting their assigned problems and provide a model to follow when working them out, thus easing anxiety and helping the student do the problems correctly.
• With each new formula there is a boxed concise statement of the formula in words. This is important for helping students who fear symbols and math to see the underlying principle embedded in the formula, and keeps this verbal understanding directly available to them as they become accustomed to working with the symbols.
2. Writing. We have once again in this revision thoroughly reviewed every sentence, simplifying constructions and terminology wherever possible and sometimes rewriting from scratch entire paragraphs or sections. It is hard enough to learn statistics without having to read complicated sentences.
3. Updating examples. We have replaced over 60 examples from the second edition with new ones published in the last year or two. This is particularly important for the sections on how to understand and evaluate statistics in research articles.
4. Updating content and controversies. Most obvious to those familiar with earlier editions will be the discussion of the APA Task Force report and the new APA Publication Manual's statements on data analysis. But the updates are everywhere in subtle ways‑even with newly identified anecdotes about historical figures in the boxes!
5. Reworking of some specific topics students had found difficult. We have substantially reworked our treatment of a few topics that some students were struggling with, including grouped frequency tables, raw‑score regression, confidence intervals, and effect size in analysis of variance. We have also made some changes in emphasis and coverage in response to instructors' suggestions, including more on the issue of causality and correlation and a fuller treatment of multiple comparisons in analysis of variance.
6. There is now a unique Web page available to instructors who adopt the book and to their students. We are particularly excited about the potential of the Web for aiding learning of statistics. Elliot Coups, has created an outstanding, dramatically innovative site. Some unique features (in addition to the usual chapter outline and objectives) include:
• For instructors: Powerpoint presentation materials for teaching the course, including examples from .the text and examples from the Instructor's Resource Manual that are not in the text.
• Downloadable mini‑chapter for students on applying statistics in their own research projects.
• Downloadable mini‑chapter for students on repeated measures analysis of variance.
Chapter objectives

• Downloadable mini‑chapter on the logic and language of research (this was Appendix A in the earlier editions)
• Tips for Success: What to practice, and what to study.
• Learn More! sections: Practice problems that include tables from the text on the Web, giving the students the opportunity to use the tables to work through problems.
• On-line student study guide, including practice problems, true/false ques­tions, and fill in the blanks.
Flash card exercises for each chapter's key terms. All formulas
• Links to statistic sites

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