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# A first course in business statistics

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• 1. Normal Curve AreasILSource: Abridged from Table I of A. Hald, Statistrcal Tables and Formulas (New York: Wiley), 1952. Reproduced bypermission of A. Hald.
• 2. Critical Values of t - 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645Source: . IUC with the ~dpermission of the Itustees of Biometril from E. S. Pearson and ,edH. 0. Tablesfor Stat~st~crans, 1,3d ed ,Biometrika, 1966. Hartley (eds), The B~ometnka Vol.
• 3. A FIRST C O U R S E IN BUSINESS ...................................................................................................................................................................,.............,.......................- Eighth Edition J A M E S T. M c C L A V E Info Tech, Inc. University of FloridaiI P. GEORGE B E N S O N Terry College of BusinessI University of GeorgiaII TERRY S l N C l C H University of South FloridaII PRENTICE HALL Upper Saddle River, NJ 07458
• 4. "W PROBABILITY 11 7 3.1 Events, Sample Spaces, and Probability 118 3.2 Unions and Intersections 130 3.3 Complementary Events 134 3.4 The Additive Rule and Mutually Exclusive Events 135 3.5 Conditional Probability 140 3.6 The Multiplicative Rule and Independent Events 144 3.7 Random Sampling 154 Statistics in Action: Lottery Buster 158 Quick Review 158 *"ss""ms,"ms%"- W".""""" * " .bb " "* "" . " m . * P RIABLES AND PROBABILITY DISTRIBUTIONS 167 4.1 Two Types of Random Variables 168I 4.2 Probability Distributions for Discrete Random Variables 171 4.3 The Binomial Distribution 181 4.4 The Poisson Distribution (Optional) 194 4.5 Probability Distributions for Continuous RandomI Variables 201 4.6 The Uniform Distribution (Optional) 202 4.7 The Normal Distribution 206 4.8 Descriptive Methods for Assessing Normality 219 4.9 Approximating a Binomial Distribution with a Normal Distribution (Optional) 225 4.10 The Exponential Distribution (Optional) 231 4.11 Sampling Distributions 236 4.12 The Central Limit Theorem 242 Statistics in Action: IQ, Economic Mobility, and the Bell Curve 251 Quick Review 252 Real-World Case: The Furniture Fire Case (A Case Covering Chapters 3-4) 257
• 5. CONTENTS vii sms-*w " "bums INFERENCES BASED ON A SINGLE SAMPLE: ESTIMATION WITH CONFIDENCE INTERVALS 259 5.1 Large-Sample Confidence Interval for a Population Mean 260 5.2 Small-Sample Confidence Interval for a Population Mean 268I 5.3 Large-Sample Confidence Interval for a PopulationI Proportion 279I 5.4 Determining the Sample Size 286I Statistics in Action: Scallops, Sampling, and the Law 292 Quick Review 293I ma"-mum""-"% " m-"us m"-w--t-""m m ""a","- ""rn *""m%*"-"I ASED ON A SINGLEI+I TESTS OF HYPOTHESIS 299 6.1 The Elements of a Test of Hypothesis 300if a 6.2 Large-Sample Test of Hypothesis About a Population Mean 306 6.3 Observed Significance Levels:p-Values 313 6.4 Small-Sample Test of Hypothesis About a Population Mean 319 6.5 Large-Sample Test of Hypothesis About a PopulationI Proportion 326I 6.6 A Nonparametric Test About a Population Median (Optional) 332i Statistics in Action: March Madness-Handicapping Quick Review 338 the NCAA Basketball Tourney 338 "s,,,m*x- -"a"I COMPARING POPULATION MEANS 345 7.1 Comparing Two Population Means: Independent Sampling 346 7.2 Comparing Two Population Means: Paired Difference Experiments 362I 7.3 Determining the Sample Size 374
• 6. viii CONTENTS 7.4 Testing the Assumption of Equal Population Variances (Optional) 377 7.5 A Nonparametric Test for Comparing Two Populations: Independent Sampling (Optional) 384 7.6 A Nonparametric Test for Comparing Two Populations: Paired Difference Experiment (Optional) 393 7.7 Comparing Three or More Population Means: Analysis of Variance (Optional) 400 Statistics in Action: On the Trail of the Cockroach 41 6 Quick Review 418 Real-World Case: The Kentucky Milk Case-Part I I (A Case Covering Chapters 5-7) 426 8.1 Comparing Two Population Proportions: Independent Sampling 428 8.2 Determining the Sample Size 435 8.3 Comparing Population Proportions: Multinomial Experiment 437 8.4 Contingency Table Analysis 445 Statistics in Action: Ethics in Computer Technology and Use 458 Quick Review 461 Real-World Case: Discrimination in the Workplace (A Case Covering Chapter 8) 468 9.1 Probabilistic Models 472 9.2 Fitting the Model: The Least Squares Approach 476- 9.3 Model Assumptions 489 9.4 An Estimator of a2 490- 9.5 Assessing the Utility of the Model: Making Inferences About the Slope PI 494 9.6 The Coefficient of Correlation 505- 9.7 The Coefficient of Determination 509
• 7. 9.8 Using the Model for Estimation and Prediction 516 9.9 Simple Linear Regression: A Complete Example 529 9.10 A Nonparametric Test for Correlation (Optional) 532Statistics in Action: Can "Dowsers" Really Detect Water? 540 Quick Review 544 INTRODUCTION TO MULTIPLE REGRESSION 557 10.1 Multiple Regression Models 558 i / 10.2 The First-Order Model: Estimating and Interpreting the p Parameters 559 10.3 Model Assumptions 565 10.4 Inferences About the P Parameters 568 10.5 Checking the Overall Utility of a Model 580 10.6 Using the Model for Estimation and Prediction 593 10.7 Residual Analysis: Checking the Regression Assumptions 598 10.8 Some Pitfalls: Estimability, Multicollinearity, and Extrapolation 614Statistics in Action: "Wringing" The Bell Curve 624 Quick Review 626 Real-World Case: The Condo Sales Case (A Case Covering Chapters 9-1 0) 634 11.1 Quality, Processes, and Systems 638 11.2 Statistical Control 642 11.3 The Logic of Control Charts 651 11.4 A Control Chart for Monitoring the Mean of a Process: The T-Chart 655 11.5 A Control Chart for Monitoring the Variation of a Process: The R-Chart 672 11.6 A Control Chart for Monitoring the Proportion of Defectives Generated by a Process: The p-Chart 683
• 8. Statistics in Action: Demings 14 Points 692 Quick Review 694 Real-World Case: The Casket Manufacturing Case (A Case Covering Chapter 11) 699 APPENDIXB Tables 707 AP PEN D l X C Calculation Formulas for Analysis of Variance: Independent Sampling 739 ANSWERS TO SELECTED EXERCISES 741 References 747 Index 753
• 9. ",: r This eighth edition of A First Course in Business Statistics is an introductory business text emphasizing inference, with extensive coverage of data collection and analysis as needed to evaluate the reported results of statistical studies and to make good decisions. As in earlier editions, the text stresses the development of statistical thinking, the assessment of credibility and value of the inferences made from data, both by those who consume and those who produce them. It assumes a mathematical background of basic algebra. A more comprehensive version of the book, Statistics for Business and Eco- nomics (8/e), is available for two-term courses or those that include more exten- sive coverage of special topics. NEW IN THE EIGHTH EDITION Major Content Changes Chapter 2 includes two new optional sections: methods for detecting outliers (Section 2.8) and graphing bivariate relationships (Section 2.9). Chapter 4 now covers descriptive methods for assessing whether a data set is ap- proximately normally distributed (Section 4.8) and normal approximation to the binomial distribution (Section 4.9). Exploring Data with Statistical Computer Software and the Graphing Calculator- Throughout the text, computer printouts from five popular Windows-based statistical software packages (SAS, SPSS, MINITAB, STATISTIX and EXCEL) are displayed and used to make decisions about the data. New to this edition, we have included instruction boxes and output for the TI-83 graph- ing calculator. Statistics in Action-One feature per chapter examines current real-life, high- profile issues. Data from the study is presented for analysis. Questions prompt the students to form their own conclusions and to think through the statistical issues involved. Real-World Business Cases-Six extensive business problem-solving cases, with real data and assignments. Each case serves as a good capstone and review of the material that has preceded it. Real-Data Exercises-Almost all the exercises in the text employ the use of cur- rent real data taken from a wide variety of publications (e.g., newspapers, magazines, and journals). Quick Review-Each chapter ends with a list of key terms and formulas, with ref- erence to the page number where they first appear. Language Lab-Following the Quick Review is a pronunciation guide for Greek letters and other special terms. Usage notes are also provided.
• 10. xii TRADITIONAL STRENGTHS We have maintained the features of A First Course in Business Statistics that we believe make it unique among business statistics texts. These features, which assist the student in achieving an overview of statistics and an understanding of its rel- evance in the business world and in everyday life, are as follows: The Use of Examples as a Teaching Device Almost all new ideas are introduced and illustrated by real data-based applica- tions and examples. We believe that students better understand definitions, gen- eralizations, and abstractions after seeing an application. Many Exercises-Labeled by Type The text includes more than 1,000 exercises illustrated by applications in almost all areas of research. Because many students have trouble learning the mechanics of statistical techniques when problems are couched in terms of realistic applica- tions, all exercise sections are divided into two parts: Learning the Mechanics. Designed as straightforward applications of new concepts, these exercises allow students to test their ability to comprehend a concept or a definition. Applying the Concepts. Based on applications taken from a wide variety of jour- nals, newspapers, and other sources, these exercises develop the students skills to comprehend real-world problems and describe situations to which the tech- niques may be applied. A Choice in Level of Coverage of Probability (Chapter 3) One of the most troublesome aspects of an introductory statistics course is the study of probability. Probability poses a challenge for instructors because they must decide on the level of presentation, and students find it a difficult subject to comprehend.We believe that one cause for these problems is the mixture of probability and counting rules that occurs in most introductory texts. We have included the counting rules and worked examples in a separate appendix (Appendix A) at the end of the text. Thus, the instructor can control the level of coverage of probability. Nonparametric Topics Integrated In a one-term course it is often difficult to find time to cover nonparametric tech- niques when they are relegated to a separate chapter at the end of the book. Conse- quently,we have integrated the most commonly used techniques in optional sections as appropriate. Coverage of Multiple Regression Analysis (Chapter 10) This topic represents one of the most useful statistical tools for the solution of ap- plied problems. Although an entire text could be devoted to regression modeling, we believe we have presented coverage that is understandable, usable, and much more comprehensive than the presentations in other introductory statistics texts.
• 11. ,. Footnotes Although the text is designed for students with a non-calculus background, foot- notes explain the role of calculus in various derivations. Footnotes are also used to inform the student about some of the theory underlying certain results. The foot- notes allow additional flexibility in the mathematical and theoretical level at which the material is presented. S U P P L E M E N T S FOR THE INSTRUCTOR The supplements for the eighth edition have been completely revised to reflect the revisions of the text. To ensure adherence to the approaches presented in the main text, each element in the package has been accuracy checked for clarity and freedom from computational, typographical, and statistical errors. Annotated Instructors Edition (AIE) (ISBN 0-13-027985-4) Marginal notes placed next to discussions of essential teaching concepts include:1I Teaching Tips-suggest alternative presentations or point out common stu- dent errors Exercises-reference specific section and chapter exercises that reinforce the concept H-disk icon identifies data sets and file names of material found on the data CD-ROM in the back of the book. Short Answers-section and chapter exercise answers are provided next to the selected exercises Instructors Notes by Mark Dummeldinger (ISBN 0-13-027410-0) This printed resource contains suggestions for using the questions at the end of the Statistics in Action boxes as the basis for class discussion on statistical ethics and other current issues, solutions to the Real-World Cases, a complete short answer book with letter of permission to duplicate for student usc, and many of the exercises and solutions that were removed from previous editions of this text. Instructors Solutions Manual by Nancy S. Boudreau (ISBN 0-1 3-027421 -6) Solutions to all of the even-numbered exercises are given in this manual. Careful attention has been paid to ensure that all methods of solution and notation are consistent with those used in the core text. Solutions to the odd-numbered exer- cises are found in the Students Solutions Manual. Test Bank by Mark Dummeldinger (ISBN 0-1 3-027419-4) Entirely rewritten, the Test Bank now includes more than 1,000 problems that cor- relate to problems presented in the text.
• 13. - Companion Microsoft Excel Manual by Mark Dummeldinger (ISBN 0-1 3-029347-4) Each companion manual works hand-in-glove with the text. Step-by-step keystroke level instructions, with screen captures, provide detailed help for using the technol- ogy to work pertinent examples and all of the technology projects in the text. A cross-reference chart indicates which text examples are included and the exact page reference in both the text and technology manual. Output with brief instruction is provided for selected odd-numbered exercises to reinforce the examples. A Student Lab section is included at the end of each chapter. The Excel Manual includes PHstat, a statistics add-in for Microsoft Excel (CD-ROM) featuring a custom menu of choices that lead to dialog boxes to help perform statistical analyses more quickly and easily than off-the-shelf Excel permits. Student Version of SPSS Student versions of SPSS, the award-winning and market-leading commercial and data analysis package, and MINITAB are available for student purchase. Details on all current products are available from Prentice Hall or via the SPSS Web site at http://www.spss.com. Learning Business Statistics with ~ i c r o s o f t Excel by John L. Neufeld (ISBN 0-13-234097-6) The use of Excel as a data analysis and computational package for statistics is ex- plained in clear, easy-to-follow steps in this self-contained paperback text. A MINITAB Guide to Statistics by Ruth Meyer and David Krueger (ISBN 0-1 3-784232-5) This manual assumes no prior knowledge of MINITAB. Organized to correspond to the table of contents of most statistics texts, this manual provides step-by-step instruction to using MINITAB for statistical analysis. ConStatS by Tufts University (ISBN 0-1 3-502600-8) ConStatS is a set of Microsoft Windows-based programs designed to help col- lege students understand concepts taught in a first-semester course on proba- bility and statistics. ConStatS helps improve students conceptual understanding of statistics by engaging them in an active, experimental style of learning. A companion ConStatS workbook (ISBN 0-13-522848-4) that guides students through the labs and ensures they gain the maximum benefit is also available. ACKNOWLEDGMENTS This book reflects the efforts of a great many people over a number of years. First we would like to thank the following professors whose reviews and feedback on orga- nization and coverage contributed to the eighth and previous editions of the book.
• 14. xvi PREFACE Reviewers Involved with the Eighth Edition Mary C. Christman, University of Maryland; James Czachor, Fordham-Lincoln Center, AT&T; William Duckworth 11, Iowa State University; Ann Hussein, Ph.D., Philadelphia University; Lawrence D. Ries, University of Missouri-Columbia. Reviewers of Previous Editions Atul Agarwal, GMI Engineering and Management Institute; Mohamed Albohali, Indiana University of Pennsylvania; Gordon J. Alexander, University of Min- nesota; Richard W. Andrews, University of Michigan; Larry M. Austin, Texas Tech University; Golam Azam, North Carolina Agricultural & Technical University; Donald W. Bartlett, University of Minnesota; Clarence Bayne, Concordia Uni- versity; Carl Bedell, Philadelphia College of Textiles and Science; David M. Bergman, University of Minnesota; William H. Beyer, University of Akron; Atul Bhatia, University of Minnesota; Jim Branscome, University of Texas at Arlington; Francis J. Brewerton, Middle Tennessee State University; Daniel G. Brick, Uni- versity of St. Thomas; Robert W. Brobst, University of Texas at Arlington; Michael Broida, Miami University of Ohio; Glenn J. Browne, University of Maryland, Bal- timore; Edward Carlstein, University of North Carolina at Chapel Hill; John M. Charnes, University of Miami; Chih-Hsu Cheng, Ohio State University; Larry Claypool, Oklahoma State University; Edward R. Clayton, Virginia Polytechnic Institute and State University; Ronald L. Coccari, Cleveland State University; Ken Constantine, University of New Hampshire; Lewis Coopersmith, Rider Uni- versity; Robert Curley, University of Central Oklahoma; Joyce Curley-Daly, Cal- ifornia Polytechnic State University; Jim Daly, California Polytechnic State University; Jim Davis, Golden Gate University; Dileep Dhavale, University of Northern Iowa; Bernard Dickman, Hofstra University; Mark Eakin, University of Texas at Arlington; Rick L. Edgeman, Colorado State University; Carol Eger, Stanford University; Robert Elrod, Georgia State University; Douglas A. Elvers, University of North Carolina at Chapel Hill; Iris Fetta, Clemson University; Susan Flach, General Mills, Inc.; Alan E. Gelfand, University of Connecticut; Joseph Glaz, University of Connecticut; Edit Gombay, University of Alberta; Jose Luis Guerrero-Cusumano, Georgetown University; Paul W. Guy, California State Uni- versity, Chico; Judd Hammack, California State University-Los Angeles; Michael E. Hanna, University of Texas at Arlington; Don Holbert, East Carolina Univer- sity; James Holstein, University of Missouri, Columbia; Warren M. Holt, South- eastern Massachusetts University; Steve Hora, University of Hawaii, Hilo; Petros Ioannatos, GMI Engineering & Management Institute; Marius Janson, University of Missouri, St. Louis; Ross H. Johnson, Madison College; I? Kasliwal, California State University-Los Ange1es;Timothy J. Killeen, University of Connecticut;Tim Krehbiel, Miami University of Ohio; David D. Krueger, St. Cloud State Universi- ty; Richard W. Kulp, Wright-Patterson AFB, Air Force Institute of Technology; Mabel T. Kung, California State University-Fullerton; Martin Labbe, State Uni- versity of New York College at New Paltz; James Lackritz, California State Uni- versity at San Diego; Lei Lei, Rutgers University; Leigh Lawton, University of St. Thomas; Peter Lenk, University of Michigan; Benjamin Lev, University of Michi- gan-Dearborn; Philip Levine, William Patterson College; Eddie M. Lewis, Uni- versity of Southern Mississippi; Fred Leysieffer, Florida State University; Xuan Li, Rutgers University; Pi-Erh Lin, Florida State University; Robert Ling, Clemson University; Benny Lo; Karen Lundquist, University of Minnesota; G. E. Martin,
• 15. Clarkson University; Brenda Masters, Oklahoma State University; William Q.Meeker, Iowa State University; Ruth K. Meyer, St. Cloud State University; Ed-ward Minieka, University of Illinois at Chicago; Rebecca Moore, Oklahoma StateUniversity; June Morita, University of Washington; Behnam Nakhai, MillersvilleUniversity; Paul I. Nelson, Kansas State University; Paula M. Oas, General OfficeProducts; Dilek Onkal, Bilkent University,Turkey;Vijay Pisharody, University ofMinnesota; Rose Prave, University of Scranton; P. V. Rao, University of Florida;Don Robinson, Illinois State University; Beth Rose, University of Southern Cali-fornia; Jan Saraph, St. Cloud State University; Lawrence A. Sherr, University ofKansas; Craig W. Slinkman, University of Texas at Arlingon; Robert K. Smidt, Cal-ifornia Polytechnic State University; Toni M. Somers, Wayne State University;Donald N. Steinnes, University of Minnesota at Du1uth;Virgil F. Stone,Texas A &M University; Katheryn Szabet, La Salle University; Alireza Tahai, MississippiState University; Kim Tamura, University of Washington; Zina Taran, RutgersUniversity; Chipei Tseng, Northern Illinois University; Pankaj Vaish, Arthur An-dersen & Company; Robert W. Van Cleave, University of Minnesota; Charles EWarnock, Colorado State University; Michael P. Wegmann, Keller GraduateSchool of Management; William J. Weida, United States Air Force Academy; T. J.Wharton, Oakland University; Kathleen M. Whitcomb, University of South Car-olina; Edna White, Florida Atlantic University; Steve Wickstrom, University ofMinnesota; James Willis, Louisiana State University; Douglas A. Wolfe, Ohio StateUniversity; Gary Yoshimoto, St. Cloud State University; Doug Zahn, Florida StateUniversity; Fike Zahroom, Moorhead State University; Christopher J. Zappe,Bucknell University. Special thanks are due to our ancillary authors, Nancy Shafer Boudreau andMark Dummeldinger, and to typist Kelly Barber, who have worked with us formany years. Laurel Technical Services has done an excellent job of accuracychecking the eighth edition and has helped us to ensure a highly accurate, cleantext. Wendy Metzger and Stephen M. Kelly should be acknowledged for theirhelp with the TI-83 boxes. The Prentice Hall staff of Kathy Boothby Sestak,Joanne Wendelken, Gina Huck, Angela Battle, Linda Behrens, and Alan Fischer,and Elm Street Publishing Services Martha Beyerlein helped greatly with allphases of the text development, production, and marketing effort. We acknowl-edge University of Georgia Terry College of Business MBA students Brian F.Adams, Derek Sean Rolle, and Misty Rumbley for helping us to research and ac-quire new exerciselcase material. Our thanks to Jane Benson for managing theexercise development process. Finally, we owe special thanks to Faith Sincich,whose efforts in preparing the manuscript for production and proofreading allstages of the book deserve special recognition. For additional information about texts and other materials available fromPrentice Hall, visit us on-line at http://www.prenhall.com. James T. McClave P George Benson . Terry Sincich
• 16. TO THE STUDENTThe following four pages will demonstrate how to use this text effectively to makestudying easier and to understand the connection between statistics and your world.Chapter Openers Providea RoadmapWhere Weve Been quicklyreviews how information learned SIMPLE LINEAR REGRESSIONpreviously applies to the chapterat hand. C O N T E N T S 9 1 Pmhah~litic odel M 9.2 R r l m glhc Modcl.Thc Least S q u a r e s A p p r o a c hWhere Were Going highlights 9.3 Model A u m p l ~ i m s 9.4 An E t m s t o r of oZhow the chapter topics fit into 9.5 Asscssm~ the U t i l i t y of the M<,dcl M a k m g Inferences A b o u t t h e Slope 8, 96 T h c C o c t l ~ c r oor ( w r c l a t w n lyour growing understanding Y7 nt~ (.OC~I~CIC~IIm r n o n ,>I I l c l e r r n 9.8 I l m g l l l c M o d e l h r I I ~ m i ! l ~ c m Prcdmion andof statistical inference. 9.9 Smple Lmuar I l e g r e w w A C o m p l c t c E x a m p l e 9.lU A N o n p a r a m e t n c T c r t tor C u r r e l a l l a n (Optional) S T A I N .,.. .T I S T . I C . .. . S . . ........ . ..... ........ ...........I . O . .N. A C T . Can "Dowscn"Real1y D e t e c t Water? huuw 11 wc measure ~CIUBTL.foolape ;and .ige .dl the rmm cilmc as assescd value. we can ct.hhrh I h r rr- SECTION 4.12 The Central Limit Theorem 251 l a t t o n h ~ p c l % c c n ~ h c c a n a h k - - o n e that lets us h v uc l h c r v a r l s h b klr p r e d s t m n Tht, chapter cov- ABL~B~J.~ e n Ihc implcl vlualmn-relatmp t w o varmhlrs. STATISTICS IQ, Econorn~cMobility, and the Bell Curve I n thc8r cunrrovcn#alhook l i a , I I d O r r v r I N (Free Prcsr. able having a normal dl.tnhut,on wllh mean p = I M nlld (tandad davlallonl. - I5 msd,slnb"tlon,or h r l i a l n r 8, The m o l e c o m p l e x p r o h l c m ot r c l a t m g m o r e t h a n t w o v a r ~ h l e the l o p r o i C h a p l c r I 0 8, shown In Flgurc449 In lhclr h o d ilcrrnlr#nand M u r r a y relsr l o l n c c o m l nvc cl.wc o l pioplc dillncd hv pmccnl~lc~ the oorm.il of dlrfnhul,,,n c1.nr I (rcrv hnlht)c,,n*,*sol lllore ~ 8 t lQ h s h , K ,hi ,W ,pir<rnlllc Cl,, . 1 ( Ihnell, 1 .,,c ,1,,nc ~ 8 t h 1 lo* ~ C ! U C L CC75ll1 .md lJS1li ~WCLIIIIICI Cl.i) I l l ( n o r ~I mnl) m c u d i 1 1hcluccn l l X l h .md 71111 pi#tcnl~les: 0 ~ encer drawn l a m (hem (Scc 10 crmlplr "Mllnplni. lIr< R"li<,,,L* Acnull,,n,n lnlcah<lulthc rilillll>lldlllli.imonp CI~, I V (dull) .arc ~ h mul11, 10,hctwcin (hc rll> and i 251h pi.lccmllo .idCl.M ( v i r i dull") .lrc 10, below the "Statistics in Action" Boxes. rrce, gcnc*, and 10. < h a m e Summer 1995 ) In C h . w r 5th pc#ccnt,lrlbwccl.i~~c5.~8c.41~~ ~lla*lralcd~n c 4 4 9 iiw llXSt~t#,l~o mAct#on,wc r x p l o r e ~i u 11, thsw pn,hli.m< l Explore High-Interest Issues One ol fhc man! i o s l r o v u ~ r 5prrkcd lh) I h i hmk 8, F o c u s (or the authortcnct ,ha< lcvel of ~nrclllgsncc l a d thi.rcul1 Highlight controversial, contemporary a tvfhcr whmc cnrn,ng ;,,c In Ihc 1>o,,c,m i,ru pcr<cn, of issues that involve statistics. thc [#ncomrldluuhutam hd, wmclhlng lbkuims ~h.ince8 n f w c n t y (or lesi) of n v n g lo the top fdlh ,it Iha rnoimc dl- trlhulloll and n,m,n, , lif,v-l,f,, I h.3"~~01 r;n,ns 8" the hc,, , m llhb 11c ha Ic*, 1h.i" <oni h.,"e I , In", o l m,ng , Work through the "Focus" questions *hove ercn t1,c mcil,,,,, ,na,mc peliplc .It p,ccnt M,,,, are stuck ncn, u1,err ,1,c,, p.mn,, wcrc on ihc 1nr<,medls to help you evaluate the findings. tr#huf#,m p.trl 1 h c c . 1 ~ ~ ~n Ilntcll~r~nccl.ahahhrhccnmea Integration of Real-World Data helps students see relevance to their daily lives.xviii
• 17. Shaded Boxes HighlightImportant Information Definitions, Strategies, Key Formulas, Rules, and other important information is Assigning probabilities to sample points is easy for some experiments. For example, if thc cxpcrment is to toss a fair coin and observe the face. we would highlighted in easy-to-readboxes. probably all agrcc to assign a probability of to the two sample pomts. Ohserve a bead and Obscrvc a tail. Howrver,many experlmenls have sample points whose probabilities arc more dlfflcult to assign. Prepare for quizzes and tests by reviewing the highlighted information. of each tvoc 01 PC to stock. An imuortant factor affectme the solut~on the 1s / proportion o curturners who purchase each type of PC. Show how t h ~ problem f s mieht be tonnulated in the framework of an experiment w ~ t h . . sample uolnts and a sample space. Indicate how prohahillties m~ght awgncd to the sample he points.Interesting Examples/ 5 o I u t io n If we use the term customer to refer to a uerson who uurchases one of the twowith Solutions typcs of PCs. the experment can be defined as the entrance of a customer and the obqervation ot whlch tvve of PC 1s ~urchaqed.Therc two sample points In the arc . . Examples, witfi complete sample space corresponding to thls experiment: I): (The customer purchase? a standard desktop unit) step-by-step solutions and L: (lhe customer purchases a laptop unit) explanations, illustrate every The difference between this and the coln-toss experiment becomes apparent when wc attcmpt to assign probahilit~es the two sample pointy. What prohah~li- lo concept and are numbered ty ~ h o u l d asign to the sample point I)? If you answer 5. you are awummg that wc the evcnts D and L should occur with equal I~kel~hood.~ust c ~ a m ~uolnts l ~ k thc lc for easy reference.* Solutions are carefully explained to . prepare for the section exercise set. The end of the solution is clearly marked by a symbol. Then we use* 5 o I u t i o n The SAS prmtout drbcnhlng lhc RdiU percentage dala 1s dkplaycd m Hgurr 2.19. The vnrrancc and sldnda~d dcwatcon, hlghhghlcd on thc pl~nloul, are: ,Computer Output ri = 3,922792 and r = 1.980604 Integrated Throughout Fluar 2 19 SAS printout of numerical Statistical software packages such , dercrlptlve measurer for 50 RhD percentages mmenrs as SPSS, MINITAB, SAS, and EXCEL N 50 Sun Wgts 50 Hean Std DBY 8.492 1.980604 Sun Variance 424.6 3.922792 crunch data quickly so you can Skewness "9s 0.854601 3797.92 Kurtosis CSS 0.419288 192.2168 spend time analyzing the results. CV 23.32317 std man 0.2801 T:*Ban.O 30.31778 Prob, T 0.0001 Learning how to interpret statistical S9n RanL 637.5 Prob>lSl 0.0001 Num - 0 so output will prove helpful in future Quantiles(Def-51 1004 ldax 13.5 99% 13.5 classes or on the job. 75% 93 9.6 95% 13.2 50% Hed 8.05 90% 11.2 25% 0% Q1 Mi" 7.1 5.2 10% 5% 6.5 5.9 When computer output appears 1% 5.1 8.3 in examples, the solution explains 2.5 how to read and interpret the output. xix
• 18. p v ~ o t of Exercises for Practice sLearning the Mechanics Applying the Concepts Every section in the book is 2.35 The total number of passengers handled m 1998 by2.311 (Mculalr the modc. mean, and median of the following data: e x h t cruse h p based ~n Port Canaveral (Florida) are followed by an Exercise Set divided l&d m the table below Find and interpret the mean 18 10 15 13 17 15 12 15 18 16 11 and median ot the data set into two parts:2 3 Calculate the mean and median of the following grade .1 point averages 1.2 25 21 3.7 2.8 2.0 Learning the Mechanics has2 3 Calculate the mean for samples where .2 a. n = 10.Z.t = XS b. n = 16.21 = 4W straightforward applications c. n = 45. Zx = 35 (1. n = 1 8 . 2 ~ 242 =2 3 Calculate the mean, medlan, and mode for each of the .3 tmn of these 50 womm. Numencal descnpttve statlrtics of new concepts. Test your for the data are ?how" ~n the MINITAB pnntout below. a. Find the mran, rnedlan. and modal agc of the d~strl- mastery of definitions, bution, Interpret the% values. how lhc mran compares to the median for a23 Drscr~hr .4 b. What do thc mran and the median mdicate about concepts, and basic the skewnes of the age dlstrlbutlon? d~strthutlonas follows: a. Skewed to the left b. Skewed to the right e. What percentage of these women are in their for- computation. Make sure ties? Theu flfttrs"Thelr sixtles? c. Symmetric you can answer all of these - questions before moving on. Cruise Line (Ship) Number of Passengers ........................... . . ............................. Canaveral (Dolphin) Carnival (Fantaw) 152,240 480,924 Applying the Concepts Dmey ( M a w ) Premier (Ocranlc) 71,504 270.361 tests your understanding of Royal Caribbean (Nordic Empress) lll%l6l Sun CTUZ Casinos 453.806 concepts and requires you to Strrlmg Cmses (New Yorker) 15,782 T m a r Intl Shmmne. . . . (Topaz) - Val 41. No 9,Jan 1939 - 28,280 apply statistical techniques in source llorrdu ~rend solving real-world problems. MINITAB Output for Exercise 2.36 ~escriptiveStatistics Variable N Mean Median Tr Mean StDev SE Mean Age 50 48.160 47.000 47.795 6.015 0.851 Variable Min Max 01 Q3 AW 36.000 68.000 45.000 51.aso I ~ e a l Data . Computer Output Most of the exercises contain data or information taken Computer output screens appear from newspaper articles, in the exercise sets to give you magazines, and journals. practice in interpretation. Statistics are all around you.
• 20. STATISTICS, DATA, A N D STATISTICAL T H I N K I N GC O N T E N T S.....,... ................................................ 1.1 The Science of Statistics 1.2 Types of Statistical Applications in Business 1.3 Fundamental Elements of Statistics 1.4 Processes (Optional) 1.5 Types of Data 1.6 Collecting Data 1.7 The Role of Statistics in Managerial Decision-MakingS T A T I S T I C S I N A C T I O N . A 20/20 View of Survey Results: Fact or Fiction?I Where Were Going scribe some business phenomenon, and inferentialS tatistics? Is it a field of study, a group of numbers that summarizes the statc of our national ccono-my, the performance of a stock, or the business con- statistics, which uses these numerical summaries to assist in making business decisions. The primaryditions in a particular locale? Or, as one popular theme of this text is inferential statistics. Thus, wellbook (Tanur et al., 1989) suggests, is it "a guide to concentrate on showing how you can use statisticsthe unknown"? Well see in Chapter 1 that each of to interpret data and use them to make decisions.these descriptions is applicable in understanding Many jobs in industry, government, medicine, andwhat statistics is. Well see that there are two areas of other fields require you to make data-driven deci-statistics: descriptive statistics, which focuses on de- sions, so understanding these methods offers you im-veloping graphical and numerical summaries that de- portant practical benefits.
• 22. SECTION 1.2 T y p e s o f S t a t i s t i c a l A p p l i c a t i o n s in B u s i n e s s 3,- DEFINITION 1.3 Inferential statistics utilizes sample data to make estimates, decisions, predictions, or other generalizations about a larger set of data. Although well discuss both descriptive and inferential statistics in the fol- lowing chapters, the primary theme of the text is inference. Lets begin by examining some business studies that illustrate applications of statistics. Study 1 "U.S. Market Share for Credit Cards" ( The Nilson Report, Oct. 8,1998) The Nilson Report collected data on all credit or debit card purchases in the Unit- ed States during the first six months of 1998.The amount of each purchase was recorded and classified according to type of card used. The results are shown in the Associated Press graphic, Figure 1.1. From the graph, you can clearly see that half of the purchases were made with a VISA card and one-fourth with a Master- Card. Since Figure 1.1 describes the type of card used in all credit card purchases for the first half of 1998, the graphic is an example of descriptive statistics. FIGURE 1.1 U.S. Credit Card Market Shares Source: The Nilson Report, Oct. 8, 1998. Diners Club 1% Study 2 "The Executive Compensation Scoreboard" (Business Week, Apr. 19, 1999) How much are the top corporate executives in the United States being paid and are they worth it? To answer these questions, Business Week magazine compiles its "Executive ~ o m ~ e n s a t i o n Scoreboard" each year based on a survey of executives at the highest-ranking companies listed in the Business Week 1000. The average* total pay of chief executive officers (CEOs) at 365 companies sampled in the 1998 scoreboard was \$10.6 million-an increase of 36% over the previous year. *Although we will not formally define the term average until Chapter 2, typical or middle can be substituted here without confusion.
• 24. SECTION 1.3 Fundamental Elements of Statistics 5c FINITION 1.4 A population is a set of units (usually people, objects, transactions, or events) that we are interested in studying. For example, populations may include (1) all employed workers in the United States, (2) all registered voters in California, (3) everyone who has pur- chased a particular brand of cellular telephone, (4) all the cars produced last year by a particular assembly line, ( 5 ) the entire stock of spare parts at United Airlines maintenance facility, (6) all sales made at the drive-through window of a McDonalds restaurant during a given year, and (7) the set of all accidents oc- curring on a particular stretch of interstate highway during a holiday period. Notice that the first three population examples (1-3) are sets (groups) of people, the next two (4-5) are sets of objects, the next (6) is a set of transactions, and the last (7) is a set of events. Also notice that each set includes all the units in the population of interest. In studying a population, we focus on one or more characteristics or prop- erties of the units in the population. We call such characteristics variables. For example, we may be interested in the variables age, gender, income, and/or the number of years of education of the people currently unemployed in the United States. DEFINITION 1.5 A variable is a characteristic or property of an individual population unit. The name "variable" is derived from the fact that any particular character- istic may vary among the units in a population. In studying a particular variable it is helpful to be able to obtain a numerical representation for it. Often, however, numerical representations are not readily available, so the process of measurement plays an important supporting role in statistical studies. Measurement is the process we use to assign numbers to vari- ables of individual population units. We might, for instance, measure the prefer- ence for a food product by asking a consumer to rate the products taste on a scale from 1 to 10. Or we might measure workforce age by simply asking each worker how old she is. In other cases, measurement involves the use of instruments such as stopwatches, scales, and calipers. If the population we wish to study is small, it is possible to measure a vari- able for every unit in the population. For example, if you are measuring the start- ing salary for all University of Michigan MBA graduates last year, it is at least feasible to obtain every salary. When we measure a variable for every unit of a population, the result is called a census of the population. Typically, however, the , I populations of interest in most applications are much larger, involving perhaps many thousands or even an infinite number of units. Examples of large popula- tions include those following Definition 1.4, as well as all invoices produced in the last year by a Fortune 500 company, all potential buyers of a new fax machine, and all stockholders of a firm listed on the New York Stock Exchange. For such popu- lations, conducting a census would be prohibitively time-consuming and/or costly.
• 25. 6 1 CHAPTER Statistics, Data, and Statistical Thinking A reasonable alternative would be to select and study a subset (or portion) of the units in the population. A sample is a subset of the units of a population. For example, suppose a company is being audited for invoice errors. Instead of examining all 15,472 invoices produced by the company during a given year, an auditor may select and examine a sample of just 100 invoices (see Figure 1.2). If he is interested in the variable "invoice error status," he would record (measure) the status (error or no error) of each sampled invoice. F I G U R E 1.2 Population Sample A sample o all f company invoices -7 1st invoice selected -7 2nd invoice selected r7 - 15,472 invoices - 100th invoice selected After the variable(s) of interest for every unit in the sample (or popula- tion) is measured, the data are analyzed, either by descriptive or inferential sta- tistical methods. The auditor, for example, may be interested only in describing the error rate in the sample of 100 invoices. More likely, however, he will want to use the information in the sample to make inferences about the population of all 15,472 invoices.
• 26. L SECTION 1.3 Fundamental Elements o f Statistics 7 DEFINITION 1.7 A statistical inference is an estimate or prediction or some other generaliza- tion about a population based on information contained in a sample. That is, we use the information contained in the sample to learn about the larger population." Thus, from the sample of 100 invoices, the auditor may esti- mate the total number of invoices containing errors in the population of 15,472 in- voices. The auditors inference about the quality of the firms invoices can be used in deciding whether to modify the firms billing operations. underfilled paint cans. As a result, the retailer has begun inspecting incoming shipments of paint from suppliers. Shipments with underfill prob1;ms will be returned to the supplier. A recent shipment contained 2,440 gallon-size cans. The retailer sampled 50 cans and weighed each on a scale capable of measuring weight to four decimal places. Properly filled cans weigh 10 pounds. a. Describe the population. b. Describe the variable of interest. c. Describe the sample. , . d. Describe the inference.So Iut io n a. The population is the set of units of interest to the retailer, which is the shipment of 2,440 cans of paint. I b. The weight of the paint cans is the variable the retailer wishes to evaluate. c. The sample is a subset of the population. In this case, it is the 50 cans of paint selected by the retailer. d. The inference of interest involves the generalization of the information con- tained in the weights of the sample of paint cans to the population of paint cans. In particular, the retailer wants to learn about the extent of the under- fill problem (if any) in the population. This might be accomplished by find- ing the average weight of the cans in the sample and using it to estimate the average weight of the cans in the population. s ..%""a wa"""-"mmm"" " m S" """ """*"rn ""msss"w"~mm"w m-"la%,-"t"- m""wn*-m""am"*"m-"a"" "" "Cola wars" is the popular term for the intense competition between Coca-Cola and Pepsi displayed in their marketing campaigns.Their campaigns have featured movie and television stars, rock videos, athletic endorsements, and claims of consumer preference based on taste tests. Suppose, as part of a Pepsi marketing campaign, 1,000 cola consumers are given a blind taste test (i.e., a taste test in which the two brand names are disguised). Each consumer is asked to state a preference for brand A or brand B. a. Describe the population. b. Describe the variable of interest. *The termspopulation and sample are often used to refer to the sets of measurements themselves, as well as to the units on wh~ch measurements are made. When a single variable of interest 1s the being measured, this usage causes little confusion But when the terminology is ambiguous, well refer to the measurements as populutmn dutu sets and sample dutu wtr, respectively
• 27. CHAPTER 1 Statistics, Data, and Statistical Thinking c. Describe the sample. d. Describe the inference. S o Iut i o n a. The population of interest is the collection or set of all cola consumers. b. The characteristic that Pepsi wants to measure is the consumers cola pref- erence as revealed under the conditions of a blind taste test, so cola prefer- ence is the variable of interest. c. The sample is the 1,000 cola consumers selected from the population of all cola consumers. d. The inference of interest is the generalization of the cola preferences of the 1,000 sampled consumers to the population of all cola consumers. In partic- ular, the preferences of the consumers in the sample can be used to estimate the percentage of all cola consumers who prefer each brand. The preceding definitions and examples identify four of the five elements of an inferential statistical problem: a population, one or more variables of interest, a sample, and an inference. But making the inference is only part of the story.We also need to know its reliability-that is, how good the inference is.The only way we can be certain that an inference about a population is correct is to include the entire population in our sample. However, because of resource constraints (i.e., in- sufficient time and/or money), we usually cant work with whole populations, so we base our inferences on just a portion of the population (a sample). Conse- quently, whenever possible, it is important to determine and report the reliability of each inference made. Reliability, then, is the fifth element of inferential statis- tical problems. The measure of reliability that accompanies an inference separates the sci- ence of statistics from the art of fortune-telling. A palm reader, like a statistician, may examine a sample (your hand) and make inferences about the population (your life). However, unlike statistical inferences, the palm readers inferences include no measure of reliability. Suppose, as in Example 1.1, we are interested in estimating the average weight of a population of paint cans from the average weight of a sample of cans. Using statistical methods, we can determine a bound o n the estimation error. This bound is simply a number that our estimation error (the difference between the average weight of the sample and the average weight of the population of cans) is not likely to exceed. Well see in later chapters that this bound is a measure of the uncertainty of our inference. The reliability of statistical inferences is discussed throughout this text. For now, we simply want you to realize that an inference is incomplete without a measure of its reliability. DEFINITION 1.8 A measure of reliability is a statement (usually quantified) about the degree of uncertainty associated with a statistical inference. Lets conclude this section with a summary of the elements of both descrip- tive and inferential statistical problems and an example to illustrate a measure of reliability.
• 28. i ,. SECTION 1.4 Processes ( O p t i o n a l ) 9 scriptive Statistical Proble 1. The population or sample of interest 2. One or more variables (characteristics of the population or sampl units) that are to be investigated 3. Tables, graphs, or numerical summary tools 4. Conclusions about the data based on the patterns revealed 2. One or more variables (characteristics of the population units) that are to be investigated 3. The sample of population units 4. The inference about the population based on information contained in the sample - u s " s , - ~ w ~ ~ m a , , sumers were indicated in a taste test. Describe how the reliability of an inference concerning. - the preferences of all cola consumers in the Pepsi bottlers marketing region could be measured.S o I ut io n When the preferences of 1,000 consumers are used to estimate the preferences of all consumers in the region, the estimate will not exactly mirror the preferences of the population. For example, if the taste test shows that 56% of the 1,000 consumers chose Pepsi, it does not follow (nor is it likely) that exactly 56% of all cola drinkers in the region prefer Pepsi. Nevertheless, we can use sound statistical reasoning (which is presented later in the text) to ensure that our sampling procedure will generate estimates that are almost certainly within a specified limit of the true percentage of all consumers who prefer Pepsi. For example, such reasoning might assure us that the estimate of the preference for Pepsi from the sample is almost certainly within 5% of the actual population preference. The implication is that the actual preference for Pepsi is between 51% [i.e., (56 - 5)%] and 61% [i.e., (56 + 5)%]-that is, (56 ? 5)%. This interval represents a measure of reliability for the inference. * PROCESSES (OPTIONAL) Sections 1.2 and 1.3 focused on the use of statistical methods to analyze and learn about populations, which are sets of existing units. Statistical methods are equally useful for analyzing and making inferences about processes. DEF A process is a series of actions or operations that transforms inputs to outputs. A process produces or generates output over time.
• 29. 10 1 CHAPTER Statistics, Data, and Statistical Thinking The most obvious processes that are of interest to businesses are production or manufacturing processes. A manufacturing process uses a series of operations performed by people and machines to convert inputs, such as raw materials and parts, to finished products (the outputs). Examples include the process used to produce the paper on which these words are printed, automobile assembly lines, and oil refineries. Figure 1.3 presents a general description of a process and its inputs and out- puts. In the context of manufacturing, the process in the figure (i.e., the transfor- mation process) could be a depiction of the overall production process or it could be a depiction of one of the many processes (sometimes called subprocesses) that exist within an overall production process. Thus, the output shown could be fin- ished goods that will be shipped to an external customer or merely the output of one of the steps or subprocesses of the overall process. In the latter case, the out- put becomes input for the next subprocess. For example, Figure 1.3 could repre- sent the overall automobile assembly process, with its output being fully assembled cars ready for shipment to dealers. Or, it could depict the windshield assembly subprocess, with its output of partially assembled cars with windshields ready for "shipment" to the next subprocess in the assembly line.FIGURE 1.3 INPUTS OUTPUTSGraphical depiction of a TRANSFORMATION PROCESSmanufacturing process Information Methods -, I Materials Machines People --- Implemented by people and/or mach~nes I Besides physical products and services, businesses and other organizations generate streams of numerical data over time that are used to evaluate the per- formance of the organization. Examples include weekly sales figures, quarterly earnings, and yearly profits. The U.S. economy (a complex organization) can be thought of as generating streams of data that include the Gross Domestic Product (GDP), stock prices, and the Consumer Price Index (see Section 1.2). Statisti- cians and other analysts conceptualize these data streams as being generated by processes.Typically, however, the series of operations or actions that cause partic- ular data to be realized are either unknown or so complex (or both) that the processes are treated as bluck boxes. A process whose operations or actions are unknown or unspecified is called a black box. Frequently, when a process is treated as a black box, its inputs are not spec- ified either. The entire focus is on the output of the process. A black box process is illustrated in Figure 1.4.
• 30. SECTION 1.4 Processes (Optional) 11 FFIGURE 1.4 TRANSFORMATION PROCESSA black box process withnumerical output INPUTS -I In studying a process, we generally focus on one or more characteristics, or properties, of the output. For example, we may be interested in the weight or the length of the units produced or even the time it takes to produce each unit. As with characteristics of population units, we call these characteristics variables. In studying processes whose output is already in numerical form (i.e., a stream of numbers), the characteristic, or property, represented by the numbers (e.g., sales, GDP, or stock prices) is typically the variable of interest. If the output is not nu- meric, we use measurement processes to assign numerical values to variables." For example. if in the automobile assembly process the weight of the fully assem- bled automobile is the variable of interest, a measurement process involving a large scale will be used to assign a numerical value to each automobile. As with populations, we use sample data to analyze and make inferences (es- timates, predictions, or other generalizations) about processes. But the concept of a sample is defined differently when dealing with processes. Recall that a popula- tion is a set of existing units and that a sample is a subset of those units. In the case of processes, however, the concept of a set of existing units is not relevant or ap- propriate. Processes generate or create their output over time-one unit after an- other. For example, a particular automobile assembly line produces a completed vehicle every four minutes. We define a sample from a process in the box. Any set of output (objects or numbers) produced by a process is called a sample. Thus, the next 10 cars turned out by the assembly line constitute a sample from the process, as do the next 100 cars or every fifth car produced today. considering offering a 50% discount to customers who wait more than a specified , number of minutes to receive their order. To help determine what the time limit should be, the company decided to estimate the average waiting time at a particular drive-through window in Dallas,Texas.For seven consecutive days, the worker taking customers orders recorded the time that every order was placed. The worker who handed the order to the customer recorded the time of delivery. In both cases, workers used synchronized digital clocks that reported the time to the nearest second. At the end of the 7-day period, 2,109 orders had been timed. *A process whose output is already in numerical form necessarily includes a measurement process as one of its subprocesses.
• 32. SECTION 1.5 Types of Data 13 TYPES OF DATAYou have learned that statistics is the science of data and that data are obtainedby measuring the values of one or more variables on the units in the sample (orpopulation). All data (and hence the variables we measure) can be classified asone of two general types: quantitative data and qualitative data. Quantitative data are data that are measured on a naturally occurring nu-merical scale." The following are examples of quantitative data: 1. The temperature (in degrees Celsius) at which each unit in a sample of 20 pieces of heat-resistant plastic begins to melt 2. The current unemployment rate (measured as a percentage) for each of the 50 states 3. The scores of a sample of 150 MBA applicants on the GMAT, a standardized business graduate school entrance exam administered nationwide 4. The number of female executives employed in each of a sample of 75 man- ufacturing companies DEFINITION 1.12 Quantitative data are measurements that are recorded on a naturally occur- ring numerical scale. In contrast, qualitative data cannot be measured on a natural numericalscale; they can only be classified into categoriest Examples of qualitative data are: 1. The political party affiliation (Democrat, Republican, or Independent) in a sample of 50 chief executive officers 2. The defective status (defective or not) of each of 100 computer chips manu- factured by Intel 3. The size of a car (subcompact, compact, mid-size, or full-size) rented by each of a sample of 30 business travelers 4. A taste testers ranking (best, worst, etc.) of four brands of barbecue sauce for a panel of 10 testers Often, we assign arbitrary numerical values to qualitative data for ease ofcomputer entry and analysis. But these assigned numerical values are simplycodes: They cannot be meaningfully added, subtracted, multiplied, or divided. Forexample, we might code Democrat = 1,Republican = 2, and Independent = 3.Similarly, a taste tester might rank the barbecue sauces from 1 (best) to 4 (worst).These are simply arbitrarily selected numerical codes for the categories and haveno utility beyond that.*Quantitative data can be subclassified as either interval data or ratio data. For ratio data. theorigin (i.e., the value 0) is a meaningful number. But the origin has no meaning with interval data.Consequently, we can add and subtract interval data, but we cant multiply and divide them. Ofthe four quantitative data sets listed, (1) and (3) are interval data, while (2) and (4) are ratio data.Qualitative data can be subclassified as either nominal data or ordinal data. The categories ofan ordinal data set can be ranked or meaningfully ordered. but the categories of a nominal dataset cant be ordered. Of the four qualitative data sets listed above, (1) and (2) are nominal and(3) and (4) are ordinal.
• 33. 14 C H A P T E1 S t a t i s t i c s , D a t a , a n d S t a t i s t i c a l T h i n k i n g R DEFINITION 1.13 Qualitative data are measurements that cannot be measured on a natural nu- merical scale; they can only be classified into one of a group of categories. Chemical and manufacturing plants sometimes discharge toxic-waste materials such as DDT into nearby rivers and streams. These toxins can adversely affect the plants and animals inhabiting the river and the river bank. The U.S. Army Corps of Engineers conducted a study of fish in the Tennessee River (in Alabama) and its three tributary creeks: Flint Creek, Limestone Creek, and Spring Creek. A total of 144 fish were captured and the following variables measured for each: 1. Riverlcreek where fish was captured 2. Species (channel catfish, largemouth bass, or smallmouth buffalofish) 3. Length (centimeters) 4. Weight (grams) 5. DDT concentration (parts per million) Classify each of the five variables measured as quantitative or qualitative. Solution The variables length, weight, and DDT are quantitative because each is measured on a numerical scale: length in centimeters, weight in grams, and DDT in parts per million. In contrast, riverlcreek and species cannot be measured quantitatively: They can only be classified into categories (e.g., channel catfish, largemouth bass, and smallmouth buffalofish for species). Consequently, data on riverlcreek and species are qualitative. i" As you would expect, the statistical methods for describing, reporting, and analyzing data depend on the type (quantitative or qualitative) of data measured. We demonstrate many useful methods in the remaining chapters of the text. But first we discuss some important ideas on data collection. COLLECTING DATA Once you decide on the type of data-quantitative or qualitative-appropriate for the problem at hand, youll need to collect the data. Generally, you can obtain the data in four different ways: 1. Data from apublished source 2. Data from a designed experiment 3. Data from a survey 4. Data collected observationally Sometimes, the data set of interest has already been collected for you and is available in a published source, such as a book, journal, or newspaper. For example, you may want to examine and summarize the unemployment rates (i.e., percentages of eligible workers who are unemployed) in the 50 states of the United States.You can find this data set (as well as numerous other data sets) at your library in the Sta- tistical Abstract of the United States, published annually by the US. government.Sim- ilarly, someone who is interested in monthly mortgage applications for new home
• 34. SECTION 1.6 Collecting Data 15construction would find this data set in the Survey of Current Business, another gov-ernment publication. Other examples of published data sources include The WallStreet Journal (financial data) and The Sporting News (sports information)." A second method of collecting data involves conducting a designed experi-ment, in which the researcher exerts strict control over the units (people, objects,or events) in the study. For example, a recent medical study investigated the po-tential of aspirin in preventing heart attacks. Volunteer physicians were dividedinto two groups-the treatment group and the control group. In the treatmentgroup, each physician took one aspirin tablet a day for one year, while each physi-cian in the control group took an aspirin-free placebo (no drug) made to look likean aspirin tablet. The researchers, not the physicians under study, controlled whoreceived the aspirin (the treatment) and who received the placebo. A properly de-signed experiment allows you to extract more information from the data than ispossible with an uncontrolled study. Surveys are a third source of data. With a survey, the researcher samples agroup of people, asks one or more questions, and records the responses. Probablythe most familiar type of survey is the political polls conducted by any one of anumber of organizations (e.g., Harris, Gallup, Roper, and CNN) and designed topredict the outcome of a political election. Another familiar survey is the Nielsensurvey, which provides the major television networks with information on themost watched TV programs. Surveys can be conducted through the mail, withtelephone interviews, or with in-person interviews.Although in-person interviewsare more expensive than mail or telephone surveys, they may be necessary whencomplex information must be collected. Finally, observational studies can be employed to collect data. In an obser-vational study, the researcher observes the experimental units in their natural set-ting and records the variable(s) of interest. For example, a company psychologistmight observe and record the level of "Type A" behavior of a sample of assemblyline workers. Similarly, a finance researcher may observe and record the closingstock prices of companies that are acquired by other firms on the day prior to thebuyout and compare them to the closing prices on the day the acquisition is an-nounced. Unlike a designed experiment, an observational study is one in which theresearcher makes no attempt to control any aspect of the experimental units. Regardless of the data collection method employed, it is likely that the datawill be a sample from some population. And if we wish to apply inferential statis-tics, we must obtain a representative sample. DEFINITION 1.14 A representative sample exhibits characteristics typical of those possessed by the population of interest. For example, consider a political poll conducted during a presidential elec-tion year. Assume the pollster wants to estimate the percentage of all 120,000,000registered voters in the United States who favor the incumbent president. Thepollster would be unwise to base the estimate on survey data collected for a sam-ple of voters from the incumbents own state. Such an estimate would almost cer-tainly be biased high.*With published data, we often make a distinction between the primary source and secondarysource. If the publisher is the original collector of the data, the source is primary. Otherwise, thedata are secondary source data.
• 38. SECTION 1.7 T h e R o l e of S t a t i s t i c s i n M a n a g e r i a l D e c i s i o n - M a k i n g 19 want to lose the power of information to help us make de- b. Refer to the American Association of University cisions, and thats what I worry about." Women (AAUW) study of self-esteem of high school girls. Explain why the results of the AAUW study are likely to be misleading. What data might be appropriate F o c u s for assessing the self-esteem of high school girls? a. Consider the false March of Dimes report on domestic c. Refer to the Food Research and Action Center study of violence and birth defects. Discuss the type of data re- hunger in America. Explain why the results of the study quired to investigate the impact of domestic violence on are likely to be misleading. What data would provide in- birth defects. What data collection method would you sight into the proportion of hungry American children? recommend?Reported Information (Source) Actual Study Information.,,.........................................................................................................................................................................................................................-............Eating oat bran is a cheap and easy way to reduce your Diet must consist of nothing but oat bran to achieve a cholesterol count. (Quaker Oats) slightly lower cholesterol count.150,000women a year die from anorexia. (Feminist group) Approximately 1,000 women a year die from problems that were likely caused by anorexia.Domestic violence causes more birth defects than all No study-false report. medical issues combined. (March of Dimes)Only 29% of high school girls are happy with themselves, Of 3,000 high school girls 29% responded "Always true" to compared to 66% of elementary school girls. (American the statement, "I am happy the way I am." Most answered, Association of University Women) "Sort of true" and "Sometimes true."One in four American children under age 12 is hungry or at Based on responses to the questions: "Do you ever cut the risk of hunger. (Food Research and Action Center) size of meals?" "Do you ever eat less than you feel you should?" "Did you ever rely on limited numbers of foods to feed your children because you were running out of money to buy food for a meal?" mets. Were the data obtained from a published source? Were all 50 states included in the study? That is, are you seeing sample data or population data? Furthermore, do the helmet laws vary among states? If so, can you really compare the fatality rates? These questions led the Chance group to the discovery of two scientific and statistically sound studies on helmets. The first, a UCLA study of nonfatal in- juries, disputed the charge that helmets shift injuries to the spine. The second study reported a dramatic decline in motorcycle crash deaths after California passed its helmet law. Successful managers rely heavily on statistical thinking to help them make decisions. The role statistics can play in managerial decision-making is displayed in the flow diagram in Figure 1.5. Every managerial decision-making problem begins with a real-world problem. This problem is then formulated in managerial terms and framed as a managerial question. The next sequence of steps (proceeding counterclockwise around the flow diagram) identifies the role that statistics can play in this process.The managerial question is translated into a statistical question,
• 39. 20 1 CHAPTER Statistics, Data, and Statistical Thinking F I G U R E 1.5 Flow diagram showing the role of statistics in managerial decis~on-making Source Chervany, Benson, and lyer (1 980) Managerial formulation Manager~al question Answer to relatmg to problem managerial question t t Statistical formulation of questlon STATISTICAL the sample data are collected and analyzed, and the statistical question is an- swered. The next step in the process is using the answer to the statistical question to reach an answer to the managerial question. The answer to the managerial ques- tion may suggest a reformulation of the original managerial problem, suggest a new managerial question, or lead to the solution of the managerial problem. One of the most difficult steps in the decision-making process-one that re- quires a cooperative effort among managers and statisticians-is the translation of the managerial question into statistical terms (for example, into a question about a population). This statistical question must be formulated so that, when an- swered, it will provide the key to the answer to the managerial question. Thus, as in the game of chess, you must formulate the statistical question with the end re- sult, the solution to the managerial question, in mind. In the remaining chapters of the text, youll become familiar with the tools essential for building a firm foundation in statistics and statistical thinking.Key TermsNote: Starred (*) terms are from the Measure of reliability 8 Reliability 8optional section in this chapter: Measurement 5 Representative sample 15Black box* 10 Observational study 15 Sample 6,11Census 5 n Population 5 Statistical inference 7Data 2 Process* 9 Statistical thinking 17Descriptive statistics 2 Published source 14 Statistics 2Designed experiment 16 Qualitative data 14 Survey 15Inference 3 Quantitative data 13 Unethical statistical practice 18Inferential statistics 3 Random sample 16 Variable 5
• 43. M E T H O D S FOR DESCRIBING SETS O F DATA C O N T E N T S ... ................................ 2.1 Describing Qualitative Data 2.2 Graphical Methods for Describing Quantitative Data 2.3 Summation Notation 2.4 Numerical Measures of Central Tendency 2.5 Numerical Measures of Variability 2.6 Interpreting the Standard Deviation 2.7 Numerical Measures of Relative Standing 2.8 Methods for Detecting Outliers (Optional) 2.9 Graphing Bivariate Relationships (Optional) 2.10 The Time Series Plot (Optional) 2.11 Distorting the Truth with Descriptive TechniquesI S T A T I S T I C S I N A C T I O N ..................................................................................................................................................... Car 6; Drivers "Road Test Digest" P Where Weve Been Where Were Going I n Chapter 1 we looked at some typical examples of the use of statistics and we discussed the role that statistical thinking plays in supporting manager- B efore we make an inference, we must be able to describe a data set. We can do this by using graphical and/or numerical methods, which we dis- ial decision-making. We examined the difference be- cuss in this chapter. As well see in Chapter 5, we use tween descriptive and inferential statistics and sample numerical descriptive measures to estimate described the five elements of inferential statistics: a the values of corresponding population descriptive population, one or more variables, a sample, an in- measures. Therefore, our efforts in this chapter will ference, and a measure of reliability for the infer- ultimately lead us to statistical inference. ence. We also learned that data can be of two types-quantitative and qualitative.
• 44. 26 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t a Suppose you wish to evaluate the managerial capabilities of a class of 400 MBA students based on theii Graduate Management Aptitude Test (GMAT) scores. How would you describe these 400 measurements? Characteristics of the data set include the typical or most frequent GMAT score, the variability in the scores, the highest and lowest scores, the "shape" of the data, and whether or not the data set contains any unusual scores. Extracting this information by "eye-balling" the data isnt easy. The 400 scores may provide too many bits of information for our minds to comprehend. Clearly we need some formal methods for summarizing and char- acterizing the information in such a data set. Methods for describing data sets are also essential for statistical inference. Most populations are large data sets. Con- sequently, we need methods for describing a sample data set that let us make de- scriptive statements (inferences) about the population from which the sample was drawn. Two methods for describing data are presented in this chapter, one graphical and the other numerical. Both play an important role in statistics. Section 2.1 pre- sents both graphical and numerical methods for describing qualitative data. Graphical methods for describing quantitative data are presented in Section 2.2 and optional Sections 2.8,2.9, and 2.10; numerical descriptive methods for quan- titative data are presented in Sections 2.3-2.7. We end this chapter with a section on the misuse of descriptive techniques. DESCRIBING QUALITATIVE DATA Recall the "Executive Compensation Scoreboard" tabulated annually by Business Week (see Study 2 in Section 1.2). Forbes magazine also conducts a salary survey of chief executive officers each year. In addition to salary information, Forbes collects and reports personal data on the CEOs, including level of education. Do most CEOs have advanced degrees, such as masters degrees or doctorates? To an- swer this question,Table 2.1 gives the highest college degree obtained (bachelors,- masters, doctorate, or none) for each of the 25 best-paid CEOs in 1998. For this study, the variable of interest, highest college degree obtained, is qualitative in nature. Qualitative data are nonnumerical in nature; thus, the value of a qualitative variable can only be classified into categories called class- es. The possible degree types-bachelors, masters, doctorate, and none-repre- sent the classes for this qualitative variable. We can summarize such data numerically in two ways: ( I ) by computing the class frequency-the number of observations in the data set that fall into each class; or (2) by computing the class relative frequency-the proportion of the total number of observations falling into each class. DEFINITION 2.1 A class is one of the categories into which qualitative data can be classified. The class frequency is the number of observations in the data set falling into a particular class.
• 45. S E C T I O 2.1 N Describing Qualitive Data 27 2.1TABLE Data on 25 Best-Paid Executives CEO Company Degree................................................................................................................................................................... -1. Michael D. Eisner Walt Disney Bachelors .- 2. Me1 Karmazin CBS Bachelors 3. Stephen M. Case America Online Bachelors 4. Stephen C. Hilbert Conseco None - 5. Craig R. Barrett Intel Doctorate--- 6. 7. Millard Drexler John F. Welch. Jr. Gap General Electric Masters . Doctorate 8. Thomas G. Stemberg Staples Masters 9. Henry R. Silverman Cendant JD(law) 10. Reuben Mark Colgate-Palmolive Masters 11. Philip J. Purcell Morgan Stanley Dean Witter Masters 12. Scott G. McNealy Sun Microsystems Masters 13. Margaret C. Whitman eBay Masters 14. Louis V. Gerstner, Jr. IBM Masters 15. John F. Gifford Maxim Integrated Products Bachelors 16. Robert L. Waltrip Service Corp. International Bachelors 17. M. Douglas Ivester Coca-Cola Bachelors 18. Gordon M. Binder Amgcn Masters 19. Charles R. Schwab Charles Schwab Masters 20. William R. Steere, Jr. Pfizer Bachelors 21. Nolan D. Archibald Black & Decker Masters 22. Charles A. Heimbold, Jr. Bristol-Myers Squibb LLB (law) 23. William L. Larson Network Association J D (law) 24. Maurice R. Greenberg American International Group LLB (law) 25. Richard Jay Kogan Schering-Plough Masters Source: Forbes, May 17,1999 DEFINITION 2.3 The class relative frequency is the class frequency divided by the total numbe of observations in the data set. Examining Table 2.1, we observe that 1 of the 25 best-paid CEOs did notobtain a college degree, 7 obtained bachelors degrees, 11masters degrees, and6 doctorates or law degrees. These numbers-1, 7, 11, and 6-represent theclass frequencies for the four classes and are shown in the summary table,Table 2.2. 2.2TABLE Table for Data on 25 Best-Paid CEOs ~umrnaty Class Frequency Relative Frequency ....................................................................................................................................... Highest Degree Obtained Number of CEOs Proportion.............................................................................................................................................................. None 1 .04 Bachelors 7 .28 Masters 11 .44 DoctorateILaw 6 .24 Totals 25 1.OOO
• 46. 28 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t a Table 2.2 also gives the relative frequency of each of the four degree classes. From Definition 2.3, we know that we calculate the relative frequency by dividing the class frequency by the total number of observations in the data set.Thus, the relative frequencies for the four degree types are 1 None: -= .04 25 C I Bachelors: -= .28 25 11 Masters: -= .44 25 6 Doctorate: - = .24 25 From these relative frequencies we observe that nearly half (44%) of the 25 best-paid CEOs obtained their masters degree. Although the summary table of Table 2.2 adequately describes the data of Table 2.1, we often want a graphical presentation as well. Figures 2.1 and 2.2 show two of the most widely used graphical methods for describing qualitative data- bar graphs and pie charts. Figure 2.1 shows the frequencies of "highest degree ob- tained" in a bar g r q h created using the EXCEL software package. Note that the height of the rectangle, or "bar," over each class is equal to the class frequency. (Optionally, the bar heights can be proportional to class relative frequencies.) In contrast, Figure 2.2 (also created using EXCEL) shows the relative frequencies of the four degree types in a pie chart. Note that the pie is a circle (spanning 360") and the size (angle) of the "pie slice" assigned to each class is proportional to the class relative frequency. For example, the slice assigned to masters degree is 44% FIGURE2.1 DEGREE I I EXCEL graph for data on 25 CEOs (7 Bachelors ( Masters 11 1 I I Doctor/Law 61 I I I I I None 4% FIGURE 2.2 None n I Lors I DOC t /Law EXCEL pie chart for data on 25 CEOs
• 48. 30 CHAPTER 2 M e t h o d s f o r D e s c r i b i n g Sets o f D a t a complications. But before using this information to make a decision about the drug, the physicians will need to provide a measure of reliability for the inference. That is, the physicians will want to know whether the difference between the per- centages of patients with complications observed in this sample of 114 patients is SUM OF N BY COMP GROUPED BY DRUG N SUM Cumulative Cumulative COMP Frequency Percent Frequency Percent 1 :REDO 5 8.8 5 8.8 2 :INFECT 4 7.0 9 15.8 3 :BOTH 1 1.8 10 17.5 4 :NONE 47 82.5 57 100.0 Cumulative Cumulative COMP Frequency Percent Frequency Percent ..................................................................... 1 :REDO 7 12.3 7 12.3 2 :INFECT 7 12.3 14 24.6 3 :BOTH 3 5.3 17 29.8 4 :NONE 40 70.2 57 100.0 FIGURE 2.4 SAS bar graph and summary tables for COMP by DRUG
• 49. I 2.1 SECTION Describing Qualitive Data 31 generalizable to the population of all coronary artery bypass patients. Measures of reliability will be discussed in Chapters 5-8. 80% of the wealth of a country lies with approximately 20% of the people. This discovery of "the vital few and the trivial many," called the Pareto principle, holds true even today. In his book Defect Prevention (1989), V. E. Kane reports theI following: 80% of sales are attributable to 20% of the customers; 80% of customer complaints result from 20% of the components of a product; 80% of defective items produced by a process result from 20% of the types of errors that are made in production. In industry a Pareto analysis involves the categorization of items and the determination of which categories contain the most1 observations, i.e., which are the "vital few" categories. The primary tool of Pareto analysis is the Pareto diagram. The Pareto diagram is simply a bar graph with the bars arranged in descending order of height from left to right across the horizontal axis. This arrangement locates the most important categories-thoseI with the largest frequencies-at the left of the graph. a. Table 2.3a contains summary data from the automobile industry (adapted from Kane, 1989).All cars produced on a particular day were inspected for general defects. Each of the 125 defects discovered were categorized by type: accessory, body, electrical, engine, and transmission. The table gives the number of defects of each type. Construct a Pareto diagram for the data. Identify the categories that produced the most defects. TABLE 3 2. Summary of Car Defects a. General Defect Number b. Body Defect Number .................................. ................................................. ............................................................. .................-I Accessory Body 50 70 Chrome Dents 25 2 Electrical 12 Paint 30 Engine 8 Upholstery 10 Transmission 5 Windshield 3 Total 125 Total 70I b. All 70 body defects were further classified as to whether they were chrome, dent, paint, upholstery, of windshield defects.Table 2.3b gives the number of each type. Construct a Pareto diagram for the body type defects. Interpret the results. S o I ut io n a. The Pareto diagram for the 125 car defects is a frequency bar graph, with the bars arranged in order of height. The diagram is displayed in Figure 2.5a. Clearly, body and accessory defects (the two categories at the far left of the graph) are the most frequently observed types of car defects. Management should focus their problem-solving efforts in these two areas. b. A Pareto diagram for the 70 body part defects is shown in Figure 2.5b. You can see that paint and dent defects (again, the two categories at the far left of the graph) are the most frequently observed types of body defects. These defects should be targeted for special attention by managers, engineers, and assembly-line workers. [Note:When one or more Pareto diagrams are used to decompose the original Pareto diagram in Figure 2Sa, the resulting graph (Figure 2.5b) is called exploding the Pareto diagram.]
• 50. 32 CHAPTER 2 M e t h o d s f o r D e s c r i b i n g S e t s of D a t aFIGURE 2.5 a. Car defectsPareto diagramsfor Example 2.2 70 60 \$ 50 40 2 r 30 20 10 T v ~ of defect e b. Car body defectsLearning the Mechanics a. Compute the frequency for each of the three classes. b. Compute the relative frequency for each of the three2.1 Complete the following table......................._ _ ............................................................... ............................. classes.Grade on Business c. Display the results, part a, in a frequency bar graph.Statistics Exam Frequency Relative Frequency d. Dkplay the results, part b, in a pie chart.A: 90-1 00 - .08 Applying the ConceptsB: 80-89 36 - -C: 65-79 90 2.3 Until their profitability was threatened by competition -D: 50-64 30 from Japan and Europe in the 1970s,most U.S. firms paidF: Below 50 28 - relatively little attention to customer satisfaction.Today,Total 200 1. O O customer satisfaction and loyalty are valued and moni-- tored by all world-class organizations. But are satisfied2.2 A qualitative variable with three classes (X,Y, and Z) is customers necessarily loyal customers? Harte-Hanks measured for each of 20 units randomly sampled from a Market Research surveyed customers of department target population. The data (observed class for each stores and banks and published the results (shown on unit) are listed below. p. 33) in American Demographics (Aug. 1999). Y X X Z X Y Y Y X X Z X a. Construct a relative frequency bar chart for each of Y Y X Z Y Y Y X the data sets.
• 53. SECTION 2.1 Describing Qualitive Data 350ILSPILL.DAT........................................................................................................................................................................................................... Cause of Spillage Spillage (metric tons, Fire/ HullTanker thousands) Collision Grounding Explosion Failure Unknown...........................................................................................................................................................................................................Atlantic EmpressCastillo De BellverAmoco CadizOdysseyTorrey CanyonSea StarHawaiian PatriotIndependentoUrqu~olaIrenes SerenadeKhark 5NovaWafraEpic Co/ocotronisSinclair PetroloreYuyo Maru No 10AssimiAndros PatriaWorld GloryBrzt~sh AmbassadorMetulaPericles G.C.Mandoil IIJacob MaerskBurmah AgateJ. Antonio LavallejaNapierExxon ValdezCorznthosTraderSt. PeterGinoGolden DrakeIonnis AngelicoussisChryssiIrenes ChallengeArgo MerchantHeimvardPegasusPacoceanTexaco OklahomaScorpioEllen ConwayCaribbean SeaCretan StarGrand ZenithAtheman VentureVenoilAragonOcean EagleSource: Daidola, J. C. "Tanker structure behavior during collision and grounding." Marine Technology, Vol. 32, No. 1,Jan.1995, p. 22 (Table 1). Reprinted with permission of The Society of Naval Architects and Marine Engineers (SNAME), 601Pavonia Ave., Jersey City, NJ 07306, USA, (201) 798-4800. Material appearing in The Society of Naval Architect and MarineEngineers (SNAME) publications cannot be reprinted without obtaining written permission.
• 54. 36 CHAPTER 2 M e t h o d s f o r D e s c r i b i n g Sets o f D a t a a. Construct a Pareto diagram for the 10 categories. c. In this case, are 80% of the investors concerned with b. According to your Pareto diagram, which environ- 20% of the environmental factors as the Pareto prin- mental factors most concern investors? ciple would suggest? Justify your answer. GRAPHICAL METHODS FOR DESCRIBING QUANTITATIVE DATA Recall from Section 1.5 that quantitative data sets consist of data that are record- ed on a meaningful numerical scale. For describing, summarizing, and detecting patterns in such data, we can use three graphical methods: dot plots, stem-and-leaf displays, and histograms. For example, suppose a financial analyst is interested in the amount of re- sources spent by computer hardware and software companies on research and de- velopment (R&D). She samples 50 of these high-technology firms and calculates the amount each spent last year on R&D as a percentage of their total revenues. The results are given in Table 2.4. As numerical measurements made on the sam- ple of 50 units (the firms), these percentages represent quantitative data. The an- alysts initial objective is to summarize and describe these data in order to extract relevant information. 2.4TABLE Percentage of Revenues Spent on Research and Development Company Percentage Company Percentage Company Percentage Company Percentage A visual inspection of the data indicates some obvious facts. For example, the smallest R&D percentage is 5.2% and the largest is 13.5%. But it is difficult to provide much additional information on the 50 R&D percentages without re- sorting to some method of summarizing the data. One such method is a dot plot. Dot Plots A dot plot for the 50 R&D percentages, produced using MTNTTAB software, is shown in Figure 2.6.The horizontal axis of Figure 2.6 is a scale for the quantitative variable, percent. The numerical value of each measurement in the data set is lo- cated on the horizontal scale by a dot. When data values repeat, the dots are placed above one another, forming a pile at that particular numerical location. As you can see, this dot plot shows that almost all of the R&D percentages are be- tween 6% and 12%, with most falling between 7% and 9%.
• 55. SECTION 2.2 Graphical M e t h o d s f o r Describing Q u a n t i t a t i v e Data 37FIGURE2.6A MINITAB dot plot forthe 50 R&D percentages Stem-and-Leaf Display We used STATTSTTX software to generate another graphical representation of these same data, a stem-and-leaf display, in Figure 2.7. In this display the stem is the portion of the measurement (percentage) to the left of the decimal point, while the remaining portion to the right of the decimal point is the lea8 The stems for the data set are listed in a column from the smallest (5) to the largest (13). Then the leaf for each observation is recorded in the row of the dis- play corresponding to the observations stem. For example, the leaf 5 of the first observation (13.5) in Table 2.3 is placed in the row corresponding to the stem 13. Similarly, the leaf 4 for the second observation (8.4) in Table 2.3 is recorded in the row corresponding to the stem 8, while the leaf 5 for the third observation (10.5) is recorded in the row corresponding to the stem 10. (The leaves for these first three observations are shaded in Figure 2.7.) Typically, the leaves in each row are ordered as shown in Figure 2.7.FIGURE 2.7A STATlSTlX stem-and-leaf STEM AND LEAF PLOT OF RDPCT MINIMUM 5.2000display for the 50 R&D MEDIAM 8.0500percentages MAXIMUM 13.500 LEAF DIGIT UNIT = 0.1 5 2 REPRESENTS 5.2 STEM LEAVES 3 5 269 12 6 055568999 23 7 11224557789 (9) 8 001222458 18 9 02455679 10 10 1556 6 11 137 3 12 3 13 50 CASES INCLUDED 0 MISSING CASES The stem-and-leaf display presents another compact picture of the data set. You can see at a glance that most of the sampled computer companies (37 of 50) spent between 6.0% and 9.9% of their revenues on R&D, and 11 of them spent between 7.0% and 7.9%. Relative to the rest of the sampled companies, three spent a high percentage of revenues on R&D-in excess of 13% Most statistical software packages allow you to modify the definitions of the stem and leaf to alter the graphical description. For example, suppose we had de- fined the stem as the tens digit for the R&D percentage data, rather than the ones and tens digits. With this definition, the stems and leaves corresponding to the measurements 13.5 and 8.4 would be as follows:
• 56. 38 CHAPTER 2 M e t h o d s f o r D e s c r i b i n g Sets o f D a t a Note that the decimal portion of the numbers has been dropped. Generally, only one digit is displayed in the leaf. If you look at the data, youll see why we didnt define the stem this way. All the R&D measurements fall below 13.5, so all the leaves would fall into just two stem rows-1 and 0-in this display. The resulting picture would not be nearly as informative as Figure 2.7. Histograms 3 A relative frequency histogram for these 50 R&D percentages is shown in Figure 2.8. The horizontal axis of Figure 2.8, which gives the percentage spent on R&D for each company, is divided into intervals commencing with the interval from (5.15-6.25) and proceeding in intervals of equal size to (12.85-13.95) percent. (The procedure for creating the class intervals will become clear in Example 2.3.) The vertical axis gives the proportion (or relative frequency) of the 50 percentages that fall in each interval. Thus, you can see that nearly a third of the companies spent between 7.35% and 8.45% of their revenues on research and development. This interval contains the highest relative frequency, and the intervals tend to contain a smaller fraction of the measurements as R&D percentage gets smaller or larger. FIGURE 2.8 Histogram for the 50 computer companies x .30 R&D percentages 5 & .20 9 .- * d 2 .lo .OO 5.15 6.25 7.35 8.45 9.55 10.65 11.75 12.85 13.95 R&D percentage By summing the relative frequencies in the intervals 6.25-7.35, 7.35-8.45, 8.45-9.55,9.55-10.65, you can see that 80% of the R&D percentages are between 6.25 and 10.65. Similarly, only 6% of the computer companies spent over 12.85 percent of their revenues on R&D. Many other summary statements can be made by further study of the histogram. Dot plots, stem-and-leaf displays, and histograms all provide useful graphic descriptions of quantitative data. Since most statistical software packages can be used to construct these displays, we will focus on their interpretation rather than their construction. Histograms can be used to display either the frequency or relative frequen- cy of the measurements falling into specified intervals known as measurement classes. The measurement classes, frequencies, and relative frequencies for the R&D percentage data are shown in Table 2.5.
• 57. 2.2 SECTION Graphical M e t h o d s f o r Describing Q u a n t i t a t i v e Data 39 2.5 TABLE Measurement Classes, Frequencies, and Relative Frequencies for the R&D Percentage Data Class Measurement Class Class Frequency Class Relative Frequency Totals 50 1.00 By looking at a histogram (say, the relative frequency histogram in Figure 2.8), you can see two important facts. First, note the total area under the histogram and then note the proportion of the total area that falls over a particular interval of the horizontal axis. Youll see that the proportion of the total area above an interval is equal to the relative frequency of measurements falling in the interval. For example, the relative frequency for the class interval 7.35-8.45 is .28. Consequently, the rec- tangle above the interval contains .28 of the total area under the histogram. Second, you can imagine the appearance of the relative frequency histogram for a very large set of data (say, a population). As the number of measurements in a data set is increased, you can obtain a better description of the data by decreas- ing the width of the class intervals.When the class intervals become small enough, a relative frequency histogram will (for all practical purposes) appear as a smooth curve (see Figure 2.9).F I G U R E2.9Effect o the size o a data set f fon the outline of a histogram Measurement classes Measurement classes Measurement classes a. Small data set b. Larger data set c. Very large data set While histograms provide good visual descriptions of data sets-particularly very large ones-they do not let us identify individual measurements. In contrast, each of the original measurements is visible to some extent in a dot plot and clear- ly visible in a stem-and-leaf display. The stem-and-leaf display arranges the data in ascending order, so its easy to locate the individual measurements. For example, in Figure 2.7 we can easily see that three of the R&D measurements are equal to 8.2, but we cant see that fact by inspecting the histogram in Figure 2.8. However, stem-and-leaf displays can become unwieldy for very large data sets. A very large number of stems and leaves causes the vertical and horizontal dimensions of the display to become cumbersome, diminishing the usefulness of the visual display.
• 58. 40 CHAPTER 2 M e t h o d s f o r D e s c r i b i n g Sets o f D a t a < < lost because of the long time the firm takes to develop price quotes for potential customers. To investigate this possibility, 50 requests for price quotes were randomly selected from the set of all quotes made last year, and the processing time was determined for each quote. The processing times are displayed in Table 2.6, and each quote was classified according to whether the order was "lost" or not (i.e., whether or not the customer placed an order after receiving a price quote). TABLE 2.6 Price Quote Processing Time (Days) Request Number Processing Time Lost? Request Number Processing Time Lost? No No No No No No Yes Yes No No No No No No No Yes No No No No Yes Yes No No No No No No No No No No Yes No No No No Yes Yes No No No No Yes Yes No No Yes No No a. Use a statistical software package to create a frequency histogram for these data. Then shade the area under the histogram that corresponds to lost orders. i a b. Use a statistical software package to create a stem-and-leaf display for these data. Then shade each leaf of the display that corresponds to a lost order. c. Compare and interpret the two graphical displays of these data. S o Iut ion a. We used SAS to generate the frequency histogram in Figure 2.10. SAS, like most statistical software, offers the user the choice of accepting default class intervals and interval widths, or the user can make his or her own selections. After some experimenting with various numbers of class inter- vals and interval widths, we used 10 intervals. SAS then created intervals of width 2 days, beginning at 1 day, just below the smallest measurement of
• 59. Graphical Methods for Describing Quantitative Data 41FIGURE 2.10 FrequencySAS frequency histogram for 14the quote processing time data 13 4 6 8 10 12 1 1 16 18 20 Processing time (days) 1.25 days, and ending with 21 days, just above the largest measurement of 20.2 days. Note that SAS labels the midpoint of each bar, rather than its endpoints. Thus, the bar labeled "2" represents measurements from 1.00 to 2.99, the bar labeled "4" represents measurements from 3.00 to 4.99, etc. This histogram clearly shows the clustering of the measurements in the lower end of the distribution (between approximately 1 and 7 days), and the relatively few measurements in the upper end of the distribution (greater than 12 days). The shading of the area of the frequency histogram corresponding to lost orders clearly indicates that they lie in the upper tail of the distribution. We used SPSS to generate the stem-and-leaf display in Figure 2.11. Note that the stem consists of the number of whole days (units and tens digits), and the leaf is the tenths digit (first digit after the decimal) of each mea- surement." The hundredths digit has been dropped to make the display more visually effective. SPSS also includes a column titled Frequency show- ing the number of measurements corresponding to each stem. Note, too, that instead of extending the stems all the way to 20 days to show the largest measurement, SPSS truncates the display after the stem corresponding to 13 days, labels the largest four measurements (shaded) as Extremes, and simply lists them horizontally in the last row of the display. Extreme observations that are detached from the remainder of the data are called outliers, and they usually receive special attention in statistical analyses. Although out- liers may represent legitimate measurements, they are frequently mistakes: incorrectly recorded, miscoded during data entry, or taken from a popula- tion different from the one from which the rest of the sample was selected. Stem-and-leaf displays are useful for identifying outliers. Note that like the histogram, the stem-and-leaf display shows the shaded "lost" orders in the upper-tail of the distribution.*In the examples in this section, the stem was formed from the digits to the left of the decimal.This is not always the case. For example, in the following data set the stems could be the tenthsdigit and the leaves the hundredths digit: .12, .15, .22, .25, .28, .33.
• 60. 42 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t aF I G U R E 2.1 1SPSS stem-and-leaf display for Frequency Stem & Leafthe quote processing time data 5.00 1 24789. 5.00 2 03569. 8.00 3 . 23344699 6.00 4 . 034469 6.00 5 . 114579 5.00 6 01467. 5.00 7 24557. 1.00 8 . 2 3.00 9 049 . 1.00 10 0 . .oo 11 . .oo 12 . 1.00 13 4 . 4.00 Extremes (14.1), (14.31, (16.31, ( 2 0 . 2 ) Stem width: 1.00 Each leqf: 1 case(s) c. As is usually the case for data sets that are not too large (say, fewer than 100 measurements), the stem-and-leaf display provides more detail than the histogram without being unwieldy. For instance, the stem-and-leaf display in Figure 2.11 clearly indicates not only that the lost orders are associated with high processing times (as does the histogram in Figure 2.10), but also exactly which of the times correspond to lost orders. Histograms are most useful for displaying very large data sets, when the overall shape of the distribution of measurements is more important than the identification of individual measurements. Nevertheless, the message of both graphical dis- plays is clear: establishing processing time limits may well result in fewer lost orders. Most statistical software packages can be used to generate histograms, stem- and-leaf displays, and dot plots. All three are useful tools for graphically describ- ing data sets. We recommend that you generate and compare the displays whenever you can. Youll find that histograms are generally more useful for very large data sets, while stem-and-leaf displays and dot plots provide useful detail for smaller data sets. I USING THE T I - 8 3 G R A P H I N G C A L C U L A T O R Making a Histogram on the TI-83 Step 1 Enter the data Press STAT 1 for STAT edit Use the arrow and ENTER keys to enter the data set into L1. Step 2. Set up the histogram plot Press 2nd Y = for STAT PLOT Press 1 for Plot 1 (continued)
• 61. SECTION 2.2 Graphical Methods f o r Describing Quantitative Data 43 Use the arrow and ENTER keys to set up the screen as shown below. Note: Press 2nd 1 for L1 Step 3 Select your window settings Press WINDOW and adjust the settings as follows: Xmin = lowest class boundary Xmax = highest class boundary Xscl = class width Ymin = 0 Ymax = greatest class frequency Yscl = approximately YmaxIlO Xres = 1 Step 4 View the graph Press GRAPH 0 p t io na l Read class frequencies and class boundaries Step You can press TRACE to read the class frequencies and class boundaries. Use the arrow keys to move between bars. Example The figures below show TI-83 window settings and histogram for the fol- lowing sample data: 86,70,62,98,73,56,53,92,86,37,62,83,78,49,78,37,67,79,57 WINDOW Xmin= 5 Xmax=100.5 -. Xscl=10 Ymin=0 Ymax=5 Yscl=l Xres=l Making a Histogram From a Frequency Table Step 1 Enrer the data Enter a value from the middle of each class in L1 Enter the class frequencies or relative frequencies in L2 Step 2 Set u p the histogram plot Press 2nd Y = for STAT PLOT Press 1 for Blot 1 (continued)
• 62. 44 CHAPTE2 R M e t h o d s for D e s c r i b i n g S e t s of D a t a Use the arrow and ENTER keys to set up the screen as shown below. Steps 3-4 Follow steps 3-4 given above. Learning the Mechanics c Re-create all the numbers in the data set and con- struct a dot plot. 2.12 Graph the relative frequency histogram for the 500 2.15 MINITAB was used to generate the following histogram: meawrements summari~edin the accompanying rela- a. Is this a frequency histogram or a relative frequency tive frequency table.Y histogram? Explain. ................................................................................. b. How many measurement classes were used in the Measurement Class Relative Frequency construction of this histogram? .............................................................................. c. How many measurements are there in the data set S2.5 .10 described by this histogram? 2.5-4.5 .15 4.5-6.5 .25 MINITAB output for Exercise 2.1 5 6.5-8.5 .20 - 8.5-10.5 .05 MIDDLE OF NUMBER OF 10.5-12.5 .10 12.5-14.5 .10 INTERVAL OBSERVATIONS 14.5-16.5 .05 20 1 * 22 3 *** 24 2 ** 2.13 Refer to Exercise 2.12. Calculate the number of the 26 3 *** 500 measurements falling into each o l the measure- 28 q **** ment clahses. Then graph a frequency histogram for 30 7 ******* these data. 32 11 *********** 2.14 SAS was used to generate the stem-and-leaf display 34 6 ****** shown here. N o t e that SAS arranges t h e slems in 36 2 ** descending order. 38 3 *** 40 3 *** 42 2 ** 44 1 * 46 1 * 2.16 The graph on p. 45 summarizes the scores obtained by 100 students on a questionnaire designed to measure managerial ability. (Scores are integer values that range from 0 to 20.A high score indicates a high level of ability.) a. How many observations were in the original data set? a. Which measurement class contains the highest pro- b. In the bottoni row of the stern-and-leaf display, iden- portion of test scores? tify the \$tern, t h e leaves, and the nurntxrs in the b. What proportion of the scores lie between 3.5 and 5.5? orignal data set represented by this stem and its c. What proportion of the scores are higher than 11.5? leaves. d. How many students scored less than 5.5:
• 63. SECTION 2.2 Graphical Methods for Describing Quantitative Data 45 by their respective teams in games in which McGwire and Sosa hit home runs (Journal of Statistics Education, Vol. 6, 1998). The data are reproduced in the table below. (An asterisk indicates a game in which McGwire or Sosa hit multiple home runs.) a. Construct a stem-and-leaf display for the number of runs scored by St. Louis during games when McGwire hit a home run. b. Repeat part a for Chicago and Sosa. c. Compare and contrast the two distributions. d. On the stem-and-leaf display, circle the games in which McGwire or Sosa hit multiple home runs. Do you detect any patterns? 1.5 3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 2.19 Bonds can be issued by the federal government, state Managerial ability and local governments, and US corporations. A mort- gage bond is a promissory note in which the issuingApplying the Concepts company pledges certain real assets as security in2.17 The table (p..46) reports the one-year percentage exchange for a specified amount of money. A debenture change in stock price for all technology companies and is an unsecured promissory note, backed only by the all industrial companies on Fortune (Sept. 6,1999) mag- general credit o l the issuer. The bond price of either a azines ranking of the "One Hundred Fastest-Growing mortgage bond or debenture is negotiated between the- Companies." a. Use a statistical software package to construct a asked price (the lowest price anyone will accept) and the bid price (the highest price anyone wants to pay) stem-and-leaf display for the stock price changes of (Alexander, Sharpe, and Bailey, Fundamentals of technology companies. Do the same for industrial Investments, 1993). The table on page 47 contains the companies. bid prices for a sample of 30 publicly traded bonds b. Use the results of part a to compare and contrast the issued by utility con~panies. stock price changes of the techrialogy and industrial a. A frequency histogram (p. 47) was generated using companies. SPSS. Note that SPSS labels the midpoint of each c. What percentage of the technology companies in the measurement class rather than the two endpoints. data set had stock prices that more than doubled? Interpret thc histogram.2.18 Mark McGwire of the St. Louis Cardinals and Sammy b. Use the histogram to determine the number of Sosa of the Chicago Cubs hit 70 and 66 home runs, bonds in the sample that had a bid price greater than respectively, during the 1998 Major League Baseball \$96.50. What proportion of the total number of season, breaking the record held by Roger Maris (61 bonds is this group? home runs) since 1961. J.S. Simonoff of New York c. Shade the area under the histogram that corresponds University collected data on the number of runs scored to the proportion in part b. STLRUNS.DAT CUBSRUNS.DAT ............................................................... ... ............................. St. Louis Cardinals chicago Cubs ................................................................ 3 6* 8* 6 7* 4 5 2 7 2 1 8* 6 13 1 9* 6 8 4 2 3* 10 3 6 9 6* 5 4 10 5 5 4 4 9 2 9 5 7 5* 5 4 8 5* 6 5 15 lo* 9 8 ll* 5 3 11 6
• 64. 2 CHAPTER M e t h o d s f o r D e s c r i b i n g Sets o f D a t a - FORT100.DAT ............................................................................................................................................................. .................. _....__. Technology Companies % Change Industrial Companies % Change ....................................................................................................................................................................................... - SIEBEL SYSTEMS MERITAGE THQ NETWORK APPLIANCE CITRIX SYSTEMS r SALTON VERITAS SOFTWARE VITESSE SEMICONDUCTOR CONSOLIDATED GRAPHICS DELL COMPUTER CREE RESEARCH QUALCOMM ARGUSS HOLDINGS KELLSTROM INDUSTRIES ARMOR HOLDINGS TEKELEC LEGATO SYSTEMS MONACO COACH ASPECT DEVELOPMENT COMPUWARE HAUPPAUGE DIGITAL ZOMAX AVT MERCURY INTERACTIVE SUMMA INDUSTRIES WHITTMAN-HART AAVID THERMAL TECH. i2 TECHNOLOGIES TOWER AUTOMOTIVE CADE INDUSTRIES INSIGHT ENTERPRISES ANALYTICAL SURVEYS HNC SOFTWARE CIBER TSR DYCOM INDUSTRIES MODTECH HOLDINGS MORRISON KNUDSEN SCHULER HOMES TECH DATA POMEROY COMPUTERS RES. Source: Fortune, Sept. 6,1999.2.20 Production processes may be classified as make-to-stock processes or make-to-order processes. Make-to-stock processes are designed to produce a standardized prod- uct that can be sold to customers from the firms inven- tory. Make-to-order processes are designed to produce products according t o customer specifications (Schroeder, Operations Management, 1993). In general, The MINITAB stem-and-leaf display of these data is performance of make-to-order processes is measured by shown at the bottom of page 47. delivery time-the time from receipt of an order until a. Circle the individual leaves that are associated with the product is delivered to the customer. The following customers who did not place a subsequent order. data set is a sample of delivery times (in days) for a par- b. Concerned that they are losing potential repeat cus- ticular make-to-order firm last year. The delivery times tomers because of long delivery times, the manage- marked by an asterisk are associated with customers ment would like to establish a guideline for the who subsequently placed additional orders with the firm. maximum tolerable delivery time. Using the stem-
• 65. - 48 C H A P T E2 R M e t h o d s f o r D e s c r i b i n g S e t s of D a t a Q ............................................................................................................................................................................................................................... CLEANAIR.DAT Company Company Identification Number Penalty Law* Identification Number Penalty Law* ............................................................................................................................................................................................................................... 01 \$ 930,000 CERCLA 16 90,000 RCRA 02 10,000 CWA 17 20,000 CWA 03 90,600 CAA 18 40,000 CWA 04 123,549 CWA 19 20,000 CWA 05 37,500 CWA 20 40,000 CWA 06 137,500 CWA 21 850,000 CWA 07 2,500 SDWA 22 35,000 CWA 08 1,000,000 CWA 23 4,000 CAA 09 25,000 CAA 24 25,000 CWA 09 25,000 CAA 25 40,000 CWA 10 25,000 CWA 26 30,000 CAA 10 25,000 RCRA 27 15,000 CWA 11 19,100 CAA 28 15,000 CAA 12 100,000 CWA 29 105,000 CAA 12 30,000 CWA 30 20,000 CWA 13 35,000 CAA 31 400,000 CWA 13 43,000 CWA 32 85,000 CWAA 14 190,000 CWA 33 300,000 CWNRCRAICERCLA 15 15,000 CWA 34 30,000 CWA *CAA: Clean Air Act; CERCLA: Comprehensive Environmental Response, Compensation, and Liability Act; CWA: Clean Water Act; RCRA: Resource Conservation and Recovery Act; SDWA: Safe Drinking Water Acl. Source: Tabor, R. H., and Stanwick, S. D. "Arkansas: An environmental perspective." Arkansas Business and Economic Review,Vol. 28, NO.2, Summer 1995,pp. 22-32 (Table 4). (Note: Some companies were involved in more than one Air Act violations relative to the other types of vio- civil action.) lations reported in the table? Explain. a. Construct a stem-and-leaf display for all 38 penalties. 2-22 in a manufacturing plant a work center is a specific pro- ( b. Circle the individual leaves that are associated with duction facility that consists of one or more people penalties imposed for violations of the Clean Air Act. andlor machines and is treated as one unit for the pur- c. What does the pattern of circles in part b suggest poses of capacity requirements planning and job sched- about the severity of the penalties imposed for Clean uling. If jobs arrive at a particular work center at a l W0RKCTR.DAT ..................................................................................................................................................................................... Number of ltems Arriving at Work Center per Hour ..................................................................................................................................................................................... 155 115 156 150 159 163 172 143 159 166 148 175 151 161 138 148 129 135 140 152 139 Number of ltems Departing Work Center per Hour MINITAB Output for Exercise 2.22 . . . . . ... . . . . . . . . -----+---------+---------+---------+-----------+----------- + - ARRIVE ......... ... . . . -----+---------+---------+-----------+---------+--------- +-DEPART 105 12 0 135 150 165 180 i
• 67. 50 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t a d. The companies that were reorganized through a lever- mathematical equation that involves a variable, called aged buyout are identified by an asterisk (*) in the a sufficient norm constraint. A histogram for realizations table. Identify these firms on the stem-and-leaf display, of I produced using simulation, is shown on page 49. ! part a, by circling their bankruptcy times. Do you ob- a. Estimale the percentage of realizations of V with serve any pattern in the graph? Explain. values ranging from .425 to .675.2.25 Its not uncommon for hearing aids to malfunction and b. Cancellation of the desired signal is limited by se- cancel the desired signal. IEEE Transactions on Speech lecting a norm constraint I:Find the value of V for a . and Audio Processing (May 1995) reported on a new company that wants to market the new hearing aid audio processing system designed to limit the amount of so that only 10% of the realizations have values signal cancellation that may occur. The system utilizes a below the selected level.&a BANKRUPT.DAT ............................................................................................................. Pre-filing Time in Bankruptcy ................................................................................................................. Pre-filing Time in Bankruptcy mpany Votes (months) Company Votes (months) ................................................................................................................. AM International None LIVE Entertainment Joint 1.4 Anglo Energy Prepack Mayflower Group* Prepack 1.4 Memorex Telex* Prepack 1.1,, Arizona Biltmore* Astrex Barrys Jewelers Prepack None None Munsingwear Natl Environmental None Joint 2.9 5.2 Calton Prepack Petrolane Gas Prepack 1.2 Cencor Joint Price Communications None 2.4 Charter Medical* Prepack Republic Health* Joint 4.5 Cherokee* Joint Resorts Intl* None 7.8 Circle Express Prepack Restaurant Enterprises* Prepack 1.5 Cook Inlet Comm. Prepack Rymer Foods Joint 2.1 Crystal Oil None SCI TV* Prepack 2.1 Divi Hotels None Southland* Joint 3.9 Edgell Comm.* Prepack Specialty Equipment* None 2.6 Endevco Prepack SPI Holdings* Joint 1.4 Gaylord Container Joint Sprouse-Reitz Prepack 1.4 Great Amer. Comm.* Prepack Sunshine Metals Joint 5.4 Hadson Prepack TI EiCommunications None 2.4 In-Store Advertising Prepack Trump Plaza Prepack 1.7 JPS Textiles* Prepack Trump Taj Mahal Prepack 1.4 Kendall* Prepack Trumps Castle Prepack 2.7 Kinder-Care None USG Prepack 1.2 Kroy* Prepack Vyquest Prepack 4.1 Ladish* Joint West Point Acq.* Prepack 2.9 LaSalle Energy* Prepack *Leveraged buyout. Source: Betker, B.L. "An empirical examination of prepackaged bankruptcy." Financial Management, Vol. 24, No. 1, Spring 1995, p. 6 (Table 2). 1 SUMMATION NOTATION Now that weve examined some graphical techniques for summarizing and de- scribing quantitative data sets, we turn to numerical methods for accomplishing this objective. Before giving the formulas for calculating numerical descriptive measures, lets look at some shorthand notation that will simplify our calcula- tion instructions. Remember that such notation is used for one reason only-to avoid repeating the same verbal descriptions over and over. If you mentally substitute the verbal definition of a symbol each time you read it, youll soon get used to it. We denote the measurements of a quantitative data set as follows: XI,x2,x3,.... X where xl is the first measurement in the data set, x2 is the second , measurement in the data set, x3 is the third measurement in the data set,. .., and
• 68. e SECTION 2.3 Summation Notation 51 x, is the nth (and last) measurement in the data set. Thus, if we have five mea- surements in a set of data, we will write x,,x2,x3,x,, xs to represent the mea- surements. If the actual numbers are 5, 3, 8, 5, and 4,we have xl = 5, x2 = 3, x, = 8,x, = 5,and X, = 4.I , Most of the formulas we use require a summation of numbers. For example, one sum well need to obtain is the sum of all the measurements in the data set, orI x l + x2 + x3 + + x,. TO shorten the notation, we use the symbol 2 for theI . . a summation. That is, xl + x2 + x3 + ... + x, = n Verbally translate 2 x, as x,. r=l r=l follows: "The sum of the measurements, whose typical member is x,,beginning with the member xl and ending with the member x. ," Suppose, as in our earlier example, x, = 5,x2 = 3,x, = 8, x, = 5, and x, = 4. 5 Then the sum of the five measurements, denoted x,, obtained as follows: is r=lI Another important calculation requires that we square each measurementI n and then sum the squares. The notation for this sum is 2 x:. For the five mea- r=l surements above, we have 5 = 5 + 32 + S2 + 52 + 42 = 25 + 9 + 64 + 25 + 16 = 139 In general, the symbol following the summation sign 2 represents the vari- able (or function of the variable) that is to be summed. e Meaning of Summation Notatio Sum the measurements on the the right of the sum- mation symbol, beginning with d ending with the nth measurement.I Learning the Mechanics n Note: In all exercises, x represents ,=I . 2.27 Suppose a data set contains the obsenations 3 , 8 , 4 , 5 , 3 , 2.26 A data set contains the observations 5 , 1 , 3 , 2 , 1 .Find: 4,6. Find: a. Cx b. 2 x 2 c. C ( X- 1 ) a. C x b. E x 2 C. C ( X - 5 ) 2 d. C ( x - 1) e. (EX) d. E ( x - 2 ) 2 e. (EX)
• 69. 52 C H A P T E2 , M e t h o d s f o r D e s c r i b i n g S e t s of D a t a R2.28 Refer to Exercise 2.26. Find: 2.29 A data set contains the observations 6,0, -2, -1,3. Find: NUMERICAL MEASURES OF CENTRAL TENDENCY When we speak of a data set, we refer to either a sample or a population. If sta- tistical inference is our goal, well wish ultimately to use sample numerical de- scriptive measures to make inferences about the corresponding measures for a population. As youll see, a large number of numerical methods are available to de- scribe quantitative data sets. Most of these methods measure one of two data characteristics: 1. The central tendency of the set of measurements-that is, the tendency of the data to cluster, or center, about certain numerical values (see Figure 2.12a). 2. The variability of the set of measurements-that is, the spread of the data (see Figure 2.12b). In this section we concentrate on measures of central tendency. T the next n section, we discuss measures of variability. The most popular and best-understood measure of central tendency for a quantitative data set is the arithmetic mean (or simply the mean) of a data set. The mean of a set of quantitative data is the sum of the measurements divid- ed by the number of measurements contained in the data set. In everyday terms, the mean is the average value of the data set and is often used to represent a "typical" value. We denote the mean of a sample of measure- ments by T (read "x-bar"), and represent the formula for its calculation as shown in the box. Numerical descriptive measures t Center -I Spread 4 - a. b.
• 70. C * SECTION 2.4 Numerical Measures of Central Tendency 53 Calculate the mean of thc following five samplc mcasurcmcnts: 5 , 3 , 8 , 5 , 6 .S o I ut io n Using the definition of sample mean and the summation notation, we find Thus, the mean of this sample is 5.4.* * r the R&D expenditure percenS o I ut io n The mean R&D percentage for the 50 companies is denoted Rather than compute F by hand (or calculator), we entered the data of Table 2.4 into a computer and employed SPSS statistical software to compute the mean. The SPSS printout is shown in Figure 2.13. The sample mean, high- - - lighted on the is i = 8.492. RDEXP Valid cases: 50.0 Missing cases: .O Percent missing: .O Mean 8.4920 Std Err .2801 Min 5.2000 Skewness .8546 Median 8.0500 Variance 3.9228 Max 13.5000 S E Skew .3366 5% Trim 8.3833 Std Dev 1.9806 Range 8.3000 Kurtosis .4193 IQR 2.5750 S E Kurt .6619FIGURE 2.1 3SPSS printout of numerical descriptive measures for 50 R&D percentages Given this information, you can visualize a distribution of R&D percent- ages centered in the vicinity of T = 8.492. An examination of the relative fre- quency histogram (Figure 2.8) confirms that i does in fact fall near the center of thc distribution. The sample mean i will play an important role in accomplishing our objec- tive of making inferences about populations based on sample information. For this reason we need to use a different symbol for the mean of a population-the mean x *In the examples given here, is sometimes rounded to the nearest tenth, sometimes the nearest hundredth, sometimes the nearest thousandth, and so on.There is no specific rule for rounding when calculating T because i is specifically defined to be the sum of all measurements divided by x n; that is, it is a specific fraction. When is used for descriptive purposes, it is often convenient to round the calculated value of i to the number of significant figures used for the original measurements. When i is to be used in other calculations, however, it may be necessary to retain more significant figures.
• 71. C 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t a of the set of measurements on every unit in the population. We use the Greek let- ter p (mu) for the population mean. ek letters to represent mean are: Well often use the sample mean, i, estimate (make an inference about) the to population mean, p. For example, the percentages of revenues spent on R&D by the population consisting of all US companies has a mean equal to some value, p. Our sample of 50 companies yielded percentages with a mean of 2 = 8.492. If, as is usu- ally the case, we dont have access to the measurements for the entire population, we could use 2 as an estimator or approximator for p. Then wed need to know something about the reliability of our inference.That is, wed need to know how ac- curately we might expect 2 to estimate p. In Chapter 5 , well find that this accuracy depends on two factors: 1. The size of the sample. The larger the sample, the more accurate the estimate will tend to be. 2. The variability, or spread, of the data. All other factors remaining constant, the more variable the data, the less accurate the estimate. Another important measure of central tendency is the median. The median of a quantitative data set is the middle number when the mea- surements are arranged in ascending (or descending) order. The median is of most value in describing large data sets. If the data set is characterized by a relative frequency histogram (Figure 2.14), the median is the point on the x-axis such that half the area under the histogram lies above the me- dian and half lies below. [Note: In Section 2.2 we observed that the relative fre- quency associated with a particular interval on the horizontal axis is proportionalFIGURE 2.14Location o the median f Median
• 72. SECTION 2.4 Numerical Measures of Central Tendency 55 - to the amount of area under the histogram that lies above the interval.] We denote the median of a sample by m. 8 - Consider the following sample of n = 7 measurements: 5,7,4,5,20,6,2. a. Calculate the median m of this sample. b. Eliminate the last measurement (the 2) and calculate the median of the remaining n = 6 measurements.SoIution a. The seven measurements in the sample are ranked in ascending order: 2,4, 5,5,6,7,20 Because the number of measurements is odd, the median is the middle mea- surement. Thus, the median of this sample is m = 5 (the second 5 listed in the sequence). b. After removing the 2 from the set of measurements (the second 5 listed in the sequence), we rank the sample measurements in ascending order as follows: 4,5,5,6,7,20 Now the number of measurements is even, so we average the middle two measurements.The median ism = (5 + 6)/2 = 5.5. + In certain situations, the median may be a better measure of central ten- dency than the mean. In particular, the median is less sensitive than the mean to extremely large or small measurements. Note, for instance, that all but one of the measurements in part a of Example 2.6 center about x = 5. The single relatively large measurement, x = 20, does not affect the value of the median, 5 , but it caus- es the mean,? = 7, to lie to the right of most of the measurements. As another example of data from which the central tendency is better de- scribed by the median than the mean, consider the salaries of professional athletes (e.g., National Basketball Association players). The presence of just a few athletes (e.g., Shaquille ONeal) with very high salaries will affect the mean more than the median. Thus, the median will provide a more accurate picture of the typical salary for the professional league. The mean could exceed the vast majority of the sample measurements (salaries),making it a misleading measure of central tendency.S 0 I U t i 0 n For this large data set, we again resort to a computer analysis. The SPSS printout is reproduced in Figure 2.15, with the median highlighted. You can see that the median is 8.05.This value implies that half of the 50 R&D percentages in the data set fall below 8.05 and half lie above 8.05.
• 73. 56 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t a RDEXP Valid cases: 50.0 Missing cases: .O Percent missing: .O Mean 8.4920 Std Err .2801 Min 5.2000 Skewness .8546 Median 8.0500 Variance 3.9228 Max 13.5000 S E Skew .3366 5% Trim 8.3833 Std Dev 1.9806 Range 8.3000 Kurtosis .4193 IQR 2.5750 S E Kurt .6619 FIGURE 2.15 SPSS printout o numerical descriptive measures for 50 R&D percentages. f Note that the mean (8.492) for these data is larger than the median. This fact indicates that the data are skewed to the right-that is, there are more extreme measurements in the right tail of the distribution than in the left tail (recall the histogram, Figure 2.8). In general, extreme values (large or small) affect the mean more than the median since these values are used explicitly in the calculation of the mean. On the other hand, the median is not affected directly by extreme measurements, since only the middle measurement (or two middle measurements) is explicitly used to calculate the median. Consequently, if measurements are pulled toward one end of the distribution (as with the R&D percentages), the mean will shift to- ward that tail more than the median. 6 i A comparison of the mean and median gives us a general method for de- tecting skewness in data sets, as shown in the next box. 4 1
• 74. SECTION 2.4 Numerical Measures of Central Tendency 57 the data set is skewed to e mean is less than (to the left of) th A third measure of central tendency is the mode of a set of measurements. DEFINITION 2.6 The mode is the measurement that occurs most frequently in the data set. ~ ~ m ~ - s ~ ~ ~ ~ ~ w Each of 10 taste testers rated a new brand of barbecue sauce on a 10 point scale, where 1 = awful and 10 = excellent. Find the mode for the 10 ratings shown below. * 8 7 9 6 8 1 0 9 9 5 7 S o I ut io n Since 9 occurs most often, the mode of the 10 taste ratings is 9. Note that the data in Example 2.8 are actually qualitative in nature (e.g., "awful," "excellent"). The mode is particularly useful for describing qualitative data. The modal category is simply the category (or class) that occurs most often. Because it emphasizes data concentration, the mode is also used with quantitative data sets to locate the region in which much of the data is concentrated. A retail- er of mens clothing would be interested in the modal neck size and sleeve length of potential customers. The modal income class of the laborers in the United States is of interest to the Labor Departnlent. For some quantitative data sets, the mode may not be very meaningful. For example, consider the percentages of revenues spent on research and develop- ment (R&D) by 50 companies,Table 2.4. A reexamination of the data reveals that three of the measurements are repeated three times: 6.5%,6.9%, and 8.2%. Thus, there are three modes in the sample and none is particularly useful as a measure of central tendency. A more meaningful measure can be obtained from a relative frequency histogram for quantitative data. Thc measurement class containing the largest relative frequency is called the modal class. Several definitions exist for locat- ing the position of the mode within a modal class, but the simplest is to define the mode as the midpoint of the modal class. For example, examine the relative frequency histogram for the R&D expenditure percentages, reproduced below in Figure 2.16. You can see that the modal class is the interval 7.35-8.45. The mode (the midpoint) is 7.90. This modal class (and the mode itself) identifies-.. the area in which the data are most concentrated, and in that sense it is a mea- sure of central tendency. However, for most applications involving quantitative
• 75. 58 C H A P T E2 R M e t h o d s f o r D e s c r i b i n g S e t s of D a t a 1 F I G U R E2.16 Relative frequency histoqram for the computer companies 9 .30 R&D percentages: The modal class 3 .2o \$ - .- * 2 .lo .oo 5.15 6.25 7.35 8.45 9.55 10.65 11.75 12.85 13.95 R&D percentage data, the mean and median provide more descriptive information than the mode. :inding One-Variable Descriptive Statistics J S l N G THE T I - 8 3 G R A P H I N G CALCULATOR itep 1 Enter the data , Press STAT 1 for STAT Edit Enter the data into one of the lists. Step 2 Calculate descriptive statistics Press STAT Press the right arrow key to highlight CALC Press ENTER for 1-Var Stats Enter the name of the list containing your data. Press 2nd 1 for L1 (or 2nd 2 for L2 etc.) Press ENTER You should see the statistics on your screen. Some of the statistics are off the bottom of the screen. Use the down arrow to scroll through to see the remaining statistics. Use the up arrow to scroll back up. Example The descriptive statistics for the sample data set 86,70,62,98,73,56,53,92,86,37,62,83,78,49,78,37,67,79,.57 are shown below, as calculated by the TI-83. (continued, -. -c
• 76. SECTION 2.4 Numerical M e a s u r e s of C e n t r a l T e n d e n c y 59 - r Sorting The descriptive statistics do not include the mode. To find the mode, sort Data your data as follows: Press STAT Press 2 for SORTA( Enter the name of the list your data is in. If your data is in L1, press 2nd 1 Press ENTER The Tcreen will say: DONE To see the sorted data, press STAT 1 for STAT EditLearning the Mechanics Applying the Concepts2.30 Calculate the mode, mean, and median of the following 2.35 The total number of passengers handled in 1998 by data: eight cruise shlps based in Port Canaveral (Florida) are 18 10 15 13 17 15 )2 4.5 18 16 11 listed in the table below. Find and interpret the mean and median of the data set.2.31 Calculate the mean and median of tde following grade ? Fortune (Oct. 25,1999) magazines second annual ranking polnt averages: 3.2 <3 2.5 2.1 , "3.7 - 2.8 * ? 2.0 l othe 50 moTt powerful women in America is provided in the table on page 60. In addition to salary, position, and2.32 Calculate the mean for samples where profile, Fortune based these rankings on the womans a. n = 10, Ex = 85 b. n = 16, Ex = 40 influence within her company and her companys effect c. n = 45, Ex = 35 d. n = 18, Ex = 242 on culture in society. Cons~der age (in years) distribu- the2.33 Calculate the mean, median, and mode for each of th8- tion of these 50 women. Numerical descriptive statistics followmg samples: for the data are shown in the MINITAB printout below. a.7,-2,3,3,0,4 b. 2,3,5,3,2,3,4,3,5,1,2,3,4 a. Find the mean, median, and modal age of the distri- c. 51,50,47,50,48,41,59,68,45,37 bution. Interpret these values.2.34 Descr~be how the mean compares to the median for a b. What do the mean and the median indicate about distribution as follows: the skewness of the age distribution? a. Skewed to the left b. Skewed to the right c. What percentage of these women are in their for- c. Symmetric ties? Their fifties? Their sixt~es? CRUISE.DAT ....................................................................................................................... Cruise Line (Ship) Number of Passengers ....................................................................................................................... Canaveral (Dolphin) 152,240 Carnival (Fantasy) 480,924 Disney (Magic) 73,504 Premier (Oceanic) 270,361 Royal Caribbean (Nordic Empress) 106,161 Sun Cruz Casinos 453,806 Sterling Cruises (New Yorker) 15,782 Topaz Intl. Shipping (Topaz) 28,280 Source Florida Trend, Vol 41, No. 9, Jan. 1999. MINITAB Output for Exercise 2.36 Descriptive Statistics Variable N Mean Median Tr Mean StDev SE Mean 50 48.160 47.000 47.795 6.015 0.851 Variable Min Max Q1 Q3 Age 36.000 68.000 45.000 51.250
• 77. 60 2 CHAPTER M e t h o d s f o r D e s c r i b i n g Sets o f D a t a , q - WOMENPOW.DAT .......... .................................................................................................................................................................................................. . . .. . Rank Name Age Company Title Carly Fiorma Hewlett-Packard CEO Heidi Miller CItigroup CFO Mary Meeker Morgan Stanley Managing Director Shclly Lazarus Ogilvy & Mather CEO Meg Whitman eBay CEO Debby Hopkins Boeing CFO Marjorie Scardino Pearson CEO Martha Stewart Martha Stewart Living CEO Nancy Peretsman Allen & Co. Ex. V.P. Pat RUSSO Lucent Technologies Ex. V.P. Patricia Dunn Barclays Global Investors Chairman Abby Joseph Cohen Goldman Sachs Managing Director Ann Livermore Hewlett-Packard CEO Andrea Jung Avon Products COO Sherry Lansing Paramount P~ctures Chairman Karen Katen Pfizer Ex. V.P. Marilyn Carlson Nelson Carlson Cos. CEO Judy McGrath MTV & M2 Presidcnt Lois Juliber Colgate-Palmolive COO Gerry Laybourne Oxygen Med~a CEO Jud~th Estrin Cisco Systems Sr. V.P. Cathleen Black Hcarst Magazines President Linda Sandford IBM General Manager Ann Moore Time Inc. President Jill Barad Matte1 CEO Oprah Winfrey Harpo Entertainment Chairman Judy Lewent Mcrck Sr. V.P. Joy Covey Amazon.com COO Rebecca Mark Azurix CEO Deborah Willingham Microsoft v .P. Dina Dubion Chase Manhattan Ex. V.P. Patricia Woertz Chevron President Lawton Fitt Goldman Sachs Managing Director Ann Fudge Kraft Foods Ex. V.P. Carolyn Ticknor Hcwlett-Packard CEO Dawn Lepore Charles Schwab CIO Jeannine Rivet UnitedHealthcare CEO Jamie Gorelick Fannie Mae Vice Chairman Jan Brandt America Online Mar. President Bridget Macaskill OppenheimerFunds CEO Jeanne Jackson Banana Republic CEO Cynthia Trudell General Motors v . P, Nina DiSesa McCann-Erickson Chairman Linda Wachner Warnaco Chairman Darla Moore Rainwater Inc. President Marion Sandler Golden West Co-CEO Michelle Anthony Sony M u w Ex. V.P. Orit Gadlesh Bain & Co. Chairman Charlotte Beers J. Walter Thompson Chairman Abigail Johnson Fidelity Investments v . P. Source: Fortune, October 25,1999. d. Use a statistical software package to construct a rel- 2.37 The Superfund Act was passed by Congress to encour- ative frequency histogram. What is thc modal age age state participation in the implementation of laws class? Compare this to the modal age found in part a. relating to the release and cleanup of hazardous sub- e. Locate the mean and median on the histogram, part d. stances. Hazardous waste sites financed by the Superfund
• 78. SECTION 2.4 N u m e r i c a l Measures of C e n t r a l T e n d e n c y 61 - Act are called Superfund sites. A total of 395 Superfund drugs that effectively inhibit PAF, keeping it from bind- sites are operated by waste management companies in ing to human cells, may be successful in treating these Arkansas (Tabor and Stanwick, Arkansas Bl~sinessand disorders. A bioassay was undertaken to investigate the Economic Review, Summer 1995).The number of these potential of 17 traditional Chinese herbal drugs in PAF Superfund sites in each of Arkansas 75 counties is inhibition (H. Guiqui, Progress in Natural Science, June shown in the table. Numerical descriptive measures for 1995).The prevention of the PAF binding process, mea- the data set are provided in the EXCEL printout. sured as a percentage, for each drug is provided in the accompanying table. DRUGP*E.D*T .......................................................... Drug PAF Inhibition (%) Hai-feng-teng (Fuji) Hai-feng-teng (Japan) Shan-ju Zhang-yiz-hu-jiaoSource: Tabor, R.H.. and Stanwick. S.D. "Arkansas: An environmental Shi-nan-tengperspective. "Arkansas Business and Economic Krvirw, Vol. 28, No. 2, Huang-hua-hu-jiaoSummer 1995,pp. 22-32 (Table 1). Hua-nan-hu-jiao Xiao-yie-pa-ai-xiang Mao-ju EXCEL Output for Exercise 2.37 Jia-,ju Xie-yie-ju SITES Da-yie-ju Bian-yie-hu-jiao Mean 5.24 Bi-bo Standard Error 0.836517879 Duo-mai-hu-jiao Yan-sen Median 3 Jiao-guo-hu-jiao Mode 2 Standard Deviation 7.244457341 Source: Guiqui, H . " P A F receptor antagonistic principles from Chinese traditional drugs." Progress in Sample Variance 52.48216216 Natural Science, Vol. 5, No. 3, June 1995, p. 301 (Table 1). Kurtosis 16 .dl176573 Skewness 3.468289878 Range 48 1 a. Construct a stem-and-leaf display for the data. Minimum 0 Maximum 48 b. Compute the median inhibition percentage for the Sum 393 17 herbal drugs. Interpret the result. Count 75 c. Compute the mean inhibition percentage for the 17 Confidence Leve1(95.000%) 1.639542488 herbal drugs. Interpret the result. d. Compute the mode of the 17 inhibition percentages. Interpret the result. e. Locate the median, mean, and mode on the stem- a. Locate the measures of central tendency on the and-leaf display, part a. Do these measures of central printout and interpret their values. tendency appear to locate the center of the data? b. Note that the data set contains at least one county 2.39 The salaries of superstar professional athletes receive with an unusually large number of Superfund sites. much attcntion in the media. The multimillion-dollar Find the largest of these measurements, called an long-term contract is now commonplace among this outlier. elite group. Nevertheless, rarely does a season pass c. Delete the outlier, part b, from the data set and re- without negotiations between one or more of the calculate the measures of central tendency. Which players associations and team owners for additional measure is most affected by the elimination of the salary and fringe benefits for all players in their par- outlier? ticular sports.2-38 Platelet-activating factor (PAF) is a potent chemical a. If a players association wanted to support its argu- that occurs in patients suffering from shock, inflamma- ment for higher "average" salaries, which measure tion, hypotension. and allergic responses as well as res- of central tendency do you think it should use? piratory and cardiovascular disorders. Consequently, Why?
• 79. 62 2 CHAPTER M e t h o d s f o r D e s c r i b i n g Sets o f D a t a - b. To refute the argument, which measure of central a. Find and interpret the median for each of the data sets.C tendency should the owners apply to the players b. For each city, calculate the ratio of the number of salaries? Why? rooms to the number of hotels. Then find the average 2.40 Major conventions and conferences attract thousands number of rooms per hotel in each city. of people and pump millions of dollars into the local c. Re-rank the cities based on your answer to part b. economy of the host city. The decision as to where to 2.41 Refer to the Financial Management (Spring 1995) study hold such conferences hinges to a large extent on the of prepackaged bankruptcy filings,Exercise 2.24 (p. 49). availability of hotel rooms. The table, extracted from Recall that each of 49 firms that negotiated a reorgani- The Wall Street Journal (Nov. 17,1995), lists the top ten zation plan with its creditors prior to filing for bank- U.S. cities ranked by the number of hotel rooms. ruptcy was classified in one of three categories: joint exchange offer with prepack, prepack solicitation only. and no pre-filing vote held. A n SPSS printout of ..........................................,.................,.................................................... . .. . descriptive statistics for the length of time in bankrupt- City No. of Rooms No. of Hotels cy (months), by category, is shown below. a. Locate the measures of central tendency on the Las Vegas printout and interpret their values. Orlando Los Angeles-Long Beach b. Is it reasonable to use a single number (e.g., mean or Chicago median) to describe the center of the time-in-bank- Washington, D.C. ruptcy distributions? Or should three "centers" be New York City calculated, one for each of the three categories of Atlanta prepack firms? Explain. San Diego 2.42 The U.S. Energy Information Association tracks the Anaheim-Santa Ana prices of regular unleaded gasoline in the United States San Francisco each year. The table (p. 63) lists the average 1998 prices Source: Smith Travel Research. September 1995. (in cents) in each of a sample of 20 states. SPSS Output for Exercise 2.41 TIME By CATEGORY Joint Valid cases: 11.0 Missing cases: .O Percent missing: .O Mean 2.6545 Std Err .5185 Min 1.2000 Skewness .7600 Median 1.5000 Variance 2.9567 Max 5.4000 S E Skew .6607 5% Trim 2.5828 Std Dev 1.7195 Range 4.2000 Kurtosis -1.4183 IQR 3.1000 S E Kurt 1.2794 Valid cases: 11.0 Missing cases: .O Percent missing: -0 Mean 4.2364 Std Err .7448 Min 2.4000 Skewness 1.8215 Median 3.2000 Variance 6.1025 Max 10.1000 S E Skew .6607 5% Trim 4.0126 Std Dev 2.4703 Range 7.7000 Kurtosis 2.6270 1QR 1.6000 S E Kurt 1.2794 .............................................................................. TIME By CATEGORY Prepack Valid cases: 27.0 Missing cases: .O Percent missing: .O Mean 1.8185 Std Err .I847 Min 1.0000 Skewness 1.4539 Median 1 .4000 Variance .9216 Max 4.1000 S E Skew .4479 5% Trim 1.7372 Std Dev .9600 Range 3.1000 Kurtosis .9867 1QR .9000 S E Kurt .8721
• 80. Numerical Measures of Variability 63 -a. Calculate the mean, median, and mode of this data set. ........................................................ GASPRICE.DAT .b. Eliminate the highest price from the data set and re- State Price peat part a. What effect does dropping this mcasure- - - ment have on the measures of central tendency Arkansas 85.2 calculated in part a? Connecticut 112.2c. Arrange the 20 prices in order from lowest to highest. Delaware 94.6 Next, eliminate ihe lowest two prices and the highest Hawaii 117.9 two prices from the data set and calculate the mean of Louisiana 85.7 Maine 90.1 the remaining prices. The result is called a 10% Massachusetts 95.6 trimmed meun, since it is calculated after dropping the Michigan 84.4 highest 10% and the lowest 10% of the values in the Missouri 85.9 data set. An advantage of the trimmed mean is that it is Nevada 102.1 not as sensitive as the arithmetic mean to extreme ob- New Hampshire 96.6 servations in the data set. New Jersey 81.7 New York 96.4 North Dakota 90.4 Oklahoma 81.4 Oregon 100.2 Pennsylvania 91.1 Texas 83.8 Wisconsin 95.6 Wyoming 84.4 Source: Statistical Abstract o f the United States: 1998. U.S. Energy Information Administration, Petroleum Marketing Monthly. NUMERICAL MEASURES OF VARIABILITY Measures of central tendency provide only a partial description of a quantitative data set. The description is incomplete without a measure of the variability, or spread, of the data set. Knowledge of the datas variability along with its center can help us visualize the shape of a data set as well as its extreme values. For example, suppose we are comparing the profit margin per construction job (as a percentage of the total bid price) for 100 construction jobs for each of two cost estimators working for a large construction company.The histograms for the two sets of 100 profit margin measurements are shown in Figure 2.17. If you examine the two histograms, you will notice that both data sets are symmetric with equal modes, medians, and means. However, cost estimator A (Figure 2.17a) has profit margins spread with almost equal relative frequency over the measure- ment classes, while cost estimator B (Figure 2.17b) has profit margins clustered about the center of the distribution. Thus, estimator Bs profit margins are less variable than estimator As. Consequently, you can see that we need a measure of variability as well as a measure of central tendency to describe a data set. Perhaps the simplest measure of the variability of a quantitative data set is its range. EFI The range of a quantitative data set is equal to the largest measuremen minus the smallest measurement. The range is easy to compute and easy to understand, but it is a rather in- sensitive measure of data variation when the data sets are large. This is because
• 81. 64 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t ac FIGURE 2.17 4 t Profit margin histograms for two cost estimators 60 60 9" 0" 0- * 45 3 45 0 C 8 9 g 30 5 13 30 z z 15 15 -10 0 10 20 30 40 -10 0 10 20 30 40 Profit (%) Profit (%) a. Cost estimator A b. Cost estimator B two data sets can have the same range and be vastly different with respect to data variation. This phenomenon is demonstrated in Figure 2.17. Although the ranges are equal and all central tendency measures are the same for these two symmet- ric data sets, there is an obvious difference between the two sets of measure- ments. The difference is that estimator Bs profit margins tend to be more stable-that is, to pile up or to cluster about the center of the data set. In contrast, estimator As profit margins are more spread out over the range, indicating a higher incidence of some high profit margins, but also a greater risk of losses. Thus, even though the ranges are equal, the profit margin record of estimator A is more variable than that of estimator B, indicating a distinct difference in their cost estimating characteristics. Lets see if we can find a measure of data variation that is more sensitive than the range. Consider the two samples in Table 2.7: Each has five measure- ments. (We have ordered the numbers for convenience.) 2.7 TABLE Two Hypothetical Data Sets Sample 1 Sample 2 Measurements 1,2,3,4,5 2,3,3,3,4 - 1 + 2 + 3 + 4 + 5 - -= 3 - 15 - 2 + 3 + 3 + 3 + 4 - -= 3 15 Mean x = x = 5 5 5 5 Deviations of measure- (1 - 3), (2 - 3), (3 - 3), (4 - 3), (2 - 3), (3 - 3), (3 - 3), (3 - 3), ment values from i (5 - 3), or -2, -I,(), 1 , 2 (4 - 3), or -1,0,0,0,1 Note that both samples have a mean of 3 and that we have also calculated the distance and direction, or deviation, between each measurement and the mean. What information do these deviations contain? If they tend to be large in magnitude, as in sample 1, the data are spread out, or highly variable. If the de- viations are mostly small, as in sample 2, the data are clustered around the mean, - x, and therefore do not exhibit much variability. You can see that these devia- tions, displayed graphically in Figure 2.18, provide information about the vari- ability of the sample measurements.
• 82. SECTION 2.5 N u m e r i c a l Measures o f V a r i a b i l i t y 65 - FIGURE 2.18 Dot plots for two data ..... - x .. . .. - x sets I I I I I I * X I I I I I I * X 0 1 2 3 4 5 0 1 2 3 4 5 a. Sample 1 b. Sample 2 The next step is to condense the information in these deviations into a single 4 numerical measure of variability. Averaging the deviations from i wont help be- cause the negative and positive deviations cancel; that is, the sum of the deviations (and thus the average deviation) is always equal to zero. Two methods come to mind for dealing with the fact that positive and nega- tive deviations from the mean cancel. The first is to treat all the deviations as- though they were positive, ignoring the sign of the negative deviations. We wont pursue this line of thought because the resulting measure of variability (the mean of the absolute values of the deviations) presents analytical difficulties beyond the scope of this text. A second method of eliminating the minus signs associated with the deviations is to square them. The quantity we can calculate from the squared deviations will provide a meaningful description of the variability of a data set and presents fewer analytical difficulties in inference-making. To use the squared deviations calculated from a data set, we first calculate the sample variance. I DEFINITION 2 8 . The sample variance for a sample of n measurements is equal to the sum of the squared deviations from the mean divided by ( n - 1).In symbols, using s2 to represent the sample variance, 2 - 5(x, i=l - S - n-1 Note: A shortcut formula for calculating s2 is n - l Referring to the two samples in Table 2.7, you can calculate the variance for sample 1 as follows: The second step in finding a meaningful measure of data variability is to cal- culate the standard deviation of the data set.
• 83. 66 2 CHAPTER M e t h o d s f o r D e s c r i b i n g S e t s of D a t a The sample standard deviation, s, is defined as the positive square root of the sample variance, s2.Thus, s = G. The population variance, denoted by the symbol a2(sigma squared), is the average of the squared distances of the measurements on all units in the popula- tion from the mean, p, and a (sigma) is the square root of this quantity. Since we 7 never really compute a2or a from the population (the object of sampling is to avoid this costly procedure), we simply denote these two quantities by their re- spective symbols. Notice that, unlike the variance, the standard deviation is expressed in the original units of measurement. For example, if the original measurements are in dollars, the variance is expressed in the peculiar units "dollar squared," but the standard deviation is expressed in dollars. You may wonder why we use the divisor (n - 1)instead of n when calculat- ing the sample variance. Wouldnt using n be more logical, so that the sample. variance would be the average squared deviation from the mean? The trouble is, using n tends to produce an underestimate of the population variance, a2.So we use (n - 1)in the denominator to provide the appropriate correction for this ten- dency.* Since sample statistics like s2 are primarily used to estimate population parameters like a 2 ,( n - 1 ) is preferred to n when defining the sample variance. Calculate the variance and standard deviation of the following sample: 2,3,3,3,4. Solution As the number of measurements increases, calculating s2 and s becomes very tedious. Fortunately, as we show in Example 2.10, we can use a statistical software package (or calculator) to find these values. If you must calculate these quantities by hand, it is advantageous to use the shortcut formula provided in Definition 2.8. To do this, we need two summations: Ex and Ex2.These can easily be obtained from the following type of tabulation: *"Appropriaten here means that s2with the divisor of (n - 1)is an unbiased estimator of 0 We . define and discuss "unbiasedness" of estimators in Chapter 4. - .
• 84. SECTION 2.5 N u m e r i c a l Measures o f Variability 67 - Then we use* Use the computer to find the sample variance s2 and the sample standard deviation s for the 50 companies percentages of revenues spent on R&D. Solution The SAS printout describing the R&D percentage data is displayed in Figure 2.19. The variance and standard deviation, highlighted on the printout, are: s2 = 3.922792 and s = 1.980604. *FIGURE 2.19 UNIVARIATE PROCEDURESAS printout of numericaldescriptive measures for 50 MomentsR&D percentages N 50 Sum Wgts 50 Mean .492 Sum Std 0604 Varia Skewness 0.854601 Kurto USS 3797.92 CSS 192.2168 CV 23.32317 Std Mean 0.2801 T:Mean=O Sgn Rank Num " = 0 30.31778 637.5 50 Prob> T Prob> S I1 0.0001 0.0001 Quantiles(Def=5) 100% Max 13.5 99% 13.5 75% Q3 9.6 95% 13.2 50% Med 8.05 9 0% 11.2 25% Q1 7.1 10% 6.5 0% Min 5.2 5% 5.9 1% 5.2 Range 8.3 Q3-Ql 2.5 Mode 6.5 You now know that the standard deviation measures the variability of a set of data and how to calculate it. But how can we interpret and use the standard de- viation? This is the topic of Section 2.6. *When calculating sZ,how many decimal places should you carry? Although there are no rules for the rounding procedure, its reasonable to retain twice as many decimal places in s2as you ultimately wish to have ins. If you wish to calculates to the nearest hundredth (two decimal places), for example, you should calculate s2to the nearest ten-thousandth (four decimal places).