Practical Guide to DesignedExperimentsDK1182_half-series-title.qxd10/8/041:28 PMPage AMECHANICAL ENGINEERINGA Series of Textbooks and Reference BooksFounding EditorL. L. FaulknerColumbus Division, Battelle Memorial Instituteand Department of Mechanical EngineeringThe Ohio State UniversityColumbus, Ohio1. Spring Designers Handbook, Harold Carlson2. Computer-Aided Graphics and Design, Daniel L. Ryan3. Lubrication Fundamentals, J. George Wills4. Solar Engineering for Domestic Buildings, William A. Himmelman5. Applied Engineering Mechanics: Statics and Dynamics, G. Boothroyd and C. Poli6. Centrifugal Pump Clinic, Igor J. Karassik7. Computer-Aided Kinetics for Machine Design, Daniel L. Ryan8. Plastics Products Design Handbook, Part A: Materials and Components;Part B: Processes and Design for Processes, edited by Edward Miller9. Turbomachinery: Basic Theory and Applications, Earl Logan, Jr.10. Vibrations of Shells and Plates, Werner Soedel11. Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni12. Practical Stress Analysis in Engineering Design, Alexander Blake13. An Introduction to the Design and Behavior of Bolted Joints, John H.Bickford14. Optimal Engineering Design: Principles and Applications, James N. Siddall15. Spring Manufacturing Handbook, Harold Carlson16. Industrial Noise Control: Fundamentals and Applications, edited by Lewis H. Bell17. Gears and Their Vibration: A Basic Approach to Understanding Gear Noise,J. Derek Smith18. Chains for Power Transmission and Material Handling: Design and Applications Handbook, American Chain Association19. Corrosion and Corrosion Protection Handbook, edited by Philip A. Schweitzer20. Gear Drive Systems: Design and Application, Peter Lynwander21. Controlling In-Plant Airborne Contaminants: Systems Design and Calculations, John D. Constance22. CAD/CAM Systems Planning and Implementation, Charles S. Knox23. 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Steam Plant Calculations Manual, V. Ganapathy35. Design Assurance for Engineers and Managers, John A. Burgess36. Heat Transfer Fluids and Systems for Process and Energy Applications,Jasbir Singh37. Potential Flows: Computer Graphic Solutions, Robert H. Kirchhoff38. Computer-Aided Graphics and Design: Second Edition, Daniel L. Ryan39. Electronically Controlled Proportional Valves: Selection and Application,Michael J. Tonyan, edited by Tobi Goldoftas40. Pressure Gauge Handbook, AMETEK, U.S. Gauge Division, edited by Philip W. Harland41. Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R. P. Donovan42. Design of Mechanical Joints, Alexander Blake43. CAD/CAM Dictionary, Edward J. Preston, George W. Crawford, and Mark E. Coticchia44. Machinery Adhesives for Locking, Retaining, and Sealing, Girard S. Haviland45. Couplings and Joints: Design, Selection, and Application, Jon R. Mancuso46. Shaft Alignment Handbook, John Piotrowski47. 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Mechanical Properties of Polymers and Composites: Second Edition, Revised and Expanded, Lawrence E. Nielsen and Robert F. Landel91. Mechanical Wear Prediction and Prevention, Raymond G. Bayer92. Mechanical Power Transmission Components, edited by David W. South and Jon R. Mancuso93. Handbook of Turbomachinery, edited by Earl Logan, Jr.94. Engineering Documentation Control Practices and Procedures, Ray E. Monahan95. Refractory Linings Thermomechanical Design and Applications, Charles A. Schacht96. Geometric Dimensioning and Tolerancing: Applications and Techniques for Use in Design, Manufacturing, and Inspection, James D. Meadows97. An Introduction to the Design and Behavior of Bolted Joints: Third Edition,Revised and Expanded, John H. Bickford98. Shaft Alignment Handbook: Second Edition, Revised and Expanded, John Piotrowski99. Computer-Aided Design of Polymer-Matrix Composite Structures, edited by Suong Van Hoa100. Friction Science and Technology, Peter J. Blau101. 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Murray114. Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka115. Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M. Lepi116. Applied Computational Fluid Dynamics, edited by Vijay K. Garg117. Fluid Sealing Technology, Heinz K. Muller and Bernard S. Nau118. Friction and Lubrication in Mechanical Design, A. A. Seireg119. Influence Functions and Matrices, Yuri A. Melnikov120. Mechanical Analysis of Electronic Packaging Systems, Stephen A. McKeown121. Couplings and Joints: Design, Selection, and Application, Second Edition,Revised and Expanded, Jon R. Mancuso122. Thermodynamics: Processes and Applications, Earl Logan, Jr.123. Gear Noise and Vibration, J. Derek SmithDK1182_half-series-title.qxd10/8/041:28 PMPage E124. Practical Fluid Mechanics for Engineering Applications, John J. Bloomer125. Handbook of Hydraulic Fluid Technology, edited by George E. Totten126. Heat Exchanger Design Handbook, T. Kuppan127. Designing for Product Sound Quality, Richard H. Lyon128. Probability Applications in Mechanical Design, Franklin E. Fisher and Joy R. Fisher129. Nickel Alloys, edited by Ulrich Heubner130. Rotating Machinery Vibration: Problem Analysis and Troubleshooting,Maurice L. Adams, Jr.131. Formulas for Dynamic Analysis, Ronald L. Huston and C. Q. Liu132. Handbook of Machinery Dynamics, Lynn L. Faulkner and Earl Logan, Jr.133. Rapid Prototyping Technology: Selection and Application, Kenneth G.Cooper134. Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S.Rangwala135. Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions, edited by John D. Campbell and Andrew K. S. Jardine136. Practical Guide to Industrial Boiler Systems, Ralph L. Vandagriff137. Lubrication Fundamentals: Second Edition, Revised and Expanded, D. M. Pirro and A. A. Wessol138. Mechanical Life Cycle Handbook: Good Environmental Design and Manufacturing, edited by Mahendra S. Hundal139. 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Ganapathy150. The CAD Guidebook: A Basic Manual for Understanding and ImprovingComputer-Aided Design, Stephen J. Schoonmaker151. Industrial Noise Control and Acoustics, Randall F. Barron152. Mechanical Properties of Engineered Materials, Wol Soboyejo153. Reliability Verification, Testing, and Analysis in Engineering Design, Gary S. Wasserman154. Fundamental Mechanics of Fluids: Third Edition, I. G. Currie155. Intermediate Heat Transfer, Kau-Fui Vincent WongDK1182_half-series-title.qxd10/8/041:28 PMPage F156. HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and Operation, Herbert W. Stanford III157. Gear Noise and Vibration: Second Edition, Revised and Expanded, J. Derek Smith 158. Handbook of Turbomachinery: Second Edition, Revised and Expanded,edited by Earl Logan, Jr., and Ramendra Roy159. Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, George A. Antaki160. Turbomachinery: Design and Theory, Rama S. R. 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Puttlitz and Kathleen A. Stalter171. Vehicle Stability, Dean Karnopp172. Mechanical Wear Fundamentals and Testing: Second Edition, Revised and Expanded, Raymond G. Bayer173. Liquid Pipeline Hydraulics, E. Shashi Menon174. Solid Fuels Combustion and Gasification, Marcio L. de Souza-Santos175. Mechanical Tolerance Stackup and Analysis, Bryan R. Fischer176. Engineering Design for Wear, Raymond G. Bayer177. Vibrations of Shells and Plates: Third Edition, Revised and Expanded, Werner Soedel178. Refractories Handbook, edited by Charles A. Schacht179. Practical Engineering Failure Analysis, Hani M. Tawancy, Anwar Ul-Hamid,and Nureddin M. Abbas180. Mechanical Alloying and Milling, C. Suryanarayana181. Mechanical Vibration: Analysis, Uncertainties, and Control, Second Edition,Revised and Expanded, Haym Benaroya182. Design of Automatic Machinery, Stephen J. Derby183. Practical Fracture Mechanics in Design: Second Edition, Revised and Expanded, Arun Shukla184. Practical Guide to Designed Experiments: A Unified Modular Approach, Paul D. FunkenbuschDK1182_half-series-title.qxd10/8/041:28 PMPage GAdditional Volumes in PreparationMechanical Engineering SoftwareSpring Design with an IBM PC, Al DietrichMechanical Design Failure Analysis: With Failure Analysis System Soft-ware for the IBM PC, David G. UllmanDK1182_half-series-title.qxd10/8/041:28 PMPage HMarcel Dekker New YorkPaul D. FunkenbuschUniversity of RochesterRochester, New York, U.S.A.Practical Guideto DesignedExperimentsA Unified Modular ApproachDK1182_half-series-title.qxd10/8/041:28 PMPage iAlthough great care has been taken to provide accurate and current information, neither theauthor(s) nor the publisher, nor anyone else associated with this publication, shall be liable forany loss, damage, or liability directly or indirectly caused or alleged to be caused by this book.The material contained herein is not intended to provide specic advice or recommendationsfor any specic situation.Trademark notice: Product or corporate names may be trademarks or registered trademarksand are used only for identication and explanation without intent toinfringe.Library of Congress Cataloging-in-Publication DataA catalog record for this book is available from the Library of Congress.ISBN: 0-8247-5388-7HeadquartersMarcel Dekker, 270 Madison Avenue, New York, NY 10016, U.S.A.tel: 212-696-9000; fax: 212-685-4540Distribution and Customer ServiceMarcel Dekker, Cimarron Road, Monticello, New York 12701, U.S.A.tel: 800-228-1160; fax: 845-796-1772World Wide Webhttp://www.dekker.comThepublisher oers discounts onthis bookwhenorderedinbulkquantities. For moreinformation, write toSpecial Sales/Professional Marketingat the headquarters addressabove.Copyright n n n n n 2005 by Marcel Dekker. All Rights Reserved.Neitherthisbooknoranypartmaybereproducedortransmittedinanyformorbyanymeans, electronic or mechanical, including photocopying, microlming, and recording, or byanyinformationstorage andretrieval system, without permissioninwriting fromthepublisher.ISBN 0-203-99731-X Master e-book ISBN(Print Edition)This edition published in the Taylor & Francis e-Library, 2005.To purchase your own copy of this or any of Taylor & Francis or Routledgescollection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.For Ming Tian, wife, partner, friend, and inspirationMD:FUNKENBUSCH,JOB:04363,PAGE:iii5388-7_Funkenbusch_Dedication_R1_0916045388-7_Funkenbusch_Dedication_R1_091604MD:FUNKENBUSCH,JOB:04363,PAGE:ivPrefaceManygraduatingengineersandscientistsreceivelittleornotrainingindesignedexperiments. Whencourseworkisprovideditisoftenabstractanddivorcedfromthepractical considerationsof interest inindustrialapplication. One result has been the growth of a short-course industry,whichprovidesin-housetrainingat thejobsite, oftenemphasizingtheTaguchi method. These courses help practicing engineers and scientists getstarted with the techniques, but often lack the rigor and depth to developtrue long-term understanding.Theobjectiveofthisbookistobridgethisgap, bypresentingtheessentialmaterialina fashionthatpermitsrapidapplicationto practicalproblems, but provides the structure andunderstandingnecessaryforlong-term growth.The book covers two-level and three-level full and fractional factorialdesigns. In addition, the L12 and L18 designs popularized by Taguchiareincluded.Theroleandselectionofthesystemresponseformeasure-ment and optimization are described. Both conventional and Taguchi (S/Nratio)approachesarediscussedandtheirsimilaritiesanddierencesdescribed. Strategiesforincorporatingreal worldvariationintotheex-perimental design (e.g., use of noise factors) are presented anddescribed. Data analysis by analysis of means, analysis of variance(ANOVA), and the use of normal probability plots will be covered.The text presents the material using a modular or building blockapproach. In the rst section a simple but complete design is presented andanalyzed. Readers see how the entire structure ts together and learn theMD:FUNKENBUSCH,JOB:04363,PAGE:vv5388-7_Funkenbusch_Preface_R2_091404essential techniques andterminolgyneededtodevelopmore complexdesignsandanalyses. Inthesecondsection(Chapters47), readersaretaughtmorecomplexconceptsanddesigns.Theindividualchaptersarebased on three essential building blocks: array design, response selection,andincorporationofnoise.Thethirdsectionofthetext(Chapter8)deals with experimental analysis and follow-up in more detail.Thebookis at alevel suitablefor thosewithabasicscienceorengineeringbackgroundbut little or noprevious exposure tomatrixexperimentsorotherelementsofplannedexperimentation.Theprimaryaudienceisupper-level(juniorandsenior)undergraduatesandrst-yeargraduatestudents. Thebookcanalsobeusedasaself-studyguideforscientists and engineers in industry.Essential features of the book are:Clear,factualinstructionexplainingbothhowandwhyPresentationofadvancedtechniquesBalanceddiscussionofalternativeapproachesExamplesandcasestudiesfrommanydisciplinesHomework/reviewproblemsPaul D. FunkenbuschMD:FUNKENBUSCH,JOB:04363,PAGE:viPreface vi5388-7_Funkenbusch_Preface_R2_091404AcknowledgmentsI would like rst to thank Mr. John Corrigan and the other good folks atMarcel Dekkerforprovidingtheopportunityandsupportnecessarytomakethisbookareality.ProfessorRichardBensonwasmydepartmentchair when I rst became interested in the eld of designed experimentationandprovidedmewitharst chancetoteach, explore, anddevelopanorganizational structure for this material. Similarly, Profs. J.C.M. Li,Stephen Burns, and my other senior colleagues in the ME department atthe University of Rochester provided the intellectual freedom and supportnecessary to further develop my approach.Writingtakes time. For this bookthat time came largelyat theexpense of my wife and children. They put up with my odd hours and (morethan usually) eccentric behavior without quibble or complaint. I amdeeplygrateful.Many people provided concepts and insight that helped in develop-ing the approachpresentedhere. Inlieuof a long, andnecessarilyincomplete, listing, Icanonlyexpressmysinceregratitudetothemanycolleagues, students, clients, andinstructors whohelpedme tobothunderstandthedetailsandseetheunifyingstructure. Thisbookismyhumble attempt to pass this understanding onto present the beauty andpower of these designs inapackage that makes themavailable inasystematic and readily accessible form.MD:FUNKENBUSCH,JOB:04363,PAGE:viivii5388-7_Funkenbusch_Acknowledgments_R1_0916045388-7_Funkenbusch_Acknowledgments_R1_091604MD:FUNKENBUSCH,JOB:04363,PAGE:viiiContentsPreface vAcknowledgments viiIntroduction1 HowaDesignedExperimentWorks 1Overview 1I. PolishingFluidDeliverySystem 1II. ControlFactors,Levels,andCoding 2III. ArraySelectionandSetup 3IV. NoiseFactorsandConditions 5V. ResponsesandCharacteristicResponses 8VI. AnalysisofMeans 10VII. AnalysisofVariance 12VIII. Conclusion 142 FundamentalConcepts 16Overview 16I. FoodforThought 16A. Is this a Case of Poor Design? Improper Use?Who is at Fault? 17B. IsthisaPoorQualityMachine? 17C. HowWouldYouGoAboutFixingtheCopier?HowWouldYouGoAboutDesigninganImprovedCopier? 18MD:FUNKENBUSCH,JOB:04363,PAGE:ixix5388-7_Funkenbusch_Contents_R2_091404II. Over-the-WallEngineering 18III. ContributionsofTaguchi 19IV. StagesofDesign 20V. DeningQuality(TraditionalApproach) 21VI. QuadraticLossFunction 23VII. TheP-Diagram 24VIII. Conclusion 27Homework/DiscussionProblems 273 StatisticalConcepts 30Overview 30I. ReviewofBasicConcepts 30A. PurposeofReview 30B. Distribution 31C. Position 32D. Dispersion 33E. DegreesofFreedom 35F. TheNormalDistribution 35G. EectofFiniteSampleSize 37II. ANOMandANOVA 39A. GoalsofAnalysis 39B. ColumnIdentication 39C. ANOMAlgebra 40D. ANOMComparisons 41E. DeterminingtheBestLevelforEachFactor 42F. ANOVAConcept 42G. PreparingtheANOVATable 43H. CalculationforEachColumn 43I. CalculationfortheTotal 44J. FillingintheANOVATable 44K. CheckontheTotals 46L. RankingandSignicance 46M. SignsofTrouble 47N. DeterminingBestLevelswithInteractions 48O. Model 49Homework/DiscussionProblems 50MD:FUNKENBUSCH,JOB:04363,PAGE:xContents x5388-7_Funkenbusch_Contents_R2_091404BuildingBlocks4 ArrayDesign(Two-LevelFactors) 55Overview 55I. BasicConcepts 55A. ArrayDesign 55B. EquivalenceofArrays 57C. CatapultExample 59D. ComparisonwithOne-At-A-TimeExperiments 60II. Interactions 61A. DenitionofanInteraction 61B. InteractionGraph 62C. LocationofInteractionTerms(FactorialArraysAppendixA) 63D. Confounding 65E. WhatIsWrongwithInteractions? 66F. ReducingtheStrengthofInteractions 67G. Resolution 69H. LocationofInteractionTerms(ScreeningArraysAppendixC) 73III. DesignTrade-OsandStrategies 73A. FourDesignStrategies 73B. TheExperimentalTriangle 74C. FullFactorialStrategy 76D. High-ResolutionDesignStrategy 78E. SpeciedInteractionStrategy 81F. Low-Resolution(Saturated)Strategy 83G. RandomizationoftheRunOrder 84Homework/DiscussionProblems 865 SelectionoftheCharacteristicResponse 89Overview 89Responseandcharacteristicresponse 89Thenaturalchoice 90I. GoodExperimentalPractice 90A. EasilyandAccuratelyMeasurable 90B. ContinuousRatherThanDiscrete 91C. TiedCloselytoFunctionality 92MD:FUNKENBUSCH,JOB:04363,PAGE:xiContents xi5388-7_Funkenbusch_Contents_R2_091404II. LinkedtoLong-TermImprovement 92A. MeasurableEarlyintheDesignProcess 93B. TiedCloselytoEnergyorMassFlow 93C. TracksFunctionNotDysfunction 94D. DynamicResponses 94III. EaseofOptimizationandImplementation 96A. ExploitsaPhysicallySignicantReferenceorZeroPoint 96B. Completeness 97C. OptimizedbyDrivingtoanExtremeValue 98D. Two-StepOptimization 99IV. TaguchisS/NRatios 101A. WhatisanS/NRatio? 101B. Smaller-the-BetterS/N 102C. Larger-the-BetterS/N 104D. Signed-TargetS/N 104E. Nominal-the-BestS/N 105F. S/NRatioApplication 107G. FinalCommentsonTaguchiS/NRatios 108Homework/DiscussionProblems 1086 InclusionofReal-WorldVariability 112Overview 112I. NoiseConditions 112A. RoleofNoiseConditions 112B. ClassicationofNoise 113C. GoalsandTrade-Os 115II. AMenuofStrategies 115A. OuterArray 115B. StressTest(Extremes) 117C. Surrogate 119D. RepeatedMeasurements 122E. SingleArray(Traditional) 123Homework/DiscussionProblems 1287 AdditionalArrayDesigns(Three-LevelFactorsandModications) 129Overview 129I. StandardDesignsforFactorswithThreeLevels 129MD:FUNKENBUSCH,JOB:04363,PAGE:xiiContents xii5388-7_Funkenbusch_Contents_R2_091404A. BasicTrade-OsforThree-LevelDesigns 129B. StandardThree-LevelDesigns 131C. Analysis of Variance for Three-Level Factors 134D. Analysis of Variance for the 18 TC ScreeningArray 135II. DesignStrategieswithThree-LevelFactors 135A. FourDesignStrategies 135B. FullFactorialStrategy 136C. High-ResolutionDesignStrategy 138D. SpeciedInteractionStrategy 138E. LowResolution(Saturated) 139III. ModicationofColumns 141A. MotivationandLimitations 141B. CombiningColumnsTechnique 141C. Virtual(Dummy)LevelsTechnique 143D. UseofBothTechniques 144Homework/DiscussionProblems 144AnalysisofResults8 AdditionalAnalysisTechniques 147Overview 147Choosingananalysistechnique 147I. PoolingofUnused(Empty)Columns 148II. PoolingofSmallestEects 149III. Replication 152IV. NormalProbabilityPlots 156A. Concept 156B. Procedure 157C. ApplicationExample 157Homework/DiscussionProblems 161AppendixA:Two-LevelFactorialArrays 163AppendixB:Three-LevelFactorialArrays 176AppendixC:ScreeningArrays 181AppendixD:CriticalValuesofF 185References 189Index 191MD:FUNKENBUSCH,JOB:04363,PAGE:xiiiContents xiii5388-7_Funkenbusch_Contents_R2_091404MD:FUNKENBUSCH,JOB:03198,PAGE:xiv5388-7_Funkenbusch_Contents_R2_0914041How a Designed Experiment WorksOverviewThischapterprovidesanoverviewofhowthecomponentsofanexper-imental design t and work together. This is performed by walking throughthe design and analysis of a simple experiment. If you are not familiar withexperimentaldesign,thischapterwillprovidethenecessaryroadmapforplacingthesubsequentchapters(whichdescribedierentdesignandanalysis components in detail) into proper context.I. POLISHING FLUID DELIVERYSYSTEMIn recent years, the processing of precision optics has undergone a trans-formationfromacraftsman-basedprocesstooneincreasinglydrivenbycomputer numerical control (CNC) systems. Computer numerical controlpolishing relies on the generation of a consistent polishing spot, so thatthecontrollercanpredictthedwelltimerequiredtoproducethedesiredremoval andsmoothingateachlocationontheoptical surface. Duringpreliminarytestingofanewpolishingmachineprototype, concernwasexpressedovertheconsistencyofthepolishinguidbeingsupplied. InMD:FUNKENBUSCH,JOB:04363,PAGE:115388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:2particular, the concentrationof abrasive suppliedbythe systemwasfrequently much lower than the nominal value. An undergraduate studentteam[1] was recruited to develop a newuid delivery systemand to identifysettings that coulddeliver concentrations close tothe nominal. Thisexample is primarily based on data collected during their work.In building the prototype machine, an existing machine platformthathad a coolant system, but no provision for including abrasives, was used.Therefore auiddeliverysystem, capable of supplyinganabrasive-containingpolishinguidmixture, hadbeenexternallyaddedtothemachine. Figure 1illustrates the basic designof this external system.Polishing uid of the desired composition is loaded into the bottle, alongwith a magnetic stirring bar. A magnetic stirring plate causes rotationofthe bar and stirring of the uid. Fluid is extracted from the bottle using aperistaltic pumpandtravels throughaexible plastic tube until it isdeposited at the polishing zone.II. CONTROL FACTORS,LEVELS, AND CODINGThenumberof possibleparametersforexperimentationisoftenmuchlargerthanthenumberwecanpracticallytest inasingleexperimentaldesign. Therefore the experimental teammust dosome preliminarywinnowing, basedontheir assessment of likelyimportance. For thepolishinguiddeliverysystem, theteamidentiedfourparametersfortesting: bottle design, diameter of the supply tube, stir bar design, and owratethroughthetube. Werefertotheseasthecontrol factorsforthisFIGURE1 Schematicillustrationofthepolishinguiddeliverysystem.Chapter1 25388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:3experiment. In addition, the team was concerned about how the value ofonecontrol factormightaecttheresultsobtainedforanothercontrolfactor, a phenomena referred to as an interaction. Two interactions wereidentiedasbeingofparticularconcernthatbetweenthedesignofthestir bar and the design of the bottle in which it moves, and that between theuid ow rate and the diameter of the tube through which it ows.To see the eect of a control factor on performance, it is necessary tochangeitsvaluewithintheexperimentaldesign.Forexample,toseetheeect ofthebottledesign, someexperimentsarerunwithonepossibledesign and some with another. We then judge the importance of the designfromthedierenceintheresultsbetweenthesetwosets. Thedierentvalues given to a control factor within an experimental design are referredtoasitslevels.Fortheuiddeliverysystem,twolevelswereselectedforeach of the control factors.For convenience, it is useful to set up a simple shorthand system torefer to dierent factors and their levels. Control factors will be designatedby capital letters (e.g., A, B) and levels by integers (e.g., 1, +1). Figure 2shows this system applied to the uid delivery system. Thus, for example,Factor B, level 1 corresponds to using a tube diameter of 3/32 in.III. ARRAYSELECTION AND SETUPAcrucial choiceinexperimental designistheselectionandsetupoftheexperimental array. For the uid delivery experiment, the array selected isshown in Fig. 3. (This particular array is the 16 TC Factorial design fromAppendix A, where TCstands for treatment condition. Toset upa specicexperimental design using this array, each of the four factors is assigned to aspecic column. You will learn how to create a design like this in Chap. 4).Although the array in Fig. 3 may look a little overwhelming at rst, this ismoreafunctionofbulkthanofcomplexity.Theunderlyingconceptsarequite simple, as can be seen by focusing on individual elements in the design.FIGURE2 Controlfactorsandlevelsforuiddeliveryexperiment.HowaDesignedExperimentWorks 35388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:4Eachnumberedrowinthearraydesignatesaparticulartreatmentcondition (TC). A treatment condition is a specic combination of controlfactor levels to be run (tested) during the experiment. Because there are 16treatment conditions inthis particular design, theexperiment involvesrunning 16 dierent combinations of control factor levels.The numbered columns in the array have had one of three dierenttypes of designationadded. Four of thecolumns arelabeledwiththecontrol factorcodes(i.e., A, B, C, D). Thesefourcolumnsprovidetheinformationactuallynecessarytoconducttheexperiment.Inparticular,theydesignate the settings of the control factors tobe usedfor eachtreatment condition. For example, for TC 2, the A column has a value of1. This means that factor A should be set to level 1 for this treatmentcondition. Referring back to Fig. 2, this means that a convex bottle shouldbe used. Decoding the rest of the information for TC 2, it can be seen thatthe tube diameter should be 3/32 in. (Bat level 1), the stir bar should havea pill geometry (C at level 1), and a ow rate of 12 ml/min (Dat level +1)should be used.FIGURE3 Arraydesignandfactorassignmentforuiddeliveryexperiment.Chapter1 45388-7_Funkenbusch_Ch01_R2_091404Theremainingcolumns inFig. 3arenot neededtoperformtheexperiment, but will be used in analyzing the results. Six of the remainingcolumns(labeledA B, A C, A D, B C, B D, andC D)represent interactionterms betweenthe various combinations of twofactors. Thedesignationonthesecolumns means that wecanusethecoding in the column to help calculate the eect of the interaction when weanalyze the experiment. The remaining columns in the design, labeled withthe small letter e, will be used to estimate the error in the experiment.IV. NOISE FACTORS AND CONDITIONSAcommonfeature inmanydesignandengineeringproblems is thepresence of parameters that we believe are important, but that cannot becontrolled in the eld. It is important that such parameters be included inthe experimental design so that recommendations based on the results willbevalidunderrealisticconditions.Forexample,theenvironmentalcon-ditions (temperature, humidity) will inuence the performance of anautomobilestarter. Toensurethestarterworkswell wheninstalledoncustomerscars, itis, therefore, importantthatexperimentationbecon-ducted at dierent values (levels) of temperature and humidity. This type ofparameter, which is varied in a controlled fashion during experimentation,butwillbeuncontrolledinthenalapplication,isreferredtoasanoisefactor. For experimental convenience, it is useful to have a designation torefertospeciccombinationsofnoisefactorlevels.Thesecombinationsare designated as noise conditions (NC). Use of this terminology parallelsthe use of treatment condition (TC) to represent a specic combination ofcontrol factor levels. (Strategies for identifying noise factors and levels, andsetting up noise conditions are discussed in Chap. 6). Extending our codingsystemtocovernoisefactorsandlevels,itisoftenhelpfultousecapitalletters near the end of the alphabet (e.g., U, V, W) to designate noise factorsand again designate levels in terms of 1 and +1. Noise conditions will benumbered sequentially (1, 2, 3, . . .).For the uid delivery system, concern was focused on the possibilitythattheuidcompositiondeliveredmightvarywith theamountofuidMD:FUNKENBUSCH,JOB:04363,PAGE:5FIGURE4 Noiseconditionsforuiddeliveryexperiment.HowaDesignedExperimentWorks 55388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:6FIGURE5Additionofcolumnsfornoiseconditionsandcharacteristicresponsetoexperimentaldesignfortheuiddeliveryexperiment.Chapter1 65388-7_Funkenbusch_Ch01_R2_091404remaininginthebottle. Tocapturethispossibility, itwasdecidedtosample the uid supplied near the beginning and near the end of each bottleused in testing. Specically, the uid was tested after 10% of the bottle hadbeenpumpedandagainafter70%. Hencetheamount ofuidalreadypumped was taken as a noise factor with two levels, 10% and 70%. Withonlyonenoisefactor,thereisnoneedtosetupafullcodingsystemoffactors andlevels for this example. Therefore informationonnoiseconditions can be summarized in a single table as shown in Fig. 4.Figure5showstheexperimental designwiththenoiseconditionsadded. Each noise condition is set up as a separate column on the right sideof the matrix. Hence the noise conditions (columns) run perpendicular to thetreatment conditions (rows), dening a 16 2 matrix of all TC/NC combi-nations. (The nal column in this gure, a is discussed in the next section.)Datais recordedfor eachTC/NCcombinationandthe resultsrecordedonthechart.ThisisshowninFig.5usingthecodingyi, jtode-signate the result obtained with treatment condition i and noise condition j.MD:FUNKENBUSCH,JOB:04363,PAGE:7FIGURE 6 TableforuseinsettingupTC/NCcombinationsandrecordingdataduringexperimentation.HowaDesignedExperimentWorks 75388-7_Funkenbusch_Ch01_R2_091404Thus, for example, y2,1 is the result recorded for TC 2 and NC 1; in otherwords, with a convex bottle (A at level 1), a tube diameter of 3/32 in. (Batlevel 1), apillgeometryforthestirbar(Catlevel 1), aowrateof12 ml/min (Dat level +1), and 10%of the uid pumped (noise condition 1).Figure6showsastrippeddownversionof Fig. 5, withonlytherequired control and noise settings for each measurement displayed. Thistype of table is useful for recording data during actual experimentation.V. RESPONSES AND CHARACTERISTICRESPONSESTo analyze the eects of the various control parameters, it is necessary todevelopanumberthatcharacterizestheperformanceofthesystemforeach set of control factors (i.e., for each treatment condition). This involvestwoimportant decisions: (1) what specicresponse(i.e., yi,j) shouldbemeasured for each TC/NC combination and (2) how should the responsesfor the dierent NC be combined to give a single value that characterizestheperformancefortheTC.Thissinglevalueiscalledthecharacteristicresponse for the treatment condition.1. Fortheuiddelivery system,itwas decided to focusonasinglepolishing solution, consisting of 9.8 wt.%1-Amalumina powder inwater.Theresponseforeach TC/NCcombination(i.e.,yi, j)wasmeasured as the wt.%of alumina inthe solution actually delivered.2. Inpreliminarytesting, themainperformanceconcernwas thefailure of the delivery systemto deliver the nominal concentration,as a result of loss (settling) of alumina fromthe water. Oversupply(i.e., a concentration greater than the nominal) was much less of aproblem. Therefore it was decided to initially focus on increasingthe concentration supplied throughout the polishing process.The average concentration was chosen as the characteristic responsefor analysis:ai yi;1yi;22where ai is the characteristic response for treatment condition i. (You willlearn more about how to identify and choose a response for measurementand a characteristic response for analysis in Chap. 5.)Figure7shows theresponses measuredintheactual experimentalongwiththevaluesofthecharacteristicresponse(ai)calculatedfromMD:FUNKENBUSCH,JOB:04363,PAGE:8Chapter1 85388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:9FIGURE7Measuredresponsesandvaluesofthecharacteristicresponseforuiddeliveryexperiment.HowaDesignedExperimentWorks 95388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:10them. Todemonstrate howtocompute a, we choose the responsesproduced for the second uid delivery treatment condition:y2;1 2:56andy2;2 1:94These two responses are combined to give a single value of the character-istic response for the second treatment condition:a2 y2;1y2;222:56 1:942 2:25This calculation is repeated for each value of i, producing 16 characteristicresponses, oneforeachtreatment condition. Thecalculatedresultsareshown in Fig. 7.VI. ANALYSIS OFMEANSANalysisOf Means(ANOM) isusedtodeterminewhichlevel ofeachfactor is the best andcanalsoprovide arelative rankingof theirimportance.Asthenameimplies,ANOMinvolvescomparingthemean(average)valuesproducedbythedierentlevelsofeachfactor.Speci-cally,thecharacteristicresponsesforalltreatmentconditions,wherethefactor was at level 1, are averaged and compared to the average obtainedwith the factor at level +1.Figure8showsthisanalysisappliedtoeachofthecolumnsintheuiddeliveryexperiment,withm+1andm1denedastheaveragesforlevels+1and 1,respectively. Disthedierencebetweenm+1andm1(i.e., m+1m1), which we will call the eect for the column. To illustratethese calculations, consider the column for Factor B (tube diameter). Thiscolumn indicates a level of +1 for treatment conditions 58 and 1316 anda value of 1 for treatment conditions 14 and 912. Thusm1 a5a6a7a8a13a14a15a168 6:16m1 a1a2a3a4a9a10a11a128 5:39andD m1m1 6:16 5:39 0:77These results are shown at the bottom of the column in Fig. 8 along withthose for each of the other columns in the design. Figure 9 shows some ofthese same results graphically. Note rst of all that the dierences observedChapter1 105388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:11FIGURE8Calculatedvaluesofeachcolumnseectfortheuiddeliveryexperiment.5388-7_Funkenbusch_Ch01_R2_091404HowaDesignedExperimentWorks 11MD:FUNKENBUSCH,JOB:04363,PAGE:12for factors A and C are much larger than those for the other factors andinteractions. So attention should be primarily focused on these two factors.Factor levels that produce larger values were desired because the goal wasto increase the concentration of the uid delivered by the system. Hence,from ANOM analysis, the best factor levels are determined to be 1 forfactor A (convex bottle geometry) and +1 for factor C (modied stir bar).Settings for the other factors and interactions are less important.VII. ANALYSIS OFVARIANCEANalysis Of VAriance (ANOVA) provides a second level of analysis. It isuseful for judging the statistical signicance of the factor and interactionFIGURE 9 Graphical presentation of ANOM results (m1 and m+1) for the uiddeliveryexperiment.Chapter1 125388-7_Funkenbusch_Ch01_R2_091404eects observed. As the name implies, ANOVAinvolves studyof thesourcesof varianceinthedata. Inessence, atermproportional tothesquareofthedierencesrevealedbyANOManalysis(i.e.,D=m+1 m1) is divided by a similar termestimated for error to produce a ratio F. Alarge value of F indicates a variance much larger than that anticipated fromerror. Depending on the condence level desired and some other details ofthe experimental design and analysis, a critical F can be dened. Factors orinteractions that produceFvalues larger thanthecritical Fvaluearejudged statistically signicant.For the uiddeliveryexperiment, it was decidedtouse a90%condencelevel tojudgesignicance, whichgaveacritical Fvalueof4.06. Figure 10 summarizes the results of ANOVAfor the experiment. Thistable is written in standard format that includes information not needed atpresent.(DetailsofthisandotheraspectsofbothANOMandANOVAanalysis are presented in Chap. 3.) However, focusing on the last column,we can obtain the central ndings of the ANOVA analysis.All of the interactions and two of the factors (B and D) have F valuesmuch lower than the critical value of 4.06. This does not necessarily meantheydonotinuencetheconcentrationdeliveredbythesupplysystem.However, if they did have an eect, it was too small to be resolved againstthe background of experimental error. Two of the factors (A and C) haveF values much larger than the critical F value, and are judged signicant.MD:FUNKENBUSCH,JOB:04363,PAGE:13FIGURE10 ANOVAtablefortheuiddeliveryexperiment.HowaDesignedExperimentWorks 135388-7_Funkenbusch_Ch01_R2_091404TheseresultsareverysimilartothoseobtainedfromtheANOManalysis. In particular, two of the factors are singled out from the others asimportant. What is added by ANOVAis an assessment of whether theseeects are real (statistically signicant) or not. In particular, for the uiddelivery system, the ANOVA analysis provides condence that the selec-tion of the proper levels of factors A and C really will aid in increasing theconcentration of uid supplied.Testsoftheuiddeliverysystemwererunwiththerecommendedsettingsbotho-lineandinstalledonthepolishingsystem. ResultsaresummarizedinFig. 11. There is clearlystill roomfor improvement.However, the system was able to consistently deliver a polishing uid withaconcentrationnearthe9.8%nominal.(Comparetheseresultswiththerange of values seen in Fig. 8.)VIII. CONCLUSIONThis chapter provided, by way of a working example, an introduction tothekeyelementsofpracticaldesignedexperimentsandaperspectiveonhow they t together to make a complete experiment. These key elementsincludedesignoftheexperimentalarray,selectionofasuitableresponseandcharacteristicresponse, incorporationofreal-worldvariability, andstatistical analysis of theresults. Thefollowingchapters providemoredetailed information on these key elements, both detailed instructions andlibraries of alternative approaches and strategies.Chapter 2provides basic backgroundoncontemporaryproductdesignandqualityconcepts, important forproperlyunderstandingtheapproaches and strategies described in the following chapters.MD:FUNKENBUSCH,JOB:04363,PAGE:14FIGURE11 Resultsofsubsequenttestingoftheuiddeliverysystemusingtherecommendedsettings.Chapter1 145388-7_Funkenbusch_Ch01_R2_091404Chapter 3 explains how to analyze designed experiments and inter-prettheirresults, includingamoredetailedexplanationofANOMandANOVA.Chapter 4 deals with experimental array design for factors with twolevels. Itbeginswithadescriptionofthearraysanddiscussionoftheirfundamental trade-osandlimitations. It thenexplainshowtoset uparraystomatchspecicapplications,includingalibraryofbasicexper-imental design strategies.Chapter 5 describes how to select and rene a response and charac-teristic response to meet specic experimental and project goals.Chapter6dealswithincorporationofreal-world(eldandmanu-facturing) variability into the experimental design. This includes discussionof dierent types of noisefactors andalibraryof strategies for in-corporating them into an experiment.Chapter 7 continues discussion of experimental array design, extend-ing it to arrays with three-level factors, as well as explaining howarrays canbe modied to include factors with nonstandard numbers of levels.Chapter 8 extends discussion of howto analyze designed experimentsandinterpret their results, bypresentingseveral additional techniques.Theseincludepoolingofsmallesteects,replication,anduseofnormalprobability plots.MD:FUNKENBUSCH,JOB:04363,PAGE:15HowaDesignedExperimentWorks 155388-7_Funkenbusch_Ch01_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:162FundamentalConceptsOverviewInthischapter,youwillgainsomebasiccontextontheroleandusesofdesignedexperimentationfor solvingpractical problems. Muchof thelayout and analysis that you will see in subsequent chapters is motivated byphilosophical considerations such as project organization, design goals,andthe denitionof quality. Gaininganunderstandingof these isimportant in understanding why particular denitions and structures areused. It will alsoguideyouinmakingrational choicesasyoudevelopexperimental designs to t your own projects specic goals.I. FOOD FORTHOUGHTThe sign shown in Fig. 1 appeared on my departments copier several yearsago. (The sign is reproduced verbatim except that, for obvious reasons, thecopiermakersnamehasbeendeleted.)Pleasereaditandthen, beforereadingthesubsequentdiscussion, considerforafewsecondshowyoumight respond to each of the following questions.165388-7_Funkenbusch_Ch02_R2_091404A. Is this aCaseofPoorDesign?Improper Use?Who isat Fault?Discussion of whos at fault can growquite contentious. At various times, Ihave heardblame placedonthe designteam, the factory, the salesrepresentative,andthecustomer.Arelativelycommonthemeistolookfor writtenrequirements for the roomsize or airowinthe ownersmanual.Iftheroomisbigenoughaccordingtothese,thenthecopiermust beat fault. If, however, it is evenasinglecubicfoot under therequirement, then it is the customers fault. Fundamentally, of course, sucharguments miss the most important point. From a customers perspective,acopierthatdoes notworkwell isunacceptableregardlessof whetheritreectsaproblemwithdesign, ormanufacturing, ortheenvironmentalconditions (room). This is unlikely to change based on the ne print in themanual.Conversely,acustomerwhoplacedanidenticalcopierinalessstressfulenvironmentmightbesatisedwithitsperformanceevenifthecopier is sensitive tooverheating. The performance obtainedfromaproduct, and therefore its perceived quality, depends both on the productitself and on the conditions under which it is used.B. Is this aPoorQuality Machine?This question really asks you to make a false choice. Bad quality or not? Inthe rst place, we have not dened quality in a meaningful way. Moreover,the situationwe are consideringis clearlymore complexthancanbeaccounted for by a simple yes or no answer. This particular machine worksMD:FUNKENBUSCH,JOB:04363,PAGE:17FIGURE1 Customerfeedback.Whoistoblame?FundamentalConcepts 175388-7_Funkenbusch_Ch02_R2_091404well most of the time (during the day) but needs some extra care at night.Clearly, it could be better, but it also might be worse (e.g., if it overheatedduring the day). Dierent customers (with dierent environments) mightalsohaveverydierentexperiences. Indiscussingquality,therefore, weneedtobe specic inour denitionandtoconsider the inuence ofvariability on performance.C. HowWouldYouGoAbout Fixingthe Copier?HowWouldYouGoAbout DesigninganImproved Copier?Responsesastohowtoxaproblemaregenerallymorepositiveandcreative thanthose involvingassignment of blame. Possible remediesincludefans, heat pumps, insulation, providingalowerenergysleepmode, and improving the energy eciency of the copy process (which usesheat to permanently fuse the image onto the paper). In general, two broadcategories can be distinguished: those that treat the symptom (accumula-tion of heat) and those that treat the source (the amount of heat produced).In dealing with an existing problem, treating the symptom, sometimes, isan eective approach. For example, it might be easier to retrot a fan intotheabovecopierthantotrytoinstallacompletelynewfusersystem.Inproductdesignandmanufacturing,however,anemphasisonthefunda-mentals(sources)isoftenmostvaluable.Careshouldbetakentoavoidletting the design process be dominated by the practice of identifying andxing specic problems, at the expense of making fundamentalimprovements in function and eciency.II. OVER-THE-WALL ENGINEERINGMachinists, especially those whoworkinacademia, get tosee manyinterestingblueprints: solidblockswithspherical cavitiessomehowma-chined inside, threads in the form of a mobius loop, and so on. These areextremeexamplesofagenericproblem,atendencytoovercompartmen-talize designandmanufacturing processes. Figure 2a illustrates thisproblem. The design division passes its blueprint over-the-wallseparatingit frommanufacturing, relyingonthemtosomehowoutputtheproductasdesigned.Figure2bshowsarelatedproblem,inwhichaproductdesignedinalaboratoryenvironmentispassedover-the-walltothecustomer, whereitmustsomehowbemadetofunctioninaverydierentenvironment.MD:FUNKENBUSCH,JOB:04363,PAGE:18Chapter2 185388-7_Funkenbusch_Ch02_R2_091404A wide range of skills is necessary to bring a complex product fromthe design board to the market. So it is reasonable to organize the processintermsofdivisions/eldssuchasdesign,manufacturing,andcustomerservice, with expertise in specic areas. The problem is, therefore, not withthe compartmentalization itself. It is with how well information about themanufacturingandoperatingenvironments canbepassedupstream(from the eld to testing to manufacturing and then to design) and withhow that information can best be incorporated into design decisions.III. CONTRIBUTIONSOFTAGUCHIGenichi Taguchi (1924)hasplayedamajorroleintheeldofqualityengineeringandimprovement,includingboththedevelopmentofexper-imental designandinitsapplicationtosolvingpractical problems. Hisworkhas stimulateda greatdealofthoughtand analysisand therehavebeensome sharpcontroversies. However, eventhose critical of somespecics in his approach generally recognize the fundamental importanceof theunderlyingconcepts. Wewill belearningfromandinterpretingTaguchis work in many places in this chapter and throughout this book.Inthefollowingsectionsofthischapter,someofthemoreimportantoftheconceptsintroducedbyTaguchi aredescribed. Insubsequentchap-ters, we will use these ideas to help guide the development of experimentaldesigns.MD:FUNKENBUSCH,JOB:04363,PAGE:19FIGURE2 Over-the-walldesign.FundamentalConcepts 195388-7_Funkenbusch_Ch02_R2_091404IV. STAGESOFDESIGNOneofTaguchisimportantcontributionshas beentostimulate thoughtabout how the design process can be structured to avoid problems, such asthose inherentin over-the-wall design, and to look for additional oppor-tunitiesforproduct improvement. Figure3illustratesadesignprocessbasedonsomeofhisideas. Itconsistsofthreedistinctstages: concept,parameter, and tolerance design.In concept design, the basic architecture, assembly, and manufactur-ing process are dened. The fundamental emphasis is making sure that theunderlyingphysics works. This stage establishes the frameworkfor allsubsequent development of theproduct. Creativityandinnovationareoften important parts of this process and can provide big rewards. On theotherhand,mistakesmaybedicult,impossible,orcostlytocorrectatlater stages. For example, we might choose aninternal combustion,electric,orhybridpowerplantforavehicleintheconceptdesignstage.Subsequent design can be used to modify the particular choice made, buttheinitialconceptselectionwilllargelydeterminethedesignspace(e.g.,range of weight, fuel eciency, pollution, and cost) in which they can work.Inparameter design, thenominal values of keyspecications areselectedsothat the product works consistently well inspite of thevariability inherent in the manufacturing and/or operating environments.The fundamental emphasis is on making the physics work under real-worldconditions. For example, by matching the thermal expansion coecients ofparts in intimate contact, a design less sensitive to operating temperaturecanbe produced.Thesetypesofchoicesoftenhave onlyaminorimpact(positive or negative) on production costs, but can lead to major improve-ments in nal performance.Intolerancedesign,practicaltrade-osaremadebetweenperform-ance and manufacturing cost. The fundamental emphasis is on making theeconomics work. As the name implies, tolerance designofteninvolvesdetermination of the manufacturing tolerances about the specied nominalvalues. The tighter the tolerances, the closer the actual product will be toMD:FUNKENBUSCH,JOB:04363,PAGE:20FIGURE3 Three-stagedesignprocess.Chapter2 205388-7_Funkenbusch_Ch02_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:21the ideal specied in the blueprints, but the higher the cost. So componentsmust be prioritized so that tolerances are tightened where they will producethe most impact for the least cost.To illustrate these stages, consider the design of a simple mechanicaldevice such as a catapult. At the end of the concept design stage, the basicphysics will have been worked out and there will be a drawing of the device,but manyspecications will still be open. For example, arange ofpossible launch angles may have been identied, but the specic angle maystill be open (launch angle =H). At the end of the parameter design stage,thedrawingwouldincludenominal dimensionsandotherspecications(launch angle = 30j). At the end of the tolerance design stage, tolerancesareaddedtothedrawingproducingablueprint suitableforassembly/manufacturing (launch angle = 30 F 5j).In a large organization, these dierent stages could be dierent boxeson an organizational chart, with dierent scientists and engineers assignedto each. In a small organization, the same few people may be involved ineach of the stages. The important point is not the formal structuring, butthe push to look at a design problem in several distinctly dierent ways. Inparticular, parameter design is a process that can deliver large benets, butis often neglected. Without parameter design, specications may be set attheendoftheconceptdesignstageandpassedthroughfortolerancing,withoutthechancetoexperimentallytestalternativesortoevaluatetheeects of variability in the manufacturing and operating environments.Designedexperiments are of particular importance insuccessfulimplementation of parameter design,and manyof the examples you willseeinthistextbookmaybethought of asparameterdesignproblems.Designed experiments are also important in tolerance design, specically toexperimentally determine which manufacturing tolerances have the largesteects on the consistency of product performance. Concept design, whichgenerally involves choices among fundamentally dierent design paths, isless amenable to this type of experimental analysis.V. DEFINING QUALITY(TRADITIONALAPPROACH)In the everyday world, quality is often used as a synonym for good ornice. Wearesoldhigh-quality electronicsbyquality employeeswho do only good quality work. This may be good public relations, butit does not tell us very much about what is really happening.Amoremeaningfulapproachtoqualitybeginsbyconsideringthecost eects of having a component (or product) that does not exactly matchFundamentalConcepts 215388-7_Funkenbusch_Ch02_R2_091404its ideal specications/requirements. As a simple example, consider a metalpinthatisdesignedtotsnugglyintoamatchingholeduringproductassembly. There is anidealdiameterforthis pin, forwhich the assemblyworksperfectlyasdesigned.If,however,thepindiameterdeviatesfromthis value, it will not work as well. If it is too large, it will be dicult (orimpossible) to insert and may cause unwanted friction and wear. If it is toosmall,itwillbewobblyorevenfallout.Asameasureofthequality,wedenetheloss,L,asthedollarcostofhavinganactualvalue,y,whichdeviates fromthe ideal value,m. This canbe expressed in equation formas follows:Ly fy m 1where f stands for is a function of. Note that when y = m, the qualityisperfectsothereshouldbenolossandL=0. Theimportantpartofthedenitionis,however,whatweassumehappenswhenymovesawayfrom m.Historically,industryhastendedtolookatqualityintermsofthetolerances needed to assemble a product from mass-produced parts. Eachcomponent is given an ideal specication (m) and a plus/minustolerance(D). For example, the diameter of the pin might be specied as 1 F 0.01 cm.Anindividualpartwithintheselimits(i.e.,between0.99and1.01cmindiameter)givesanacceptableresultandcanbeused. Apartoutsideofthese limits (1.01 cm) does not produce an acceptable resultand should be rejected. Expressing this in equation form gives:Ly 0 mD < y < mD2Ly ksy < mD; y > mDwhere ksis a constant reecting the cost of remanufacturing (scrapping andreplacing) the part.MD:FUNKENBUSCH,JOB:04363,PAGE:22FIGURE4 Traditionalpass/failsystemforassessingquality.Chapter2 225388-7_Funkenbusch_Ch02_R2_091404Thispass/fail(accept/reject)systemforjudgingqualityproducesastep function for L and is illustrated in Fig. 4. It ties quality directly to thetraditionalengineeringuseoftolerancesandprovidesarelativelysimplewayofestimatingthecost associatedwithhavingsomedistributionofimperfect parts. However, dening quality this way also has its drawbacks.Most fundamentally, this system provides a poor representation of actualperformance as a measure of quality. A part which only marginally withintolerances is rated equally with one that has exactly the desired dimensions,althoughtheir performancemaydier markedly. Moreover, this samemarginally in tolerance part is rated completely dierently from one that isonly marginally out of tolerance, although their performance is probablyindistinguishable.Beyondtheseemingunfairness of this system, thepoor tietoperformancecanhaveother negativeconsequences. Considertwo-partsuppliers, bothofwhomdeliverequallyhighpercentageswithintoler-ance,as illustratedinFig.5. Therstsupplier accomplishesthiswithatightlycontrolledprocess, whose distributionis sharplyspikedat theoptimumvalue,m.Thesecondsupplierhasapoorlycontrolledprocesswith a wide distribution, but gets the necessary percentage within toleranceby screening out parts beyond the allowable limits. In terms of the pass/failsystem, they will be ranked equally. However, clearly, the rst supplier isdelivering a superior quality product.VI. QUADRATIC LOSSFUNCTIONRecognizingsome of the problems withthis traditional approachtodening quality, Taguchi proposed the quadratic loss function as an alter-native.Ifaparthasavaluethatisdierentfromtheideal(i.e.,y p m),its performance will be inferior even if it is within tolerance. The cost ofthis reduced performance may be considered a measure of the quality loss.MD:FUNKENBUSCH,JOB:04363,PAGE:23FIGURE5 Partdistributionfromtwopossiblesuppliers.FundamentalConcepts 235388-7_Funkenbusch_Ch02_R2_091404The quadratic loss function assumes that this loss scales with the square ofthe distance from the ideal. ThusLy kqy m23where kq is a constant. This relationship is shown in Fig. 6.The selection of a quadratic function can be rationalized in dierentways, but obviously is somewhat arbitrary. One nice feature of Eq. (3) isthat it produces a very simple form when applied to a distribution of parts.Averaging the loss from Eq. (3) over a distribution with an average of l,and a standard deviation of r, gives:Q kql m 2 r2h i4where Qis the average quality loss (cost) per part. This formula emphasizesthe fact that both average and variability are important in determining thequality of a product. Applying this formula to the case shown in Fig. 5, forexample, the superior performance expectedfromthe rst suppliersproduct would be reected in a lower valueof Q, as a result of a smallervalue for the r2term.Finally, note that, while we have so far discussed quality in terms ofvariability in the product, the same concepts are applicable to variability inproductperformancecausedbyvariationsinoperatingenvironment.Inthis case, rwouldreect the spreadinperformance causedby theenvironment.VII. THE P-DIAGRAMTheP(productorprocess)diagramisanalconceptualtool,usefulforunderstandinghowperformanceandenvironmentarerelated. Figure7MD:FUNKENBUSCH,JOB:04363,PAGE:24FIGURE6 Thequadraticlossfunctionforassessingquality.Chapter2 245388-7_Funkenbusch_Ch02_R2_091404shows the rststep in constructing a P-diagram. Underideal conditions,weinput asignal factor(s) totheproduct andoutput acorrespondingresponse. A signal factor is a parameter set by the end user to control theproduct performance. As shown in the graph, under ideal conditions, thereisaone-to-onecorrespondencebetweentheselectedvalueofthesignal(input)andtheresponse(output). Signal factorsaremosteasilyunder-stood in terms of literal signals, such as the setting on a knob or the voltageon an electronic controller. However, it is also sometimes helpful to thinkin terms of signal factors in terms of physical inputs to the system, such aspower supplied or raw material ow. In this case, the ideal signalresponserelationshiprepresentsthesuccessful conversionofthisinputintonalform/product, e.g., conversion of input power into motion (kinetic energy).InFig. 8, theeectsofvariationsintheproductortheoperatingenvironment are introduced in the form of noise factors. A noise factor isdened as a parameter that is left uncontrolled or cannot be controlled inactual operation. Therefore any inuence of noise factors on the responseresultsinanundesirablevariationwhentheproductisactuallyused.Asshown in Fig. 8, random scatter in the response is introduced and the slopeof the signal/response curve may be changed.Finally, Fig. 9shows thecompletedP-diagram, inwhichcontrolfactorshavebeenselectedtooptimizethedesign. Acontrol factorisaparameterthatistestedduringthedesignprocessinordertoidentifyitsbestvalue. Thisvalueisthenxedinthenal design. AsshownintheMD:FUNKENBUSCH,JOB:04363,PAGE:25FIGURE7 Idealresponseofproductorprocessresponsefollowssignalfactorwithdesiredslope.FundamentalConcepts 255388-7_Funkenbusch_Ch02_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:26FIGURE8 Uncontrolled(noise) factors candistort theresponse, changingtheslopefromtheidealvalueandcausingscatter.FIGURE 9 Completed P-diagram, illustrating the role of control factors inrestoringthedesiredcorrespondencebetweensignalandresponse.Chapter2 265388-7_Funkenbusch_Ch02_R2_091404graph, proper selection of control factors is used to reduce the variability intheresponseandadjusttheslopeofthesignal/responsecurveontothedesired value.Toillustrateuseof theP-diagram, consideritsapplicationtothedesign of an industrial furnace. The desired response for the furnace is theinternal temperature, which is used in heat-treating metal parts. The signalfactor is simply the setting on the temperature controller (or, possibly, thepower input to the heating elements). Ideally, the temperature everywhereinsidethefurnacewouldcorrespondidenticallytothevalueset bythetemperature controller. However, we know that the actual temperature inthefurnacemaybeinuencedbyawidevarietyofparametersthataredicult or impossibletocontrol inservice. Theseconstitutethenoisefactorsforthefurnace. Examplesincludetheexternal temperatureandhumidity, the power supply voltage, the number and size of parts (thermalload) placedinthefurnace, thelocationof individual partswithinthefurnace, and the aging of the heating elements. The design engineer needsto be aware of these parameters and their potential inuence when he or sheselects specic values for the parameters that canbe specied. Theseparameters that the designengineer canspecifyrepresent the controlfactors. Examplesforthefurnaceincludethetype, number, andlayoutof the heatingelements, the number andlocationof the temperaturesensors,thecontroller(e.g.,on/ovs.proportional,responsetime,etc.),and the shape and size of the chamber.VIII. CONCLUSIONThis chapter introduced some important current concepts in the elds ofmanufacturing, quality, and design, including the need to avoid over-the-wall engineering, theconcept of parameterdesign, theimportanceofvariability inassessing quality, andthe use of the P-diagramforvisualizingtherolesofnoiseandcontrol factorsindeterminingperfor-mance. These concepts have played an important role in the developmentof the overall frameworkfor experimental designinindustry. Theirinuence on many specic aspects of the experimental design process willbe seen in the following chapters.Homework/DiscussionProblems1) Describearecentproductorprocessfailurethatyouhaveeither witnessed or been involved in (e.g., your car not starting). Who doMD:FUNKENBUSCH,JOB:04363,PAGE:27FundamentalConcepts 275388-7_Funkenbusch_Ch02_R2_091404you feel was to blame for the failure? Do you think it was an example ofover-the-wall design? Explain why or why not.2) Describeanexampleofsomethingyoufeelrepresentsover-the-wall design, preferablysomethingfromyour ownexperience. Bespecic about what seemed to have been missing in the design process.3) Describe howuse of a three-stage design process as described inthis chapter (Fig. 3) can help avoid problems with over-the-wall design.4) Assumethatyouhavebeenassignedtothedesignteamforanewcoeemaker. Describethemajordecisionsthatyouwouldneedtomake during each of the three design stages shown in Fig. 3. Be specic.5) Many manufacturers, particularly small ones, still prefer to usestrictly pass/fail designspecications todetermine the acceptabilityof parts. For example, theywant the customer tospecifyhis or herrequirementsstrictlyintermsofapartsizeandxedtolerancesuchas2 F 0.02 cm.a. Explain, fromthe standpoint of such a manufacturer, why he orshe may prefer such a system.b. What advantages does this systemalso oer to you, thecustomer?c. Describe some of the incentives for the customer to change thissystem(i.e., what are some of the problems and limitations).6) Refer toFig. 5. Manufacturingthe distributionshownforsupplier2(rightside)mayactuallybemorecostlythanthatthetighterdistributionshownfor supplier 1. For example, consider theneedtoscreen parts to obtain an acceptable distribution. Describe at least twoways that such screening could raise costs.7) Thequadraticlossfunction[Eq. (4)] emphasizestherolesofboththeaverageandvarianceindeterminingquality. Advocatessome-timesapplythisequationdirectlytoproduct dataasawayof makingquantitativecomparisons. Skepticsgenerallyaccepttheconcept behindthequadraticlossfunction, butdonotbelieveit shouldbeinterpretedliterally.Amanufacturingprocessforpowdermetallurgypartsisdesignedaroundanideal linear shrinkage of 8.9%. (Bothhigher andlowershrinkages are possible but undesirable.) Sample powders suppliedbytwo dierent vendors give dierent results when used for this process.Vendor 1: 8.9%average shrinkage with a standard deviationof 1.5%.MD:FUNKENBUSCH,JOB:04363,PAGE:28Chapter2 285388-7_Funkenbusch_Ch02_R2_091404Vendor2: 9.1%averageshrinkagewithastandarddeviationof0.25%.a. Apply the quadratic loss function to determine whichvendors powder produces better results.b. How would a skeptic view this assessment and why?c. Propose an alternative way to make this comparison.Contrastyourproposal withthequadraticlossfunction(advantages and disadvantages).8) DrawtheP-diagramanduseittoexplaintherelationshipbetween control factors and noise factors.9) Select a product from your home or work (e.g., a refrigerator).Draw a P-diagram for it and identify a signal factor, response, and at leastve control and noise factors.10) Clearlyandconciselydistinguishbetweenthefollowingtypesof factors. (That is, how would you determine the category to which afactor belongs?)a. Noise factor vs. control factorb. Signal factor vs. the responsec. Signal factor vs. control factor11) Inanalyzingtheresponseof acar, theextent towhichtheaccelerator is depressed is modeled as a factor.a. Give an example of a response for which the depression of theaccelerator would be a signal factor. Explain.b. Giveanexampleof aresponsefor whichdepressionof theaccelerator is best considered a noise factor. Explain.MD:FUNKENBUSCH,JOB:04363,PAGE:29FundamentalConcepts 295388-7_Funkenbusch_Ch02_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:303StatisticalConceptsOverviewThis chapter is divided into two sections. In the rst, some basic statisticalconceptsarereviewedandillustrated. Inthesecond, theprocedureforusinganalysisof means(ANOM) andanalysisof variance(ANOVA),which you explored in application in Chap. 1 is developed in more detail.Upon successfully completing this chapter you will be prepared to performANOM and ANOVA and interpret the results from these analyses.I. REVIEW OFBASIC CONCEPTSA. Purposeof ReviewMost engineers and scientistshave had someexposureto basicstatistics,such as calculating a standard deviation or computing a condence limitwith a normal distribution. In contrast, relatively few have performed ananalysis of variance, which may appear quite complex in contrast. In fact,however, ANOVA isbased on manyof the samefundamental principlesasthemorefamiliartechniques.Thissectionprovidesaquickoverview305388-7_Funkenbusch_Ch03_R2_091404of somebasicstatistics, includingmanypointsuseful inunderstandinganalogous operations in ANOM and ANOVA.B. DistributionAnyreal measurement contains somedegreeof variability, either as aconsequence of real dierences in the measurand (the quantity being mea-sured)oruncertaintyinthemeasurementsystem.Forexample,amanu-facturing process may be designed to produce sections of plastic pipe 1 minlength, but the measured length of any individual section will undoubtedlydeviate somewhat from this specication.Thevariabilityinmeasuredvalues ismost convenientlycapturedusingafrequencydistributionsuchas thoseillustratedinFig. 1. Thehorizontal axis shows the measured value, while the vertical axis denes therelative probabilitythat agivenmeasurement will lie withinacertainrange.ForFig.1a,thepossiblemeasuredvaluesaresortedintodiscretebins, sothatadiscretedistributionisshown. Theprobabilitythat arandomly chosen measurement is within the range dened by a particularbin may be read o the vertical axis directly. Alternatively, one can say thatifalargenumberofmeasurementsaretaken, thevertical axisgivesthefraction expected to be within each bin (i.e., the frequency of occurrence).Note that the sum of frequencies for all bins must add up to 1.Figure 1b shows a continuous distribution. In this case, the widthofeachbinisinnitesimallysmall, sotheprobabilityofhavingexactlysomevalueisalsovanishinglysmall. Instead, itisnecessarytothinkintermsof theprobabilityof havingameasuredvaluewithinaspeciedrange(ineecttodeneabinsothattheprobabilitymaybedeter-mined). Thisisgivenbytheareaunderthecurvebetweenthespeciedvaluesandcanbeobtainedbyintegratingthe(equationforthe) curvebetween the limits specied for the range. Note that in this case the totalarea under the curve must again be equal to one.Thecompletefrequencydistributioncurvefullycapturesinfor-mation about the distribution of the measured value. In practice, however,it is common to summarize this information in simpler forms. This may bedone because full information is lacking or simply for convenience. Aretailpurchaser of the plastic pipe, e.g., would probably be more confused thanenlightened by a frequency distribution curve. Average length is probablyenough for the purchaser. On the other hand, the manufacturer probablyneedstotrackvariability. Inproduct manufacturing, thecompletefre-quency distribution, including the shape, is of interest.MD:FUNKENBUSCH,JOB:04363,PAGE:31StatisticalConcepts 315388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:32C. PositionThe average, or mean, is the sum of the individual measurements dividedby the total number of measurements. It can be found from the followingformula:x Xni1xin1aFIGURE1 (a) Discrete and (b) continuous distributions.Chapter3 325388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:33where x is the average for the sample, xi are the individual measured datapoints, and n is the total number of data points in the sample. This is themost commonwaytoquantifycurveposition, andgenerallythemostmathematically useful.If datafromacomplete populationare used, the correspondingequation is:l Xni1xiN1bwhere lis the true average for the population, xiare the measuredindividual datapoints, andNisthetotal numberofdatapointsinthepopulation. x is an estimate of the true population value, l.Alternatives to using the average do exist. The median is the value forwhich the number of measurements below the median is equal to the num-ber of measurements above it. Since the median is based only on balancingnumbersaboveandbelowitisnotasdistortedbyextremevaluesastheaverage,apropertythatissometimesveryuseful.Forexample,asinglebillionairemovingintoacityof1millionpeoplewouldraisethecitysaverage net worth per person by more than $1000, but have essentially noaect on the median. Finally, the position is also sometimes quantied interms of the mode(s), dened as the value(s) at which the frequency curvereachesamaximum.Thisiseasytodographicallyandcanbeusefulinidentifying subpopulations.D. DispersionDispersionis ameasure of the scatter about the mean. The simplestmeasureofdispersionistherange,denedasthedierencebetweenthemaximumandminimumvaluesmeasured. Rangeissometimesusedinstatistical process control applications andtoprovide error bars onexperimental data. However, it is dicult to use quantitatively and subjectto distortion by outliers.Standard deviation is one of the most useful measures of dispersion.Standard deviation is often estimated from a sample of the total popula-tion by using the following formula:s Xni1xix2n 1vuuut2aStatisticalConcepts 335388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:34where s is an estimate of the true (population) standard deviation based ona nite sample of data, xi are the individual data points, xis the average ofthe sample, and n is the number of data points in the sample. If, instead of asample, all members of thepopulationareused, thetrue(population)standard deviation can be obtained as:r XNi1xil2Nvuuuut2bwhereristhetrue(population)standarddeviation, listhepopulationaverage, andNis the number of points inthe population. Fromanengineeringstandpoint, standarddeviationis aconvenient measureofdispersion because it has the same units as the value being measured.Variance is given by the square of the standard deviation:s2Xni1xix2n 13ar2XNi1xil2N3bOnekeyadvantageofvarianceisthatuncorrelated(independent)vari-ancescanbeadded, whereasstandarddeviationscannot. Toillustrate,consideralargedistributionof plasticpipes1mlongwithastandarddeviation of 1 cm. If these pipes are randomly selected and placed together(end to end) in sets of two, what is the average and standard deviation ofthe resulting distribution?Theaverageisclearly2mbutthestandarddeviationisnot2cm.Instead, the standarddeviationmust be calculatedby rst addingthe variances and then converting to the standard deviation. In this case,thevarianceofthenewdistributionis(1cm)2+(1cm)2=2cm2.Thestandard deviation is the root of the variance, M2 cm c1.4 cm.Notethat thistypeof calculationonlyworksifthevariancesareuncorrelated. Inthisexample, thiswasassuredbystatingthatthetotalnumber of pipes is large and the selection of two pipes to be placed togetheris random. Hence, the distribution of lengths fromwhich the second pipe isselectedisnotinuencedbytheselectionoftherstpipe.Ifwetriedtoplace large and short pipes together to compensate for each other, thisChapter3 345388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:35calculation would be invalid. Being able to add (uncorrelated) variances isvery important in tolerancing. It also plays a key role in statistical analysis.One additional measure of the dispersion that is often encountered isthe coecient of variance, Cv. This is simply the standard deviation dividedbytheaverage. It isnatural toexpect tighterdistributions, inabsoluteterms, on smaller parts than on larger parts. For example, a 1-mmstandarddeviation is quite large for a part with an average length of 1 cm, but seemssmall fora1-m-longpart. Thecoecientofvariancenormalizesthedispersion against the average to obtain a relative measure:Cv rlcsx4E. Degrees ofFreedomThenumberofimpendentquantitiesthatcanbecalculatedfromexper-imental data cannot exceed the number of data points. Degrees of freedom(dof) are a useful way of accounting for the amount of data available, andused, in analyzing an experiment.For example, in Eq. (3a), the square of the dierence between a datapoint andthe average maybe thought of as anestimate of the true(population) variance. In the numerator of Eq. (3a) the terms for each ofthe data points are summed. In the denominator, this sum is divided by thenumber of independent estimates involved. Althoughtherearendatapoints, these same points were used to calculate the estimate of the averagevalue (x), which also appears in Eq. (3a). Therefore, there are only n 1independent estimates of the variance.F. The Normal DistributionFigure 2 shows the familiar bell-shaped curve associated with a normaldistribution.Anormaldistributionisoftenobservedexperimentally andalsooftenassumedinstatistical analyses. Whyshouldthis beso?If ameasured value reects the sumof a large number of small randomevents itturnsout that it will tendtobedistributednormally, regardlessof thedistribution associated with each of the individual events.Asimple example is useful to illustrate this eect. If we roll a standard(fair)six-sideddie, thefrequencydistributionoftheoutcomeshouldbeuniform,withanequal(1/6)probabilityofproducinganyintegervaluefrom 1 to 6. This is illustrated in Fig. 3a. The outcome is determined solelyStatisticalConcepts 355388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:36FIGURE 3 Distribution of values obtained from rolling fair six-sided dice. (a) onedie, (b) two dice, (c) three dice, (d) four dice.FIGURE2 Normal distribution.Chapter3 365388-7_Funkenbusch_Ch03_R2_091404bythesingle,randomeventandsoreectsthefrequencydistributionofthat event. If we roll two dice (Fig. 3b), the outcome is associated with tworandomevents. Eachisimportantandthedistributionhasadistinctiveshapereectingthis. Withthreedice(Fig. 3c), theimportanceof eachsingle, randomevent is reduced and the curve begins to assume a somewhatrounded shape. By the time we reach four dice (Fig. 3d) the curve alreadybegins to have a quite bell-shaped, normal-distribution-like appearance.Thispropertyturnsouttobeimportantinmanyapplications.Forexample, observationof anonnormal distributioninamanufacturingprocessisan indicationthattheprocessmaybestrongly inuenced byafew uncontrolled parameters.Elimination of theseshould result in muchlessvariability.Similarly,inawell-controlledexperiment,oncethemainpotential sources of error are eliminated, the remaining error is often thesum of many small errors from dierent sources. It is common, therefore,toassumethatexperimentalerrorproducesanormaldistributionaboutthe average as a basis of comparison in determining whether an observedeect is statistically signicant.If the average and standard deviation of a distribution of data pointsare known, and the distribution is assumed to be normal, it is possible toestimatetheprobabilitythatameasurementthatispartofthisdistribu-tion will be within certain limits. Standardized tables for this are availablein most statistics textbooks. For example, the probability of the measure-ment being within limits of F1r of the average is about 68.3%. Similarly,theprobabilitiesofbeingwithinF2randF3rareabout95.4%and99.7%, respectively.Inaddition,thesepercentagescanalsobeusedtojudgewhetherameasurement diers signicantly fromthat expected fromthe distribution.For example, thereis only a 0.3% (100% 99.7%) chance that randomerror would cause an individual measurement to be more than 3r from themean. If we take a measurement and observe that its value is outside of thisrange, we would probably conclude that the value diers signicantly fromthat expected based on random(experimental) error. If the measured valuewere only F2r from the average (100%95.4% = 4.6% probability) wemight still conclude that it was signicantlydierent, althoughwithsomewhat less condence.G. Effect ofFiniteSample SizeThe above comparisons withthe normal distributionassume that thepopulation variance is precisely known. Generally, however, it is necessaryMD:FUNKENBUSCH,JOB:04363,PAGE:37StatisticalConcepts 375388-7_Funkenbusch_Ch03_R2_091404to estimate the population variance, r2, from the variance obtained with anitesample, s2. Theadditional uncertaintyintroducedbytheneedtomake this estimate causes a spread in the frequency distribution curve. Itbecomes more probable that a measurement will be obtainedmanystandard deviations from the average value because of uncertainty in thestandard deviation itself.Theeectsof nitesamplesizeinestimatingthevariancecanbeaccountedfor by replacing the normal distributionwith Studentst-distribution, as illustrated in Fig. 4. There are actually many t-distribu-tions, based on the number of degrees of freedom available for estimatingthe variance. With only 1 degree of freedom (i.e., one independent estimateofthevariance), theestimateisnotverygoodandsothet-distributioncurve is very broad. With more degrees of freedom, the estimate improvesandthet-distributionisnarrower.Figure4showscurvesfor1degreeoffreedomand10 degreesoffreedomalongwithanormaldistributionforcomparison. In the limit (that is innite degrees of freedom) thet-distribution is the same as the normal distribution.MD:FUNKENBUSCH,JOB:04363,PAGE:38FIGURE 4 Students t-distribution with 1 dof (broadest curve) and 4 dof (middlecurve) compared to normal distribution (narrowest curve).Chapter3 385388-7_Funkenbusch_Ch03_R2_091404II. ANOM AND ANOVAYou have already walked through ANOMand ANOVAanalyses in Chap.1, and you will see additional examples later. In this section, you will learnhowtocalculatethenecessaryterms for ANOMandANOVAonanexperimental array. Moststatistical softwarepackageswill performthealgebra involved in these calculations and in practice you will probably endupusingoneof themtodoyouractual numbercrunching. Therefore,although the necessary equations will be provided, the main emphasis is onunderstandingtheunderlyinglogicandassumptions.Indoingthis,youmay nd it helpful to draw on the review material in the rst section foranalogies and comparisons.A. Goals ofAnalysisAnalysisoftheresultsfromadesignedexperimentisnormallydirectedtoward one or more of the following goals:1. Identifying the best levels for the control factors.2. Ranking of factor and interaction eects in order of importance.3. Determining the (statistical) signicance of eects.4. Looking for signs of trouble in the experiment.Not every experiment has all of these goals. In particular, judgmentof statistical signicance is dependent on having an estimate of the error,which is not always available. Of these four goals, ANOM is used mainlyfor identifying the best levels, although it can also be used to rank eects.ANOVA is useful in achieving the other three goals (2 through 4).B. Column IdentificationBefore beginning either ANOM or ANOVA, it is important to conrm theidentication of each of the factors and interactions to be examined with aspecic column in the experimental array. In the uid delivery system, forexample, Factor Awas placed in column 1, Factor Bin column 2, etc. (Fig.8 in Chap. 1). Therefore any results obtained for column 1 are attributed toFactor A in the analysis, etc. (Strategies for assigning factors to particularcolumns, anddeterminingwhichcolumnscorrespondtowhichinterac-tions, are presented in Chap. 4. For nowassume that, as in the uid deliveryexample, this information will be available when you are ready to conductthe ANOM and ANOVA.)MD:FUNKENBUSCH,JOB:04363,PAGE:39StatisticalConcepts 395388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:40It is also important, to conrm the assignment of particular columnsto estimate experimental error. Any interaction terms in these columns areassumed to be zero. In the uid delivery experiment, for example, all of thehigher order interactions (i.e., ABC, ABD, ACD, BCD, A B C D) were assumed to be zero, leaving the correspondingcolumns (5, 6, 7, 9, and 12) for error estimation. To avoid bias, assignmentof columns to error estimation should generally be done prior to studyingor analyzing the results. (If no columns are assigned to error, ANOVAmaystill beperformed, but judgment of statistical signicancemaynot bepossible. See Chap. 8for adiscussionof alternativemethods for judgingsignicance.)C. ANOM AlgebraANOMaddresses the rst twopossible goals. It canbe usedto(1)determine which level of each factor is the best and (2) provide a relativeranking of their importance. ANOMinvolves comparing the mean(average)valuesproducedbythedierentlevelsofeachfactor.Speci-cally,thecharacteristicresponsesforalltreatmentconditionswherethefactor was at level 1 are averaged and compared to the average obtainedwith the factor at level +1.The rst calculation needed is to determine the overall average, m*,of the characteristic responses for the experiment:m* Xni1ain5where ai is the characteristic response for treatment condition i and n is thetotal number of characteristic responses. (Note that for the analysispresented here, it is assumed that there is one and only one characteristicresponse for each treatment condition. Thus, the number of characteristicresponses is just equal to the number of treatment conditions. This is nottrue if replication is used. Replication is discussed in Chap. 8.)Then for each column in the array, the average of the characteristicresponse obtained when the column has a level of +1, m+1, and when ithas a level of 1, m1, is calculated. Note that although two averages arecalculated for each column (m+1 and m1), they must average together togive the overall average for the experiment (m*), so that there is only oneindependent term (that is 1 degree of freedom) associated with each (two-Chapter3 405388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:41level) column. Tofacilitate comparisons amongcolumns, wecanalsocalculate the dierence between these averages:D m1m16Todemonstrate these calculations, consider the calculationforFactorCintheuiddeliveryexperiment.ThenecessarydataareshowninFig. 8, Chap. 1. Theoverall average, m*, (whichisthesameforallcolumns) is given by:m* Xni1ain a1a2a3a4a5a6a7a8a9a10a11a12a13a14a15a1616 5:775625TherewereeightcharacteristicresponsescollectedwhenFactorChad a level of 1; specically a1, a2, a5, a6, a9, a10, a13, and a14. Thus:m1 a1a2a5a6a9a10a13a1487:31 2:25 6:10 5:52 2:38 1:52 0:94 3:448 3:6825Similarly, there were 8 characteristic responses collected when FactorC had a level of +1; specically a3, a4, a7, a8, a11, a12, a15, and a16, giving:m1 a3a4a7a8a11a12a15a1688:58 8:58 10:42 9:17 5:46 7:02 7:34 6:388 7:86875Finally, for this column the eect, D, isD m1m1 7:86875 3:6825 4:19D. ANOM ComparisonsTheabsolutesizeoftheeectcalculatedforeachfactorandinteractionprovides asimple basis for comparingtheir importance. Inthe uidStatisticalConcepts 415388-7_Funkenbusch_Ch03_R2_091404MD:FUNKENBUSCH,JOB:04363,PAGE:42delivery example, relative importance of dierent factors and interactionscould be compared in terms of the relative size of D in Fig. 8 in Chap. 1.Figure9inChap. 1showedthis sameresult graphically. Thegreater,relative importance of Factors A and C was apparent.E. DeterminingtheBestLevelforEach FactorIf the experimental goal is to either minimize or maximize the characteristicresponse, selection of the desired level for each factor based on ANOM isstraightforward. Thebest level is simply the level with the smallest (tominimize) or largest (to maximize) average of the characteristic response.For example, to maximize the abrasive concentration in the uid deliverysystem, factor levels should be chosen based on which level gives the largestaverage. As seen in Fig. 8, Chap. 1, this means that level 1 should be usedfor Factor A (i.e., a convex bottle geometry) since m1>m+1. Graphi-cally thisprocessis convenientlydoneby selectingthe high pointsof theANOM graph in Fig. 9, Chap. 1.If anintermediate level of the characteristic resp