Statistics for Psychology / Edition 4 available in Hardcover
- Pub. Date:
- Prentice Hall
|Edition description:||Older Edition|
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Table of ContentsPreface to the Instructor xi
Introduction to the Student xvi
Displaying the Order in a Group of Numbers Using Tables and Graphs 1
The Two Branches of Statistical Methods 2
Some Basic Concepts 3
Important Trivia for Poetic Statistics Students 6
Frequency Tables 7
Math Anxiety, Statistics Anxiety, and You: A Message for Those of You Who Are Truly Worried About This Course 12
Shapes of Frequency Distributions 15
Controversy: Misleading Graphs 19
Frequency Tables and Histograms in Research Articles 21
Key Terms 24
Example Worked-Out Problems 24
Practice Problems 25
Using SPSS 29
Chapter Note 32
Central Tendency and Variability 33
Central Tendency 34
The Sheer Joy (Yes, Joy) of Statistical Analysis 51
Controversy: The Tyranny of the Mean 52
Gender, Ethnicity, and Math Performance 53
Central Tendency and Variability in Research Articles 55
Key Terms 57
Example Worked-Out Problems 57
Practice Problems 59
Using SPSS 62
Chapter Notes 65
Some Key Ingredients for Inferential Statistics: Z Scores, the Normal Curve, Sample versus Population, and Probability 67
Z Scores 68
The Normal Curve 73
de Moivre, the Eccentric Stranger Who Invented the Normal Curve 74
Sample and Population 83
Surveys, Polls, and 1948's Costly "Free Sample" 86
Pascal Begins Probability Theory at the Gambling Table, Then Learns to Bet on God 89
Controversies: Is the Normal Curve Really So Normal? and Using Nonrandom Samples 93
Z Scores, Normal Curves, Samples and Populations, and Probabilities in Research Articles 95
Advanced Topics: Probability Rules and Conditional Probabilities 96
Key Terms 98
Example Worked-Out Problems 99
Practice Problems 102
Using SPSS 105
Chapter Notes 106
Introduction to Hypothesis Testing 107
A Hypothesis-Testing Example 108
The Core Logic of Hypothesis Testing 109
The Hypothesis-Testing Process 110
One-Tailed and Two-Tailed Hypothesis Tests 119
Controversy: Should Significance Tests Be Banned? 124
Jacob Cohen, the Ultimate New Yorker: Funny, Pushy, Brilliant, and Kind 126
Hypothesis Tests in Research Articles 127
Key Terms 129
Example Worked-Out Problems 129
Practice Problems 131
Chapter Notes 136
Hypothesis Tests with Means of Samples 137
The Distribution of Means 138
Hypothesis Testing with a Distribution of Means: The Z Test 146
More About Polls: Sampling Errors and Errors in Thinking About Samples 147
Controversy: Marginal Significance 153
Hypothesis Tests About Means of Samples (Z Tests) and Standard Errors in Research Articles 154
Advanced Topic: Estimation, Standard Errors, and Confidence Intervals 156
Advanced Topic Controversy: Confidence Intervals versus Significance Tests 162
Advanced Topic: Confidence Intervals in Research Articles 163
Key Terms 164
Example Worked-Out Problems 164
Practice Problems 167
Chapter Notes 173
Making Sense of Statistical Significance: Decision Errors, Effect Size, and Statistical Power 175
Decision Errors 175
Effect Size 179
Effect Sizes for Relaxation and Meditation: A Restful Meta-Analysis 184
Statistical Power 187
What Determines the Power of a Study? 191
The Power of Typical Psychology Experiments 199
The Role of Power When Planning a Study 203
The Role of Power When Interpreting the Results of a Study 205
Controversy: Statistical Significance versus Effect Size 208
Decision Errors, Effect Size, and Power in Research Articles 210
Advanced Topic: Figuring Statistical Power 212
Key Terms 215
Example Worked-Out Problems 215
Practice Problems 217
Chapter Note 221
Introduction to t Tests: Single Sample and Dependent Means 222
The t Test for a Single Sample 223
William S. Gosset, Alias "Student": Not a Mathematician, But a Practical Man 224
The t Test for Dependent Means 236
Assumptions of the t Test for a Single Sample and the t Test for Dependent Means 247
Effect Size and Power for the t Test for Dependent Means 247
Controversy: Advantages and Disadvantages of Repeated-Measures Designs 250
The Power of Studies Using Difference Scores: How the Lanarkshire Milk Experiment Could Have Been Milked for More 251
Single Sample t Tests and Dependent Means t Tests in Research Articles 252
Key Terms 254
Example Worked-Out Problems 254
Practice Problems 258
Using SPSS 265
Chapter Notes 268
The t Test for Independent Means 270
The Distribution of Differences Between Means 271
Hypothesis Testing with a t Test for Independent Means 278
Assumptions of the t Test for Independent Means 286
Monte Carlo Methods: When Mathematics Becomes Just an Experiment, and Statistics Depend on a Game of Chance 286
Effect Size and Power for the t Test for Independent Means 288
Review and Comparison of the Three Kinds of t Tests 290
Controversy: The Problem of Too Many t Tests 291
The t Test for Independent Means in Research Articles 292
Advanced Topic: Power for the t Test for Independent Means When Sample Sizes Are Not Equal 293
Key Terms 295
Example Worked-Out Problems 295
Practice Problems 298
Using SPSS 305
Chapter Notes 309
Introduction to the Analysis of Variance 310
Basic Logic of the Analysis of Variance 311
Sir Ronald Fisher, Caustic Genius of Statistics 317
Carrying Out an Analysis of Variance 319
Hypothesis Testing with the Analysis of Variance 327
Assumptions in the Analysis of Variance 331
Planned Contrasts 334
Post Hoc Comparisons 337
Effect Size and Power for the Analysis of Variance 339
Controversy: Omnibus Tests versus Planned Contrasts 343
Analyses of Variance in Research Articles 344
Advanced Topic: The Structural Model in the Analysis of Variance 345
Principles of the Structural Model 345
Key Terms 352
Example Worked-Out Problems 353
Practice Problems 357
Using SPSS 364
Chapter Notes 368
Factorial Analysis of Variance 370
Basic Logic of Factorial Designs and Interaction Effects 371
Recognizing and Interpreting Interaction Effects 376
Basic Logic of the Two-Way Analysis of Variance 386
Personality and Situational Influences on Behavior: An Interaction Effect 387
Assumptions in the Factorial Analysis of Variance 389
Extensions and Special Cases of the Analysis of Variance 389
Controversy: Dichotomizing Numeric Variables 391
Factorial Analysis of Variance in Research Articles 393
Advanced Topic: Figuring a Two-Way Analysis of Variance 395
Advanced Topic: Power and Effect Size in the Factorial Analysis of Variance 406
Key Terms 411
Example Worked-Out Problems 412
Practice Problems 415
Using SPSS 426
Chapter Notes 431
Graphing Correlations: The Scatter Diagram 434
Patterns of Correlation 437
The Correlation Coefficient 443
Galton: Gentleman Genius 446
Significance of a Correlation Coefficient 452
Correlation and Causality 456
Issues in Interpreting the Correlation Coefficient 458
Illusory Correlation: When You Know Perfectly Well That If It's Big, It's Fat-and You Are Perfectly Wrong 460
Effect Size and Power for the Correlation Coefficient 464
Controversy: What Is a Large Correlation? 466
Correlation in Research Articles 467
Key Terms 471
Example Worked-Out Problems 471
Practice Problems 474
Using SPSS 482
Chapter Notes 485
Predictor (X) and Criterion (Y) Variables 488
The Linear Prediction Rule 488
The Regression Line 492
Finding the Best Linear Prediction Rule 496
The Least Squared Error Principle 498
Issues in Prediction 503
Multiple Regression 506
Limitations of Prediction 508
Controversy: Unstandardized and Standardized Regression Coefficients; Comparing Predictors 509
Clinical versus Statistical Prediction 510
Prediction in Research Articles 511
Advanced Topic: Error and Proportionate Reduction in Error 514
Key Terms 519
Example Worked-Out Problems 519
Practice Problems 524
Using SPSS 532
Chapter Notes 535
Chi-Square Tests 536
Karl Pearson, Inventor of Chi-Square and Center of Controversy 537
The Chi-Square Statistic and the Chi-Square Test for Goodness of Fit 538
The Chi-Square Test for Independence 546
Assumptions for Chi-Square Tests 554
Effect Size and Power for Chi-Square Tests for Independence 554
Controversy: The Minimum Expected Frequency 558
Chi-Square Tests in Research Articles 559
Key Terms 561
Example Worked-Out Problems 561
Practice Problems 565
Using SPSS 572
Chapter Notes 576
Strategies When Population Distributions Are Not Normal: Data Transformations and Rank-Order Tests 577
Assumptions in the Standard Hypothesis-Testing Procedures 578
Data Transformations 580
Rank-Order Tests 585
Comparison of Methods 589
Controversy: Computer-Intensive Methods 591
Where Do Random Numbers Come From? 594
Data Transformations and Rank-Order Tests in Research Articles 595
Key Terms 597
Example Worked-Out Problems 597
Practice Problems 597
Using SPSS 602
Chapter Notes 609
The General Linear Model and Making Sense of Advanced Statistical Procedures in Research Articles 611
The General Linear Model 612
Two Women Make a Point About Gender and Statistics 616
Partial Correlation 617
Multilevel Modeling 620
Factor Analysis 622
Causal Modeling 625
The Golden Age of Statistics: Four Guys Around London 627
Procedures That Compare Groups 634
Analysis of Covariance (ANCOVA) 634
Multivariate Analysis of Variance (MANOVA) and Multivariate Analysis of Covariance (MANCOVA) 635
Overview of Statistical Techniques 636
Controversy: Should Statistics Be Controversial? 637
The Forced Partnership of Fisher and Pearson 638
How to Read Results Using Unfamiliar Statistical Techniques 639
Key Terms 642
Practice Problems 642
Using SPSS 654
Chapter Notes 662
Answers to Set I Practice Problems 673
Glossary of Symbols 708
Our approach was developed over three decades of successful teachingsuccessful not only in the sense that students have consistently rated the course (a statistics course, remember) as a highlight of their major, but also in the sense that students come back to us later saying, "I was light-years ahead of my fellow graduate students because of your course," or "Even though I don't do research, your course has really helped me read the journals in my field."
The response to the first and second edition has been overwhelming. We have received hundreds of thank-you e-mails and letters from instructors (and from students themselves!) from all over the English-speaking world. Of course, we were also delighted by the enthusiastic review in Contemporary Psychology (Bourgeois, 1997).
In this third edition we have tried to maintain those things that have been especially appreciated, while reworking the book to take into account the feedback we have received, our own experiences, and advances and changes in the field. We have also added new pedagogical features to make the book even more accessible for students. However, before turning to the third edition, we want to reiterate what we said in thefirst edition about how this book from the beginning has been quite different from other statistics texts.
A BRIEF HISTORY OF THE STATISTICS TEXT GENRE
In the 1950s and 1960s statistics texts were dry, daunting, mathematical tomes that quickly left most students behind. In the 1970s, there was a revolutionin 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 ark, 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.
WHAT WE HAVE DONE DIFFERENTLY
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 t test with a large data setcomputers 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 variancethe 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 = Σ(I M)2/N. 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: Variance = ΣX2 (ΣX)2/N/N. Repeatedly working problems using this formula does nothing but teach the student the difference between ΣX2 and (ΣX)2!
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 numericallyand 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 multiplechoice, 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 multiple-choice questions will be particularly useful for those of you who do not have the resources to grade essays.
INFLUENCES ON THE THIRD EDITION
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. (For overviews of our main research programs, see A. Aron et al., 2001; E. Aron, 2000.) 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.
SPECIFIC CHANGES IN THE THIRD EDITION
- 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, an4 divide the chapter into more accessible "chunks."
- Doubling the number of practice problems. Each chapter now bas 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.
- 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.
- 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.
- 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 wayseven with newly identified anecdotes about historical figures in the boxes!
- 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.
- 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
Some changes we have not made. The 11 points listed earlier in this Preface remain as the central, unique features of this book. Also, except in a few cases where we felt we could make a significant improvement in pedagogy, we have not changed each chapter's major teaching examples. Instructors using the second edition told us they have built their lectures around these examples and don't want to have to start from scratch with new ones.
KEEP IN TOUCH
Our goal is to do whatever we can to help you make your course a success. If you have any questions or suggestions, please write or e-mail. Also, if you should find an error somewhere, for everyone's benefit, please let us know right away. When errors have come up in the past, we have usually been able to fix them in the very next printing.