Please discuss, elaborate, and reflect on the following from chapter 7. I have listed the important topics and some of them are questions.
Give examples and elaborate on the applications of the topic. After reading your textbook, I want you to have a good understanding of the fundamentals of each chapter and show it to me. Please don’t copy & paste from your textbook or some other online source. In other words, don’t plagiarize. You can read online material if it helps to understand the material, but you have to write your own sentences.
Chapter-7 topics and questions:
· What is a normal distribution?
· List the characteristics of a normal distribution.
· How do you convert empirical distributions to standard normal distribution?
· How do you find the proportion of a normal distribution that lies between two scores?
· How do you find the scores between which a certain proportion of a normal distribution falls?
Exploring Statistics Tales of Distributions
Outcrop Publishers Conway, Arkansas
Exploring Statistics: Tales of Distributions 12th Edition Chris Spatz
Cover design: Grace Oxley Answer Key: Jill Schmidlkofer Webmaster & Ebook: Fingertek Web Design, Tina Haggard Managers: Justin Murdock, Kevin Spatz
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Photo Credits – Chapter 1 Karl Pearson – Courtesy of Wellcomeimages.org Ronald A. Fisher – R.A. Fisher portrait, 0006973, Special Collections Research Center, North Carolina State
University Libraries, Raleigh, North Carolina Jerzy Neyman – Paul R. Halmos Photograph Collection, e_ph 0223_01, Dolph Briscoe Center for American History,
The University of Texas at Austin Jacob Cohen – New York University Archives, Records of the NYU Photo Bureau
Printed in the United States of America by Walsworth ® 2 3 4 5 6 7 24 23 22 21 20
Online study guide available at http://exploringstatistics.com/studyguide.php
v About The Author
Chris Spatz is at Hendrix College where he twice served as chair of the Psychology Department. Dr. Spatz’s undergraduate education was at Hendrix, and his PhD in experimental psychology is from Tulane University in New Orleans. He subsequently completed postdoctoral fellowships in animal behavior at the University of California, Berkeley, and the University of Michigan. Before returning to Hendrix to teach, Spatz held positions at The University of the South and the University of Arkansas at Monticello.
Spatz served as a reviewer for the journal Teaching of Psychology for more than 20 years. He co-authored a research methods textbook, wrote several chapters for edited books, and was a section editor for the Encyclopedia of Statistics in Behavioral Science.
In addition to writing and publishing, Dr. Spatz enjoys the outdoors, especially canoeing, camping, and gardening. He swims several times a week (mode = 3). Spatz has been an opponent of high textbook prices for years, and he is happy to be part of a new wave of authors who provide high-quality textbooks to students at affordable prices.
About The Author
With love and affection,
this textbook is dedicated to
Thea Siria Spatz, Ed.D., CHES
vii Brief Contents
Preface xiv 1 Introduction 1 2 Exploring Data: Frequency Distributions and Graphs 29 3 Exploring Data: Central Tendency 45 4 Exploring Data: Variability 59 5 Other Descriptive Statistics 77 6 Correlation and Regression 94 7 Theoretical Distributions Including the Normal Distribution 127 8 Samples, Sampling Distributions, and Confidence Intervals 150 9 Effect Size and NHST: One-Sample Designs 175 10 Effect Size, Confidence Intervals, and NHST:
Two-Sample Designs 200 11 Analysis of Variance: Independent Samples 231 12 Analysis of Variance: Repeated Measures 259 13 Analysis of Variance: Factorial Design 271 14 Chi Square Tests 303 15 More Nonparametric Tests 328 16 Choosing Tests and Writing Interpretations 356
A Getting Started 371 B Grouped Frequency Distributions and Central Tendency 376 C Tables 380 D Glossary of Words 401 E Glossary of Symbols 405 F Glossary of Formulas 407 G Answers to Problems 414
References 466 Index 472
chapter 1 Introduction 1 Disciplines That Use Quantitative Data 5 What Do You Mean, “Statistics”? 6 Statistics: A Dynamic Discipline 8 Some Terminology 9 Problems and Answers 12 Scales of Measurement 13 Statistics and Experimental Design 16 Experimental Design Variables 17 Statistics and Philosophy 20 Statistics: Then and Now 21 How to Analyze a Data Set 22 Helpful Features of This Book 22 Computers, Calculators, and Pencils 24 Concluding Thoughts 25 Key Terms 27
Transition Passage to Descriptive Statistics 28
chapter 2 Exploring Data: Frequency Distributions and Graphs 29 Simple Frequency Distributions 31 Grouped Frequency Distributions 33 Graphs of Frequency Distributions 35 Describing Distributions 39
The Line Graph 41 More on Graphics 42 A Moment to Reflect 43 Key Terms 44
chapter 3 Exploring Data: Central Tendency 45 Measures of Central Tendency 46 Finding Central Tendency of Simple Frequency Distributions 49 When to Use the Mean, Median, and Mode 52 Determining Skewness From the Mean and Median 54 The Weighted Mean 55 Estimating Answers 56 Key Terms 58
chapter 4 Exploring Data: Variability 59 Range 61 Interquartile Range 61 Standard Deviation 63 Standard Deviation as a Descriptive Index of Variability 64 ŝ as an Estimate of σ 69 Variance 73 Statistical Software Programs 74 Key Terms 76
chapter 5 Other Descriptive Statistics 77 Describing Individual Scores 78 Boxplots 82 Effect Size Index 86 The Descriptive Statistics Report 89 Key Terms 92
Transition Passage to Bivariate Statistics 93
chapter 6 Correlation and Regression 94 Bivariate Distributions 96 Positive Correlation 96 Negative Correlation 99 Zero Correlation 101 Correlation Coefficient 102 Scatterplots 106
Interpretations of r 106 Uses of r 110 Strong Relationships but Low Correlation Coefficients 112 Other Kinds of Correlation Coefficients 115 Linear Regression 116 The Regression Equation 117 Key Terms 124 What Would You Recommend? Chapters 2-6 125
Transition Passage to Inferential Statistics 126
chapter 7 Theoretical Distributions Including the Normal Distribution 127 Probability 128 A Rectangular Distribution 129 A Binomial Distribution 130 Comparison of Theoretical and Empirical Distributions 131 The Normal Distribution 132 Comparison of Theoretical and Empirical Answers 146 Other Theoretical Distributions 146 Key Terms 147
Transition Passage to the Analysis of Data From Experiments 149
chapter 8 Samples, Sampling Distributions, and Confidence Intervals 150 Random Samples 152 Biased Samples 155 Research Samples 156 Sampling Distributions 157 Sampling Distribution of the Mean 157 Central Limit Theorem 159 Constructing a Sampling Distribution When σ Is Not Available 164 The t Distribution 165 Confidence Interval About a Population Mean 168 Categories of Inferential Statistics 172 Key Terms 173
Transition Passage to Null Hypothesis Significance Testing 174
chapter 9 Effect Size and NHST: One-Sample Designs 175 Effect Size Index 176 The Logic of Null Hypothesis Significance Testing (NHST) 179 Using the t Distribution for Null Hypothesis Significance Testing 182 A Problem and the Accepted Solution 184 The One-Sample t Test 186 An Analysis of Possible Mistakes 188 The Meaning of p in p < .05 191 One-Tailed and Two-Tailed Tests 192 Other Sampling Distributions 195 Using the t Distribution to Test the Significance of a Correlation Coefficient 195 t Distribution Background 197 Why .05? 198 Key Terms 199
chapter 10 Effect Size, Confidence Intervals, and NHST: Two-Sample Designs 200 A Short Lesson on How to Design an Experiment 201 Two Designs: Paired Samples and Independent Samples 202 Degrees of Freedom 206 Paired-Samples Design 208 Independent-Samples Design 212 The NHST Approach 217 Statistical Significance and Importance 222 Reaching Correct Conclusions 222 Statistical Power 225 Key Terms 228 What Would You Recommend? Chapters 7-10 229
Transition Passage to More Complex Designs 230
chapter 11 Analysis of Variance: Independent Samples 231 Rationale of ANOVA 233 More New Terms 240 Sums of Squares 240 Mean Squares and Degrees of Freedom 245 Calculation and Interpretation of F Values Using the F Distribution 246 Schedules of Reinforcement—A Lesson in Persistence 248 Comparisons Among Means 250 Assumptions of the Analysis of Variance 254 Random Assignment 254 Effect Size Indexes and Power 255 Key Terms 258
chapter 12 Analysis of Variance: Repeated Measures 259 A Data Set 260 Repeated-Measures ANOVA: The Rationale 261 An Example Problem 262 Tukey HSD Tests 265 Type I and Type II Errors 266 Some Behind-the-Scenes Information About Repeated-Measures ANOVA 267 Key Terms 270
chapter 13 Analysis of Variance: Factorial Design 271 Factorial Design 272 Main Effects and Interaction 276 A Simple Example of a Factorial Design 282 Analysis of a 2 × 3 Design 291 Comparing Levels Within a Factor—Tukey HSD Tests 297 Effect Size Indexes for Factorial ANOVA 299 Restrictions and Limitations 299 Key Terms 301
Transition Passage to Nonparametric Statistics 302
chapter 14 Chi Square Tests 303 The Chi Square Distribution and the Chi Square Test 305 Chi Square as a Test of Independence 307 Shortcut for Any 2 × 2 Table 310 Effect Size Indexes for 2 × 2 Tables 310 Chi Square as a Test for Goodness of Fit 314
Chi Square With More Than One Degree of Freedom 316 Small Expected Frequencies 321 When You May Use Chi Square 324 Key Terms 327
chapter 15 More Nonparametric Tests 328 The Rationale of Nonparametric Tests 329 Comparison of Nonparametric to Parametric Tests 330 Mann-Whitney U Test 332 Wilcoxon Signed-Rank T Test 339 Wilcoxon-Wilcox Multiple-Comparisons Test 344 Correlation of Ranked Data 348 Key Terms 353 What Would You Recommend? Chapters 11-15 353
chapter 16 Choosing Tests and Writing Interpretations 356 A Review 356 My (Almost) Final Word 357 Future Steps 358 Choosing Tests and Writing Interpretations 359 Key Term 368
Appendixes A Getting Started 371 B Grouped Frequency Distributions and Central
Tendency 376 C Tables 380 D Glossary of Words 401 E Glossary of Symbols 405 F Glossary of Formulas 407 G Answers to Problems 414
References 466 Index 472
Exploring Statistics: Tales of Distributions (12th edition) is a textbook for a one-term statistics course in the social or behavioral sciences, education, or an allied health/nursing field. Its focus is conceptualization, understanding, and interpretation, rather than computation. Designed to be comprehensible and complete for students who take only one statistics course, it also includes elements that prepare students for additional statistics courses. For example, basic experimental design terms such as independent and dependent variables are explained so students can be expected to write fairly complete interpretations of their analyses. In many places, the student is invited to stop and think or do a thought exercise. Some problems ask the student to decide which statistical technique is appropriate. In sum, this book’s approach is in tune with instructors who emphasize critical thinking in their course.
This textbook has been remarkably successful for more than 40 years. Students, professors, and reviewers have praised it. A common refrain is that the book has a conversational, narrative style that is engaging, especially for a statistics text. Other features that distinguish this textbook from others include the following:
• Data sets are approached with an attitude of exploration. • Changes in statistical practice over the years are acknowledged, especially the recent
emphasis on effect sizes and confidence intervals. • Criticism of null hypothesis significance testing (NHST) is explained. • Examples and problems represent a variety of disciplines and everyday life. • Most problems are based on actual studies rather than fabricated scenarios. • Interpretation is emphasized throughout. • Problems are interspersed within a chapter, not grouped at the end. • Answers to all problems are included. • Answers are comprehensively explained—over 50 pages of detail. • A final chapter, Choosing Tests and Writing Interpretations, requires active responses to
Even if our statistical appetite is far from keen, we all of us should like to know enough to understand, or to withstand, the statistics that are constantly being thrown at us in print or conversation—much of it pretty bad statistics. The only cure for bad statistics is apparently more and better statistics. All in all, it certainly appears that the rudiments of sound statistical sense are coming to be an essential of a liberal education.
– Robert Sessions Woodworth
• Effect size indexes are treated as important descriptive statistics, not add-ons to NHST. • Important words and phrases are defined in the margin when they first occur. • Objectives, which open each chapter, serve first for orientation and later as review
items. • Key Terms are identified for each chapter. • Clues to the Future alert students to concepts that come up again. • Error Detection boxes tell ways to detect mistakes or prevent them. • Transition Passages alert students to a change in focus in chapters that follow. • Comprehensive Problems encompass all (or most) of the techniques in a chapter. • What Would You Recommend? problems require choices from among techniques in
For this 12th edition, I increased the emphasis on effect sizes and confidence intervals, moving them to the front of Chapter 9 and Chapter 10. The controversy over NHST is addressed more thoroughly. Power gets additional attention. Of course, examples and problems based on contemporary data are updated, and there are a few new problems. In addition, a helpful Study Guide to Accompany Exploring Statistics (12th edition) was written by Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, all of Hendrix College. The study guide is available online at exploringstatistics.com.
Students who engage in this book and their course can expect to:
• Solve statistical problems • Understand and explain statistical reasoning • Choose appropriate statistical techniques for common research designs • Write explanations that are congruent with statistical analyses
After many editions with a conventional publisher, Exploring Statistics: Tales of Distributions is now published by Outcrop Publishers. As a result, the price of the print edition is about one-fourth that of the 10th edition. Nevertheless, the authorship and quality of earlier editions continue as before.
The person I acknowledge first is the person who most deserves acknowledgment. And for the 11th and 12th editions, she is especially deserving. This book and its accompanying publishing company, Outcrop Publishers, would not exist except for Thea Siria Spatz, encourager, supporter, proofreader, and cheer captain. This edition, like all its predecessors, is dedicated to her.
Kevin Spatz, manager of Outcrop Publishers, directed the distribution of the 11th edition, advised, week by week, and suggested the cover design for the 12th edition. Justin Murdock now serves as manager, continuing the tradition that Kevin started. Tina Haggard of Fingertek Web Design created the book’s website, the text’s ebook, and the online study guide. She provided advice and solutions for many problems. Thanks to Jill Schmidlkofer, who edited the extensive answer section again for this edition. Emily Jones Spatz created new drawings for the text. I’m particularly grateful to Grace Oxley for a cover design that conveys exploration, and to Liann Lech, who copyedited for clarity and consistency. Walsworth® turned a messy collection of files into a handsome book—thank you Nathan Stufflebean and Dennis Paalhar. Others who were instrumental in this edition or its predecessors include Jon Arms, Ellen Bruce, Mary Kay Dunaway, Bob Eslinger, James O. Johnston, Roger E. Kirk, Rob Nichols, Jennifer Peszka, Mark Spatz, and Selene Spatz. I am especially grateful to Hendrix College and my Hendrix colleagues for their support over many years, and in particular, to Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, who wrote the study guide that accompanies the text.
This textbook has benefited from perceptive reviews and significant suggestions by some 90 statistics teachers over the years. For this 12th edition, I particularly thank
Jessica Alexander, Centenary College Lindsay Kennedy, Hendrix College Se-Kang Kim, Fordham University Roger E. Kirk, Baylor University Kristi Lekies, The Ohio State University Jennifer Peszka, Hendrix College Robert Rosenthal, University of California, Riverside
I’ve always had a touch of the teacher in me—as an older sibling, a parent, a professor, and now a grandfather. Education is a first-class task, in my opinion. I hope this book conveys my enthusiasm for it. (By the way, if you are a student who is so thorough as to read even the acknowledgments, you should know that I included phrases and examples in a number of places that reward your kind of diligence.)
If you find errors in this book, please report them to me at [email protected] I will post corrections at the book’s website: exploringstatistics.com.
O B J E C T I V E S F O R C H A P T E R 1
After studying the text and working the problems in this chapter, you should be able to:
1. Distinguish between descriptive and inferential statistics 2. Define population, sample, parameter, statistic, and variable as they are
used in statistics 3. Distinguish between quantitative and categorical variables 4. Distinguish between continuous and discrete variables 5. Identify the lower and upper limits of a continuous variable 6. Identify four scales of measurement and distinguish among them 7. Distinguish between statistics and experimental design 8. Define independent variable, dependent variable, and extraneous variable
and identify them in experiments 9. Describe statistics’ place in epistemology 10. List actions to take to analyze a data set 11. Identify a few events in the history of statistics
WE BEGIN OUR exploration of statistics with a trip to London. The year is 1900. Walking into an office at University College
London, we meet a tall, well-dressed man about 40 years old. He is Karl Pearson, Professor of Applied Mathematics and Mechanics. I ask him to tell us a little about himself and why he is an important person. He seems authoritative, glad to talk about himself. As a young man, he says, he wrote essays, a play, and a novel, and he also worked for women’s suffrage. These days, he is excited about this new branch of biology called genetics. He says he supervises lots of data gathering.
2 Chapter 1
Pearson, warming to our group, lectures us about the major problem in science—there is no agreement on how to decide among competing theories. Fortunately, he just published a new statistical method that provides an objective way to decide among competing theories, regardless of the discipline. The method is called chi square.1 Pearson says, “Now, arguments will be much fewer. Gather a thousand data points and calculate a chi square test. The result gives everyone an objective way to determine whether or not the data fit the theory.”
Exploration Notes from a student: Exploration off to good start. Hit on a nice, easy-to- remember date to start with, visited a founder of statistics, and had a statistic called chi square described as a big deal.
Our next stop is Rothamsted Experiment Station just north of London. Now the year is 1925. There are fields all around the agricultural research facility, each divided into many smaller plots. The growth in the fields seems quite variable.
Arriving at the office, the atmosphere is congenial. The staff is having tea. There are two topics—a new baby and a new book. We get introduced to Ronald Fisher, the chief statistician. Fisher is a small man with thick glasses and red hair.
He tells us about his new child2 and then motions to a book on the table. Sneaking a peek, we read the title: Statistical Methods for Research Workers. Fisher becomes focused on his book, holding forth in an authoritative way.
He says the book explains how to conduct experiments and that an experiment is just a comparison of two or more conditions. He tells us we don’t need a thousand data points. He says that small samples, randomly selected, are the way for science to progress. “With an experiment and my technique of analysis of variance,” he exclaims, “you can determine why that field out there”—here he waves toward the window—“is so variable. We can find out what makes some plots lush and some mimsy.” Analysis of variance,3 he says, works in any discipline, not just agriculture.
Exploration Notes: Looks like statistics had some controversy in it.4 Also looks like progress. Statistics is used for experiments, too, and not just for testing theories. And Fisher says experiments can be used to compare anything. If that’s right, I can use statistics no matter what I major in.
1 Chi square, which is explained in this book in Chapter 14, has been called one of the 20 most important inventions in the 20th century (Hacking, 1984). 2 (in what will become a family with eight children). 3 explained in Chapters 11-13 4 The slight sniping I’ve built into this story is just a hint of the strong animosity between Fisher and Pearson.
Ronald A. Fisher
Next we go to Poland to visit Jerzy Neyman at his office at the University of Warsaw. It is 1933. As we walk in, he smiles, seems happy we’ve arrived, and makes us feel completely welcome.
Motioning to an envelope on his desk, he tells us it holds a manuscript that he and Egon Pearson5 wrote. “The problem with Fisher’s analysis of variance test is that it focuses exclusively on finding a difference between groups. Suppose the statistical test doesn’t detect a difference. Does that prove there is no difference? No, of course not. It may be that the test was just not sensitive enough to detect the difference. Right?”
At his question, a few of us nod in agreement. Seeing uncertainty, he notes, “Maybe a larger sample is needed to find the difference, you see? Anyway, what we’ve done is expand statistics to cover not just finding a difference, but also what it means when the test doesn’t find a difference. Our approach is what you people in your time will call null hypothesis significance testing.”
Exploration Notes: Statistics seems like a work in progress. Changing. Now it is not just about finding a difference but also about what it means not to find a difference. Also, looks like null hypothesis significance testing is a phrase that might turn up on tests.
Our next trip is to libraries, say, anytime between 1940 and 2000. For this exploration, the task is to examine articles in professional journals published in various disciplines. The disciplines include anthropology, biology, chemistry, defense strategy, education, forestry, geology, health, immunology, jurisprudence, manufacturing, medicine, neurology, ophthalmology, political science, psychology, sociology, zoology, and others. I’m sure you get the idea—the whole range of disciplines that use quantitative measures in their research. What this exploration produces is the discovery that all of these disciplines rely on a data analysis technique called null hypothesis significance testing (NHST).6 Many different statistical tests are employed. However, for all the tests in all the disciplines, the phrase, “p < .05” turns up frequently.
Exploration Notes: It seems that all that earlier controversy has subsided and scientists in all sorts of disciplines have agreed that NHST is the way to analyze quantitative data. All of them seem to think that if there is a comparison to be made, applying NHST is a necessary step to get correct conclusions. All of them use “p < .05,” so I’ll have to be sure to find out exactly what that means.
5 Egon Pearson was Karl Pearson’s son. 6 Null hypothesis significance testing is first explained in Chapters 9 and 10.
4 Chapter 1
Our next excursion is a 1962 visit with Jacob Cohen at New York University in New York City. He is holding his article about studies published in the Journal of Abnormal and Social Psychology, a leading psychology journal. He tells us that the NHST technique has problems. Also, he says we should be calculating an effect size statistic, which will show whether the differences observed in our experiments are large or small.
Exploration Notes: The idea of an effect size index makes a lot of sense. Just knowing there is a difference isn’t enough. How big is the difference? Wonder what “problems with NHST” is all about.
Back to the library for a final excursion to check out recent events. We come across a 2014 article by Geoff Cumming on the “new statistics.” We find things like, “avoid NHST and use better techniques” (p. 26) and “we should not trust any p value” (p. 13). This seems like awfully strong advice. Are researchers taking this advice? Looking through more of today’s research in journals in several fields, we find that most statistical analyses use NHST and there are many instances of “p < .05.”
Exploration Notes, Conclusion: These days, it looks like statistics is in transition again. There’s a lot of controversy out there about how to analyze data from experiments. The NHST approach is still very common, though, so it’s clear I must learn it. But I want to be prepared for changes. I hope knowing NHST will be helpful for the future.7
Welcome to statistics at a time when the discipline is once again in transition. A well- established tradition (null hypothesis significance testing) has been in place for almost a century but is now under attack. New ways of thinking about data analysis are emerging, and along with them, a collection of statistics that do not include the traditional NHST approach. As for the immediate future, though, NHST remains the method most widely used by researchers in many fields. In addition, much of the thinking required for NHST is required for other approaches.
Our exploration tour is over, so I’ll quit supplying notes; they are your responsibility now. As your own experience probably shows, making up your own summary notes improves retention of what you read. In addition, I have a suggestion. Adopt a mindset that thinks growth. A student with a growth mindset expects to learn new things. When challenges arise, as they
7 Not only helpful, but necessary, I would say.
Disciplines that Use Quantitative Data
inevitably do, acknowledge them and figure out how to meet the challenge. A growth mindset treats ability as something to be developed (see Dweck, 2016). If you engage yourself in this course, you can expect to use what you learn for the rest of your life.
The main title of this book is “Exploring Statistics.” Exploring conveys the idea of uncovering something that was not apparent before. An attitude of searching, wondering, checking, and so forth is what I want to encourage. (Those who object to traditional NHST procedures are driven by this exploration motivation.) As for this book’s subtitle, “Tales of Distributions,” I’ll have more to say about it as we go along.
Which disciplines use quantitative data? The list is long and more variable than the list I gave earlier. The examples and problems in this textbook, however, come from psychology, biology, sociology, education, medicine, politics, business, economics, forestry, and everyday life. Statistics is a …
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