six sigma workbook for dummies (isbn - 0470045191)
TRANSCRIPT
by Craig GygiBruce Williams
Terry Gustafson
Six Sigmareg
WorkbookFOR
DUMmIESpermil
01_045191 ffirsqxd 81606 1135 PM Page iii
01_045191 ffirsqxd 81606 1135 PM Page ii
Six Sigmareg
WorkbookFOR
DUMmIESpermil
01_045191 ffirsqxd 81606 1135 PM Page i
01_045191 ffirsqxd 81606 1135 PM Page ii
by Craig GygiBruce Williams
Terry Gustafson
Six Sigmareg
WorkbookFOR
DUMmIESpermil
01_045191 ffirsqxd 81606 1135 PM Page iii
Six Sigmareg Workbook For Dummiesreg
Published byWiley Publishing Inc111 River StHoboken NJ 07030-5774wwwwileycom
Copyright copy 2006 by Wiley Publishing Inc Indianapolis Indiana
Published by Wiley Publishing Inc Indianapolis Indiana
Published simultaneously in Canada
No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any meanselectronic mechanical photocopying recording scanning or otherwise except as permitted under Sections 107 or 108 ofthe 1976 United States Copyright Act without either the prior written permission of the Publisher or authorization throughpayment of the appropriate per-copy fee to the Copyright Clearance Center 222 Rosewood Drive Danvers MA 01923978-750-8400 fax 978-646-8600 Requests to the Publisher for permission should be addressed to the Legal DepartmentWiley Publishing Inc 10475 Crosspoint Blvd Indianapolis IN 46256 317-572-3447 fax 317-572-4355 or online athttpwwwwileycomgopermissions
Trademarks Wiley the Wiley Publishing logo For Dummies the Dummies Man logo A Reference for the Rest of Us TheDummies Way Dummies Daily The Fun and Easy Way Dummiescom and related trade dress are trademarks or registeredtrademarks of John Wiley amp Sons Inc andor its affiliates in the United States and other countries and may not be usedwithout written permission All other trademarks are the property of their respective owners Wiley Publishing Inc is notassociated with any product or vendor mentioned in this book
LIMIT OF LIABILITYDISCLAIMER OF WARRANTY THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONSOR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK ANDSPECIFICALLY DISCLAIM ALL WARRANTIES INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PAR-TICULAR PURPOSE NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS THEADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION THIS WORK IS SOLDWITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL ACCOUNTING OROTHER PROFESSIONAL SERVICES IF PROFESSIONAL ASSISTANCE IS REQUIRED THE SERVICES OF A COMPETENTPROFESSIONAL PERSON SHOULD BE SOUGHT NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FORDAMAGES ARISING HEREFROM THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK ASA CITATION ANDOR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR ORTHE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMEN-DATIONS IT MAY MAKE FURTHER READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORKMAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ
For general information on our other products and services please contact our Customer Care Department within the USat 800-762-2974 outside the US at 317-572-3993 or fax 317-572-4002
For technical support please visit wwwwileycomtechsupport
Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available inelectronic books
Library of Congress Control Number 2006927762
ISBN-13 978-0-470-04519-0
ISBN-10 0-470-04519-1
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 1
1OQVQZQWIN
01_045191 ffirsqxd 81606 1135 PM Page iv
About the AuthorsCraig Kent Gygi began studying and applying the elements of Six Sigma well before theywere formalized into todayrsquos renowned breakthrough method As an engineering graduatestudent at Brigham Young University in the early 1990s he integrated these then-unorganizedimprovement techniques into his research and coaching of student product developmentteams Upon beginning his career in 1994 at Motorolarsquos Advanced Manufacturing ResearchLab in Florida he was formally introduced to the just-formalizing Six Sigma method It res-onated deeply with his previous findings From that time Craig has applied taught and ledSix Sigma in all his endeavors including management and technical capacities at MotorolaIomega and General Atomics
In 1998 Craig founded TolStack Inc to develop commercial Six Sigma software tools He alsoworked for several years as a technical colleague of Dr Mikel J Harry the original consultantof Six Sigma co-developing and teaching new advances in its theory and application In 2002Craig co-founded Savvi International into which TolStack merged Savvi provides solutionsfor business performance improvement using Six Sigma Lean and Business ProcessManagement techniques Craig acted as the director of Savvirsquos products service and toolsToday Craig works with companies in the USA and internationally to integrate Six Sigmapractices directly into their core operations
A Master Black Belt Craig has wielded Six Sigma techniques now for over 13 years spanningprojects from design to manufacturing to business management He is also an expert teacherhaving instructed and mentored at all levels of Six Sigma from executives to White Belts
Bruce David Williams has been fascinated with complex systems since the launch of Sputnikon his third birthday With undergraduate degrees from the University of Colorado in Physicsand Astrophysics he entered a career in aerospace systems where he first encountered Six Sigma after Motorola won the inaugural Baldridge Award in 1988 Later with graduatedegrees in technical management and computer science from Johns Hopkins University andColorado and as a member of the Hubble Telescope development team he was intrigued by how breakdowns in the smallest components could lead to colossal system failures Heentered the Six Sigma industry for good in the mid-1990s when he founded a software com-pany to pursue product life-cycle traceability
Bruce has since been founder and CEO of two Six Sigma research and technology firms and isnow Chairman and CEO of Savvi International a provider of solutions for business perform-ance improvement using Six Sigma Lean and Business Process Management techniques Heresides in the highly-variable environment of the desert foothills of North Scottsdale Arizonawith his wife two children and a normal distribution of dogs cats birds and horses
Terry James Gustafson comes out of the world of business and finance and brings a practi-cal and entrepreneurial perspective to Six Sigma After an undergraduate degree in financefrom Baldwin-Wallace and an MBA degree from Kent State he began his career in the field ofaccounting with KPMG Peat Marwick in 1969 and advanced to a Partner position in auditingAfter leaving public accounting in 1990 Terry helped found build and operate a series oftechnology-based entrepreneurial ventures including venture-backed companies as well asa public company
In 2002 Terry co-founded Savvi International which provides solutions for business perform-ance improvement using Six Sigma Lean and Business Process Management techniquesTerry serves as Savvirsquos chief finance and operations officer
Since founding Savvi Terry has been immersed in Six Sigma techniques helping to designand develop Savvirsquos training courses in Six Sigma and Lean In addition he has extensivelytaught Six Sigma courses both in a classroom and as an online instructor
01_045191 ffirsqxd 81606 1135 PM Page v
DedicationCraig Gygi To Darren a true brother
Bruce Williams To my mom and dad Jane and Coe When it comes to cause and effect theywrote my book
Terry Gustafson To my incredible wife Sherrie for putting up with all my entrepreneurialnonsense without once using the term ldquodummyrdquo
Publisherrsquos AcknowledgmentsWersquore proud of this book please send us your comments through our Dummies online registration form located atwwwdummiescomregister
Some of the people who helped bring this book to market include the following
Acquisitions Editorial and Media Development
Project Editor Natalie Faye Harris
Acquisitions Editor Kathy Cox
Copy Editor Jessica Smith
General Reviewer Tom Pearson
Editorial Manager Christine Beck
Editorial Assistants Erin Calligan David Lutton Nadine Bell
Cartoons Rich Tennant (wwwthe5thwavecom)
Composition
Project Coordinator Jennifer Theriot
Layout and Graphics Carrie A Foster Denny HagerStephanie D Jumper Lynsey Osborn
Proofreaders Debbye Butler John Greenough
Indexer Dakota Indexing
Publishing and Editorial for Consumer Dummies
Diane Graves Steele Vice President and Publisher Consumer Dummies
Joyce Pepple Acquisitions Director Consumer Dummies
Kristin A Cocks Product Development Director Consumer Dummies
Michael Spring Vice President and Publisher Travel
Kelly Regan Editorial Director Travel
Publishing for Technology Dummies
Andy Cummings Vice President and Publisher Dummies TechnologyGeneral User
Composition Services
Gerry Fahey Vice President of Production Services
Debbie Stailey Director of Composition Services
01_045191 ffirsqxd 81606 1135 PM Page vi
Contents at a GlanceIntroduction1
Part I Getting Started in Six Sigma5Chapter 1 Getting Ready for Six Sigma The Effects of Variation 7Chapter 2 Forming a Six Sigma Initiative13Chapter 3 Leading and Managing a Six Sigma Initiative 27
Part II Defining a Six Sigma Project 43Chapter 4 Putting the Right Foot Forward Defining a Six Sigma Project45Chapter 5 Brainstorming the Inputs to Your Process53Chapter 6 Prioritizing Which Inputs to Address 69
Part III Mastering Measuring 85Chapter 7 Categorizing Data and Calculating Measures of Variation 87Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs 107Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly 129
Part IV Assessing the Right Approach to Your Analysis 141Chapter 10 Mastering Measurement System Analysis (MSA) 143Chapter 11 Capability Matching Performance to Need155Chapter 12 Narrowing Your Inputs with Confidence 169
Part V Improving and Controlling 183Chapter 13 Quantifying Variable Relationships185Chapter 14 Planning and Conducting 2k Factorial Experiments 209Chapter 15 Constructing Control Plans and Charts 239
Part VI The Part of Tens281Chapter 16 Ten Implementation Myths of Six Sigma 283Chapter 17 Ten Tips for Finishing a Six Sigma Project Successfully287
Index291
02_045191 ftocqxd 81606 1141 PM Page vii
02_045191 ftocqxd 81606 1141 PM Page viii
Table of ContentsIntroduction 1
About This Book1Conventions Used in This Book 2What Yoursquore Not to Read2Foolish Assumptions 2How This Book Is Organized2
Part I Getting Started in Six Sigma 2Part II Defining a Six Sigma Project 3Part III Mastering Measuring3Part IV Assessing the Right Approach to Your Analysis 3Part V Improving and Controlling 3Part VI The Part of Tens 4
Icons Used in This Book4Where to Go from Here4
Part I Getting Started in Six Sigma 5
Chapter 1 Getting Ready for Six Sigma The Effects of Variation 7Recognizing Variation around You7Evaluating Variation and Business Performance with Y = f(X) + f 9
Breaking down Y = f(X) + f to a simple process9Applying Y = f(X) + f A practice example 9
Assessing the Impact of Variation on Business Performance 11
Chapter 2 Forming a Six Sigma Initiative13Planning Your Key Business Objectives (KBOs)13
Establishing a KBO checklist 14Verifying your process alignment15
Determining the Proper Training Program 16Who will manage the training program 16Finding the expertise within your organization 16Scoping the training program17Recognizing cultural predisposition19Defining the training plan20
Deciding Who Will Conduct the Training21Teaming for Success Shopping for Your Implementation Partners24Pricing and Contracting Approaches 26
Chapter 3 Leading and Managing a Six Sigma Initiative27Selecting Your Leadership Team27
Approaching the selection process 27Go team Unifying individuals to create a team30
Implementing Your Communications Plan 31Understanding the two communications plans 31Elements of your communications plan31Whatrsquos to communicate 32Who does the communicating 33Timing is everything 36
02_045191 ftocqxd 81606 1141 PM Page ix
Using communications tools 36Where oh where have the communications gone 36Writing your communications plans38
Selecting Software Products and Integrating Information Technology Architectures 39Yours mine or ours Platform questions 39Practitioner tools 40Management tools40Enterprise Integration SOA and BPM41
Defining and Implementing Your Management Plan41
Part II Defining a Six Sigma Project 43
Chapter 4 Putting the Right Foot Forward Defining a Six Sigma Project45Getting Project Ideas by Using the Business Case Writing Tool 45Prioritizing and Aligning Projects with Business-Customer-Process Scorecards46Project Definition I Writing a Problem Statement 48Project Definition II Writing an Objective Statement49Launching a Project 51Solutions to Defining a Six Sigma Project52
Chapter 5 Brainstorming the Inputs to Your Process53All Together Now Brainstorming with Your Team to Create Affinity Diagrams53Dem Bones Creating Fishbone Diagrams 57Examining Your Processes with Process Flow Maps60Finding Critical Fruit in the CT Tree 62Solutions to Brainstorming the Inputs Problems 65
Chapter 6 Prioritizing Which Inputs to Address 69Weeding and Pruning the Input Garden Using Pareto Diagrams 69Cementing the Foundation Creating SIPOC Diagrams71Untangling Webs Creating a Cause-and-Effect Matrix 73Performing a Failure Modes Effects Analysis (FMEA) 76Solutions to Prioritizing Inputs Problems80
Part III Mastering Measuring85
Chapter 7 Categorizing Data and Calculating Measures of Variation87Differentiating Data Types 87Calculating Measures of Variation Location 89Variety Is the Spice of Life Measuring Variation Spread92Time Warp Separating Short-Term and Long-Term Variation94Solutions to Categorizing Data and Calculating Measures of Variation Problems 100
Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs 107Putting Dot Plots or Histograms to Use 107Setting Up Box and Whisker Plots 110Seeing Spots Using Scatter Plots114Hindsight Is 2020 Using Process Behavior or Time Series Charts 119Solutions to Exercises for Yields and Defects Problems121
x Six Sigma Workbook For Dummies
02_045191 ftocqxd 81606 1141 PM Page x
Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly 129Get Real Creating Realistic Specifications 129Getting It Right the First Time Calculating First Time Yield (FTY)131Rolling Many into One Calculating Rolled Throughput Yield (RTY) 132ldquoHow Bad Is It Docrdquo Calculating Defect Rates134Whatrsquos Missing Linking Yield to Defects136Solutions to Yield and Defects Problems138
Part IV Assessing the Right Approach to Your Analysis141
Chapter 10 Mastering Measurement System Analysis (MSA)143Your Basic Sanity Check Auditing Measurement Systems143Do We Agree Performing an Attribute Measurement System Analysis 146Gauging Gages Analyzing Continuous Variable Measurement Systems 150Solutions to MSA Problems 153
Chapter 11 Capability Matching Performance to Need155Calculating and Interpreting Sigma (Z) Scores155Shift Happens Transforming Between Short- and Long-Term Performance158Calculating and Interpreting Capability Indices160Prescribing an Improvement Plan160Solutions to Capability Problems165
Chapter 12 Narrowing Your Inputs with Confidence 169Creating Confidence Intervals for Means170Calculating Confidence Intervals for Standard Deviations173Four Out of Five Recommend Using Confidence Intervals for Proportions176Solutions to Narrowing Inputs Problems179
Part V Improving and Controlling 183
Chapter 13 Quantifying Variable Relationships 185Quantifying Correlation between Variables185Fitting Lines to Variable Relationships190Assessing the Adequacy of a Fitted Line 192Solutions to Quantifying Variable Relationships Problems197
Chapter 14 Planning and Conducting 2k Factorial Experiments 209Donrsquot Be a Frankenstein Planning Experiments 209Managing Those Pesky Nuisance Variables212Calculating Main Effects214Calculating Interaction Effects 219Determining Which Effects Are Significant 224The Ultimate Power Trip Forming Y = f(X) Equations227Solutions to 2k Factorial Experiment Problems229
Chapter 15 Constructing Control Plans and Charts 239Doing the Poka-Yoke Mistake-Proofing Products or Processes 239Forming Control Plans to Maintain Your Improvements 241Selecting the Right Control Chart for Your Situation 242Interpreting Your Control Charts 245
xiTable of Contents
02_045191 ftocqxd 81606 1141 PM Page xi
Constructing an Individuals and Moving Range (I-MR) Chart249Raising the Bar for Small Samples Averages and Ranges ( X-R) Charts252Making a p Chart for Your Attribute Data 256Creating a u Chart for Your Attribute Data 260Solutions to Control Plan and Chart Problems 265
Part VI The Part of Tens 281
Chapter 16 Ten Implementation Myths of Six Sigma 283Six Sigma Is about Achieving ldquoSix Sigmardquo 283Six Sigma Will Make Us Start All Over Again with Something Different 284Six Sigma Stifles Creativity284Modeling Processes Is Too Complicated and Doesnrsquot Go Anywhere284Six Sigma Is Another ldquoProgram of the Monthrdquo 285Six Sigma Is Just a Quick-hit Cost-Reduction Initiative 285Six Sigma Is Too Onerous and Prescriptive 285You Canrsquot Implement Six Sigma Yourself285The Six Sigma Approach Is Way Too Expensive and Disruptive 286If Yoursquore Not Doing Black Belt Projects Yoursquore Not Really Doing Six Sigma 286
Chapter 17 Ten Tips for Finishing a Six Sigma Project Successfully 287Properly Scoping Your Project 287Anticipating Scope Creepy-Crawlies287Charting the Entire Course 288Making Sure the Right People Are Aboard288Remembering That Short Is Sweet288Setting Achievable Goals289Communicating for Success 289Satisfying the Stakeholders289Maintaining Active and Unwavering Support289Applying Formal Project Management 290
Index 291
xii Six Sigma Workbook For Dummies
02_045191 ftocqxd 81606 1141 PM Page xii
Introduction
Six Sigma is the single most effective problem-solving methodology for improving busi-ness and organizational performance Therersquos not a business technical or process
challenge that canrsquot be improved with Six Sigma The worldrsquos top corporations have used itto increase their profits collectively by more than $100 billion over the past ten years Incertain corporations Six Sigma proficiency on your resume is now a prerequisite to movinginto a management position
If yoursquore part of a Fortune 500 company mdash particularly a manufacturing company mdash chancesare yoursquove heard about Six Sigma You may even have been through a training regimen andbeen part of a corporate initiative or an improvement project If so you know the capabilitiesof Six Sigma you have witnessed its power and achievements firsthand
But if like most people yoursquore outside of the upper echelons of big business Six Sigma isnrsquotwell known It has been too expensive and complicated for small- and medium-sized busi-nesses public institutions not-for-profit organizations educational environments and aspir-ing individuals Its potential has remained out of reach for the vast majority of professionalsand organizations world-wide
Fortunately all this is changing As the methods and tools of Six Sigma have spread it hasbecome easier to understand less expensive to learn and more straightforward to imple-ment The mysteries of Six Sigma have been revealed
Simply stated Six Sigma is about applying a structured scientific method to improve anyaspect of a business organization process or person Itrsquos about engaging in disciplined datacollection and analysis to determine the best possible ways of meeting your customersrsquoneeds while satisfying your own and by minimizing wasted resources and maximizing profitin the process
About This BookThis workbook is unique What used to only be available through expensive consultant-ledprofessional training is laid out in simplicity here This workbook includes step-by-step expla-nations and examples of the tools methods formulas and tactics of Six Sigma Exercises andpractice problems build your mastery in applying the tools and techniques And ready-to-usetemplates and worksheets provide you immediate access to the power of Six Sigma
Corporate Six Sigma training also uses expensive calculation software like Minitab Usingexpensive calculation software is fine if your company can provide you with $1000 softwareprograms but everyone else is priced out Not to worry In this workbook wersquove providedall the formulas and calculations so that with simple explanations you can perform thesesame calculations quickly by hand The only price is the cost of this workbook (and a box ofNumber 2 pencils) Or you can automate them using your favorite spreadsheet software
Together with the more conceptual Six Sigma For Dummies Six Sigma Workbook For Dummiestruly forms a 2-volume box set for ldquoSix Sigma in a boxrdquo Just add a little practice and yoursquoreready to join the growing ranks of professionals who list Six Sigma as a critical competency totheir success And be sure to keep a notebook handy for working out problems as some ofthem will require a bit more ldquoscrap paperrdquo than even a workbook can provide
03_045191 introqxd 81606 1138 PM Page 1
Conventions Used in This BookMathematical formulas and variables used in this workbook are written in an italicizedfont This will help you pick them out from the rest of the text and explanations
What Yoursquore Not to ReadEven after all the hard work wersquove put into this workbook we donrsquot expect you to readevery word of it Its purpose is application so read only what you need to gain mas-tery of the skill or topic You donrsquot have to do every practice problem Wersquove includedseveral problems on each concept just in case you need the practice
Foolish AssumptionsSix Sigma has a lot of math and statistics in it We assume yoursquore familiar with basicarithmetic operations mdash adding subtracting multiplying and dividing At some pointsit does get a little more complicated than that but not much and not very often
We also assume that you have a context in which to apply the tools and techniques ofSix Sigma It may be a bona fide Six Sigma project you have been assigned to and areworking on Or it may be an improvement effort of your own creation Or it may even bean aspect of your personal life you want to improve In any case as you read throughthis workbook try out the tools yoursquore discovering Itrsquos the quickest path to proficiency
How This Book Is OrganizedThis workbook is organized along the lines of DMAIC mdash thatrsquos Define Measure AnalyzeImprove and Control mdash the problem-solving methodology of all Six Sigma thinking andworking If you need help with a specific task you can jump into the workbook at thatspecific point along the DMAIC roadmap to get the focused practice or guidance youneed Or if you want to gain a comprehensive mastery of all the skills of Six Sigma youcan follow through from the first page to the last
Part I Getting Started in Six SigmaIn Part I you find guidance checklists and templates for organizing your launch intoSix Sigma This part could be for yourself or it could be for getting an entire organiza-tion ready to head down the Six Sigma path From aligning with key business objec-tives to forming a management communications plan therersquos a lot of preparation whenstarting a breakthrough improvement journey
Chapter 1 provides a first-hand live introduction to the issues of variation andimprovement Chapter 2 gives you tools and worksheets for forming and aligning theimprovement work with your organizationrsquos key business objectives Chapter 3 pro-vides checklists templates and examples for setting up an organization-wide SixSigma program
2 Six Sigma Workbook For Dummies
03_045191 introqxd 81606 1138 PM Page 2
Part II Defining a Six Sigma ProjectPart II is all about defining the improvement work you do in Six Sigma If yoursquore start-ing a Six Sigma project this is the part for you You practice generating viable projectideas find templates for scoping your work and complete exercises for setting break-through objectives and goals
Chapter 4 contains examples and templates for correctly defining each Six Sigma proj-ect Chapter 5 gives you expertise in all the tools for identifying the potential causes ofpoor performance Going through Chapter 6 you discover how to whittle down a hostof potential causes to a handful of the ldquocritical fewrdquo
Part III Mastering MeasuringPart III is full of exercises and practice for mastering the skills of measuring If you donrsquotmeasure it you canrsquot know it If you donrsquot know it you canrsquot control it If you canrsquot con-trol it yoursquore at the mercy of chance And nobody wants to be in that position
In Chapter 7 you find exercises for discovering how to collect data and calculate itsstatistical characteristics Chapter 8 is a guide to the powerful skills of creating andinterpreting charts graphs and plots The practice problems in Chapter 9 show youhow to measure the capability of your system or process
Part IV Assessing the Right Approach to Your AnalysisUnderstanding what your measurements mean is the focus of Part IV In this part youfind exercises that show you how to perform critical analyses of your collected meas-urements uncover root causes and confirm hypotheses
In Chapter 10 you find calculation templates for analyzing your measurement systemChapter 11 covers the topic of capability mdash you find out how to calculate whetheryour process meets its performance requirements Throughout Chapter 12 you prac-tice calculating confidence intervals to statistically analyze differences among inputsand outputs
Part V Improving and ControllingThe final stages of DMAIC are Improve and Control Skills for synthesizing improve-ments to solve problems can be mastered by anyone through the exercises and work-sheets provided in this part And you canrsquot neglect Control mdash be sure to becomeproficient in the skills of controlling and maintaining the improvements you make
Chapter 13 gives you the skills to quantify how one factor affects another This is donethrough correlation and curve fitting Chapter 14 shows you all the aspects of design-ing conducting and analyzing 2k factorial experiments In Chapter 15 you practice cre-ating Poka-Yokes control plans and statistical process control charts
3Introduction
03_045191 introqxd 81606 1138 PM Page 3
Part VI The Part of TensIn this part we provide you with two helpful top ten lists Chapter 16 debunks the ten most common myths about Six Sigma and Chapter 17 lists the ten most criticalthings to do to complete a Six Sigma project And be sure to check out wwwdummiescomgosixsigmaworkbook for bonus forms that you can print for your own use
Icons Used in This BookThroughout the workbook yoursquoll see symbols in the margins called icons These iconshighlight special types of information When you see any of the following icons this iswhat they mean
Each section of this workbook begins with a brief overview of the topic After the introyou see an example problem with a fully worked solution for use as a reference whenyou work the practice problems You can quickly locate the example problems by look-ing for this icon
These are handy points that help you perform and apply some of the trickier parts ofSix Sigma more quickly and correctly
This icon is used in almost every section of the workbook It lets you know that theassociated text summarizes the key application principles formulas or proceduresneeded for that particular skill
When you see this icon it alerts you to be aware of a particular risk or pitfall thatcould cause you trouble
Where to Go from HereThe beauty of a For Dummies book is that you donrsquot have to start at the beginning andwork your way through every page Instead each chapter is self-contained you canstart with whichever chapter interests you the most and then jump to wherever youwant to go next
Here are some suggestions on where to start
If you already have a Six Sigma project picked out or assigned to you start inChapter 4 Yoursquoll find practice there on properly defining your problem and itsintended solution
If you have a specific Six Sigma task to perform and just need some practice onit start in the Table of Contents Find the topic you need and go to it In no timeyour skills for that task will be honed and ready to go
If yoursquore forming a Six Sigma initiative for your organization start in Chapters 2and 3
4 Six Sigma Workbook For Dummies
03_045191 introqxd 81606 1138 PM Page 4
Part IGetting Started in Six Sigma
04_045191 pt01qxd 81606 1138 PM Page 5
In this part
Six Sigma starts with initiative mdash either your own oryour organizationrsquos This part provides you and your
colleagues with useful exercises and templates for recog-nizing organizing and launching your journey to break-through improvement
04_045191 pt01qxd 81606 1138 PM Page 6
Chapter 1
Getting Ready for Six Sigma The Effects of Variation
In This Chapter Realizing that variation is everywhere
Mastering the Six Sigma breakthrough equation
Exploring the effect of variation on business performance
The characterization measurement analysis and control of variation is the central themeof Six Sigma Every process and every product is affected by variation Variation mdash
within limits mdash is okay and is even desirable However you can actually have too muchvariation If undesirable variation is out of control failure is the result
The key goals of Six Sigma are to fix undesirable variation ignore variation that doesnrsquotmatter and allow for variation that canrsquot be fixed Many of the tools and techniques in thisworkbook help you determine whether your variation is desirable or undesirable Thesetools also show you how to fix the variation that can actually be fixed so that your effortsare concentrated where you can make the most improvement impact
Before you go any further in this workbook however you must accept two undeniable truthsabout variation Every output varies and every input varies Donrsquot you feel better now thatyoursquove accepted the fact that variation happens Itrsquos simply a fact of life You can now focuson finding and correcting as much of that variation as possible mdash making your processes and products the best they can be Please also check out wwwdummiescomgosixsigmaworkbook for some useful forms you can print out
Recognizing Variation around YouTo get started recognizing the variation around you use the variation journal in Figure 1-1to chronicle the variation you encounter in a day of normal activities Because you haveaccepted that everything has variation your daily journal could exceed the size of theLibrary of Congress if you try to include everything that happens Instead try to concen-trate on more significant events mdash events that if impacted by unacceptable variation couldhave a negative effect on your life or job Figure 1-2 is an example of the entries you mightmake Try to record at least 20 key items As indicated on the worksheet consider the typeof failure that the variation could cause and whether the variation can be controlled
05_045191 ch01qxd 81606 1106 PM Page 7
Variation Journal
Date
Activity Undesirable Variation Effect of Variation
Arriving at work Arriving late YesLoss of wages lost business lost job
Returning phone calls Calls returned late YesLost businessirritation of boss
Accessing servercomputer
Server slow to respond or notavailable MaybeInefficiency lost data
Is VariationControllable
YesNo
Figure 1-2Entry
examplesfor the
variationjournal
Variation Journal
Date
Activity Undesirable Variation Effect of Variation
Is VariationControllable
YesNo
Figure 1-1The SixSigma
variationjournal
8 Part I Getting Started in Six Sigma
05_045191 ch01qxd 81606 1106 PM Page 8
Evaluating Variation and BusinessPerformance with Y = f(X) + f
The word ldquobreakthroughrdquo is bandied about all the time particularly when advertisingsome product mdash a breakthrough shaving system breakthrough hair gel a break-through mousetrap and on and on So you have to be a bit cautious when claimingsomething is a breakthrough product But Six Sigma practitioners donrsquot hesitate to callthe basic equation Y = f(X) + f a breakthrough equation When you apply the conceptthat all outcomes (Ys) are the result of some number of inputs (Xs) that interact insome way mdash f or the function mdash to produce that outcome and that there are alwayssome other factors either known or unknown mdash f or epsilon mdash that will impact theoutcome then you are well on your way to breakthrough improvements
The elegance of the Y = f(X) + f equation is that it applies to anything and everything mdashfrom the simplest process such as mixing a drink to the most complex such as build-ing a space shuttle After a process is broken down into primary elements you canthen identify the desired outcome find the inputs that contribute to that outcomeidentify those inputs that really matter recognize where error and variation mightoccur and plan improvements that will have a positive impact This workbook givesyou guidance in applying these concepts and conducting improvement activities
Breaking down Y = f(X) + fto a simple processFollowing is an example of a simple process with which almost everyone is familiar mdashcooking eggs If you apply Y = f(X) + f to this process the results look like this
Outcome (Y) mdash properly cooked eggs
Inputs (Xs) mdash eggs oil heat pan timer
Function (f) mdash shells are removed oil is added to pan eggs are placed in panheat is applied for a specific time eggs are removed
Epsilon (f) mdash size of eggs age of eggs temperature of eggs thickness of panamount of heat timer accuracy type of oil altitude
Some of these factors can be quantified and controlled but others canrsquot The trick is todetermine which if any of these inputs have a significant impact on the outcome andcan be controlled One of the basic tenets of Six Sigma is focusing efforts only on thoseinputs that have a substantial impact and that are practical to address Time andresources would be wasted if you tried to improve the egg-cooking process by chang-ing the altitude
Applying Y = f(X) + f A practice exampleAs practice apply the Y = f(X) + f equation to a more complicated process one muchlike many business processes Think about the equation as you read the story about ahouse fire on Elm Street Then use the worksheet in Figure 1-3 to match elements ofthe story to the components of the equation Herersquos the story problem
9Chapter 1 Getting Ready for Six Sigma The Effects of Variation
05_045191 ch01qxd 81606 1106 PM Page 9
A house catches fire on Elm Street and a neighbor calls 911 to report the fire andgive the address The 911 operator triggers an alarm in the nearest fire station andtransmits the address When the alarm sounds the firefighters dress load thetruck and leave the firehouse Using a current map and an established route planthey find and take the most direct route to Elm Street Halfway to Elm Street theyencounter a major traffic jam which is normal for that time of day and have todetour Shortly thereafter a freak sleet storm forces the truck to slow to a crawlWhen the truck finally arrives the men hook up the hoses but find that waterpressure is low and that water flow to the fire is significantly less than normalEventually the flames are extinguished but only after the house is a total lossFortunately all the residents of the house escaped without injury
From the perspective of the fire department identify a primary Y at least ten impor-tant Xs and the elements of error
Figure 1-4 is a solution to the practice house fire exercise Even if your solutionincludes different items itrsquos okay You have at least started thinking how the break-through equation applies to business processes
Y = f(X) + ε Worksheet
Output (Y) Input (X) Error
Figure 1-3The Y = f(X) + f
exerciseworksheet
10 Part I Getting Started in Six Sigma
05_045191 ch01qxd 81606 1106 PM Page 10
Assessing the Impact of Variation on Business Performance
After you realize that variation is prevalent in all processes you have to determine theeffect variation has on your process and then you have to assess if itrsquos really a prob-lem at all After all some variation is inevitable and you can live with it right Wellmaybe or maybe not
Suppose yoursquore the general manager of Widgets International the undisputed marketleader Yoursquore rightfully proud of your market position and believe that producing thehighest quality widgets has lead you to where you are today So yoursquore taken abackwhen an upstart Six Sigma Yellow Belt dares ask you about the true reliability of yourwidget production process ldquoWe produce the best darn widgets possiblerdquo you retortldquoWe just canrsquot do much to improve that process Our rigorous assessments have deter-mined that each and every step in the process is 95 percent reliable Therefore thewhole process is 95 percent reliable How can we do better than thatrdquo
The Yellow Belt persists so you decide to humor her even though yoursquore confident in your figures ldquoHow many steps are there in the production processrdquo she asksConsulting your most recent production chart you answer ldquoWe have a total of 20 dis-crete stepsrdquo The Yellow Belt then says ldquoLetrsquos do some calculations If each of the firsttwo steps is 95 percent reliable the chance of a widget making it through both stepswithout a defect is 095 times 095 or 9025 percentrdquo ldquoUhhhrdquo you respond eloquentlyldquo90 percent is still pretty good rightrdquo ldquoWell surerdquo she says ldquobut we arenrsquot done yetThe chance of a widget making it through the first three steps without a defect is 095times 095 times 095 or 8574 percentrdquo You begin to get a sinking feeling in the pit ofyour stomach Sweat begins to run down your face You begin to see the big pictureand stammer ldquoAre you telling me that this calculation should be made through all 20steps What is the bottom line hererdquo She whips out her calculator and pounds in afew numbers and then she gives you the bad news ldquoWell if each of the 20 steps is 95percent reliable the chance of a widget making it all the way through the processwithout defect is 36 percentrdquo
Y = f(X)+ ε Worksheet
Output (Y) Input (X) ErrorPut out the fire quickly
or Prevent injuries or loss of life
911 callFire station alarmAddress of fireMapRoute planTrainingFirefighterFire truckWaterHosesTraffic
TrafficSleet stormLow water pressure
Figure 1-4The solution
to the practice
house fireexercise
11Chapter 1 Getting Ready for Six Sigma The Effects of Variation
05_045191 ch01qxd 81606 1106 PM Page 11
You feel like yoursquove been whacked with a 2 x 4 ldquoHow did you get so smartrdquo you askMiss Yellow Belt ldquoThis type of analysis is called lsquorolled throughput yield or RTYrsquo andis part of any basic Six Sigma trainingrdquo she replies She again pounds her calculatorbriefly and says ldquoAnd another interesting calculation we can make from this data isDefects Per Unit or DPU In this case the DPU is 102rdquo Now yoursquore really stunned ldquoSowhat yoursquore telling me is that only about one-third of our widgets make it through theprocess without a defect and we can expect on average more than one defect perwidget But I know from our latest production report that our final inspection is at 95percent Wouldnrsquot that mean we have a tremendous amount of expense rework over-head and excess inventory to make up the difference between the calculated 36 per-cent rolled throughput yield and our observed 95 percent final inspectionrdquo The YellowBelt smiled ldquoNow you know why Irsquom hererdquo
Finally you ask the Yellow Belt ldquoCan you give me a template that I can use to calculatethese key ratios for other processesrdquo Figure 1-5 is a worksheet you can use to calcu-late the RTY and DPU for your own processes
ProcessStep
1234567891011121314151617181920
DescriptionYield or
Reliability RTY
DPU
Instructions Start with process step 1 whose RTY is the same as its yield for step 2 and enter the cumulative RTY Continue for all process steps to calculate the RTY for the entire process
Calculating DPU (From Six Sigma For Dummies p 138) DPU = -1n(RTY)
Remember 1n is the natural logarithm and can be obtained from any scientific calculator as well as from math tables
Figure 1-5The work-
sheet forcalculating
RTY andDPU
12 Part I Getting Started in Six Sigma
05_045191 ch01qxd 81606 1106 PM Page 12
Chapter 2
Forming a Six Sigma InitiativeIn This Chapter Dealing with key business objectives
Determining your training program
Choosing your Six Sigma trainers
Selecting your implementation partners
Believe it or not Six Sigma can be applied in several ways mdash not just at the executivelevel of a major corporation At the personal level you can use the methods and tools
of Six Sigma in your everyday life and work to address personal and professional challengesand to solve problems At the team level you can conduct formalized improvement projectswhere a project leader mdash usually a certified Six Sigma Black Belt mdash leads a team through adefined improvement process which can improve a key metric by 70 percent or more Butat the organizational level Six Sigma is deployed across your enterprise in such a way thatthe projects and practices of Six Sigma become part of the organizational culture and func-tional routine This chapter helps you understand the activities and tasks needed to suc-cessfully complete an organizational-level deployment initiative You can also print outmany forms used in this chapter from wwwdummiescomgosixsigmaworkbook
A Six Sigma deployment initiative requires you to align both your business objectives andyour organizational framework in support of the management and operational methods of SixSigma practice You must determine what you can and canrsquot do yourself and you must aug-ment your resources with the right assistance to go forward This chapter and Chapter 3guide you through a deployment initiative If you arenrsquot performing a deployment initiativeand are focused on Six Sigma projects skip to Chapter 4
Doing business the Six Sigma way is easier after yoursquore familiar with it Six Sigma is like mas-tering anything else Until you figure out how to do it itrsquos difficult But after yoursquore comfort-able the task at hand is just so much simpler For example consider skiing Before youactually discover the key to skiing well itrsquos all a struggle You fight your way down the hillyou teeter out of balance and you exhaust yourself as you wrestle the laws of physics But ifyou take a few lessons to really master skiing what happens Suddenly you find that doing itright takes much less effort and your performance and efficiency improve Business works inthe same way Itrsquos difficult but only until mdash with Six Sigma mdash you figure out how to do itright at which time it becomes much easier
Planning Your Key Business Objectives (KBOs)Six Sigma is the most powerful problem-solving tool in business because you wonrsquot find aproblem or challenge area that canrsquot be improved significantly through the proper applica-tion of Six Sigma methods and tools But donrsquot forget the risk of turning your team loose withthe Six Sigma arsenal They can charge off and solve the wrong problems Pursuing pathsthat arenrsquot important is wasteful for any business For example your team would be wrong toinvest in improving the production quality of a product that the market doesnrsquot want or toimprove the sales efficiency for a product with negative margins
06_045191 ch02qxd 81606 1142 PM Page 13
You can find countless cases of whatrsquos known as functional sub-optimization which isimproving areas that donrsquot contribute to whatrsquos most important Obviously improve-ment is improvement but in the Six Sigma world the improvement has to really meansomething in the big picture or it doesnrsquot count
Your improvements must be tied to key business objectives (KBOs) such as growth inmarket share improvements in customer satisfaction or reduction in product defectsAs you form your Six Sigma deployment initiative and develop a corps of Black BeltsGreen Belts and other practitioners yoursquoll be able to identify candidate projects forSix Sigma improvement But before these projects are approved they should bevetted for alignment with your organizationrsquos KBOs
To ensure alignment with your objectives you need to develop a system and ulti-mately a culture that looks at the bigger picture If you do this the projects them-selves generate gains of significant organizational value the staff becomes aware ofwhat constitutes alignment and your company culture naturally and consistentlyfocuses on the principle of aligning efforts and performance with KBOs
Establishing a KBO checklistYour organization has KBOs which are identified in places such as annual reportsstrategic plans and operating plans at the organizational level the business unit leveland the functional level These KBOs form the basis of your alignment efforts
Your organizational KBOs may not necessarily be well-defined or even consistent Theeffort to define optimize and align KBOs across your organization is outside the scopeof a Six Sigma deployment initiative mdash but your alignment effort depends on themGoing forward in your alignment efforts yoursquoll constantly rely on a set of KBOs Besure theyrsquore the right ones
Before performing the alignment exercise be sure you have all of your companyrsquosKBOs in hand These KBOs are sourced from several areas Use the following checklistto verify your KBO set
Find out overall corporate goals These goals are identified in annual reportscorporate tag lines missions and vision and values statements Theyrsquore alsoarticulated in speeches and position papers by key executives Often these goalsare not identified in measurable objective terms You may need to translateamorphous goal statements into quantifiable metrics
Find out Voice of the Business objectives (VOB) These objectives are defined inoperating plans budgets sales or other productivity targets quotas and otheroperating parameters that reflect the goals of the organization Theyrsquore in boththe core business product and service line areas as well as the support processareas such as IT HR Facilities and Finance
Find out Voice of the Customer objectives (VOC) These objectives which arecritical to the success of your initiative are typically expressed through the mar-keting and research efforts that reflect the wishes of the market and of your cus-tomers Certain exercises in Design for Six Sigma (DFSS) which is the practice ofapplying Six Sigma to designing products services or business processesstrictly define quantifiable VOC measures
Make sure objectives are consistent Check to verify that your objectives areinternally consistent and prioritized and that they tie together logically Makesure that any inconsistencies have been addressed and rectified
14 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 14
Make sure objectives are quantifiable Be sure your objectives are in quantifiedmeasurable terms They should be expressed numerically and in such a way thatthe results of the projects can be measured and compared to the original values
Make sure objectives are current KBOs change and in some organizations theychange often Put an update mechanism in place to ensure that you have the cur-rent set in front of you
Verifying your process alignmentWith the definition of your key business objectives in hand you now have the basis forthe future evaluation of your project goals and the verification that your processimprovement initiative remains aligned with whatrsquos most important to the organiza-tion Alignment will be an ongoing cause-and-effect exercise requiring you to under-stand the Critical Xrsquos that influence the Significant Yrsquos mdash or in other words the KBOs
Use the worksheet in Figure 2-1 to assist you in identifying KBOs From sources includ-ing your organizationrsquos mission and vision statements business unit operating plansand local functional unit plans identify the key business objectives Then for eachobjective identify what quantifiable measure represents these objectives Upon com-pleting this table examine the objectives and the metrics for alignment and consistency
Business Objectives Alignment Worksheet
Date
Process Area Key Business Objective
MissionVision Statement
Business Unit Operating Plan
Functional Unit Plans
Key Metric(s)
Figure 2-1The KBO
alignmentworksheet
15Chapter 2 Forming a Six Sigma Initiative
06_045191 ch02qxd 81606 1142 PM Page 15
Determining the Proper Training ProgramThe first step in your deployment initiative is training You must get your staff up tospeed on how to apply the methods and tools of Six Sigma The training programneeds to be precise and the various training courses need to be delivered in a specificmanner and order
While the initial training period may last from months (for a small organization) toyears (for a large corporation or agency) a Six Sigma company never stops trainingThe initial training for an organization of several hundred people for example couldbe completed in less than six months For a large corporation like Bank of America ora huge government agency like the US Air Force even the initial training can take sev-eral years But all Six Sigma organizations regardless of size continuously follow upwith new courses refresher training and new-hire training This section gives you arundown on how to proceed with your companyrsquos own training
Who will manage the training programYou must determine who will be responsible for managing your companyrsquos trainingprogram Each option in this decision has consequences in terms of the orientationconsistency and buy-in from your staff The options include the following
The leaders of your profit and loss (PampL) areas and major functional organiza-tions Place the ownership of the training program directly within the PampL andfunctional organizations if you want to orient the overall initiative more directlyat bottom-line performance These organizations directly manage their work andby managing the Six Sigma training initiative they focus more directly on resultsThey can also tailor the training more specifically to their work area The risksassociated with this option include inconsistency in training and some losses inthe economy of scale in the training program But most companies deploy inthis manner and have great success
The Six Sigma Deployment Leader who typically reports directly to the CEOHave your Deployment Leader manage the training program if you want to cen-tralize the training and keep it aligned directly to the overall initiative If you useoutside trainers the Deployment Leader will be in the best position to align thetraining program with the company KBOs
In larger organizations the training department Some companies choose tomanage their Six Sigma training program from within their training department
You may think that an in-house training department is the most logical optionbut thatrsquos not necessarily the case True your training department is in the train-ing business but itrsquos best to manage the training program through your trainingdepartment only if you want to generalize your program into a more educationaland fundamental skills-based initiative
Based on the nature of your organization and your goals determine where yoursquoll base the management of your training program Communicate this decision to thestakeholders
Finding the expertise within your organizationMany thousands of people have been involved in Six Sigma initiatives As a resultcapable practitioners are showing up in organizations large and small across all
16 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 16
industries and all around the globe Chances are that someone close to you has knowl-edge of Six Sigma initiatives
Before you go any further in your deployment initiative take the time to assess thelevel of Six Sigma deployment initiative expertise in your organization Using Figure2-2 identify all individuals who have been formally trained in Six Sigma and specifi-cally document their roles the number of projects they were involved in and theresults they achieved Meet individually or in a group with these individuals and listento their Six Sigma stories They will have experienced some form of Six Sigma trainingand more importantly they may be Green Belts Black Belts and perhaps even MasterBlack Belts If so theyrsquore capable of defining your program and even conducting sometraining sessions Enroll them in the process from this point forward
Scoping the training programYour Six Sigma training program is directly tied to performance Trainees conduct proj-ects and begin influencing your organization and returning results as part of their ini-tial training regimen and they continue onward after their training is completed Thenumber of individuals practicing at the varying levels of Six Sigma capability withinyour organization is directly a matter of how many are trained and active at any givenpoint in time You must determine the scope of your training program which meansyou have to know the number of staff members who will be trained the level of profi-ciency theyrsquoll be trained to and in which functions of the organization theyrsquoll betrained in
Because trainees have such a strong influence on behavior and performance you mustscope your training program carefully Overtraining results in hoards of practitionersundertaking projects faster than they can be properly aligned or mdash worse mdash results inpractitioners sitting idly awaiting quality projects On the other hand undertrainingmeans insufficient momentum lost savings and disenfranchised staff members
To scope the program properly note the roles filled in a Six Sigma enterprise
Six Sigma Skills Assessment Worksheet
Date
Name Training Projects ResultsSix Sigma Role(s)
Figure 2-2Six Sigma
skillsassessmentworksheet
17Chapter 2 Forming a Six Sigma Initiative
06_045191 ch02qxd 81606 1142 PM Page 17
Executives participate in a workshop that explains the Six Sigma deploymentprocess introduces the concepts used by practitioners and develops the imple-mentation plan
Senior managers participate in a workshop that in addition to an overview ofthe deployment process and practitioner concepts sets out the implementationplan and assigns the roles and responsibilities for each of the functional areas ofthe business
Black Belts as full-time Six Sigma practitioners are dedicated to working onprojects that solve your toughest challenges Black Belts are necessary only forlarge and complex issues Rarely does the Black Belt population exceed 1 to 2percent of an organizationrsquos total staff
Green Belts participate in Six Sigma projects on a part-time basis either as lead-ers or contributors Anyone can be a Green Belt staff management and evenexecutives The Green Belt population is typically 5 to 10 percent of total staffbut wersquove seen cases where an organization trains its entire professional staffNote All supervisors and managers should be trained Green Belts
Yellow Belts may participate in projects but the intention of training to theYellow Belt level is to develop working Six Sigma practices within the generalpopulation You may train 50 percent or more mdash even 100 percent mdash of yourstaff to the level of Yellow Belt proficiency
Designers receive specialty training in a subject area known as Design for SixSigma or DFSS A Six Sigma deployment will include DFSS training for all designers
Master Black Belts act as hands-on experts teach Six Sigma courses and mentorothers
Table 2-1 summarizes these roles and indicates the levels of training involved in anaverage Six Sigma training program
Table 2-1 Standard Six Sigma Training RegimenRole How Many Standard Training Training Group Size
Executive All executives 2- to 3-day workshop Up to 15
Senior All senior managers 2- to 3-day workshop Up to 15managers
Deployment 1 Workshop plus mentoring 1Leader
Champions 1 per organization 5-day intensive instructions Up to 15
Black Belts Typically 1 to 2 4 weeks over 4 months 15 to 20
Green Belts Typically 5 to 10 2 weeks over 2 months 15 to 20
Yellow Belts Typically 25 to 50 1 week over 1 month 20
Awareness Everyone 1frasl2-day overview 20 to 100 (or more)
Designers All designers Multiple 2- to 3-day courses 12 to 15
Master Black 1 to 2 per Black Belt training plus 5 or fewerBelts organization 2 weeks
Black Belts are dedicated full time during the four-month training process They develop and complete a SixSigma improvement project as part of their training program
18 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 18
The ranges indicated in Table 2-1 are typical but not absolute because no set standardexists Your Six Sigma deployment initiative may vary because every organization isdifferent and each organization has its unique needs and drivers The drivers thataffect your training regimen include
Breadth How broad must you go in your organization to achieve the resultsyoursquore looking for If yoursquore looking for immediate tactical impact in a single workarea or line of business you need only a small team of specialists Conversely ifyoursquore looking to change the company culture yoursquoll want a broad initiative
Depth How serious are your problems and how complex are your challengesThe more complex and problematic the more Black Belts yoursquoll need Howevermany companies operate at a level of complexity below the need for Black BeltsNote that the Six Sigma training industry tends to overtrain
Many organizations fail to recognize the level and importance of applying Six Sigma tothe many areas of design work in an organization Itrsquos not just the design of your prod-ucts or services but also the design of your systems processes and proceduresDesign for Six Sigma addresses all elements of the design process Donrsquot shortchangeyour initiative by undertraining designers in DFSS
Your next exercise is to define the scope of the initiative within your organizationDonrsquot worry about the timing and schedule of training at this point mdash just focus on theoverall scope Using the worksheet in Figure 2-3 define the number of courses andidentify who you expect to undergo training at each level
Recognizing cultural predispositionSystems and organizations resist change And some are more resistant than othersPutting it the other way some organizations are more ready willing and adaptable tochange than others The degree of cultural predisposition toward or against a changeinitiative like Six Sigma is a key factor in rolling out the training program If you misjudgethe culture and implement the program the wrong way your initiative may backfire
Consider the following items when trying to decipher the cultural predisposition ofyour organization
Six Sigma Training Program
Date
Six Sigma Role
ExecutivesSenior managersDeployment LeaderChampionsBlack BeltsGreen BeltsYellow BeltsAwarenessDesignersMaster Black Belt
1 1
Participants Courses Names
Figure 2-3Trainingprogram
worksheet
19Chapter 2 Forming a Six Sigma Initiative
06_045191 ch02qxd 81606 1142 PM Page 19
Urgency How urgent is the business situation How much does everyone sharethat sense of urgency The more urgent the business drivers the more aggres-sively yoursquoll want to implement the training program celebrate early wins andstimulate the momentum that moves the key business metrics Just rememberthat in this approach everyone has to be feeling the same degree of pressure
Complacency How complacent is the environment The more complacent theenvironment is the more tools yoursquoll need to stimulate change These toolsinclude leadership and support strong communications the training of key indi-viduals strategic program wins celebrated individual successes and the use ofreward mechanisms
Resistance How outright will the resistance be What will be the basis for theresistance You may have staff members who were involved elsewhere in poorlyexecuted Six Sigma programs or who have otherwise heard negative thingsabout Six Sigma programs You may have business unit leaders whose empiresare at risk once your Six Sigma program begins to shine light in dark placesOften the most vocal detractors become the most ardent supporters aftertheyrsquove been through training and completed a project Remember to move adistribution you not only move the lead but also the tail
Metrics-averse (aka ldquolazyrdquo) Most pre-Six Sigma organizations prefer to com-municate and base decisions on such notable tools as politics opinion traditionand intuition as opposed to DMAIC (Define Measure Analyze Improve andControl) Without the intellectual toolset of Six Sigma itrsquos just easier to work andthink in terms of opinion and intuition Depending on how deep-seated this cul-tural tumor is in your organization yoursquoll hear varying degrees of how difficult itis to quantify and measure things
Defining the training planYoursquove determined who will manage the program identified your organizationrsquos inter-nal capabilities scoped the program and assessed the cultural predispositions Nowitrsquos time to define the training program plan Follow these guidelines
Determine your training schedule First determine the order of your trainingclasses Use the worksheet in Figure 2-4 as a guide Wersquove entered the typical firstset of courses and workshops but the order and timing are up to you
Decide what facilities yoursquoll use Training courses are usually conducted ingroups of about 15 Executive sessions may be smaller while awareness sessionsare larger events In addition you may be using an e-learning environment(known as a Learning Management System or LMS) for some of your training
Often Six Sigma training is held off-site using the alternate venue psychologywhich means you take people out of their ordinary work environments andaccompanying mind-sets The executive sessions may be held in a retreat envi-ronment and the regular stand-up courses at a local hotel or training center Ofcourse you may also want to use in-house facilities
Decide whether your Six Sigma training should be held on-site or off-site by con-sidering how important the alternative venue psychology is to your addressingyour cultural predisposition factors
Choose your resources Now itrsquos time to decide who will conduct your com-panyrsquos Six Sigma training Read on to find out how
20 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 20
Deciding Who Will Conduct the TrainingUntil recently the only viable option for Six Sigma training was to hire an outside train-ing and consulting firm Because of this it was simply a matter of selecting the firmthat best met your requirements This is no longer the case Due to the growth of SixSigma across industry lines and coupled with the commoditization of material younow have choices
Thatrsquos the good news The bad news is that one of your choices includes conductingthe training yourself which today is a viable option Choosing to conduct the trainingyourself means that yoursquore now facing the ldquoNot Invented Hererdquo syndrome and thatyoursquore confronting a classic MakeBuy decision
You have four general options as to who will conduct the training
Contract a Six Sigma systems shop These shops are comprehensive providersthat contract with you to do it all They perform the training for all courses bothin stand-up and e-learning mode and they also provide consulting and mentor-ing If they donrsquot have the expertise in-house they contract it for you Thesefirms also provide or recommend the software tools mdash both their own and third-party tools mdash and conduct applications workshops
Contract several boutique Six Sigma trainers The Six Sigma industry hasnumerous specialty trainers that focus on specific niches such as Belt TrainingDesign for Six Sigma Executive Coaching Project Management and so on Sometraining companies focus on vertical markets such as health care financial serv-ices or manufacturing With this boutique trainer approach you hire the expert-ise and buy the software you need acting as the systems integrator to put thepieces together
Order
123456789
1011121314151617181920
Executive WorkshopManagement Initialization
Leader Champion1st Wave Black Green Belt
Staff Awareness ndash Basics2nd Wave Black Green Belt
1st Wave Yellow Belt1st Wave Design for Six Sigma
Course Workshop
Six Sigma Training Schedule
Timeframe ndash Month Week
Figure 2-4Six Sigma
trainingschedule
21Chapter 2 Forming a Six Sigma Initiative
06_045191 ch02qxd 81606 1142 PM Page 21
Hire on Six Sigma training expertise and buy materials Yoursquoll find many quali-fied Six Sigma Black Belts and Master Black Belts available for hire In additionseveral companies offer training materials for sale You can hire the instructortalent you need buy the training materials and do it all yourself
Hire on Six Sigma training expertise and develop materials You can do this ifnecessary Hire experts and have them develop a custom set of training materi-als from a combination of books and other readily available sources of informa-tion Yoursquore still likely to buy and not produce some of the software tools mdashtherersquos no return on investment calculation that justifies developing your ownprocess modeling portfolio management or analytical software package mdash butthere may be a role for custom connectivity and reporting software
In order to make the choices between these options you apply the same decisionprocess you would apply for any development and staffing challenge By consideringboth the value of having this talent reside permanently within your organization aswell as the uniqueness of the need Study Figure 2-5 to understand these tradeoffs Byfinding the quadrant that best fits your situation yoursquoll understand whether to makeor buy your training talent
Follow this checklist to determine which training option is the best fit for your organization
Is your Six Sigma training regimen relatively typical or will it be uniquely differentfrom standard industry training Will you need to extensively customize materialsbefore you use them as opposed to effectively using the industry trainers andmaterials already available in the marketplace If you answered yes to either ques-tion you may need to interview some industry trainers and examine curriculummaterials to better qualify your companyrsquos needs Note however that most organi-zations are fully satisfied with the industryrsquos standard curriculum
Can contracted industry trainers effectively teach the material to your staff ordo you want to have in-house staff trainers perform the instruction
If your needs are so unique that standard training and materials wonrsquot work foryou and you want to have in-house trainers perform the instruction hire (or use your existing) Black Belts and Master Black Belts They can create custommaterials and train your staff
Value
Uniqueness
High
High
Low
Low
Hire expertiseand buy the
materials
Hire expertiseand develop
materials
Contract piecesand integratethem yourself
Contract asystems shop
Figure 2-5Make-buy
decisionspace
22 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 22
If you need customized material but believe that the average marketplace experience is sufficient contract boutique firms and secure an experiencedDeployment Leader to be your systems integrator
If your needs are standard and you can base your program on existing industrymaterials but you want in-house trainers to deliver the instruction buy thematerials and have your instructors deliver the training
If standard industry materials and trainers will work for you and you want tooutsource your Six Sigma training program contract with a systems shop
Even if yoursquove led or been part of a Six Sigma training initiative before and yoursquovedecided to take on the entire training initiative in-house bring in an experiencedmentor mdash someone to act as a sounding board and who can provide you objectiveopinions and advice Whatever you do donrsquot go down this path alone
You need to consider the following concepts as you make your decision
Uniqueness of material Every organization is unique in many ways mdash theuniqueness is how the organization differentiates itself But is your organizationso different that the fundamental methods and tools of Six Sigma must be exten-sively modified and tailored before theyrsquore applicable in your organizationProbably not
Uniqueness of instruction You may have unique deployment needs such thatstandard materials are applicable but your instructional approach may need to be different Perhaps you need a significant e-learning component multiple-language versions or industry-specific instruction
Empire-building The purpose of Six Sigma training is to educate and empoweryour team with the methods and techniques of Six Sigma so that you canimprove the bottom-line performance of your enterprise Be careful to avoid cre-ating a self-serving Six Sigma training bureaucracy
Plethora of expertise You can now readily hire industry-specific Six Sigmaexperts who can effectively train your staff as well as immerse themselves aspractitioners in your business
Consultant-averse Many organizations are culturally (and financially) opposedto using outside consultants Recognize however that not only is the cost oftraining small compared to the value returned to the business but also thatexperience matters in Six Sigma
Material and instruction quality Every Six Sigma training regimen has commonancestry that traces back to Motorola in the 1980s but not all material in themarketplace has the same development and production quality Instructionaldesign is a discipline all its own but knowing the material doesnrsquot equate togood material and instruction quality
Outsourcing You canrsquot outsource leadership Regardless of your approach yoursenior-most leaders and managers must be engaged intimately and they must bethe driving force behind the program and its progress These duties canrsquot be con-tracted hired-out or delegated
After yoursquove chosen your trainer yoursquore ready to complete the training program makebuy worksheet shown in Figure 2-6 For each category on the worksheet select eitherin-house or contractprocure
23Chapter 2 Forming a Six Sigma Initiative
06_045191 ch02qxd 81606 1142 PM Page 23
Teaming for Success Shopping for Your Implementation Partners
Now that yoursquove decided what yoursquore buying from contractors yoursquore ready to shopAnd lucky for you like in any other established market the Six Sigma training market-place has a healthy selection Yoursquoll find a few great providers many good ones manynot-so-good ones and a few really awful ones Itrsquos no trick to find the right partner butyou need to know what to look for
Six Sigma isnrsquot a regulated industry You wonrsquot find any IEEE or ISO standards or anygovernment academic or nonprofit regulatory bodies Itrsquos also not policed or reviewedby a consumer watch group Instead itrsquos a cottage industry with a broad and looselydefined set of conventions and practices The uninformed buyer can waste consider-able time and effort and go down paths that lead to frustration and loss However on the other hand the informed buyer can make outstanding choices and reap quickbenefits
The worksheet in Figure 2-7 guides you in selecting and evaluating your implementa-tion partners
Category
Training CurriculaTraining CurriculaTraining CurriculaTraining CurriculaTraining CurriculaTraining CurriculaTraining Curricula
e-learninge-learningFacilities
InstructionInstructionInstructionInstructionInstructionInstruction
SoftwareSoftwareSoftware
MentorshipMentorshipMentorship
Belt Training CurriculaBelt Certification Criteria
Executive WorkshopManagementInitialization Workshop
LeaderChampion CurriculaAll Hands Awareness CurriculaDesign for Six Sigma Curricula
CurriculaLearning Management System
Classroom FacilitiesBelt Instruction
Executive InstructionFacilitationLeaderChampion Instruction
ManagementInitialization FacilitationAll Hands Awareness InstructionDesign for Six Sigma Instruction
AnalyticsProcess Management
ProgramProject TrackingExecutive Mentoring
Deployment LeaderChampionBelt Project Mentoring
XXX
Donrsquot even think about making these software tools Great tools already exist in the market and are available at reasonable prices
Item Develop In-House ContractProcure
Six Sigma Training Program MakeBuy Worksheet
Figure 2-6Makebuy
worksheet
24 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 24
Cat
egor
y
Tra
inin
g Fi
rms
Rel
evan
ce
Cul
tura
l Alig
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t
Con
trac
tual
Alig
nmen
t
Com
pre
hen
sive
ness
Fu
lfillm
ent
Tra
inin
g P
ract
itio
ners
C
apab
ility
Ef
fect
iven
ess
C
ost
Tra
inin
g M
ater
ials
C
ours
e C
atal
og
Mat
eria
l Qua
lity
M
ater
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edig
ree
C
usto
miz
atio
n
Emb
edd
ed S
oftw
are
C
omp
lete
ness
C
ost
e-Le
arni
ng
Cou
rse
Cat
alog
Le
arni
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ype
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der
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hat
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ners
kno
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uff
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th
e tr
aine
r ef
fect
ivel
y tr
ain
our
staf
fA
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pay
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righ
t p
rice
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is m
ater
ial c
over
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at w
e ne
ed
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ater
ials
wel
l des
igne
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evel
oped
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th
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ater
ial v
ette
d fo
r au
then
tici
ty
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der
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e th
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s w
e ne
ed
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s th
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ater
ial u
se t
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tool
s w
e w
ant
Are
th
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ater
ials
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ple
te
Am
I ge
ttin
g th
e m
ater
ials
for
the
bes
t p
rice
Doe
s th
is c
over
wh
at w
e ne
ed
Is t
his
th
e le
arni
ng m
odal
ity
we
wan
tA
re t
he
mat
eria
ls w
ell d
esig
ned
amp d
evel
oped
Is
th
is t
he
righ
t p
latf
orm
for
e-le
arni
ng d
eliv
ery
Wh
ere
shou
ld w
e so
urce
our
sof
twar
eD
oes
this
pro
per
ly m
eet
our
need
sD
oes
this
pro
per
ly m
eet
our
need
sD
oes
this
pro
per
ly m
eet
our
need
sD
oes
this
pro
per
ly m
eet
our
need
s
Are
we
gett
ing
the
advi
ce a
nd c
ouns
el w
e ne
ed
Item
Eva
luat
ion
Cri
teri
aK
ey Q
ues
tion
Six
Sigm
a T
rain
ing
Pro
gram
Sou
rce
Eva
luat
ion
Wor
ksh
eet
25Chapter 2 Forming a Six Sigma Initiative
Figu
re 2
-7
Sour
ceev
alua
tion
wor
kshe
et
06_045191 ch02qxd 81606 1142 PM Page 25
Six Sigma training isnrsquot rocket science Donrsquot let yourself be bamboozled or slick-talkedinto anything by being made to feel uninformed uneducated or inferior Yoursquore thebuyer and you should shop for Six Sigma services just like you would shop for anythingelse
Pricing and Contracting ApproachesThe Six Sigma training industry has experimented with a number of pricing and con-tractual approaches mdash some of which succeeded while others failed Consider the fol-lowing options and decide which approach is right for you
Price per wave This is the most common pricing approach With this approachyou pay by the course which is also known in the industry as a wave of stu-dents This approach is common enough that you can use it effectively as a basisfor comparison in both standup and e-learning environments
Time and materials In this consulting-oriented approach you pay for the timeand materials used by the instructor as he teaches the courses
Shared savings In this model which was first pioneered at General Electric inthe mid-1990s the provider receives a percentage of the savings realized by SixSigma projects This approach has the benefit of tying the providerrsquos efforts toyour bottom-line performance However the shared savings approach suffersfrom potential challenges in accurately assigning value to the projects Thisapproach may also allow providers to reap a tremendous bonus as a result ofyour large savings
Volume discounts In this approach you pay less for each additional course orservice you use based on a total volume of goods and services that you use
Expenses As a rule the customer pays all expenses for travel and incidentalsincurred by the providers If you want to do this differently yoursquoll be runningagainst the grain of the industry
Milestone-based Because Six Sigma training programs are lengthy mdash lasting formonths and years mdash itrsquos commonplace for the program plan to contain reviewpoints or milestones at which time performance and customer satisfaction areassessed
Establish your contracts with sufficient measurement points metrics and out-clausesIf your first selected partner isnrsquot working out donrsquot hesitate to make the change
26 Part I Getting Started in Six Sigma
06_045191 ch02qxd 81606 1142 PM Page 26
Chapter 3
Leading and Managing a Six Sigma Initiative
In This Chapter Choosing a team to lead your Six Sigma initiative
Understanding the importance of communication
Working with a Six Sigma Management Plan
If yoursquore leading or participating in an initiative to deploy the knowledge and apply themethods and tools of Six Sigma across your organization this chapter like Chapter 2
helps you through the deployment process Even if yoursquore not part of a formal deploymentinitiative and you instead simply intend to perform Six Sigma projects donrsquot skip over thischapter You need to understand how Six Sigma is deployed In this chapter you figure outhow to form the team that leads and manages your initiative Chapter 2 shows you how totake aim mdash aligning the players and determining your implementation partners Now itrsquostime to fire mdash itrsquos time to begin the initiative deployment itself
The word ldquomayberdquo doesnrsquot exist in a Six Sigma initiative You canrsquot simply stick your toe in totest the water (but we guarantee that the uncharted waters are going to feel at least a littlebit uncomfortable) After you start a Six Sigma initiative you quickly pass the point of noreturn But the good news is that a well-proven and time-tested prescriptive formula for yourdeployment is available In this chapter you apply this formula to your organization and situation and as a result you and your organization are well-prepared for the changes andactivities to come Be sure to visit wwwdummiescomgosixsigmaworkbook for someuseful forms you can print out
Selecting Your Leadership TeamYour Leadership Team should have both the authority and responsibility to ensure thateveryone across the organization is properly empowered and supported throughout the lifeof your Six Sigma initiative This team is directly accountable for producing the results mdash the measurable quantifiable and significant improvements in key metrics and benchmarks mdashthrough facilitation of the initiative process The makeup of the Leadership Team definesyour success Accordingly you leave nothing in your initiative to chance mdash a few things areleft to random variation perhaps but none are left to chance mdash and neither should the selec-tion of your Leadership Team
Approaching the selection processPart of the selection process is simple Certain business unit leaders and executives aremembers of the team automatically These team members include those who own financialresponsibility and key functional processes such as leaders and executives from the humanresources and information technology departments However another part of the process is
07_045191 ch03qxd 81606 1141 PM Page 27
more complicated mdash particularly in the selection of the Deployment Leader and theSenior Champion or Champions You also select certain key stakeholders as membersof the Leadership Team Careful selection of these stakeholders is important to thesuccess of your initiative
Depending on the size of your organization you may have a large Leadership Team mdasha dozen or more mdash which is okay because the Leadership Team isnrsquot a deliberationbody that must reach consensus on difficult issues The entire Team doesnrsquot evenalways meet as a group The Leadership Team is more of a SWAT team that fans outacross your organization mdash as well as across your various constituencies mdash to ensurethat your initiative deploys successfully
Follow these three steps in your selection process
1 Identify central organizational roles
To identify these roles begin by identifying the major direct and supportingprocesses in the organization and the leaders or owners of those processes andfunctions These leaders include
bull All leaders accountable for financial performance Because financial per-formance areas are the most important target areas for Six Sigma processimprovement these leaders are by definition part of the Leadership Team
bull All leaders of the major enabling process areas Examples of these areasinclude Finance Accounting Legal Procurement Human ResourcesInformation Technology Facilities and other similar processes that enableyour core business processes Identify these leaders in your organizationalstructure Also determine how budgets are assigned and how accountabil-ity is measured
2 Identify central Six Sigma initiative roles
Here are the three such roles that are part of the Leadership Team
bull Communications Leader Because communication is so vitally importantto a Six Sigma initiative and because the initiative will have a communica-tions plan you may consider naming a specific individual to lead the com-munications activities
bull Deployment Leader This leader is the individual charged with ensuringthat the initiative is deployed across the organization per the deploymentplan Even though this role is traditionally titled as a leadership role itrsquosmore related to management than leadership
bull Champion This is the true Six Sigma leadership role The Champion is thesenior evangelist who must understand what Six Sigma can and canrsquot doThis person must champion or support the initiativersquos adoption
3 Identify other key participants
In addition to the organizational and Six Sigma initiative leaders you may haveother key individuals who belong on your Leadership Team Identifying theseother key individuals is critically important to your success particularly wherechange management is concerned Additional stakeholders may include the following
bull An outside board member Six Sigma is going to shake things up in yourcompany Business will be done differently from now on which results inchanges to the way you communicate the status and performance of yourcompany If you need an advocate who can effectively ldquocarry the waterrdquo for you to key shareholders analysts sponsors donors or other externalconstituents you want to include an outside board member on yourLeadership Team
28 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 28
bull An outside advisor Your business will be different after Six Sigma You maymodify organization structures reporting alignments and other internalinstitutions These changes impact the company culture
Throughout the Six Sigma process you may want to have an external advisor mdash someone whorsquos been through this process before This outsideadvisor is someone whom members of the Leadership Team can confide inand seek advice and solace from You should consider seeking an advisorwho isnrsquot part of your training team
bull Key suppliers Your initiative will reach into your supply chain In fact youmay be implementing Six Sigma now because someone reached into his orher supply chain and there you were Suppliers will be directly affected byyour initiative so you may want to include them
bull Key customers Why are you deploying a Six Sigma initiative in the firstplace To help serve customers better of course The direct beneficiariesof your efforts certainly have a keen interest in how your initiative isaccomplished so keep them informed
bull Social network leaders Remember that all organizations have leaders whodonrsquot appear at the top of an organization chart The cultural changes fromthe initiative will affect areas of the organization that may be invisible tothe official leadership channels mdash so much so that you would be remiss if you didnrsquot recognize and include individuals who represent the socialorganization
bull A training team leader If you have selected an outside implementationpartner (see Chapter 2 for details) you may want to include a seniormember on your Leadership Team
Many Six Sigma training and consulting firms are rich with statisticians methodolo-gists analysts and engineers but are short on senior business leadership skills andexperience For this reason you have to be careful whorsquos chosen for your LeadershipTeam For example your training partner may be an effective facilitator but he maynot belong on your Leadership Team
Membership on the Leadership Team isnrsquot optional Everyone chosen must participatebecause the initiative depends on the performance of every member of the organiza-tion All members must fulfill their roles and responsibilities and the members of theLeadership Team are no exception The Team is only as strong as its weakest link
Because full participation is crucial many organizations modify the compensationstructure of the internal Leadership Team participants and apply incentives to ensurethat the job is taken seriously These incentive changes include the top executives Itrsquosnot uncommon for organizations to tie a significant percentage mdash up to 50 percent mdashof incentive compensation for their executives directly to Six Sigma performanceimprovements
Even if the top management fully supports the initiative as a group not every individ-ual will be on board Six Sigma initiatives bring about bold change which can worrysome top management team members In its relentless pursuit of improvement SixSigma shines bright lights into the dark corners of the enterprise where many seniorleaders have personal agendas hidden The successful initiative finds those darkplaces and exposes the agendas which often become targets for improvement
Use the worksheet in Figure 3-1 to make your Leadership Team selections
29Chapter 3 Leading and Managing a Six Sigma Initiative
07_045191 ch03qxd 81606 1141 PM Page 29
Go team Unifying individuals to create a teamAfter yoursquove selected the individuals for your team you must forge a team out of themUnfortunately individuals donrsquot spontaneously combust to create a team Creating ateam takes work The dynamics and psychology of teaming are a science and disci-pline unto itself and you should keep the following guidelines in mind
Rarely do teams come together simply because a leader calls them together Inother words simply declaring a group of individuals a team isnrsquot enough Suchan occurrence would be a statistical oddity for sure
Many times certain members of the team will perceive a Six Sigma initiative as agreat risk to themselves or their constituents These fears must be carefullyaddressed or the team wonrsquot come together
The team leader mdash usually the organizationrsquos top executive mdash may not be the rightindividual to energize and inspire the team Recognize this fact and figure out howto take the appropriate action perhaps with the designated Senior Champion
Within teams cooperation coalitions and dissent all change over timeCommunications and anticipative action are critical to maintaining team cohesiveness
Leadership Team Membership Worksheet
Date
Area Role
Operations ExecutivePampL Business UnitPampL Business UnitPampL Business UnitPampL Business UnitOther Business Units
FinanceAccountingProcurementLegal - ContractsHuman ResourcesInformation TechnologyFacilitiesOther Support Functions
Communications LeaderDeployment LeaderSenior Champion
Outside Board MemberOutside AdvisorKey Supplier(s)Key Customer(s)Social Network Leader(s)Training Team LeaderOthers
Business Unit LeaderBusiness Unit LeaderBusiness Unit LeaderBusinessUnit LeaderBusiness Unit LeaderBusiness Unit Leader
Enabling Unit LeaderEnabling Unit LeaderEnabling Unit LeaderEnabling Unit LeaderEnabling Unit LeaderEnabling Unit LeaderEnabling Unit LeaderEnabling Unit Leader
Six Sigma LeaderSix Sigma LeaderSix Sigma Leader
StakeholderStakeholderStakeholderStakeholderStakeholderStakeholder
Individual
Figure 3-1Leadership
Team worksheet
30 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 30
Implementing Your Communications PlanCommunications are the tugboats of change They nudge your supertanker of anorganization to a new course and direction In fact many organizations have discov-ered through their Six Sigma communications initiatives how to better communicate ingeneral Without consistent messages active listening and strong reinforcements noregimen of training or army of consultants can align everyone in the same new way
Communicating is easy but itrsquos also difficult at the same time For example communi-cating is so easy compared to the vexing details of the business problems that itrsquos oftenoverlooked The tools of communication are so routine that theyrsquore often taken forgranted The need for repetition is so fundamental to changing habits and behaviorsand yet leaders often ignore it for fear of sounding redundant or pedantic After the ini-tiative feedback says that everyonersquos tired of hearing about the new program itrsquos oftentoned down
Because a Six Sigma initiative is a retooling of the thinking and behaviors of everyonein the organization communications are so important that they warrant plans leader-ship and measurements all their own Donrsquot underestimate the importance of manag-ing communications
Understanding the two communications plansYou actually have two communications plans to choose from for your Six Sigma initia-tive The Deployment Initiative Communications Plan identifies and directs how every-one in the organization knows what is occurring during the life cycle of the programCommunications are occurring in all directions from the top-down and bottom-upinternally and externally laterally and informally and formally With this plan youensure that all these channels and forms of communication are enabled and active
The second type of plan is the Project Communications Plan Because projects are thecritical implementation tools of a Six Sigma initiative (see Chapter 4 for more on projecttools) they warrant a communications plan all their own Each project team adheres tothe protocol as called for in its Project Communications Plan to ensure that the activi-ties resources and focus of the project team are reaching the intended results
Together these two communications plans serve as the basis for applying the many toolsof Six Sigma management to build momentum and sustain progress By working withthese plans you can keep everyone aligned confirm key tenets of the initiative fightproject scope creep head off rumors and consistently improve business performance
Elements of your communications planBoth the Deployment Initiative Communications Plan and the Project CommunicationsPlan address seven distinct elements You must be sure that your plans address theseThe seven elements are
Why The purpose for the communication which is to formally establish andenforce the commitment to communicate certain information
What The item of communication Refer to Figure 3-2 for a summary of the manyitems of communication in a Six Sigma initiative
By Whom The individual responsible for ensuring that the communication occurs(see Figure 3-5)
To Whom The audience or recipients of the communication
31Chapter 3 Leading and Managing a Six Sigma Initiative
07_045191 ch03qxd 81606 1141 PM Page 31
When The time and frequency at which you deliver the communication
How The tool or delivery mechanism that you use to communicate Refer toFigure 3-6 for a list of the many ways you can communicate
Where The location mdash physical or virtual mdash where the recipients find the com-municated information
The nature of communication varies over the life cycle of the initiative including thecontent of whatrsquos communicated to whom and by whom itrsquos communicated when ittakes place and what tool of choice is most applicable How differently would youcommunicate during the initialization phase as opposed to the deployment phaseHow would you change the nature of your communications during the expansionphase And finally what would your communications be like during the sustainingphase Keep these questions in mind as you read mdash you put them to use in an exerciseat the end of the chapter
The following sections cover these elements in more detail In the section ldquoWritingyour communications planrdquo you build your own communications plan so you can seehow the process includes these seven elements
Whatrsquos to communicateSix Sigma initiatives communicate considerable amounts of information across theorganization Figure 3-2 is a list of the major items communicated So that you betterunderstand the contents and purpose of each item in your plan complete the figure bywriting a brief explanation of each
Items of Six Sigma Communication
Where Used ExplanationItem
Purpose
Plans
Methods
Lexicon
Metrics
Analyses
Experiments
Reports
Budgets Projects
Action Items Projects
Schedules Projects
Success Stories Initiatives Projects Controls
War Stories Initiatives Projects Controls
FAQs Initiative
Elevator Pitch Initiative
Indicators Processes and Management
Attitude Across the Business
Initiatives Projects Controls
Infrastructure Projects Experiments Controls
Measurements Experiments Analyses
Both Formal and Informal Language
Within Projects and Across the Business
Projects Controls
Projects Studies
Initiatives Projects Controls
Figure 3-2Items
of com-munication
32 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 32
Before the initiative can proceed beyond the Leadership Team one critical item ofcommunication must be prepared mdash the elevator pitch The Six Sigma elevator pitchwhich is a common way of defining and describing your initiative in a smooth briefpassionate and consistent manner is a cornerstone of the initiativersquos communicationsplan Brevity is important here mdash the elevator pitch got its name because you shouldbe able to give the entire spiel during a short elevator ride If you create anythinglonger you risk losing your audience
In Figure 3-3 you craft your own Six Sigma elevator pitch Take your time on this task mdashitrsquos not easy to boil down your entire initiative to a crisp and clever definition You onlyhave to address four topics to complete the elevator pitch Theyrsquore laid out for you inthe figure
Herersquos a purposefully generic example of an elevator pitch It covers each of the topicsfrom Figure 3-3 in a handful of sentences that can be spoken in about a minute
Wersquore embarking on a long-term continuous improvement effort called Six Sigma Itrsquosnot a fad itrsquos a well-proven approach to reducing inefficiencies and increasing effec-tiveness across all aspects of our business Our industry is faced with increasing globalcompetition against both the quality and price of our products and services Our SixSigma program will give us the tools to constantly eliminate the things that are holdingus back This will require all of us to learn new tools and ways of doing things but as aresult wersquoll be a stronger company and each of us will be more agile and valuable
Now try creating your own elevator pitch Even though ours is generic make yourspassionate and specific to your situation
Great work mdash yoursquove completed one of the most difficult communications The nextcommunication the frequently asked questions (FAQ) list is much easier In Figure 3-4you see a set of commonly asked questions about Six Sigma Complete the worksheetby answering the questions Feel free to add new questions that you think may beasked in your organization
Who does the communicatingEveryone communicates in Six Sigma Communications happen from the top-down mdashfrom the highest levels of leadership down to the last hourly worker or volunteerThey also occur from the bottom-up from anyone in the organization who sees aproblem that needs fixing or a way to do something better up to those who can actu-ally do something about it Communications are also lateral mdash which means thatmethods and tools are shared and results are verified across and between projectsand departments And theyrsquore both outside-in with inputs from customers suppliersand stakeholders and inside-out with inputs going from the organization to theseconstituents Finally communications are also very much bidirectional with every-one listening as much as he or she is being listened to
The Six Sigma Elevator Pitch
Item StatementFloor
Ground Floor
Mezzanine
Balcony
Penthouse
Identify the initiative and what itiacutes about
Explain why its so important to be doing it
Articulate the benefits for the organization
Show what the initiative means to the individual
Figure 3-3The elevator
pitch
33Chapter 3 Leading and Managing a Six Sigma Initiative
07_045191 ch03qxd 81606 1141 PM Page 33
Figure 3-5 shows a matrix of each general class of participant Because communica-tions are bidirectional each class is listed on both axes You may identify others thatyou can include for your organization Take the time to consider what type of commu-nication goes in each square Ask the following questions
Which communications are most important to the effectiveness of the initiative
Are there any communications that are unlikely to ever occur Why Should they
Which communication channels are likely to be exercised the most
Which communications are the most unusual mdash but critical to the initiativenonetheless
Do any communications pose a risk to the success of the initiative What shouldbe done about it
Which communications are most vulnerable to being disrupted What would youdo to protect them
Six Sigma FAQs
Question AnswerSubject
BasicInformation
What is Six Sigma all aboutIsnt Six Sigma just for manufacturingWeve seen all these various quality intiativescome and go What makes this one differentWhats with all the goofy beltsIsnt this really just a job-reduction initiativeDoes this mean more consultants
How ItAffects Me
How will this affect my jobHow will this affect my department or teamHow does this affect my career opportunitiesWere so busy now how can we possibly take on a new initiativeDo we all have to become math geeks
How ItAffects theCompany
How does this make us a better companyDoes this mean another reorganizationWill the managers understand and support thisHows it going to roll out Where do we startHow does this affect our customersHow does this affect our suppliers
How ItWorks
Where did the term ndash and the movement ndash come fromWhat has this meant for other businesseslike oursIs there a great book that describes itDo we really have to achieve Six SigmaHow will this affect our company culture
Figure 3-4Six Sigmafrequently
asked questions
34 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 34
35Chapter 3 Leading and Managing a Six Sigma Initiative
Six
Sigm
a C
omm
un
icat
ors
To
From
Ch
ief E
xecu
tive
Top
Exe
cuti
ves
Pamp
L O
wne
rs
Fina
nce
amp A
dm
in
Hum
an R
esou
rces
Info
rmat
ion
Tec
hno
logy
Pro
cure
men
t
Lega
l ndash C
ontr
acts
Faci
litie
s
Oth
er S
upp
ort
6S D
eplo
ymen
t Le
ader
6S S
enio
r C
ham
pio
n
6S C
omm
unic
atio
ns L
ead
er
Ch
amp
ions
Out
sid
e B
oard
Mem
ber
Out
sid
e A
dvi
sor
Key
Sup
plie
r(s)
Key
Cus
tom
er(s
)
Soci
al N
etw
ork
Lead
er(s
)
Tra
inin
g T
eam
Lea
der
Ch
amp
ions
Mas
ter
Bla
ck B
elts
Bla
ck B
elts
Gre
en B
elts
Non
-Bel
ts
Oth
ers
Chief E
xecu
tive
Top Exe
cutiv
esPampL O
wners
Finan
ce amp
Adm
in
Human
Res
ources
Info
rmat
ion T
echnolo
gy
Procu
rem
ent
Legal
ndash Contra
ctsFa
ciliti
es
Other
Support
6S D
eplo
ymen
t Lea
der
6S Se
nior C
hampio
n
6S C
omm
unicatio
ns Lea
derCham
pions
Outsid
e Boar
d Mem
ber
Outsid
e Advis
or
Key Su
pplier(s
)
Key C
ustom
er(s
)
Socia
l Net
work Lea
der(s
)
Train
ing T
eam
Leader
Champio
ns
Mas
ter B
lack B
elts
Black B
elts
Green
Belt
sNon-Belt
sOth
ers
Figu
re 3
-5
Com
mun
i-ca
tors
07_045191 ch03qxd 81606 1141 PM Page 35
Timing is everythingYou should communicate at four different times
At a milestone event or point of achievement
At a periodic time interval
At an impromptu time thatrsquos right for communication
At an impromptu time thatrsquos wrong for communication
Like with any major initiative all four cases will come into play during your Six Sigmadeployment
Using communications toolsYou can communicate in countless ways with countless types of tools Figure 3-6 catego-rizes many standard tools of communication into a list including the typical applications ofeach tool Study this table to discover how many tools are available to you This table alsoshows you the different powers and applicability of each tool You canrsquot use all the tools allthe time Instead itrsquos a matter of the right combination
As an exercise consider and identify when itrsquos most appropriate to use each toolWhich tools are most useful to each of your communicators from Figure 3-5 When inthe process of your initiative or project would you use each tool Which are mostpowerful and efficient in your organization Using the worksheet in Figure 3-6 makenotes in the ldquoWhenrdquo column that help you identify and organize the uses of thesecommunication tools
Where oh where have the communications goneEnvironment dramatically affects the nature of communications which is why formalpresentations are staged in formal environments and why off-site retreats are thebackdrop for new out-of-the-box thinking One of the original Six Sigma training firmstook corporate executives to a ranch in rural Arizona and had them riding the rangeand herding cattle as part of their reorientation Donrsquot underestimate the role andpower of environment in your communications
Follow these guidelines as you prepare your Six Sigma initiative
To command everyonersquos attention senior management should use the podium ofits most revered settings to announce and present information at key timesAnything less will denigrate the importance of the program
Off-site retreats have proven to be effective venues for concentrating change effortswith managers and other organizational leaders
Senior managers and process owners should regularly venture into the organizationand conduct formal communications events as well as informal encounters withinthe environment of the Six Sigma projects and work processes that are in the midstof change
One-on-one sessions in non-threatening environments enable key communications
Because online ldquoenvironmentsrdquo are vital in todayrsquos technological world tools such asan internal Web site and a newsletter which include all key communications and alink to repositories should be developed and maintained throughout the initiative
36 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 36
37Chapter 3 Leading and Managing a Six Sigma Initiative
Form
al D
ocum
ents
The
Com
mun
icat
ions
Too
lkit
App
licat
ion
Tool
Whe
n
Das
hboa
rds
(BPM
App
licat
ions
)Fo
rmal
Pre
sent
atio
nsRe
posi
tori
esSi
x Si
gma
New
slet
ters
Colu
mn
in O
ther
New
slet
ters
Mem
os a
nd L
ette
rsSu
rvey
sV
ideo
tape
sFa
ce-T
o-Fa
ceIm
prom
ptu
Pres
enta
tions
Info
rmal
Dis
cuss
ions
Mee
tings
Tele
phon
e Ca
llsSt
orie
sA
dver
tisem
ents
Trin
kets
and
Mem
ento
s
Cong
ratu
latio
nsE-
mai
lTo
wn
Hal
lsCE
O M
emo
to E
mpl
oyee
sPo
ster
s an
d Ba
nner
sFA
Qs
Pam
phle
tsBr
own
Bag
Lunc
hes
Intr
anet
Pos
ting
Upd
ates
Phon
e H
otlin
eM
ilest
one
Reco
gniti
on E
vent
s
Sugg
estio
n an
d Q
uest
ion
Box
Qua
lity
Qui
z or
Cro
ssw
ord
Puzz
le
The
Bullh
orn
e-
mai
ls a
re n
ever
com
plet
ely
priv
ate
Proj
ect D
efin
ition
Sta
tus
Rep
orts
Whi
te P
aper
s P
lans
Pro
cedu
res
Nea
r Re
al-ti
me
Busi
ness
and
Sys
tem
s St
atus
Sco
reca
rdin
gA
war
enes
s B
rief
ings
Rev
iew
s T
rain
ing
App
rova
lsRe
fere
nce
Info
rmat
ion
Tem
plat
es a
nd T
ools
Ach
ieve
men
ts S
ucce
ss S
tori
es I
nitia
tive
Stat
usA
chie
vem
ents
Suc
cess
Sto
ries
Ini
tiativ
e St
atus
Dir
ectiv
es Q
uest
ions
amp C
once
rns
Ans
wer
sEm
ploy
ee M
oral
e P
rogr
am M
omen
tum
Cus
tom
er O
pini
on S
uppl
ier
Info
rmat
ion
Pers
onal
ized
Rec
ords
Tea
m-B
uild
ing
Lev
ityQ
uest
ions
amp A
nsw
ers
Und
erst
andi
ng amp
Em
path
y R
einf
orce
men
tK
now
ledg
e Tr
ansf
erK
now
ledg
e Tr
ansf
erCo
nsen
sus
Build
ing
Gro
up D
ecis
ion-
Mak
ing
Priv
ate
Dis
cuss
ions
At-A
-Dis
tanc
eTe
am D
evel
opm
ent
Cons
ensu
s-Bu
ildin
gSa
les
Mom
entu
m A
lignm
ent
Mom
entu
m-B
uild
ing
Rew
ards
Rec
ogni
tion
Tea
m-B
uild
ing
Publ
ic R
ecog
nitio
n S
uppo
rt M
omen
tum
Priv
ate
Dis
cuss
ions
At-A
-Dis
tanc
e D
irec
tives
Gro
up C
omm
unic
atio
nsCo
nsen
sus-
Build
ing
Cat
hars
is D
iver
sity
Gro
up C
omm
unic
atio
ns
Aw
aren
ess
Con
sens
us-B
uild
ing
Adv
ertis
ing
Mom
entu
m A
lignm
ent
Crea
te a
Com
mon
Lan
guag
e A
ddre
ss Is
sues
and
Con
cern
sM
ini-T
utor
ials
- Re
min
ders
Ref
resh
ers
Kno
wle
dge-
Shar
ing
Con
sens
us-B
uild
ing
Stat
us K
now
ledg
e-Sh
arin
gQ
uest
ions
amp A
nsw
ers
Und
erst
andi
ng amp
Em
path
y R
einf
orce
men
tRe
cogn
ition
of A
chie
vem
ent
Ano
nym
ous
Inpu
t Ca
thar
sys
Div
ersi
tySk
ills
Dev
elop
men
t Le
vity
Som
etim
es y
ou ju
st h
ave
to s
hout
it fr
om th
e m
ount
ain-
top
Figu
re 3
-6
Com
mun
i-ca
tions
tool
s
07_045191 ch03qxd 81606 1141 PM Page 37
Writing your communications plansAfter completing your communications planning yoursquore ready to assemble each of theelements into the two communications plans Figure 3-7 contains a template for organ-izing your Deployment Initiative Communications Plan Using the materials and resultsfrom the exercises you performed throughout the chapter write your plan
Figure 3-8 contains a template for organizing your Project Communications Plan Usingthe materials and results from the exercises you performed throughout the chapterwrite your plan
Six Sigma Deployment Initiative Communications Plan
I Cover Title Date Release Version Author Approval(s)II Role and Purpose State the role this plan will play within the organizationIII Values Identify the way your organization values and uses communications Indicate the communications responsibilities for executives managers practitioners and staffIV Items of Communication Identify the scope of the communications plan Indicate key items to be communicatedV Key Personnel List the principal author and responsible parties for this plan Confirm participation and approval from the top organization executive(s) Indicate who will be responsible for issuing and responding to the communications itemsVI Timing Indicate communications items by milestone Indicate communications items by fixed frequencyVII Tools List the different tools to be used and by whom Indicate where certain tools are inappropriate as wellVIII Environment Identify where published reports presentations letters and other pertinent communications will be archived and accessible for reference Indicate other facilities environments information technologies and channels to be usedIX Measurement amp Analysis Identify key measures of successful communications and how the analyses of these measures will lead to improvements
Figure 3-7Building the
Six Sigmacommunica-
tions plan
38 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 38
Selecting Software Products and IntegratingInformation Technology Architectures
The software tools for Six Sigma sort into two major categories practitioner tools andmanagement tools In this section you identify your needs and determine which ofthese tools is right for you
Yours mine or ours Platform questionsThe first issue to address is the extent to which you need your data and softwareapplications to be designed and implemented for enterprise use which depends pri-marily on the size and scope of your organization and your initiative If yoursquore deploy-ing Six Sigma across a large organization you need enterprise-class tools If yourimplementation is in a small organization or single department desktop tools shouldbe sufficient To decide whether you need to use enterprise-class tools answer thequestions in Figure 3-9
Did you answer ldquoYesrdquo to all the questions in Figure 3-9 If any of your answers wereldquoNordquo chances are great that you wonrsquot be using enterprise tools
Six Sigma Enteprise vs Desktop WorksheetYes No
Is your Six Sigma intiative being deployed across your entire organization
Is your organization presently enabled with enterprise type applications
Do you intend to use outsourced Web applications as your programsoftware
Is your Six Sigma program going to affect more than 1000 staff memebers
Do you have customers who require integrated data systems
Do you anticipate having more than 20 Black Belts
Figure 3-9Enterprisecomputingworksheet
Six Sigma Project Communications Plan
What By Whom To Whom When How Where
Project TeamDocument and
Project Charter Black BeltProject Lead Process Owner Prior to Project Approval Program ArchivesPresentation
Six Sigma Champion
Project Team Meeting Project Team Other Notices CalendarsBlack BeltProject Lead ltDateDayTimeDurationgt ltMeeting LocationgtNotices and Agendas Invitees Documents
Meeting Minutes Black BeltTeam Scribe Distribution List Day After Meeting e-mail Program Archives
Project TeamTeam WorkAction Items Black BeltProject Lead TBD e-mail Phone Call Program Archives
Six Sigma Champion
Project TeamStatus Reports andBlack BeltProject Lead Process Owner ltper Milestone Eventsgt e-mail Program ArchivesProgress Reports
Six Sigma Champion
Process OwnerBlack Belt or Project Six Sigma
Project Budget ltper Project Budgetgt e-mail Program ArchivesFinancial Analyst Champion Project
Analyst
Process Owner
Project Reviews Black BeltProject Lead Six Sigma ltper Milestone EventsgtNotices Calendars
ltMeeting LocationgtChampion Project DocumentsAnalyst
Figure 3-8Project
Communi-cations
Plan
39Chapter 3 Leading and Managing a Six Sigma Initiative
07_045191 ch03qxd 81606 1141 PM Page 39
Practitioner toolsSix Sigma practitioners use a variety of tools when performing the measurementsanalysis and improvements of business processes These tools are summarized inFigure 3-10 Use this checklist to verify that you have selected tools that fulfill each of these needs
Management toolsManagement tools enable and assist project management facilitate communicationsaid learning and retention and provide a repository for future reference Figure 3-11 isa checklist of the management tools required Verify that you have capabilities in placefor performing these functions
Management Software Tools
Tool Applications Software
Communications
Program and PortfolioManagement
Initiative-level tracking and management of programperformance Ideation Project Selection Alignment Prioritization
Project ManagementProject-level budgeting schedules resources milestonesmethodology benefits risks controls
KnowledgeManagement
Methods and Tools repositories Best Practices ReferenceDatabases Collaboration Plans History and Archives
Leadership Motivation Statusing Informing ExplainingListening Direction Correction Rewarding Celebration
ReportingStatusing Actions Results RecommendationsAchievements Failures Challenges Lessons-Learned BestPractices Dashboards Balanced Scorecard
Learning Tools Lectures e-Learning Handbooks Guides
Figure 3-11Manage-
ment soft-ware tools
Practitioner Software Tools
Tool Applications Software
Process Modeling
Simulation Process Timing Resource Consumption Patterns Bottlenecks
Analytics
Measurement Systems Analysis Graphical Analysis CampE MatrixTime-Series Descriptive Statistics ANOVA Advanced StatisticsProcess Capability and Capability-Complexity Analysis ToleranceAnalysis Regression Exploratory Analysis Mutivariate Analysis
Process Definitions Material Flow Value Stream IdentificationResources Cycle Times Functional Alignments (Swimlanes)
Design of ExperimentsRobust Experiments Multi-Factorial Designs Response SurfaceDesigns Taguchi Designs
DesignAxiomatic Design QFD Kano Modeling Robust Design TRIZPugh Concept Selection
Figure 3-10Practitioner
softwaretools
40 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 40
Enterprise Integration SOA and BPMIn addition to the practitioner and management software tools specific to Six Sigmaprogram execution there are other information technology contributions to a SixSigma initiative These contributions are based on leveraging the existing technologyinfrastructure and application programs already in use across your organizationThese programs may include such critical systems as financial and accounting applica-tions customer management software supply chain and delivery management pro-curement software design tools and more Many existing business processes areencoded formally within these systems or informally in the way people use them andas a result your Six Sigma initiative will eventually address these systems
Make sure your IT department uses the following technologies
Enterprise Application Integration and Service-Oriented Architectures YourSix Sigma reporting tools dashboards and other measures of business perform-ance will require you to extract information from these systems Use connectiv-ity tools middleware and other forms of Enterprise Application Integration (EAI)to extract the data Increasingly Service-Oriented Architectures (SOAs) are usedas an abstraction layer to make data extraction simpler
Business Process Management Business Process Management (BPM) systemsextend the functionality of process management and process analysis tools intoproactive simulation and control of key components of the business
Defining and Implementing Your Management Plan
Your Six Sigma initiative is a long-term commitment to a new way of performing every-day business Six Sigma involves both a shift in thinking to a measurable basis for set-ting objectives and performance as well as the development of a project orientation toapproaching and solving problems Your organization will undergo an extended pro-grammatic effort to deploy the knowledge methods and tools of Six Sigma and willdevelop the infrastructure communications incentives and control systems
Your Six Sigma initiative has a life cycle with four distinct phases
Initialization This is the phase yoursquore in right now mdash the process of preparingand developing the infrastructure and support systems needed to begin the initiative
Deployment In this phase the many elements of the infrastructure aredeployed Training and project work begin and the first results come in
Expansion After the initial deployment phase is completed you expand the pro-gram to encompass the entire organization including key suppliers and cus-tomers
Sustaining This is the most challenging phase of all because you have to sustainthe gains mdash and these arenrsquot the gains in bottom-line performance Theyrsquore thegains in cultural agility This is the phase of transitioning from ldquothe new initia-tiverdquo to ldquothe way we workrdquo
Unfortunately Six Sigma doesnrsquot happen magically To accomplish a successful SixSigma initiative you need to create and work with an Initiative Management Plan
41Chapter 3 Leading and Managing a Six Sigma Initiative
07_045191 ch03qxd 81606 1141 PM Page 41
The Six Sigma Initiative Management Plan is similar to other initiative plans because itrequires a complete and thorough consideration of all aspects of the initiative and afully documented and traceable set of measures and controls
Figure 3-12 is a checklist for your plan As you prepare your plan be certain that eachof the elements in the checklist is included
Phase
Initiative Management Plan Checklist
Management Task Complete
Initialization Business Objectives
Scope of Initiative
Leadership Team Selection Assignments
Communications Plan Definition
Human Resources
Definition of Training Regimen
Critical Success Factors
Timeline of Major Milestones
Risk Factors and Contingency Plans
Revision and Update Management
Policies
Deployment Leadership Team Initiation Workshop
Infrastructure Readiness
Communications Elements
Information Technology
Human Resources Alignment
Initial Training Waves
First Projects Process Improvement Areas
Results Reports Communications
Expansion Lessons Learned ndash System Plan Updates
Scope Adjustments
Training Waves
Additional Projects Process Improvements
Results Reports Communications
ltrepeatgtSustaining Lessons Learned ndash System Plan Updates
Infrastructure Adjustments
Cultural Incentive Realignments
Institutionalization Elements
Additional Projects Process Improvements
Figure 3-12The
InitiativeManage-
ment Planchecklist
42 Part I Getting Started in Six Sigma
07_045191 ch03qxd 81606 1141 PM Page 42
Part IIDefining a Six Sigma Project
08_045191 pt02qxd 81606 1128 PM Page 43
In this part
From the topmost to the lowest level applicationsSix Sigma uses the Define-Measure-Analyze-Improve-
Control (DMAIC) project methodology to achieve break-through improvement results In this part you go throughexercises and worksheets that build your mastery of defin-ing the problem to be solved and the project effort thatwill solve it Consistent improvement requires your carefulstudy and precise scoping of what the problem actually is
08_045191 pt02qxd 81606 1128 PM Page 44
Chapter 4
Putting the Right Foot Forward Defining a Six Sigma Project
In This Chapter Identifying problem areas with a business case
Creating scorecards to prioritize projects
Defining projects with problem statements
Using objective statements for project definition
Following a launch checklist
Prior chapters show you how to form an overall Six Sigma initiative in your organizationhow to gather the necessary resources and how to make sure your initiative meets
the goals of your organization Through these chapters you discover how to lead andmanage your initiative and how to make sure your team lines of communication and tech-nologies are all in place So now you think yoursquore ready to leap in with both feet franticallyattacking every problem in sight and slashing and burning your way through all that varia-tion error and waste right Not so fast
Correctly defining your project is a critical tipping point in your Six Sigma initiative mdash a mis-step here can totally derail the whole shebang Selecting and scoping projects particularly inthe early stages of an initiative must be done in a rational orderly process because nothingcauses a loss of confidence in an initiative more than a failure at the outset Your first proj-ects must have extremely high chances for clear success These first wins build confidenceand momentum in your improvement efforts Anything less is the kiss of death
This chapter provides you with the tools and techniques to ensure that your project selec-tion process gets your initiative off on the right foot
Getting Project Ideas by Using the Business Case Writing Tool
After yoursquove reined in your sense of urgency to get started yoursquore ready to kick-off an orderlyproject selection process The first step is to look around for projects right Not exactlythatrsquos still a bit premature The first step is to identify the problem areas of the business By using a tool called a business case you can identify where in the business problems areoccurring provide a summary description of the situation and estimate the potential valueof improvement efforts The intent here isnrsquot to define a Six Sigma project but to clearly illu-minate where projects are needed the most
The description of the business problem doesnrsquot need excessive detail at this point mdash thedetail comes with the definition of the associated projects
09_045191 ch04qxd 81606 1130 PM Page 45
1 Herersquos a business case example Find and underline each of the five required business case elements
Procedures call for all shipments to be turned over to the carrier within four hours of receipt atthe dock Shipping dock A has an average time of more than 12 hours resulting in storage issuesand customer complaints We had to add 100 square feet of storage to the dock and have lostthree customers this month due to late shipments
Solve It
In order to stimulate you to find problem areas in your business following is a checklistof red flag items any one of which could indicate a business problem area to address
Product returns Low quality Capacity restraints
Receivable collection issues Low yield Long cycle times
Stressful work Rework Excessive inventory
Chaotic or complicated workflow Waste Customer complaints
46 Part II Defining a Six Sigma Project
Q After yoursquove identified a potential prob-lem area itrsquos time to prepare the businesscase An effective business case mustinclude the following elements
bull The specific system or process beingscrutinized
bull The area of the business affected
bull The base goal or objective not beingmet
bull The resulting problems or issues
bull The estimated impact in dollars oranother metric
Underline each of these business case elements in the following statement
Our accounts receivable performancefor the finance invoicing area isnrsquot meeting the goal of 47 days sales out-standing Overall this poor perform-ance is causing cash flow and budgetproblems that are costing us as muchas $4 million per year
A Herersquos what the statement looks like withthe business case elements underlined
Our accounts receivable performancefor the finance invoicing area isnrsquot meeting the goal of 47 days sales outstanding Overall this poor perform-ance is causing cash flow and budgetproblems that are costing us as muchas $4 million per year
Use the following template to create business cases and to start the process of select-ing appropriate Six Sigma projects
The performance of ______________________________________________ isnrsquot meeting thegoal of ______________________________________________________________ in the area of__________________________________________________________________ This results in___________________________________________________________ causing these negativeeffects _________________________________________________________________________
Prioritizing and Aligning Projects withBusiness-Customer-Process Scorecards
Hopefully by now yoursquove identified a number of problem areas in your business andhave prepared business cases for each (if you havenrsquot check out the previous section
09_045191 ch04qxd 81606 1130 PM Page 46
ldquoGetting Project Ideas by Using the Business Case Writing Toolrdquo for tips) So now the ques-tion is ldquoWhat criteria should I use for selecting appropriate Six Sigma projectsrdquo Here are justa few options Alphabetical random highest dollar and even by throwing darts But be care-ful Even though yoursquove identified legitimate problem areas that will yield good candidatesfor projects haphazard project selection can still hamper or derail your initiative
Always keep in mind that to obtain the maximum benefit from your initiative you must linkyour Six Sigma project selection with the strategic needs of the business
Take a look at Figure 4-1 which illustrates the overall concept of aligning Six Sigma with busi-ness goals and objectives
At this critical point in the selection process you probably have a number of ldquoVoicesrdquo shout-ing at you and demanding attention the Voice of the Customer (VOC) Voice of the Process(VOP) and Voice of the Business (VOB) Often you find these Voices competing with eachother for your attention For example the VOC wants lower prices better pizza and fasterdelivery the VOP wants the best possible ingredients regardless of the cost and the VOBwants to make more money
Take the Three Sigma Pizza Emporium for example This business has a number of problemsand the Six Sigma team didnrsquot know where to start on improvement efforts So the team com-pleted a business-customer-process scorecard which is shown in Figure 4-2
After multiplying the three column rankings for each process the Six Sigma team at theThree Sigma Pizza Emporium decided that making pizza was the place to start improvementefforts The team identified several key projects in that area
Name of Ark Process
Processes that need someamount of improvement
(from Business Cases)
Impact onCustomer Needs -
Internal orExternal (VOC)The effect thisprocess has on
meeting the needsand expectationsof the customer
Improvement Needor Amount (VOP)A rating based on
existing performancelevels that are
required to meet theneeds of the business
Importance toMeeting Business
Goals andObjectives (VOB)
The effect improvingthis process willhave on existing
goals and objectivesof the business
Overall Ranking
The score resultingfrom multiplyingthe ratings in the
three columns
Inventory management 2 1 4 8
Staffing and training 3 3 2 18
Phone-in ordering 4 2 3 24
Preparing pizza 5 4 3 60
Delivering pizza 5 2 5 50
Impact ratings 1=Little 2=Somewhat 3=Moderate 4=High 5=Extreme
Figure 4-2Business-customer-
processscorecard
Objective
Link Six Sigma tobusinesspriorities
Achievebreakthroughimprovement
Integrate into day-to-day business
Project identificationand launch
Solution to theproblem and a finalreport
Implementation andfinancial benefit
Phase
Recognize
Output
Define
Measure
Analyze
Improve
Control
Realize
Figure 4-1Project
alignment tobusiness
needs
47Chapter 4 Putting the Right Foot Forward Defining a Six Sigma Project
09_045191 ch04qxd 81606 1130 PM Page 47
Figure 4-3 is a blank scorecard template you can use to prioritize the problem areas for yourpotential Six Sigma projects by considering all the important Voices in your process You canprint this form from wwwdummiescomgosixsigmaworkbook
Project Definition I Writing a Problem Statement
Before mounting your white horse and leaping right into solving your business problemsyou need to define and describe the problem by using a problem statement This tool clarifiesthe issue by specifically identifying what has to improve to meet your goal the magnitude of the problem where the problem occurs and the financial impact The problem statementcan then be used to communicate the problem to the people whose support you need
Following is a checklist that shows all the critical elements of a successful problem statement
A description of the problem and the metric used to describe it
The process name and location of the problem
The time frame over which the problem has been occurring
The size or magnitude of the problem
Name of Ark Process
Processes that need someamount of improvement
(from Business Cases)
Impact onCustomer Needs -
Internal orExternal (VOC)The effect thisprocess has on
meeting the needsand expectationsof the customer
Improvement Needor Amount (VOP)A rating based on
existing performancelevels that are
required to meet theneeds of the business
Importance toMeeting Business
Goals andObjectives (VOB)
The effect improvingthis process willhave on existing
goals and objectivesof the business
Overall Ranking
The score resultingfrom multiplyingthe ratings in the
three columns
Impact ratings 1=Little 2=Somewhat 3=Moderate 4=High 5=Extreme
Figure 4-3Business-customer-
processscorecardtemplate
48 Part II Defining a Six Sigma Project
Q The Six Sigma team at the Three SigmaPizza Emporium created a problem state-ment to pinpoint the businessrsquos problemswith pizza production The first draft ofthe problem statement read as follows
ldquoThere is a problem with the number ofundercooked pizzasrdquo
A When the team presented this problemstatement to the General Manager hisresponse was ldquoSo what I knew thatrdquo Theteam leader told the team to try againusing the checklist above The next draftread as follows
ldquoIn the last six months 5 percent ofpizzas had to be scrapped prior to
boxing due to undercooking The boxershad no procedures for sending pizzasback for additional cooking time In addi-tion 2 percent of the pizzas delivered tocustomers were undercooked resulting in 125 customer complaints Pizza scrap-ping cost the company $23550 during thelast six months in addition to a loss ofcustomersrdquo
If you compare the new statement withthe previous critical elements checklistyou see that all critical elements havebeen included The General Manager nowhas a clear picture of the problem and hewill no doubt support the Six Sigma proj-ects conducted by the team in this area
09_045191 ch04qxd 81606 1130 PM Page 48
49Chapter 4 Putting the Right Foot Forward Defining a Six Sigma Project
2 Now itrsquos your turn to help the Three Sigma Pizza Emporium team write another problem state-ment If you recall from the scorecard created in the section ldquoPrioritizing and Aligning Projectswith Business-Customer-Process Scorecardsrdquo pizza delivery was the second-ranked problem areaTake the following incomplete statement and rewrite it as an effective problem statement makingsure to include all items from the checklist
ldquoCustomers are complaining about delivery timesrdquo
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
An example solution to this practice problem is located at the end of this chapter Of course thereis no single right answer The only requirement is that your solution contains all the necessaryelements
Solve It
Following is a blank problem statement template to help you organize your thoughtsas you prepare for your next project
A description of the problem ___________________________________________________
_____________________________________________________________________________
The metric used ______________________________________________________________
_____________________________________________________________________________
Specifically where the problem is occurring______________________________________
_____________________________________________________________________________
How long the problem has been occurring _______________________________________
_____________________________________________________________________________
The size or magnitude of the problem ___________________________________________
_____________________________________________________________________________
Project Definition II Writing an Objective Statement
One more tool is important for you to make sure that your improvement projectlaunches properly This tool is called an objective statement which directly addressesthe problem statement In order to be effective the objective statement must containall of the following elements It must improve some metric from some baseline to somegoal in some amount of time with some positive impact on some corporate goal orobjective Simply put the objective statement must indicate the level of improvementexpected from improvement efforts including specific quantifiable amounts and thetime required to complete
09_045191 ch04qxd 81606 1130 PM Page 49
50 Part II Defining a Six Sigma Project
Six Sigma practitioners often use a memory jogger called SMART to help write effective objective statements Each letter reminds you of a goal to achieve in yourstatement
Specific Make sure that the specific deliverables and outcomes are stated andthat you answer the question ldquoWhatrsquos the specific purpose of this projectrdquo
Measurable Be sure that your objective is both quantifiable and verifiable andthat it includes such things as quality quantity cost and timeliness
Aggressive but Attainable A challenging objective makes the project interestingand fulfilling while also providing worthwhile returns However donrsquot try tosolve world hunger
Relevant The objective must be relevant to business goals
Time bound You must state a definitive time frame for reaching your objective
Q The Three Sigma Pizza Emporium teamcreated an objective statement The firstdraft of the statement for pizza produc-tion looked like this
ldquoRetrain employees to reduce thenumber of undercooked pizzasrdquo
A The team leader sent the team back tothe drawing board with instructions toinclude all the critical elements requiredfor a proper objective statement Theteamrsquos next effort read as follows
ldquoOur goal is to reduce the number ofpizzas scrapped from undercooking from5 percent to less than 05 percent by June30 In addition we also want to reducethe number of undercooked pizzas thatreach customers from 2 percent to zeroby the same date Doing so will helprestore our image as a quality pizzaparlor and will save the company$50000 annuallyrdquo
If you compare the new draft to therequired elements mentioned at thebeginning of this section yoursquoll find thatall are included
3 Returning again to the Three Sigma Pizza Emporium situation see if you can help the team pre-pare an effective objective statement for the pizza delivery problem Take the following incom-plete statement and rewrite it as an effective objective statement making sure to include allrequired items
ldquoImprove pizza delivery timesrdquo
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
An example solution to this practice problem is at the end of this chapter Of course there is no single right answer The only requirement is that your solution contains all of the necessaryelements
Solve It
09_045191 ch04qxd 81606 1130 PM Page 50
51Chapter 4 Putting the Right Foot Forward Defining a Six Sigma Project
Following is a blank objective statement template to help you organize your thoughtsas you prepare for your next project
The metric to be improved_____________________________________________________
_____________________________________________________________________________
The current baseline __________________________________________________________
_____________________________________________________________________________
The goal _____________________________________________________________________
_____________________________________________________________________________
The time frame for improvement _______________________________________________
_____________________________________________________________________________
The corporate goal or objective ________________________________________________
_____________________________________________________________________________
The impact on the goal or objective_____________________________________________
_____________________________________________________________________________
Launching a ProjectYoursquore probably chomping at the bit to launch your project Yoursquove written the busi-ness case yoursquove made sure your identified problem area is important to the companyand yoursquove written a problem statement and an objective statement So you say letrsquosgo However donrsquot get ahead of yourself mdash there are still a few things to remember andmanage as the project starts
To make sure the launch process is orderly follow this checklist
Identify everyone who has to approve the project
Obtain written approval
Identify the people impacted by your project
Notify the impacted people of whatrsquos to come
Get final approval from the project team leader
Identify Six Sigma skill levels (Belts) that are needed
Identify the process members who will participate
Identify the entire project team by name
Fire
09_045191 ch04qxd 81606 1130 PM Page 51
Solutions to Defining a Six Sigma Projecta Herersquos a solution to the business case example
Procedures call for all shipments (the specific process) to be turned over to the carrier within4 hours of receipt (the base goal not being met) at the dock Shipping dock A (area of the busi-ness affected) has an average time of more than 12 hours resulting in storage issues and cus-tomer complaints (the resulting problem) We had to add 100 square feet of storage to thedock and have lost three customers this month (the impact) due to late shipments
b Now itrsquos your turn to help the Three Sigma Pizza Emporium team write another problem state-ment If you recall from the scorecard created in the section ldquoPrioritizing and Aligning Projectswith Business-Customer-Process Scorecardsrdquo pizza delivery was the second-ranked problemarea Take the following incomplete statement and rewrite it as an effective problem statementmaking sure to include all items from the checklist
ldquoCustomers are complaining about delivery timesrdquo
An effective problem statement for this case reads as follows
ldquoSince June 30 the average time to deliver a pizza within our market area a 15-mile radius is35 minutes During this time we had a low of 22 minutes and a high of 66 minutes and weexceeded our advertised guarantee of 30 minutes 55 percent of the time The result of these slowdelivery times is customer complaints free pizzas to the customer and a loss of business Freeand scrapped pizzas resulted in a revenue loss of $23800 during the period Driver costs for redelivery cost the company an additional $6600 Lost customers are estimated to represent$45000 in annual businessrdquo
c Returning again to the Three Sigma Pizza Emporium situation see if you can help the team pre-pare an effective objective statement for the pizza delivery problem Take the following incom-plete statement and rewrite it as an effective objective statement making sure to include allrequired items
ldquoImprove pizza delivery timesrdquo
An effective objective statement for this case reads as follows
ldquoOur objective is to reduce the number of pizzas delivered in more than 30 minutes from 55 per-cent to less than 5 percent by June 30 of next year Doing so will further enhance our image ofexcellence and will save the company over $60000 annuallyrdquo
52 Part II Defining a Six Sigma Project
09_045191 ch04qxd 81606 1130 PM Page 52
Chapter 5
Brainstorming the Inputs to Your Process
In This Chapter Creating diagrams to identify inputs
Developing process flow maps
Sorting inputs with CT trees
Every process every product every event mdash what do these all have in common Each is comprised of or caused by some number of inputs known in Six Sigma parlance as
Xs which take their place in the equation Y= f(X) Before you can start improving your out-come or Y you have to first identify all of the inputs that created this outcome Withouttools and a rigorous process identifying inputs is a daunting task
Even something as simple as a 2 lead pencil has an astounding number of inputs mdash woodpaint metal rubber graphite and all the people machines and processes to mine refineproduce mold bend shape spindle ship receive and assemble the parts Now considersomething as complex as an automobile mdash you have steel plastic rubber paint aluminumglass well you get the picture
Six Sigma practitioners have developed tools and techniques to act as catalysts to facilitatethe identification of the inputs into a process or product Because the rest of this workbookdeals with categorizing measuring improving and controlling inputs itrsquos important thatyoursquore proficient in identifying these inputs So in this chapter you find out how to prepare aseries of diagrams each one of which digs you deeper and deeper into the world of inputs
All Together Now Brainstorming with YourTeam to Create Affinity Diagrams
An affinity diagram is a diagram thatrsquos the result of an exercise where members of a groupwrite their ideas on small paper notes and sort the ideas into logical categories Brainstorm-ing has always been an effective tool for generating a large number of ideas mdash and the firststep in creating an affinity diagram is a brainstorming session Brainstorming works becausethe ideas and knowledge generated by two or more people acting as a group are alwaysgreater than those of the sum of the people acting individually
10_045191 ch05qxd 81606 1140 PM Page 53
Herersquos the traditional cycle of a brainstorming session
1 Agree on a subject The group must first agree on the category or condition tobe the subject of the session
2 Encourage participation Each team member should be encouraged to participate
3 Discourage criticism Ensure that everyone understands that debates and criti-cism arenrsquot allowed
4 Decide on a contribution method Members can contribute in rotation or in freeflow Just make sure everyone is heard
5 Promote equality Ensure that every member has an equal opportunity to beheard
6 Listen Hear and respect the ideas of others mdash there are no stupid ideas
7 Record ideas Make sure all ideas are written down
8 Continue as long as necessary Stop only after no more new ideas are offered
9 Edit Review the list for clarity and duplicates
10 Repeat Repeat the process for all identified categories
After yoursquove generated a whole truckload of ideas you can start your affinity diagramThe process of creating an affinity diagram stimulates an effective gathering and organ-ization of the ideas into natural groupings mdash groupings that start to pinpoint theessence of a problem and provide the raw material for a breakthrough solution
Likely more than one person in the brainstorming group already knows whatrsquos causingthe problem mdash these people just donrsquot know that they know After the ideas are gener-ated and organized the light will come on and the entire group will say ldquoAhhh sothatrsquos the problemrdquo
54 Part II Defining a Six Sigma Project
Q Herersquos a checklist for generating an affinity diagram
1 Write down the problem or issue
Be sure that the topic is clear and concise enough to address in one session ldquoHow do we boilthe oceanrdquo is too broad However ldquoWhy is our server computer crashingrdquo is a more suitabletopic to delve into
2 Conduct a brainstorming session for the problem
Follow the brainstorming steps that we outlined earlier in this section Spend a minimum of 15minutes and as long as necessary discussing this issue
3 During the discussion have the participants write each of their ideas on a separate stickynote with only one idea per note
Avoid using ideas that are single words Instead use phrases or sentences For instance unac-ceptable ideas for the topic ldquoWhy is our server computer crashingrdquo would be ldquostupidityrdquo ldquojunkrdquoldquoevil spiritsrdquo or ldquowho knowsrdquo Acceptable ideas would be ldquoUsers are poorly trainedrdquo ldquoComputerhardware is obsoleterdquo and ldquoSoftware is incompatiblerdquo
Continue as long as necessary Itrsquos not unusual to generate more than 100 notes
10_045191 ch05qxd 81606 1140 PM Page 54
4 Post all of the sticky notes on a smooth surface such as a wall or whiteboard
Without discussion have the participants sort the notes into a few logical categories mdash usuallyaround five but in no case more than ten Encourage participants to move notes from categoryto category to where they fit best
Sorting in silence helps participants focus on meaning and not on the emotion or baggage thatarises in most discussions
5 After all the sorting is done instruct the participants to create titles for the categories
A category title should fit the notes bundled under it Donrsquot force titles or notes into placesthey donrsquot fit When the sorting process is done create an affinity diagram
A The affinity diagram should look something like the following
Category 1
This is the
This is the
This is the
Category 4
This is the
This is the
Category 2
This is the
This is the
This is the
This is the
This is the
This is the
This is the
This is the
Category 5
This is the
This is the
This is the
This is the
This is the
This is the
Category 3
This is the
Main Topic
This is the
This is the
This is the
This is the
55Chapter 5 Brainstorming the Inputs to Your Process
10_045191 ch05qxd 81606 1140 PM Page 55
Solve It
Why is our server crashing
56 Part II Defining a Six Sigma Project
1 Following is a short list of ideas that may have been generated in a brainstorming session on thetopic ldquoWhy is our server computer crashingrdquo Review the ideas and then enter each in one of thecategory columns in the included figure After entering all the ideas add suitable category titlesHere are the brainstorming ideas
bull Obsolete hardware bull Old virus software
bull Poor training bull Spyware
bull Large applications bull Power brownouts
bull Incompatible software bull No new-hire training
bull Power surges bull Macintosh computers
bull Weak firewall bull Linux software
bull Server room too warm bull Microsoft software
bull Excess network connections bull Server room not secure
bull Virus infection bull Dial-up connection
bull Old server software bull No budget
bull Faulty disk drive
Now that yoursquore familiar with the affinity diagram and brainstorming process pick anagging problem that you face in your job gather several other people familiar withthe issue and conduct an affinity exercise using the blank template in Figure 5-1 Youcan print it from wwwdummiescomgosixsigmaworkbook
10_045191 ch05qxd 81606 1140 PM Page 56
After you create an affinity diagram take it a step further and drill down a bit into theinputs to find the causes of variation by using a very useful tool called the cause-and-effect diagram which is also called the Ishikawa diagram or because of its appear-ance the fishbone diagram (see Figure 5-2)
The fishbone diagram which was created by Kaoru Ishikawa a quality managementpioneer in Japan is used to explore all the potential causes (inputs) that result in asingle output Even though an affinity diagram helps you group inputs into general cat-egories the fishbone diagram goes a step further and links inputs together to help youdig for the true root causes of variation
Figure 5-2The
fishbonediagram
57Chapter 5 Brainstorming the Inputs to Your Process
Dem Bones Creating Fishbone Diagrams
Figure 5-1Affinity
diagramtemplate
10_045191 ch05qxd 81606 1140 PM Page 57
Q Using Figure 5-2 create your own fishbone diagram
A In constructing the fishbone diagram the problem or condition (the output or Y) is entered at theldquoheadrdquo of the fish with the backbone extending to the left Each of the major intersecting ldquobonesrdquois a primary category of input These categories may be the same ones identified in the affinitydiagram process or they can be the often-used generic categories such as Measurement Man (or People) Method (or Process) Materials Machine (or Equipment) and Environment Theimportant thing to remember is that categories need to fit your identified output and the inputsgenerated
The smaller horizontal ldquobonesrdquo are the individual inputs previously generated Each input isexamined carefully to make sure itrsquos not only in the right category but also to determine whetheritrsquos a direct input at all If an input turns out to be only indirectly contributing to the outcome and primarily affects another input these indirect inputs are placed as ldquosub-bonesrdquo connected toanother input Of course if upon examination a generated idea has no impact at all on the out-come this idea is discarded and not entered on the fishbone diagram
The following is an example of a completed fishbone diagram
Measurement
Times
Comp DowntimeFreq of Updates
Process TimeQueue Time
Environment Material Machine
Call Volume
HolidaysTime of the day
Method
Carrier UpdatesTicket Types
Postal ServiceCall Routings
ComputersPO Damage
Traveler ProfilesCompany Profiles
Man
DeliveryDefects
Training
ResourcesShiftExperience Level
SelfInternetSystems
T1 - LinesDial-up LinePhone Service
MaintenanceServers
Terminals
58 Part II Defining a Six Sigma Project
Many of the ideas produced in an affinity diagram exercise may not be causes at all orthey may turn out to be symptoms of the root cause In a fishbone diagram exerciseeach input is explored in depth linked to associated inputs and classified as either aroot or secondary cause
10_045191 ch05qxd 81606 1140 PM Page 58
59Chapter 5 Brainstorming the Inputs to Your Process
2 Following are some of the inputs from the affinity diagram exercise from earlier in the chapterFind an appropriate spot for each input on the fishbone diagram template in the included figureLook long and hard at each one mdash if you come to the conclusion that an item simply doesnrsquot havean impact on the outcome donrsquot enter it If an item appears to be an indirect input enter it as adiagonal under another input If you find that an item does impact the outcome but doesnrsquot fitany of the five categories create a new category of your own Remember like in the affinity dia-gram exercise your output is continuous server crashes Herersquos a hint to help you The serveritself has a functioning uninterruptible power supply but the computer room doesnrsquot Here arethe inputs for this fishbone diagram exercise
bull Power surges bull Virus infection
bull Weak firewall bull Power brownouts
bull Macintosh computers bull Old virus software
bull Server room too warm bull Spyware
Solve It
Personnel Training Hardware
Why is the server computer crashing
Software Network
Now using the problem from your own company that you chose in the previous sec-tion along with the affinity exercise results you brainstormed create a fishbone dia-gram using the blank template in Figure 5-3 You can also print it from wwwdummiescomgosixsigmaworkbook
Figure 5-3Fishbonediagram
template
10_045191 ch05qxd 81606 1140 PM Page 59
60 Part II Defining a Six Sigma Project
Examining Your Processes with Process Flow Maps
After yoursquove generated a number of potential inputs into your selected outcome andhave begun to categorize and analyze these inputs itrsquos time for you to take the nextstep Even though yoursquove been looking at the possible causes of your outcome youhavenrsquot yet taken a look at the whole system or process of which your outcome is theend result Before you can make improvements to your process which will give youthe outcome that you want you have to know what the process really is One way toexamine a process is to use a graphical representation of a process called a processflow map But beware of a process flow map if you havenrsquot verified that it representswhatrsquos really occurring
The illustration in Figure 5-4 has been around for many years but itrsquos still valid todayHerersquos what it all means
ldquoWhat you think it isrdquo represents the established ldquostandardrdquo procedure or ldquocon-ventional wisdomrdquo often what the documentation if it exists says it is
ldquoWhat it actually isrdquo represents the actual current practice or whatrsquos really hap-pening It may change frequently Often each person knows his or her own partof the process but no one actually knows all of this process
ldquoWhat it should berdquo represents the potential improvement or what the idealprocess could be
Figure 5-5 shows examples of some drawing conventions used in process flow map-ping Although the exact shape and meaning of each icon is flexible itrsquos important tobe consistent throughout your organization
StartEnd
Process
Decision
Document
Data
Storage
Inspection
Transport
Figure 5-5Processmapping
icons
1
What you THINK it is What it Should be
2
3
What it ACTUALLY is
Figure 5-4Three
views of aprocess
10_045191 ch05qxd 81606 1140 PM Page 60
61Chapter 5 Brainstorming the Inputs to Your Process
Q Create an example of a simple process map for a process that starts with a customer feelinghungry and finishes with the customer eating pizza
A The following figure represents the example process flow map
CustomerHungry
Calls forOrder
PizzaCorrect
CustomerEats
TakeOrder
MakePizza
CookPizza
BoxPizza
DeliverPizza
3 Create a process flow map for the following manufacturing process This process starts with metalparts entering a machine and coating shop and ends with the finished parts moving to inventoryBelow are all the steps in this process but theyrsquore not in any particular order After reviewingthese steps create a process flow map using the symbols shown in Figure 5-5 Be sure to useevery step once Here are the steps
bull Scrap bull Store
bull Drill holes bull Final inspection
bull Initial inspection bull Scrap
bull Move to coating bull Install in drill jig
bull Clean part bull Prepare inventory report
bull Inspect holes bull Install in dip jig
bull Mark for drilling bull Move to store room
bull Scrap bull Smooth holes
bull Light sanding bull Chemical bath
bull Dip in plastic coat
Solve It
10_045191 ch05qxd 81606 1140 PM Page 61
62 Part II Defining a Six Sigma Project
For even more practice create a process flow map for a process you use every day inyour job
Finding Critical Fruit in the CT TreeA CT tree is another diagrammatic way of sorting and displaying inputs but with a dif-ferent wrinkle Affinity and fishbone diagrams sort inputs by type but a CT tree sortsinputs by what is Critical To (Six Sigma crowds are pretty original arenrsquot they) themajor contributors to the success of your desired outcome What makes the CT treeparticularly useful is that it relates inputs to specific outputs that yoursquore interested inThis is the only such tool that starts with the outcome of interest and backs into thecauses
Q Try your hand at creating your own CT tree
A Review the example CT tree in the following figure
10_045191 ch05qxd 81606 1140 PM Page 62
63Chapter 5 Brainstorming the Inputs to Your Process
Criti
cal t
o Se
lect
ion
Mod
els
Conf
igur
atio
ns
Criti
cal t
o Pr
iceCr
itica
l to
Satis
fact
ion
Com
pone
nt C
osts
Asse
mbl
y Co
sts
Ship
ping
Cos
ts
Reta
il M
arku
p
Criti
cal t
o Pe
rform
ance
Chip
Set
Butto
ns
Hing
es
Batte
ry S
treng
th
Ante
nna
Disp
lay
Reso
lutio
n
Back
light
ing
Criti
cal t
o Av
aila
bilit
y
Orde
rs to
Sup
plie
rs
Supp
liers
Bac
klog
Ship
ping
Log
istic
s
Asse
mbl
y Th
roug
hout
Reta
il Re
ceiv
ing
Colo
rs
Tone
s
Feat
ures
Leve
ls
10_045191 ch05qxd 81606 1140 PM Page 63
64 Part II Defining a Six Sigma Project
Solve It
Customer Satisfaction with Pizza
Critical to QualityCritical to Price Critical to Timeliness
4 Assume that yoursquore the general manager of Three Sigma Delivery Pizzeria and that yoursquove beengetting your crust handed to you by your competition So you conduct a brainstorming sessionwith your employees and you prepare an affinity diagram and a fishbone diagram You then mapyour process so you have a good idea how your process works
As a result of the efforts so far you decide that customer satisfaction is poor and that this is theoutcome you have to change You further identified three major contributors that are critical tocustomer satisfaction pricing quality and timeliness Following is a list of inputs identified inbrainstorming Your fishbone exercise confirmed that these inputs have a potential impact onyour outcome mdash which makes them ldquoCritical Tordquo inputs mdash so you want to further classify them bythe three major contributor categories After reviewing the following list of inputs write each onein the appropriate box in the included figure
bull Available lines bull Telephone staffing bull Delivery time
bull Taste bull Cooking time bull Telephone answering
bull Packaging bull Competition bull Ingredient freshness
bull Cost of extra ingredients bull Appearance
For further practice create your own CT tree by using the work process you selectedin the section ldquoExamining Your Processes with Process Flow Mapsrdquo Use the CT treetemplate in Figure 5-6 for guidance
Critical toCritical to Critical to
Figure 5-6CT tree
template
10_045191 ch05qxd 81606 1140 PM Page 64
The following figure shows an example solution Keep in mind that the relationships betweenpossible inputs are not hard and fast and that they depend on the perspective of the partici-pants One of the strengths of this type of exercise is the diverse views which create out-of-the-box solutions
b Following are some of the inputs from the affinity diagram exercise from earlier in the chapterFind an appropriate spot for each input on the fishbone diagram template in the problemfigure Look long and hard at each one mdash if you come to the conclusion that an item simplydoesnrsquot have an impact on the outcome donrsquot enter it If an item appears to be an indirectinput enter it as a diagonal under another input If you find that an item does impact the out-come but doesnrsquot fit any of the five categories create a new category of your own Rememberlike in the affinity diagrams exercise your output is continuous server crashes Herersquos a hint tohelp you The server itself has a functioning uninterruptible power supply but the computerroom doesnrsquot Here are the inputs for this fishbone diagram exercise
Why is our server computer crashing
Personnel
Poor training
No new-hire training
Power surges
Server roomtoo warm
Power brownouts
Server room not secure
No budget
Excess connections
Macintosh computers
Dial-up connection
Obsolete hardware
Faulty disk drive
Large applications
Incompatiblesoftware
Virus infection
Old server software
Old virus software
Spyware
Linux software
Microsoft software
Weak firewall
Environment Network Hardware Software
65Chapter 5 Brainstorming the Inputs to Your Process
Solutions to Brainstorming the Inputs Problems
a Following is a short list of ideas that may have been generated in a brainstorming session onthe topic ldquoWhy is our server computer crashingrdquo Review the ideas and then enter each in oneof the category columns in the problem figure After entering all the ideas add suitable cate-gory titles Here are the brainstorming ideas
bull Obsolete hardware
bull Poor training
bull Large applications
bull Incompatible software
bull Power surges
bull Weak firewall
bull Server room too warm
bull Excess network connections
bull Virus infection
bull Old server software
bull Faulty disk drive
bull Old virus software
bull Spyware
bull Power brownouts
bull No new-hire training
bull Macintosh computers
bull Linux software
bull Microsoft software
bull Server room not secure
bull Dial-up connection
bull No budget
10_045191 ch05qxd 81606 1140 PM Page 65
bull Power surges
bull Weak firewall
bull Macintosh computers
bull Server room too warm
bull Virus infection
bull Power brownouts
bull Old virus software
bull Spyware
See the following figure for the solution Upon closer examination you should have found that the power surges brownouts and the room temperature were all related You probably created anew category called Environment (or something similar) for these items Though the server itselfhas an uninterruptible power supply the air conditioning in the server room was often shuttingdown causing temperature increases So you probably decided that the power fluctuationsdidnrsquot affect the server directly but could be a root cause and should be further investigated
You also may have realized that the combination of the weak firewall and the old virus softwareleaves the server vulnerable to infection Even though the weak firewall itself doesnrsquot impactthe server it could be a contributing root cause which means that it should be flagged for fur-ther study and measurement
Spyware while annoying doesnrsquot cause the server to crash And Macintosh computers whichwersquore using to write this workbook obviously have nothing to do with the issue at hand andshouldnrsquot be persecuted by jealous PC users
c Create a process flow map for the following manufacturing process This process starts withmetal parts entering a machine and coating shop and ends with the finished parts moving toinventory Below are all the steps in this process but theyrsquore not in any particular order Afterreviewing these steps create a process map using the symbols shown in Figure 5-5 Be sure touse every step once Here are the steps
Why is our servercomputer crashing
Software
Old v
irus s
oftw
are
Wea
k fir
ewal
l
Environment
Server room too warm Virus infection
Browno
uts
Power
surg
es
66 Part II Defining a Six Sigma Project
bull Scrap
bull Drill holes
bull Initial inspection
bull Move to coating
bull Clean part
bull Inspect holes
bull Mark for drilling
bull Scrap
bull Light sanding
bull Dip in plastic coat
bull Store
bull Final inspection
bull Scrap
bull Install in drill jig
bull Prepare inventory report
bull Install in dip jig
bull Move to store room
bull Smooth holes
bull Chemical bath
10_045191 ch05qxd 81606 1140 PM Page 66
The following figure is an example solution Even though your process map may not be exactlythe same if the steps are in a sequence that doesnrsquot create sequential gaps the process map isuseful
d Assume that yoursquore the general manager of Three Sigma Delivery Pizzeria and that yoursquove beengetting your crust handed to you by your competition So you conduct a brainstorming sessionwith your employees and prepare an affinity diagram and a fishbone diagram You then mapyour process so you have a good idea how your process works
As a result of the efforts so far you decide that customer satisfaction is poor and that this isthe outcome you have to change You further identified three major contributors that are criti-cal to customer satisfaction pricing quality and timeliness Following is a list of inputs identi-fied in brainstorming Your fishbone exercise confirmed that these inputs have a potentialimpact on your outcome mdash which makes them ldquoCritical Tordquo inputs mdash so you want to furtherclassify them by the three major contributor categories After reviewing the following list ofinputs write each one in the appropriate box in the problem figure
bull Available lines
bull Taste
bull Packaging
bull Cost of extra ingredients
bull Telephone staffing
Begin Clean part
Scrap
AMount in drill jigInspect
End
Drill holes
Move tocoating
Smooth holes
Mount in dip jig
Prepare finalinventory
report
Finalinventory
report
Chemical bath Plasticcoating dip
Light sanding
Scrap
Inspect
Scrap
Inspect
Mark holes
B
C
C
B
A
Move tostoreroom
Store
67Chapter 5 Brainstorming the Inputs to Your Process
10_045191 ch05qxd 81606 1140 PM Page 67
bull Cooking time
bull Competition
bull Appearance
bull Delivery time
bull Telephone answering
bull Ingredient freshness
The following figure is an example solution
The important outcome from this exercise is the careful consideration of what is critical toyour desired outcome and how inputs are categorized to achieve that outcome
Customer Satisfaction with Pizza
Critical to QualityCritical to Price
Competition
Perceived value
Cost of extra ingredients
Taste
Ingredient freshness
Cooking time
Appearance Delivery time
Telephone staffing
Packaging
Available lines
Telephone Answering
Critical to Timeliness
68 Part II Defining a Six Sigma Project
10_045191 ch05qxd 81606 1140 PM Page 68
Chapter 6
Prioritizing Which Inputs to AddressIn This Chapter Narrowing down your inputs by creating Pareto diagrams
Finding critical inputs with SIPOC diagrams
Discovering root causes with a cause-and-effect matrix
Using a Failure Modes Effects Analysis (FMEA) to determine the inputs with the greatest impact
Chapter 5 shows you how to identify classify and start to prioritize the inputs into aproduct or process how to conduct a rousing group get-together how to create dia-
grams that look like dead fish and how to determine whether your system or process reallyworks like you thought it did You also discover that some inputs are critical to one thing oranother while others arenrsquot If yoursquove already perused that chapter no doubt yoursquore nowchomping at the bit ready to charge in on your white horse and fix everything in sight (Ifyou havenrsquot already perused Chapter 5 you should probably do so now)
In order to concentrate improvement efforts on only those inputs that have a significantimpact you need to prioritize your inputs so that limited resources can be applied to whatwill produce the greatest improvement This chapter shows you how You can also printsome of the forms used in this chapter from wwwdummiescomgosixsigmaworkbook
Weeding and Pruning the Input GardenUsing Pareto Diagrams
The Pareto Principle or the 80-20 rule as itrsquos sometimes called is as true today as it was 100years ago when Vilfredo Pareto observed that 80 percent of all grain in Italy came from 20percent of the farmers Applying the Pareto concept to inputs is a powerful way to sort outldquothe vital fewrdquo from ldquothe trivial manyrdquo If you remember that 20 percent of inputs produce 80percent of the results you can narrow your focus down significantly
Q Create your own Pareto diagram
A The most common way to apply Paretoanalysis in Six Sigma is to relatively rankinputs based on the amount of impact eachhas on the outcome You then create a
visual representation a Pareto diagram ofthe input ranking such as a bar chart thatquickly identifies which critical inputs areranked the highest as in the followingfigure
11_045191 ch06qxd 81606 1135 PM Page 69
1 Your BelchFire 5000 coupe is a bit of a gas hog so you decide to apply the Six Sigma principles toimprove your gas mileage In an affinity diagram exercise with your bowling buddies you brain-stormed the following list of inputs that impact your gas mileage
70 Part II Defining a Six Sigma Project
Gas brandtype
Idle time
Driving speed
Vehicle maintenance
Tire pressure
Tire brand
Passenger weight
Accessories used
Weather
Engine size
Transmission type
Aerodynamics
Follow these steps to finish this practice problem
1 Place the inputs into a fishbone diagram
Use the following figure and the generic categories shown on the template
2 After yoursquove placed the inputs in the diagram relatively rank them toclearly differentiate the impact of each on your outcome
Even though the degree of impact for some inputs may be intuitively obvi-ous the impact of others may not be Use the discussion and analysis fromthe Affinity and Fishbone exercises in Chapter 5 to help establish rankingsYou may also want to explore any data thatrsquos readily available to help con-firm or refine your rankings
People Machine Method
How can I improvegas mileage
Material Measurement Environment
Donrsquot try to apply this concept too early in your improvement process You first need toidentify all the inputs before you can start sorting out the vital few (see Chapter 5 for how-toinfo on input gathering)
Characteristic of Interest
Critical Inputs
Num
eric
al S
cale
11_045191 ch06qxd 81606 1135 PM Page 70
You can use any ranking method you want as long as the differences in impact areclear For this practice exercise use a ranking of 0 to 10 with 10 being the greatestimpact and 0 being no impact at all Place your ranking score next to each input onthe fishbone diagram
You may be tempted at this point to start taking measurements to help you rank yourinputs You could measure the change you get in gas mileage for different brands of gasor you could evaluate gas mileage for different passenger weights and for different driv-ing speeds This type of measurement is a powerful way to identify the critical inputsbut unfortunately it takes a huge effort Remember the whole point is to avoid expend-ing effort to measure an input that doesnrsquot have a significant impact on the problem
3 Put the ranked inputs into a Pareto diagram to get a clear visual indication ofwhich inputs are critical inputs
Using Microsoft Excel or any other chart-making application enter your ranked inputs into anappropriate chart format such as a bar chart Make the vertical axis the ranking scale and thehorizontal axis the inputs After the chart is finished indicate which inputs are critical by put-ting a mark such as an ldquoXrdquo on the 2 to 3 highest bars Also indicate which inputs are candi-dates for further measurement and analysis
Solve It
71Chapter 6 Prioritizing Which Inputs to Address
Cementing the Foundation Creating SIPOC Diagrams
SIPOC (pronounced sy-pok) when spoken with a guttural inflection sounds like a Klingoncurse on Star Trek In reality however itrsquos an acronym for one of the most fundamental stepsin the Six Sigma process
SIPOC mdash which stands for Suppliers-Inputs-Process-Outputs-Controls mdash is a tool for buildinghigh-level process maps that consider the impact of suppliers and the requirements of cus-tomers In this context both suppliers and customers may be external or they may bepeople systems or processes within the same organization
A SIPOC diagram is based on simple process flow maps but it delves much further mdash providing immediate feedback as to which inputs are critical to the process output A SIPOCdiagram focuses on inputs outputs customers and suppliers and provides the foundationfor significant DMAIC (Define-Measure-Analyze-Improve-Control the basic Six Sigmaprocess) improvement
Following are the steps you take to complete a SIPOC diagram
1 Identify the process you want to map and then define the scope and boundary points
Be sure to use action verbs to describe what the process is supposed to do A key tocreating a SIPOC diagram is to correctly identify the current process before moving onto the other parts of the diagram
2 Identify the outputs Describe the products or services produced by the process
3 Define the customer or customers
Name the people processes or organizations that receive the outputs As part of thisstep define the customer requirements by listing what they demand and what theyrsquoreentitled to
4 List all of the inputs to the process mdash identify the people information materials andother resources necessary for the process to produce the identified outputs
5 Identify the sources or suppliers of the inputs
11_045191 ch06qxd 81606 1135 PM Page 71
2 Practice creating a SIPOC diagram for the process of building a 2 lead pencil Penrod Pencils Incbuys all the pencil components builds the pencils and sells them to a pencil wholesaler Following isall the information you need to complete a SIPOC diagram for the pencil process The data is inrandom order so you have to carefully consider where each item fits in the included figure which isa SIPOC template Herersquos a hint The template includes the exact number of boxes in each category
72 Part II Defining a Six Sigma Project
Q Create a SIPOC diagram for the order-taking portion of a pizza order process for the Three SigmaPizza Delivery company which is struggling with poor customer satisfaction and is examining allinteractions with customers to find areas for improvement
A The following figure is an example of the completed SIPOC diagram
Suppliers
ATampT phonesOffice DepotTI calculatorsNEC cash register
Inputs
Pizza typeSizeQuantityExtra toppingsSpecial ordersDrink types amp quantitiesOther productsPhone numberAddressNameTime day and dateVolume
Process
Customer Order Level 1 process flow diagram
See below
Outputs
PriceOrder confirmationBake orderData on cycle timeOrder rate dataOrder transactionDelivery info
Customers
CookAccounting
Requirements
Complete call lt 3 minOrder to cook lt 1 minuteComplete back orderCorrect bake orderCorrect addressCorrect price
Calls forOrder
Order toCook
AnswerPhone
WriteOrder
ConfirmOrder
SetPrice
Address ampPhone
6-inch graphite rods
Finished pencil
Pencil packages
Grantrsquos Graphite
Form wood barrel
Erasers
Woodyrsquos Wood
Glue wood barrel
Wood sheets
Finished pencil assembly
Paint pencil
Stanrsquos Steel Parts
Cut graphite rods
Eraser assembly
Wood glue
Finished barrel assembly
Insert eraser in clasp
Reasonable price
Rubber rods
Oswald Office ProductDistributors
Yellow paint
Cut rubber rods
Finished wood barrel
Polly Paint
Timely delivery
Package pencils
Attach eraser assembly
Eraser clasp
Consistent quality
6-foot graphite rods
George Glue Emporium
Packaged pencils
Ralphrsquos Rubber Products
Packard Packaging Products
Insert graphite rod in woodbarrel
11_045191 ch06qxd 81606 1135 PM Page 72
73Chapter 6 Prioritizing Which Inputs to Address
Solve It
S I P O C RSupplier Input
(Use nouns)Process
(Use verbs)
1
2
3
4
5
6
7
8
Output(Use nouns)
Customer Requirements
9
Untangling Webs Creating a Cause-and-Effect Matrix
In Six Sigma Y = f(X) In other words all outcomes (Ys) are the result of certain inputs(Xs) and the forces that act on them In simplest terms this formula is based on causeand effect
The Cause and Effect Matrix or CampE Matrix helps Six Sigma practitioners make a linkbetween multiple inputs and the resulting outcomes By identifying and prioritizingthese relationships you can explore and graphically display the possible causes of aproblem or condition which allows you to then search for the root cause
In many applications the CampE Matrix relates process inputs and outputs to customerrequirements mdash donrsquot forget the Voice of the Customer In other instances the CampEMatrix relates inputs to outputs in order to focus your improvement efforts andresources
11_045191 ch06qxd 81606 1135 PM Page 73
74 Part II Defining a Six Sigma Project
Q Create a CampE Matrix for the pizza order process used in the section ldquoCementing the FoundationCreating SIPOC Diagramsrdquo
A The following figure represents a possible CampE Matrix example
NameAddressTimeDayDateTelephone numberRecipeIngredientsOven temperatureCook timeVolumePreparation timeIngredient availabilityDelivery
Hot
Piz
za
Del
iver
y T
ime
Pro
per
Coo
king
Bur
nt P
izza
Ran
k or
der
sum
Cor
rect
Ingr
edie
nts
9
554
4857
7
Customer ranking
Inputs
Outputs (Voice of the Customer)
7
1010
10
9
855
4
7891010
8
448107
10
781081
011790810
3617221823626823181
160150
3 Complete a CampE Matrix for a service with which almost everyone is familiar mdash Internet access In thisexercise yoursquore going to be the customer whorsquos using the CampE Matrix to decide on an Internet serviceprovider (ISP) Herersquos the scenario After brainstorming and preparing an affinity diagram with themembers of your family you identified seven outcomes or customer requirements for Internet serv-ice that are important to your family In addition you identified seven inputs from an ISP that impactsthese requirements Following are 14 items mdash seven inputs and seven outputs mdash in random order
Ease of use
Short setup
Access on the road
Web filter
Quick log-on
Setup time
Remote coverage
Fast downloads
Transfer rate
Log-on time
Kids can use it
ISP stability
Ability to block sites
Reliability
Following are the steps to complete the CampE matrix that will help you make an ISP selectiondecision
1 Sort these items into inputs and outputs Use the worksheet at the end ofthese steps as a template
2 Place each output (customer requirement) in a column heading
11_045191 ch06qxd 81606 1135 PM Page 74
75Chapter 6 Prioritizing Which Inputs to Address
3 Rank each customer requirement using a scale from 1 to 5 with 5 being amust-have down to a 1 which is a nice feature but isnrsquot critical
In this exercise assume the following rankings Ability to block 1 Access onroad 2 Quick log-on 3 Short setup 3 Fast downloads 4 Reliability 4 Kidscan use it 5
4 List each input in a row heading
These important inputs are often called critical-to-customer characteristics orCTCs for short
5 Rank the relationships between the inputs and outputs by placing a valueof 1 3 or 9 at the intersections
Nine indicates a strong relationship 3 a medium relationship and 1 a weakrelationship Leave a blank if no relationship exists This ranking scale issomewhat arbitrary but be sure that you clearly distinguish between strongand weak relationships
6 Calculate the rank order sum for each input to determine its relative priority
To do so multiply each inputrsquos individual relationship value by its corre-sponding customer requirement ranking value and then add the multipliedvalues across the row
For example if an input has a medium or 3 relationship with a particularoutput that has a customer ranking of 5 enter 3 times 5 or 15 in the inter-secting square
7 To complete the CampE Matrix enter the rank order sums from highest tolowest in a Pareto diagram such as a bar chart The most importantinputs are now obvious
Solve It
Ran
k or
der
sum
Customer ranking
Inputs
Outputs (Voice of the Customer)
11_045191 ch06qxd 81606 1135 PM Page 75
Performing a Failure Modes Effects Analysis (FMEA)
Failures in products or processes are inevitable However remember that the key prem-ise of Six Sigma says that variation is everywhere and is in everything Six Sigma alsosays that even though you canrsquot eliminate all variation you can detect and eliminate themost harmful variation and allow for the variation that canrsquot be totally eliminated
The fact is that every input has the potential to fail and cause problems with the out-come Remember the words of the great philosopher Kenny Rogers ldquoEvery handrsquos awinner and every handrsquos a loserrdquo Itrsquos the same with inputs Each one can be successfulin contributing to the desired outcome but any one of them also can cause a cata-strophic failure Huge companies have been destroyed by the simplest of processerrors mdash errors that could have been prevented if a Failure Modes Effects Analysis(FMEA) had been performed
76 Part II Defining a Six Sigma Project
Q Define and explain Failure Modes EffectsAnalysis (FMEA)
A An FMEA is a structured approach to iden-tifying the potential ways a product or
process can fail and to identifying how youcan detect the failure and its effects so youcan reduce the risk of either occurrence orimpact or both Keep these points in mindfor the next problem
4 As the general manager of Three Sigma Pizza you want to continue your efforts in making cus-tomers happier After preparing an affinity diagram and a CampE Matrix for your pizza preparationprocess you have chosen one input mdash pizza cooking time mdash as a candidate for a Six Sigmaimprovement project Before kicking off the project you want to prepare an FMEA to focus theproject where it will have the most impact Using the following template follow these steps tocomplete your FMEA
Process name FMEA Template
Process stepPotential
failure modePotential
failure effects Seve
rity
Occ
uren
ce
Det
ecti
onR
PNCurrent
ControlsPotentialcauses
Date
11_045191 ch06qxd 81606 1135 PM Page 76
77Chapter 6 Prioritizing Which Inputs to Address
1 Select and enter your input variables
In this exercise use ldquoPizza cooking timerdquo as the input Proper considerationat this step can eliminate trivial factors from further consideration Only listvariables that have been identified from your previous work mdash such as aCampE Matrix an affinity diagram a Pareto diagram or a SIPOC diagram mdashinstead of listing every possible factor
2 Enter potential failure modes (the consequences if the input variable isnrsquotproperly controlled) associated with each individual input variable youlist Each variable may have more than one failure mode
As a hint to get you going in this exercise use ldquoCooking time too shortrdquo andldquoCooking time too longrdquo as the input
3 Enter the potential failure effect the failure mode would have on yourproduct or process
A failure mode may have more than one effect so be sure you identify all ofthem as separate line items
4 Quantify the severity of the failure modes by assigning each a severityscore from 1 to 10
The more severe the effect the higher its score Even though all identifiedfailure modes may have some impact on your customer that impact wonrsquotbe the same for each
In the following figure are some guidelines for ranking the severity of eachfailure modersquos impact on your customer The 1-to-10 scale is most commonin industrial applications because it allows for wider variation among theresults Narrower score ranges such as 1 to 5 make it easier for teams toagree The key is to be consistent
5 Enter the potential causes of each listed failure mode and then assignoccurrence scores which rank the likelihood of a failure occurring foreach of the potential causes
At this point you can generate your identified causes in a brainstorming ses-sion with those directly involved in the system product or process
The following figure shows guidelines for ranking the likelihood that a failuremode will occur A score of one means that there is very little chance thatthe failure will occur The scoring scale goes up to ten which says that a failure is certain Your organization may modify or adopt these scoringguidelines to fit your specific market and situation but remember that con-sistency in rating factors is the key
1234
5
6
7
8910
Customer will not notice the adverse effect or it is insignificantCustomer will probably experience slight annoyanceCustomer will experience annoyance as a result of poor performanceCustomer dissatisfaction as a result of poor performanceCustomer is made uncomfortable or their productivity is reducedby the continued poor performanceCustomer complains as a result of performance issueHigh degree of customer dissatisfaction due to being unable to usea portion of the productVery high degree of dissatisfaction due to loss of performanceCustomer has lost total use of productCustomer has lost total use of product and will never return
RATING DEGREE OF SEVERITY
11_045191 ch06qxd 81606 1135 PM Page 77
6 Identify and list the controls that are currently in place to prevent eachfailure modersquos cause
7 Score each control to determine its ability to detect the failure modebefore harm or damage is done or to prevent the failure mode from everoccurring
The following figure shows guidelines to help you rate detectabilityObviously itrsquos more beneficial to your customer and your process ifyou prevent the failure rather than detect it after it has already occurredHowever when failures do occur the best case scenario is that itrsquos detectedbefore reaching the customer
8 Compile and analyze your FMEA work
Calculate a Risk Priority Number or RPN for each failure mode cause bymultiplying the severity occurrence and detectability scores together forthat mode The highest RPN numbers indicate the highest risk and show youwhere your initial efforts should be concentrated
1
2
3
4
5
6
7
8
9
10
Sure that the potential failure will be found or prevented beforereaching the next customerAlmost certain that the potential failure will be found or preventedbefore reaching the next customerLow likelihood that the potential failure will reach the nextcustomer undetectedControls may detect or prevent the potential failure from reachingthe next customerModerate likelihood that the potential failure will reach the nextcustomerControls are unlikely to detect or prevent the potential failure from reaching the next customerPoor likelihood that the potential failure will be detected orprevented before reaching the next customerVery poor likelihood that the potential failure will be detected orprevented before reaching the next customerCurrent controls probably will not even detect the potential failureAbsolute certainty that the current controls will not detect thepotential failure
RATING ABILITY TO DETECT
12345678910
Likelihood of occurrence is remoteLow failure rate with supporting documentationLow failure rate without supporting documentationOccasional failuresRelatively moderate failure rate with supporting documentationModerate failure rate without supporting documentationRelatively high failure rate with supporting documentationHigh failure rate without supporting documentationFailure is almost certain based on dataAssured of failure based on data
RATING LIKELIHOOD OF OCCURRENCE
78 Part II Defining a Six Sigma Project
11_045191 ch06qxd 81606 1135 PM Page 78
Now that yoursquove identified the highest priority items you can set proposedactions assign responsibility and agree on a due date This use of the FMEA turnsthe analysis into a proactive and actionable document
You should always attempt to reduce the RPN by preventing the failure moderather than detecting it The idea is to make it impossible for the failure to occur
After the improvement actions have been accomplished and documented enter newscores for severity occurrence and detectability A new RPN gives you a clear indi-cation of whether your efforts have been successful You can then decide to take fur-ther actions on this failure mode or move on to the next highest priority item
Solve It
79Chapter 6 Prioritizing Which Inputs to Address
11_045191 ch06qxd 81606 1135 PM Page 79
80 Part II Defining a Six Sigma Project
Solutions to Prioritizing Inputs Problemsa Following is an example solution to the Pareto Diagram exercise including a fishbone dia-
gram that identifies and ranks the inputs (see following figure)
A Pareto diagram helps visualize the critical inputs (See the following figure)
In this case driving speed tire pressure and vehicle maintenance are singled out for fur-ther measurement and analysis Even though you may be tempted to immediately startimprovement efforts in all three areas you still have no data confirming that any of theseinputs is the culprit Measurements will narrow the choices even further so you may nothave to lighten up that lead foot quite yet
Engine size transmission type and aerodynamics shouldnrsquot be ranked Even though youhave no doubt that these inputs have a very definite impact on gas mileage they arenrsquot con-trollable which means they arenrsquot points of improvement leverage For the same reasonweather should not be ranked However in some circumstances you may have the flexibilityto not drive in bad weather In that case weather would be ranked possibly as a criticalinput
Drivin
g Spe
ed
Tire
Pre
ssur
e
Vehi
cle M
aint
enan
ce
Pass
enge
r Weig
ht
Idle T
ime
Gas br
and
type
Acces
sorie
s Use
Tire
Bra
nd
Wea
ther
Trans
miss
ion
Type
Engin
e Size
Aerod
ynam
ics
10987654321
Critical Inputs
People
Passenger weight - 2
Gas brandtype - 1 Weather - 0
Aerodynamics - 0Vehicle maintenance - 7
Tire pressure - 5Tire brand - 1
Accessories use - 2Engine size - 0
Transmission type - 0Driving speed - 10
Idle time - 2
Machine Method
How can I improvegas mileage
Material Measurement Environment
11_045191 ch06qxd 81606 1135 PM Page 80
b Using the entire list of SIPOC elements the following figure shows an orderly and detailed representation of the pencil production process
SI
PO
CR
Sup
pli
er
Gra
ntiacutes
Gra
ph
ite
Ral
ph
rsquos R
ubb
er P
rod
ucts
Stan
rsquos S
teel
Par
ts
Woo
dyrsquo
s W
ood
Geo
rge
Glu
e Em
por
ium
Pol
ly P
aint
Pac
kard
Pac
kagi
ng P
rod
ucts
Osw
ald
Off
ice
Pro
duc
t D
istr
ibut
ors
Tim
ely
del
iver
y
Con
sist
ent
qua
lity
Rea
sona
ble
pri
ce
6rsquo g
rap
hit
ero
ds
6rsquo g
rap
hit
ero
ds
Cut
gra
ph
ite
rod
s
Cut
rub
ber
rod
s
Inse
rt e
rase
r in
cla
sp
Form
woo
d b
arre
l
Glu
e w
ood
bar
rel
Pai
nt p
enci
l
Pac
kage
pen
cils
Att
ach
era
ser
asse
mb
ly
Inse
rt g
rap
hit
e ro
din
woo
d b
arre
l
Rub
ber
rod
sEr
aser
s
Eras
eras
sem
bly
Fini
shed
pen
cil
Pac
kage
dp
enci
ls
Fini
shed
woo
d b
arre
l
Fini
shed
bar
rel
asse
mb
ly
Eras
er c
lasp
Woo
d s
hee
ts
Woo
d g
lue
Yello
w p
aint
Pen
cil
pac
kage
s
Inp
ut
(Use
nou
ns)
Pro
cess
(Use
ver
bs)
1 2 3 4 5 6 7 8
Ou
tpu
t(U
se n
ouns
)C
ust
omer
Req
uir
emen
ts
9
Fini
shed
pen
cil
asse
mb
ly
81Chapter 6 Prioritizing Which Inputs to Address
11_045191 ch06qxd 81606 1135 PM Page 81
c The following figure shows the CampE Matrix with the raw relationship scores
The next figure shows the same Matrix with the rank order sum calculations
In the following figure the rank order sum totals are represented in a Pareto diagram givingstrong visual evidence as to which inputs are the most critical inputs
Ease of use
Web filter
Setup time
Remote coverage
Transfer rate
Log-on time
ISP stability
Shor
t se
tup
Ab
ility
to
blo
ck
Acc
ess
on r
oad
Qui
ck lo
g-in
Fast
dow
nloa
ds
Rel
iab
ility
Ran
k or
der
sum
Kid
s ca
n us
e it
3Customer ranking
Inputs
Outputs (Voice of the Customer)
5 1 2 3 4 4
9
27
45
15
15
5
5
15
5
9
1
2
6
18
6
6
6
9
9
27
12
12
36
12
12
12
12
36
70
45
48
23
68
60
60
Ease of use
Web filter
Setup time
Remote coverage
Transfer rate
Log-on time
ISP stability
Shor
t se
tup
Ab
ility
to
blo
ck
Acc
ess
on r
oad
Qui
ck lo
g-in
Fast
dow
nloa
ds
Rel
iab
ility
Ran
k or
der
sum
Kid
s ca
n us
e it
3Customer ranking
Inputs
Outputs (Voice of the Customer)
5 1 2 3 4 4
3
9
9
3
3
1
1
3
1
9
1
1
3
9
3
3
3
3
3
9
3
3
9
3
3
3
3
9
82 Part II Defining a Six Sigma Project
11_045191 ch06qxd 81606 1135 PM Page 82
83Chapter 6 Prioritizing Which Inputs to Address
d The following figure is a solution to the FMEA exercise Though your FMEA solution may notlook much like this one you need to be sure that you made logical assumptions and conclusionsas you entered the data
Be sure that your improvement efforts are focused on the highest RPN because this iswhere you can have the most impact
Process name FMEA Template
Process step
Pizza cooking time Time too short Doughy crust
Unmelted cheese
Raw ingredients
Cold pizza delivered
Burnt crust
Burnt cheese
Burnt ingredients
Inaccurate timer
Pizza removed too early
Timer not set
Placed in oven after timer started
Staff inattention
Inaccurate timer
Pizza removed too late
Timer not set
Placed in oven before timer started
Staff inattention
8
8
8
8
9
9
9
1
6
7
4
9
1
6
7
4
9
10
4
4
4
4
10
4
4
4
4
80
192
224
128
288
90
216
288
144
324
Potentialfailure mode
Potentialfailure effects Se
veri
ty
Occ
ure
nce
Det
ecti
onR
PNCurrent
ControlsPotential
causes
Date
70
60
40
30
Ease of Use Transfer Rate Log-on Time ISP Stability Setup Time Web Filter RemoteCoverage
0
20
10
50
11_045191 ch06qxd 81606 1135 PM Page 83
84 Part II Defining a Six Sigma Project
11_045191 ch06qxd 81606 1135 PM Page 84
Part IIIMastering Measuring
12_045191 pt03qxd 81606 1140 PM Page 85
In this part
What you donrsquot measure you canrsquot know What youdonrsquot know you canrsquot control And if you canrsquot con-
trol yoursquore left to the whims of chance Measurement isthe first step in improvement This part provides you with exercises to build your expertise in measuring andknowing the critical outputs and inputs of the product orsystem yoursquore improving
12_045191 pt03qxd 81606 1140 PM Page 86
Chapter 7
Categorizing Data and CalculatingMeasures of Variation
In This Chapter Determining what kind of data you have
Calculating means modes and medians
Calculating how much variation is in a set of data
Separating variation into its short-term and long-term components
Yoursquore reading this book with one goal in mind Improving your process However youhave to remember that improving your processes requires that you understand varia-
tion And to understand variation you need to figure out how to categorize data by typeand how to summarize it with some simple tools After you understand what type of datayou have where itrsquos located and how it varies yoursquore much closer to understanding andimproving your processes Read on mdash this chapter guides you through the process ofunderstanding and categorizing your data
Differentiating Data TypesData comes in different shapes and sizes If you know what kind of data you have you canthen figure out what type of analysis and what tools or methods you need to use in your SixSigma work Your data values will either be able to be placed into named categories (thisdata is called attribute or categorical data) or will follow a continuous scale (this data is calledcontinuous or variable data)
Attribute or categorical data is used to describe a named attribute of the characteristic orprocess This type of data is discrete which means that the data can be counted ranked orsorted but not added subtracted or averaged For example if you had one red ball and oneblue one it makes no sense to try to average the color
You can ask yourself a simple question about your data that tells you whether itrsquos continuousldquoCan I meaningfully add or subtract values of this datardquo If the answer is ldquoyesrdquo you have con-tinuous data
13_045191 ch07qxd 81606 1135 PM Page 87
88 Part III Mastering Measuring
1 Packages entering a sorting facility arelabeled as having either ldquourgentrdquo orldquonormalrdquo priority What kind of data is thischaracteristic of the packages
Solve It
2 Yoursquore collecting weight measurements of aplated contact switch The measurementsare in grams What type of data is this
Solve It
Q You have a set of data for an ice creamstore that tells you what flavors are on themenu
Chocolate Chocolate chip cookiedough
Strawberry Raspberry cheesecake
Butter pecan Rocky road
Vanilla Pistachio
Neapolitan Orange sherbet
Mint chocolate Bubble gum
What type of data is this
A The ice cream flavors in this list are clearlynamed categories which makes it attributedata
Q Now consider having more detail for theice cream such as how many cartons ofeach flavor were sold last week Using thefollowing table determine what kind ofdata you have
Flavor Cartons Sold
Chocolate 22
Strawberry 14
Butter pecan 3
Vanilla 19
Neapolitan 7
Mint chocolate 8
Choc chip cookie dough 9
Raspberry cheesecake 11
Rocky road 3
Pistachio 1
Orange sherbet 2
Bubble gum 6
A The additional information shown in thetable for the flavor categories is importantAsk yourself ldquoCan I meaningfully add orsubtract values of this datardquo For examplecan you add all the carton numberstogether to get a meaningful total Yes Orcan you subtract the number of chocolatecartons from the number of vanilla cartonsto find out how many more cartons ofchocolate ice cream were sold than vanillaYes again These answers tell you that thisadditional data is continuous
13_045191 ch07qxd 81606 1135 PM Page 88
89Chapter 7 Categorizing Data and Calculating Measures of Variation
3 Each final inspection record for manufac-tured disk drives is stamped with a datecode What type of data are these datecode stamps
Solve It
4 Student grades are given as A Andash B+ andso on Is this attribute or categorical dataWhat about grades given on a numericalgrade point scale A = 400 Andash = 367 B+ =333 and so on Is this attribute or categor-ical data What is the advantage of anumerical grade point scale
Solve It
Calculating Measures of Variation LocationTo begin to understand variation in your data you have to be able to describe it withnumbers The most basic numerical measures are those that quantify the location ofthe central tendency of your variation mdash or in other words the values that are mostlikely to occur
When you have continuous data you can calculate three different measures of varia-tion location
Mode The mode of a set of data is the single value that is most frequentlyobserved and is associated with the highest peak of a distribution
Median The median of a set of data is the single value where half the data isbelow and half is above The median is the preferred measure of variation loca-tion when your collected data contains outliers or extreme data points well out-side the range of other data
You determine the median by ordering your collected data from least to greatestand counting the total number of points If you find an odd number of datapoints the median is the middle value or the one exactly halfway through thelist If you find an even number of data points the median is the average of thetwo points in the middle of the list
Mean The mean mdash or average mdash of a set of data is represented mathematicallyby the symbol x
x nx i=
where
bull x (pronounced ldquoex barrdquo) is the symbol representing the calculated mean
bull xi represents each of the individual measurement values
bull Σ the Greek capital letter sigma tells you to sum up (add) all the individualmeasurements
bull n is the number of individual measurements in your data set
13_045191 ch07qxd 81606 1135 PM Page 89
When going through the calculations for all the measures of central location remember the following
bull The mode is the data value that is observed most frequently For this pressure data you cansee that the value of 246 kPa occurs three times mdash more than any other mdash making it the mode
bull The median is the exact middle value of the rank-ordered data For the 24 points of this data setthe middle value lands between ordered points 12 and 13 That makes 246 kPa the median
bull Determining the mean is more calculator intensive But the calculation is still straightforwardYou add up all the individual data values and divide by the number of individual data points24 which gives you 2466 kPa as the mean
bull The following figure is a dot plot (see Chapter 8 for practice on creating dot plots) of the collec-tion pressure measurements You can see how symmetrical the variation is When variation issymmetrical the mode median and mean are often nearly identical
90 Part III Mastering Measuring
Q For the following collection of pressure measurements (kPa) calculate the mode median andmean
247 228 232 251
237 248 240 246
252 261 242 247
263 239 229 246
262 243 244 263
267 233 246 249
A Sorting the data from least to greatest helps you find the mode and the median
Value Order
228 1
229 2
232 3
233 4
237 5
239 6
240 7
242 8
243 9
244 10
246 11
246 12
Value Order
246 13
247 14
247 15
248 16
249 17
251 18
252 19
261 20
262 21
263 22
263 23
267 24
13_045191 ch07qxd 81606 1136 PM Page 90
91Chapter 7 Categorizing Data and Calculating Measures of Variation
5 Find the mode median and mean of thefollowing collection of process cycle timemeasurements
8 19 9 10
21 8 10 8
11 11 13 21
Solve It
6 What is the average grade point of the following class of students What is themedian
333 300 367 300
367 300 333 300
033 367 333 333
333 333 367 367
300 333 300 400
Solve It
When a set of data with variation is not symmetrical the mode median and mean are often differentDonrsquot be surprised when this happens
Dotplot of Pressure (kPa)
Pressure (kPa)
228 234 240 246 252 258 264
13_045191 ch07qxd 81606 1136 PM Page 91
92 Part III Mastering Measuring
7 For the following home purchase pricedata points does the mean or the medianbetter communicate the location of thecentral tendency of the variation Why
$184000 $168000 $174500 $177200
$292400 $181000 $172010 $169900
Solve It
8 What is the average value (or mean) for the following set of plastic film thicknessmeasurements
0044 0041 0046 0049
0049 0050 0038 0044
0048 0042 0043 0040
0039 0042 0043 0039
Solve It
Variety Is the Spice of Life Measuring Variation Spread
The central location of the variation in your data is only the first of two critical param-eters you need to quantify The second parameter is the measure of how much varia-tion spread you find in your data around its central location Table 7-1 summarizesthree different measures of variation spread
Table 7-1 Summary of Statistical Measures of Variation SpreadMeasure of Variation Spread Definition Comments
Range R x xMAX MIN= - Simple Preferred metric for sets of data with only a few (2 to 9)members Drawback Greatlyinfluenced by outliers
Variance Useful for more advanced σ n
x x1
i2
2
=-
-_ iexperimentation and analyses
Standard Most commonly used for data deviation σ n 1
x x2
i=
-
-_ isets with 10 or more membersMore accurate than the rangemetric for larger data sets
13_045191 ch07qxd 81606 1136 PM Page 92
Notice how all the squared xi ndash x terms are positive And notice how when you square them youend up with the weird units of $2 or squared dollars As strange as that may seem itrsquos ok Therersquos nosuch thing in the real world but there is in the ethereal math universe It all works out in the end
Next you add up all the squared dollar terms and divide the sum by one less than the number ofpoints in your data set (n ndash 1)
$ $ nx x
1 8 123 003 11 3 286 16
i
22
2
-
-=
-=
_ i
For now this calculated variance number is just a first step to obtaining the standard deviation (σ)Later the variance will help you determine which factors contribute to overall variation whendesigning experiments to improve your processes As you mature in statistics variance measuresbecome very useful tools
The final calculation is for the standard deviation (σ) After yoursquove found the variance itrsquosextremely easy to find the standard deviation mdash you just take the square root
$ $ σ σ 3 286 16 57 332 2= = =
As you can see with the standard deviation yoursquove come back down to Earth and are again usingthe real-world units of dollars instead of squared dollars And now you have a quantitative meas-ure of how much variation spread you have in your data
93Chapter 7 Categorizing Data and Calculating Measures of Variation
Q What is the range variance and standard deviation in the dollar values for the following set ofinvoices
$524218 $519404 $526370 $518845
$521721 $520050 $507354 $516745
A The range (R) in the dollar value in this collection of invoices is the difference between the great-est (xMAX) and the least (xMIN) By identifying these extreme values in the data itrsquos easy to plugthem into the formula and arrive at the answer
R = xMAX ndash xMIN = $526370 ndash $507354 = $19016
To calculate the variance (σ2) you must first calculate the mean x which ends up being $519338At this point laying out a calculation table is helpful
xi xi ndash x (xi ndash x)2
$525218 $4880 $2 238134
$519404 $065 $2 043
$526370 $7032 $2 494424
$518845 ndash$494 $2 2436
xi xi ndash x (xi ndash x)2
$521721 $2383 $2 56765
$520050 $711 $2 5060
$507354 ndash$11984 $2 1436215
$516745 ndash$2593 $2 67235
Many computer programs and calculators automatically perform the tedious intermediate steps tocalculate the variance and standard deviation All you have to do is enter the raw data A great exam-ple is Microsoft Excel or any other spreadsheet application Master these tools to make your Six Sigmalife much much easier
13_045191 ch07qxd 81606 1136 PM Page 93
94 Part III Mastering Measuring
9 Calculate the range variance and standarddeviation for the set of process cycle timemeasurements given in Problem 5 of thischapter
Solve It
11 Calculate the range and standard deviationfor the set of home purchase prices inProblem 7 in two ways first with all eightdata points and second by excluding the$292400 point With the elimination of justone data point how are the range and stan-dard deviation measurements affected
Solve It
10 Calculate the range variance and standarddeviation for the class grade point data inProblem 6
Solve It
12 Calculate the standard deviation for the setof plastic film thickness measurements inProblem 8
Solve It
Time Warp Separating Short-Term and Long-Term Variation
Quantifying variation spread can be tricky because it changes over time Over a shortperiod of time the variation you observe is smaller than it will be over a longer periodof time One of the first tasks of the Six Sigma practitioner is to separate observed vari-ation into these two buckets mdash short-term and long-term
The basic formula for standard deviation captures all the variation occurring within aset of data So when your data set covers enough time to include the influence of allthe special-cause variables and factors itrsquos a measure of the long-term variation
nx x
σ 1LT
i
2
=-
-_ i
To extract the short-term variation from a set of data you have to find the averagerange between the sequential data points
R R nR
σ 1 1281
1whereSTi= =
-
c m So combining together you get
nR
σ1 128 1ST
i=-
^ h
Never try to calculate the short-term standard deviation on anything other than asequential set of measurements That is only perform this calculation on a set of meas-urements thatrsquos in the order that the measurements were taken You have to use thecorrect order because the calculation of the short-term standard deviation is based onthe natural ranges that occur between the characteristicsrsquo measurements If the orderof the measurements is altered at all your answer will be smaller than it should be
13_045191 ch07qxd 81606 1136 PM Page 94
95Chapter 7 Categorizing Data and Calculating Measures of Variation
Q Water level measurements of a well have been recorded for the last 40 months (recorded in orderfrom oldest measurement to newest from left to right and top to bottom)
57 44 46 13
57 56 42 53
95 100 93 117
95 83 79 93
47 43 64 50
60 39 55 55
45 55 57 76
75 74 80 91
91 89 89 101
121 112 95 100
Chart this variation over time and calculate both the long-term and the short-term standard deviations
A First create a chart of this variation over the course of the 40 months of measurements startingwith the oldest recording (month 1) and continuing to the most recent (month 40) The followingfigure shows one chart example
You can see how the variation changes over time displaying a smaller amount of variation overany given short subset of months as compared to the large variation that occurs over the full-timescale
An immediate calculation table helps in calculating both the long-term and the short-term stan-dard deviations if yoursquore doing these by hand Start by calculating the mean (x = 72) for thevalues Notice how the intermediate calculations in the columns of the following table are laid outto enable the final results
0 500
50
100
150
10 15 20
Time (month)
Wel
l Le
vel
25 30 35 40
13_045191 ch07qxd 81606 1136 PM Page 95
Month xi xi ndash x (xi ndash x)2 R x xi i i 1= - -
1 57 ndash15 225 NA
2 44 ndash28 784 13
3 46 ndash26 676 02
4 13 ndash59 3481 33
5 57 ndash15 225 44
6 56 ndash16 256 01
7 42 ndash30 900 14
8 53 ndash19 361 11
9 95 23 529 42
10 100 28 784 05
11 93 21 441 07
12 117 45 2025 24
13 95 23 529 22
14 83 11 121 12
15 79 07 049 04
16 93 21 441 14
17 47 ndash25 625 46
18 43 ndash29 841 04
19 64 ndash08 064 21
20 50 ndash22 484 14
21 60 ndash12 144 10
22 39 ndash33 1089 21
23 55 ndash17 289 16
24 55 ndash17 289 00
25 45 ndash27 729 10
26 55 ndash17 289 10
27 57 ndash15 225 02
28 76 04 016 19
29 75 03 009 01
30 74 02 004 01
31 80 08 064 06
32 91 19 361 11
33 91 19 361 00
34 89 17 289 02
35 89 17 289 00
96 Part III Mastering Measuring
13_045191 ch07qxd 81606 1136 PM Page 96
97Chapter 7 Categorizing Data and Calculating Measures of Variation
13 For a sequential set of 70 bolt torque measurements the sum of the squared error Σ(xi ndash x )2 is3750 and the sum of the interpoint ranges is 178 What are the short-term and long-term standarddeviations for this set of data
Solve It
Month xi xi ndash x (xi ndash x)2 R x xi i i 1= - -
36 101 29 841 12
37 121 49 2401 20
38 112 40 1600 09
39 95 23 529 17
40 100 28 784 05
With the intermediate calculation table in hand yoursquore now in position to calculate the short-termand long-term standard deviations
nx x
R
σ
σ
1 40 1244 43 2 50
1 128 1 1281 29 1 15
LT
i
ST
2
=-
-=
-=
= = =
_ i
As it always should be the short-term variation is less than the long-term variation This makessense because over the long-term factors and influences enter in shifting the entire variation orcausing it to drift or fluctuate more than it does over any given short period
13_045191 ch07qxd 81606 1136 PM Page 97
14 Complete the following calculation table to determine the short- and long-term standard devia-tions for a 35-point set of plate density measurements
No xi xi ndash x (xi ndash x)2 R x xi i i 1= - -
1 103
2 95
3 113
4 92
5 112
6 97
7 90
8 81
9 89
10 77
11 98
12 86
13 86
14 83
15 90
16 84
17 84
18 87
19 53
20 62
21 63
22 77
23 85
24 86
25 75
26 86
27 85
28 89
29 94
30 103
31 93
32 109
33 91
34 96
35 90
x nx i= =
nx x
σ 1LT
i
2
=-
-=
_ i
nR
σ1 128 1ST
i=-
^ h
Solve It
98 Part III Mastering Measuring
13_045191 ch07qxd 81606 1136 PM Page 98
15 Complete the following calculation table to determine the short- and long-term standard devia-tions for a 30-point set of cycle time measurements
No xi xi ndash x (xi ndash x)2 Ri = |xi ndash xindash1|
1 42
2 42
3 44
4 39
5 40
6 48
7 47
8 61
9 49
10 57
11 57
12 47
13 46
14 55
15 55
16 56
17 60
18 47
19 40
20 41
21 47
22 36
23 45
24 39
25 46
26 28
27 54
28 52
29 50
30 53
x nx i= =
nx x
σ 1LT
i
2
=-
-=
_ i
nR
σ1 128 1ST
i=-
=^ h
Solve It
99Chapter 7 Categorizing Data and Calculating Measures of Variation
13_045191 ch07qxd 81606 1136 PM Page 99
Solutions to Categorizing Data andCalculating Measures of Variation Problems
a Packages entering a sorting facility are labeled as having either ldquourgentrdquo or ldquonormalrdquo priorityWhat kind of data is this characteristic of the packages
ldquoUrgentrdquo and ldquonormalrdquo are named attributes of the packages These values canrsquot be meaning-fully added or subtracted Clearly what you have here is attribute or category data
b Yoursquore collecting weight measurements of a plated contact switch The measurements are ingrams What type of data is this
Weight measurements in grams would look something like 245 378 123 and so on Therersquosno limit to the possible weight measurements mdash they could be anything within a range of num-bers Also this data can be meaningfully added or subtracted For example the combinedweight of a 123 gram switch and a 378 gram switch would be 401 grams Clearly what youhave here is continuous or variable data
c Each final inspection record for manufactured disk drives is stamped with a date code Whattype of data are these date code stamps
Even though they may appear to be numerical these date stamps are instead categorical Forexample have you ever tried to calculate Tuesday and a half Or can you find an answer toThursday plus Friday These situations donrsquot make sense which tells you that these datastamps are attribute or category data
However you can organize this data through the sequence of the days of the week SundayMonday Tuesday and so on When you can categorize attribute data in this way itrsquos called ordi-nal data
d Student grades are given as A Andash B+ and so on Is this attribute or categorical data Whatabout grades given on a numerical grade point scale A = 400 Andash = 367 B+ = 333 and so on Is this attribute or categorical data What is the advantage of a numerical grade point scale
When you have letter grades you have attribute data You canrsquot add an A to a B And you canrsquotmathematically find an average of an A and a C grade But if you cleverly assign a numericalvalue to each letter grade such as A = 400 Andash = 367 and so on voila you immediately trans-form your data from attribute to continuous data Now you can find an average or a numericaldifference between studentsrsquo grade point averages Imagine the nerdy teacher who came upwith this idea
e Find the mode median and mean of the following collection of process cycle time measurements
8 19 9 10
21 8 10 8
11 11 13 21
The first thing to do is to rank the data from least to greatest like this
Order Value Order Value
1 8 7 11
2 8 8 11
3 8 9 13
4 9 10 19
5 10 11 21
6 10 12 21
100 Part III Mastering Measuring
13_045191 ch07qxd 81606 1136 PM Page 100
The mode is the value in the data that occurs most frequently In this case the most frequentvalue is 8 which occurs three times
The median is the value at the midpoint of the ranked list of data For a set of 12 measurementsthe sixth and seventh ordered numbers separate the bottom half from the top half For the setof data in this problem the sixth ranked point is 10 and the seventh is 11 So the median is thepoint midway between these two values which is 105
The mean of the data set is found by taking the sum of all the measurements and then dividingby the number of measurements in the set
x nx
12150 12 5i= = =
f What is the average grade point of the following class of students What is the median
333 300 367 300
367 300 333 300
033 367 333 333
333 333 367 367
300 333 300 400
Using the formula for the average you can find the mean of these student grades
x nx
2063 99 3 20i= = =
You find the median by ranking the grade data from least to greatest
Order Value Order Value
1 033 11 333
2 300 12 333
3 300 13 333
4 300 14 333
5 300 15 367
6 300 16 367
7 300 17 367
8 333 18 367
9 333 19 367
10 333 20 400
For this 20-student data set the median falls between the 10th and 11th ordered points So themedian is 333
g For the following home purchase price data points does the mean or the median better com-municate the location of the central tendency of the variation Why
$184000 $168000 $174500 $177200
$292400 $181000 $172010 $169900
The mean for this home price data is $189876 and the median is $175850 One way to see thedifference between the mean and the median is to plot them graphically along with a dot plot ofthe raw data like in the following figure
101Chapter 7 Categorizing Data and Calculating Measures of Variation
13_045191 ch07qxd 81606 1136 PM Page 101
You can see that the median value lands within the main grouping of the data The mean how-ever is pulled outside the main cluster of the data by the one outlier point at $292400
h What is the average value (or mean) for the following set of plastic film thickness measurements
0044 0041 0046 0049
0049 0050 0038 0044
0048 0042 0043 0040
0039 0042 0043 0039
To calculate the mean or average you take the sum of all the data values and divide by thenumber of points in your data set
x nx
160 697 0 044i= = =
i Calculate the range variance and standard deviation for the set of process cycle time measure-ments given in Problem 5
The range is the maximum minus the minimum values
R = xMAX ndash xMIN = 21 ndash 8 = 13
After first calculating that x equals 124 the calculation table shown here contains the interme-diate calculations required to calculate the variance
xi xi ndash x (xi ndash x)2 xi xi ndash x (xi ndash x)2
8 ndash4 16 10 ndash3 9
19 7 49 8 ndash4 16
9 ndash3 9 11 ndash1 1
10 ndash2 4 11 ndash1 1
21 9 81 13 0 0
8 ndash5 25 21 8 64
With this table in hand you calculate the variance with the following formula
nx x
σ 1 16 1263 23 9
i2
2
=-
-=
-=
_ i
Median = $175850
Home Price
180000 198000 216000 234000 252000 270000 288000
Mean = $189876
102 Part III Mastering Measuring
13_045191 ch07qxd 81606 1136 PM Page 102
With the variance calculated you can now simply take the square root to get the standard deviation
σ σ 23 9 4 92= = =
j Calculate the range variance and standard deviation for the class grade point data in Problem 6
The range is the maximum minus the minimum values in your set of data
R = xMAX ndash xMIN = 400 ndash 033 = 367
Calculating the variance is easier if you use a calculation table like the one shown here
xi xi ndash x (xi ndash x)2 xi xi ndash x (xi ndash x)2
333 013 0017 333 013 0017
300 ndash020 0040 333 013 0017
367 047 0221 333 013 0017
300 ndash020 0040 333 013 0017
367 047 0221 367 047 0221
300 ndash020 0040 367 047 0221
333 013 0017 300 ndash020 0040
300 ndash020 0040 333 013 0017
033 ndash287 8824 300 ndash020 0040
367 047 0221 400 080 0641
After creating a calculation table use the intermediate values in the table to calculate the vari-ance and standard deviation
nx x
σ
σ σ
1 20 110 34 0 54
0 54 0 74
andi2
2
2
=-
-=
-=
= = =
_ i
k Calculate the range and standard deviation for the set of home purchase prices in Problem 7 intwo ways first with all eight data points and second by excluding the $292400 point With theelimination of just this one data point how are the range and standard deviation measurementsaffected
With all the data points the range for the home price data is
R = xMAX ndash xMIN = $292400 ndash $168000 = $124400
However if you remove the $292400 outlying point the range becomes
R = xMAX ndash xMIN = $184000 ndash $168000 = $16000
With all the data points the calculation table for the home price data looks like this
xi xi ndash x (xi ndash x)2
$184000 ndash$5876 $2 34530314
$168000 ndash$21876 $2 478570314
$174500 ndash$15376 $2 236429064
$177200 ndash$12676 $2 160687314
$292400 $102524 $2 10511119314
$181000 ndash$8876 $2 78787814
$172010 ndash$17866 $2 319202889
$169900 ndash$19976 $2 399050564
103Chapter 7 Categorizing Data and Calculating Measures of Variation
13_045191 ch07qxd 81606 1136 PM Page 103
Using this intermediate calculation table the standard deviation for eight data points can becalculated as
$ $ nx x
σ 1 8 112 218 377 588 41 779
i
22
=-
-=
-=
_ i
By simply eliminating the $292400 row from the intermediate calculation table the calculationfor the standard deviation changes to
$ $ nx x
σ 1 7 11 707 258 273 5 855
i
22
=-
-=
-=
_ i
You can see from the two different range and standard deviation calculations that both meas-ures of variation spread are sensitive to outliers
l Calculate the standard deviation for the set of plastic film thickness measurements in Problem 8
After calculating the mean (x = 0044) an intermediate calculation table is always a good nextstep for calculating the standard deviation
xi xi ndash x (xi ndash x)2 xi xi ndash x (xi ndash x)2
0044 0001 00000003 0048 0005 00000222
0041 ndash0002 00000058 0042 ndash0002 00000033
0046 0002 00000047 0043 ndash0001 00000004
0049 0005 00000277 0040 ndash0004 00000133
0049 0005 00000253 0039 ndash0004 00000169
0050 0007 00000440 0042 ndash0002 00000023
0038 ndash0005 00000262 0043 ndash0001 00000008
0044 0001 00000005 0039 ndash0005 00000242
You can then use these intermediate values to calculate the standard deviation of the data set
nx x
σ 1 16 10 0002178 0 0038
i
2
=-
-=
-=
_ i
m For a sequential set of 70 bolt torque measurements the sum of the squared error Σ(xi ndash x )2 is3750 and the sum of the interpoint ranges is 178 What are the short-term and long-term stan-dard deviations for this set of data
The short-term standard deviation is calculated from the following equation
nR
σ1 128 1ST
i=-
^ h
By knowing the sum of the interpoint ranges and the number measurements you can plugdirectly into this equation to solve for the short-term standard deviation
σ
1 128 70 1178 2 3ST =
-=
^ h
The long-term standard deviation is calculated from the equation
nx x
σ 1LT
i=
-
-_ i
By knowing the sum of the squared error and the number of measurements you can plugdirectly into this equation to solve for the long-term standard deviation
σ 70 13 750 7 4LT =
-=
104 Part III Mastering Measuring
13_045191 ch07qxd 81606 1136 PM Page 104
n Complete the following calculation table to determine the short- and long-term standard devia-tions for a 35-point set of plate density measurements
No xi xi ndash x (xi ndash x)2 R x xi i i 1= - -
1 103 149 2224 NA
2 95 69 478 8
3 113 249 6207 18
4 92 29 85 22
5 112 239 5719 21
6 97 89 795 15
7 90 19 37 7
8 81 ndash71 502 9
9 89 09 08 8
10 77 ndash111 1229 12
11 98 99 983 21
12 86 ndash21 44 12
13 86 ndash21 44 0
14 83 ndash51 259 3
15 90 19 37 7
16 84 ndash41 167 6
17 84 ndash41 167 0
18 87 ndash11 12 3
19 53 ndash351 12310 34
20 62 ndash261 6805 9
21 63 ndash251 6293 1
22 77 ndash111 1229 14
23 85 ndash31 95 8
24 86 ndash21 44 1
25 75 ndash131 1712 11
26 86 ndash21 44 11
27 85 ndash31 95 1
28 89 09 08 4
29 94 59 350 5
30 103 149 2224 9
31 93 49 242 10
32 109 209 4374 16
33 91 29 85 18
34 96 79 626 5
35 90 19 37 6
x nx
88 1i= =
nx x
σ 1 12 8LT
i
2
=-
-=
_ i
nR
σ1 128 1
8 7STi=-
=^ h
88 128 87
105Chapter 7 Categorizing Data and Calculating Measures of Variation
13_045191 ch07qxd 81606 1136 PM Page 105
o Complete the following calculation table to determine the short- and long-term standard devia-tions for a 30-point set of cycle time measurements
No xi xi ndash x (xi ndash x)2 R x xi i i 1= - -
1 42 ndash54 295 NA
2 42 ndash54 295 0
3 44 ndash34 118 2
4 39 ndash84 711 5
5 40 ndash74 553 1
6 48 06 03 8
7 47 ndash04 02 1
8 61 136 1841 14
9 49 16 25 12
10 57 96 915 8
11 57 96 915 0
12 47 ndash04 02 10
13 46 ndash14 21 1
14 55 76 573 9
15 55 76 573 0
16 56 86 734 1
17 60 126 1579 4
18 47 ndash04 02 13
19 40 ndash74 553 7
20 41 ndash64 414 1
21 47 ndash04 02 6
22 36 ndash114 1307 11
23 45 ndash24 59 9
24 39 ndash84 711 6
25 46 ndash14 21 7
26 28 ndash194 3777 18
27 54 66 431 26
28 52 46 209 2
29 50 26 66 2
30 53 56 310 3
x nx
47 4i= =
nx x
σ 1 7 7LT
i
2
=-
-=
_ i
nR
σ1 128 1
5 7STi=-
=^ h
106 Part III Mastering Measuring
13_045191 ch07qxd 81606 1136 PM Page 106
Chapter 8
A Picturersquos Worth 1000 WordsMeasuring with Charts and Graphs
In This Chapter Determining the shape of variation with dot plots and histograms
Comparing variation distributions with box and whisker plots
Setting up scatter plots to explore relationships between characteristics
Creating process behavior and time series charts to see how a characteristic changes over time
Nothing communicates an idea better than a clear picture And besides just lookingimpressive a good chart quickly reveals the important story within your data and
helps your team see previously hidden relationships between key inputs and outputs Youand your team will be able to compare and contrast what you think is happening with whatyou actually see in the charts and graphs You should master creating the basic charts inthis chapter and should religiously use them as the very first battery of tests you performon all your data Yoursquoll be amazed at the rapid insight you gain
Putting Dot Plots or Histograms to UseThe purpose of dot plots and histograms is to graphically show you where variation occurswithin a critical characteristic mdash whether itrsquos all lumped together within a narrow interval orevenly spread out over a wide range of values A dot plot or histogram will immediately tellyou how frequently various values occur in your data and will provide clues to the sources ofthe variation
You create a dot plot or histogram by following the same simple process for both Followthese steps
1 Create a horizontal line representing the scale of measure for the characteristicyoursquore charting
2 Divide the length of the horizontal scale into ten to twenty equal ldquobucketsrdquo betweenthe smallest and largest observed values
3 For each measurement in your data set place a dot in the corresponding bucketalong the horizontal axis
Stack up successive dots that occur within the same bucket To create a histogramsimply replace the stacked dots with a bar of the same height
4 Repeat Step 3 until yoursquove placed a dot on the chart for each measurement
14_045191 ch08qxd 81606 1141 PM Page 107
Interpret your dot plot or histogram by looking for the following basic patterns
What shape does the data form Do the dots (or bars) on the graph form a singlebell-shaped curve or hump Or is there a uniform distribution across a range ofvalues Are the dots clumped together with one side trailing out more than theother
Where is the mode or tallest peak of the distribution located Is there one peakor are there multiple peaks
At what point along the horizontal axis of your dot plot or histogram do thestacked dots or bars seem to balance out like a teeter-totter This point is theapproximate location of the variation mean or average
What is the range The distance along the horizontal axis between the largest dot(xMAX) and the smallest dot (xMIN) is the range (R = xMAX ndash xMIN)
Are there outliers (specific points that donrsquot seem to fit the grouping of the rest)in your data Are they either too far to the right or too far to the left of the rest ofthe data to be concluded as coming from the same set of circumstances that cre-ated all the other points
108 Part III Mastering Measuring
Q Create a dot plot chart for the following set of cycle time measurements and describe the natureof the variation
84 64 95 116
114 125 46 85
112 68 76 56
53 70 75 48
79 82 93 83
83 83 117 88
A Start by creating a horizontal line representing the cycle time scale of measure On the left endplace a tick mark for the smallest value in the data set On the right end place a tick mark for thelargest value Your line should look like this
Now divide the horizontal scale between the max and min into equal-sized buckets For thisexample try ten divisions So take the distance between the max and the min and divide it by tento find the bucket width
x x
1012 5 4 6 0 79bucket width number buckets
MAX MIN=-
= - =
Using this bucket width value calculate and draw in the remaining tick marks along the horizon-tal axis to create the ten dot plot buckets
Now for the fun part Place a dot for each data point in the corresponding bucket For examplethe 114 cycle time measurement lands between 1092 and 1171 So place a dot for this data pointin the 1092 ndash 1171 bucket like this
Cycle Time
460
539
618
697
776
855
934
101
3
109
2
117
1
125
0
Cycle Time
460
125
0
14_045191 ch08qxd 81606 1141 PM Page 108
What does this dot plot tell you about the nature of the variation Well it actually reveals quite alot Note the following
bull The shape of the variation distribution appears somewhat bell-shaped Most cycle time meas-urements are clustered around the middle of the distribution with the frequency of occurrencetrailing off as you move farther out from the middle (but as you can see there are some extrapoints near the min and max)
bull The mode (the tallest peak in the dot plot) and the mean (the pivot point on the horizontal axiswhere all the dots balance) both seem to be at about the same spot mdash in the 776 ndash 855 bucketmaybe at a value of about 82 Check this rough visual estimate with an actual calculation of themean (see Chapter 7 to find out how to calculate the mean) How close is the visual estimate
bull The spread of the variation is easily seen in the dot plot The cycle time measurements rangefrom 125 to 46
bull There arenrsquot any outlying data points that are far out on either side of the rest of the variation
109Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
Cycle Time
460
539
618
697
776
855
934
101
3
109
2
117
1
125
0
Cycle Time
460
539
618
697
776
855
934
101
3
109
2
117
1
125
0
1 Create a 10-bucket dot plot or histogramwith the following purchase order data
$111543 $ 87209 $130096 $156273
$155019 $161407 $ 79754 $113742
$153171 $ 90488 $ 99695 $ 82433
$ 81301 $ 92697 $ 97496 $ 79978
$103156 $108177 $126390 $109413
$110664 $109260 $157241 $118811
Solve It
2 Use your dot plot or histogram fromProblem 1 of this chapter to describe thevariation of the purchase order dataWhatrsquos the shape of the variation distribu-tion Wherersquos the variation centered Howspread out is the variation Are there anyoutliers in the data
Solve It
If a datapoint falls right on a bucket division place it in either one of the buckets As long as youdonrsquot put it in both yoursquore okay The min and max data points will always fall into the lowest andhighest buckets respectively Herersquos what the completed dot plot for this example looks like
14_045191 ch08qxd 81606 1141 PM Page 109
110 Part III Mastering Measuring
3 Draw a 12-bucket dot plot or histogram for the following processing time measurements Keep some scrap paperhandy
71 56 77 59
63 64 60 78
77 75 68 56
57 78 70 53
62 77 92 113
66 63 79 61
52 62 63 66
Solve It
4 Use your dot plot or histogram fromProblem 3 of this chapter to describe the variation of the process time measure-ments Whatrsquos the shape of the variationdistribution Wherersquos the variation centered How spread out is the variationAre there any outliers in the data
Solve It
Setting Up Box and Whisker PlotsSometimes you need to quickly compare two or more variation distributions of thesame characteristic Itrsquos like standing two people back-to-back to see whorsquos taller Butin this case yoursquore asking questions such as ldquoWhich distribution is more spread outrdquoand ldquoWhich has the higher central locationrdquo This is when box and whisker plots comein handy
These plots which are usually called box plots can be created by following these steps
1 Rank order each of the sets of data from smallest value to largest value thatyoursquoll be including in your box plot
2 Separately for each set of data divide your rank-ordered data into fourths orquartiles as statisticians like to say
bull From xMIN the smallest observed point to a point called Q1 is the lowerquartile of the points in your data set
bull From Q1 to the median of your data set is the second quartile of the points(see Chapter 7 to find out how to calculate the median of a data set)
bull From the median to a point called Q3 is the third quartile
bull From Q3 to xMAX the largest point in your data set is the fourth quartile
3 Draw a horizontal or vertical axis to place the box plot on
Create a numerical scale along this axis that spans from just below the smallestvalue of all the data sets to just above the largest value
4 For the first data set locate along the axis the points for xMIN Q1 the medianQ3 and xMAX
14_045191 ch08qxd 81606 1141 PM Page 110
5 Draw a box between the Q1 and Q3 points
6 Place a heavy line through the box at the point of the median
7 Draw whisker-like lines extending from the Q1 and Q3 points to xMIN and xMAXrespectively
8 Repeat these same steps for each successive subgroup distribution yoursquoreincluding for comparison with your box plot
On the same axis draw each successive distribution next to the previous oneYou now have a graphical picture for comparing the variation distributions toeach other
111Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
Q Draw a box plot for the following two sets of production volume measurements for two manufac-turing lines
Line A Line B
116788 105804 105355 110463 107819 86065 100173 78915
118669 96947 112454 114794 84479 84400 89366 96238
103414 110412 114032 109262 93802 96099 83925 121554
100161 94586 114142 92078 79970 91956 107792 104966
107654 118804 107049 100351 118956 129140 82318 87221
99974 99949 122249 108432
After you draw the plot describe the similarities and differences between each linersquos performance
A The first step in creating a box plot is to rank order each set of data Here are the ordered resultsfor each production line
Order Line A Line B Order Line A Line B
1 92078 78915 13 108432 96238
2 94586 79970 14 109262 100173
3 96947 82318 15 110412 104966
4 99949 83925 16 110463 107792
5 99974 84400 17 112454 107819
6 100161 84479 18 114032 118956
7 100351 86065 19 114142 121554
8 103414 87221 20 114794 129140
9 105355 89366 21 116788
10 105804 91956 22 118669
11 107049 93802 23 118804
12 107654 96099 24 122249
With the data rank ordered you can easily pick off the xMIN and xMAX points for each set of measure-ments For Line A xMIN = 92078 and xMAX =122249 For Line B xMIN = 78915 and xMAX = 129140
Finding the median for each data set is also easy (Refer back to Chapter 7 if you need a quickrefresher) The Line A median is 108043 and the Line B median is 92879
14_045191 ch08qxd 81606 1141 PM Page 111
112 Part III Mastering Measuring
To find Q1 and Q3 just look for the data points that correspond to the first and third quarters ofthe data For Line A with 24 data points you divide by four so each quarter consists of 6 pointsThat means that Q1 will be the average of the sixth and seventh points in the rank ordered data(100256) and Q3 will be the average of the 18th and 19th points (114087) For Line A that leadsto Q1 being 100256 and Q3 being 114087 For Line B with 20 data points Q1 is at the fifth rank-ordered point (84400) and Q3 is at the 15th (104966)
With the calculation of all these points prepared your next step is to create an axis to put yourbox plots on You can make your axis either up and down (vertical) or left to right (horizontal) Itreally doesnrsquot matter This axis represents the scale of measure of your data So if your scale ofmeasure feels natural to be seen left to right rather than up and down make your axis horizontalTime is a good example of a variable that is usually plotted left to right Temperature is one that isnormally vertical For the production volume data of this example a vertical axis will work fineDraw a vertical line for the axis create tick marks along its length to indicate the scale and plotpoints for each of the values mdash xMIN Q1 median Q3 and xMAX
For the box plot for Line A all you need to do now is draw the box between the Q1 and Q3 pointsThen draw thin whiskers from the box edges out to xMIN and xMAX You complete the box plot forLine A by putting the line through the box at the point of its median like this
Line A
Pro
du
ctio
n V
olu
me
130000
120000
110000
100000
90000
80000
70000
Line A
Pro
du
ctio
n V
olu
me
130000
120000
110000
times xMAX = 122249
times Q3 = 114087
times median = 108043
times Q1 = 100256
times xMIN = 92078100000
90000
80000
70000
14_045191 ch08qxd 81606 1141 PM Page 112
113Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
5 Create a box plot for the following set ofmonthly sales figures Make sure you havescrap paper handy
$13245 $38267 $21576 $16151
$60231 $15705 $32689 $13168
$30221 $24036 $22154 $22409
Solve It
6 Describe the variation of the monthly salesdata from Problem 5 of this chapter Nowsuppose the monthly sales data is from theprevious twelve months What amount ofsales would you expect for the next monthfollowing
Solve It
Now follow the same steps to add a box plot for the Line B data to the same plot It ends up look-ing like this
From the completed side-by-side box plots you can easily see the similarities and differencesbetween the performances of the two manufacturing lines The variation for Line B is much morespread out than Line A Line B also has its central location lower than Line A and Line Brsquos dataisnrsquot symmetrical mdash itrsquos skewed to the lower production volume values If you wanted to selectthe production line that consistently produces higher volumes yoursquod definitely pick Line A eventhough Line B had the highest single measurement
Line A Line B
Pro
du
ctio
n V
olu
me
130000
120000
110000
100000
90000
80000
70000
14_045191 ch08qxd 81606 1141 PM Page 113
114 Part III Mastering Measuring
7 Three products mdash A B and C mdash are beingcompared for warranty returns Using thedata below which shows the monthlynumber of warranty returns create boxplots for all three products
Product A
20 17 18 18
27 22 22 20
Product B
32 31 31 24
27 32 34 32
32 33 30 30
28 39 31 32
Product C
20 29 16 16
14 24 20 22
16 20 29 24
Solve It
8 Using the box plots from Problem 7 of thischapter describe the differences and simi-larities between the warranty performanceof the three products Keep scrap paper onhand
Solve It
Seeing Spots Using Scatter PlotsScatter plots help you explore the relationship between two characteristics As thevalues for Characteristic A increase for example what happens to values forCharacteristic B Do they also increase Or do they decrease Or neither
To create a scatter plot you must have pairs of measurements for each observation mdashmeasurements for each characteristic taken at the same point in time under the sameconditions Otherwise therersquos no basis for exploring the relationship between the twocharacteristics or variables
You can create a scatter plot by going through the following steps
1 Form data pairs that represent x-y points from the collected data
For each observation pair the simultaneously measured values for the two char-acteristics together to form an x-y point that can be plotted on a two-axis x-ygraph
2 Create a two-axis plotting framework
To create your framework draw two axes one horizontal and the other verti-cal Draw one axis for each of the two characteristics yoursquore exploring
The scale for each axis should be in the units that correspond to its assignedcharacteristic mdash millimeters for length pounds for weight minutes for timenumber of defects found on an inspected part or whatever the units may bethat quantify the characteristics yoursquore exploring
3 Plot each x-y pair as a point on the two-axis framework
14_045191 ch08qxd 81606 1141 PM Page 114
Scatter plots can be created when one of the two characteristics is not measured on acontinuous scale but instead consists of attribute data (see Chapter 7) For examplethe characteristic of ldquoproduction volumerdquo (measured on a continuous units-per-hourscale) can be plotted against the characteristic of ldquoproduction line IDrdquo (perhaps namedLine A and Line B)
You interpret your scatter plot by looking for relationships or patterns between thetwo plotted characteristics Look for
Graphical correlation between the two plotted characteristics or variables Iftherersquos no pattern in the plotted points mdash just a random scattering of points mdashthen therersquos no correlation or relationship between the characteristicsClustering of the plotted points or patterns that follow the shape of a line or acurve reveal correlation between the two variables
Direction of correlation between the variables When there are patterns doesan increase in one variable lead to an increase in the other Or a decrease in onelead to a decrease in the other If so there is a positive relationship between thevariables But if a change in one variable leads to a change in the opposite direc-tion for the other there is a negative relationship between the variables
Strength of effect between the variables How drastic or steep is the linear rela-tionship between the variables If a small change in one variable leads to a largechange in the other yoursquore observing a strong effect between them
115Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
Q The company yoursquore working for has had problems with its compressed air supply mdash sometimesthe compressor equipment canrsquot supply the demanded production flowrate Yoursquove found somehistorical data that includes measurements of the compressed air flowrate and another variablethe speed setting of the production line that you think may factor into the compressed air per-formance The measurements for each of these variables were taken at the same time once eachweek Herersquos a listing of the data yoursquove found Is there any relationship between productionspeed setting and compressed air pressure
Week Production Compressed Air Week Production Compressed AirSpeed Setting Pressure (psi) Speed Setting Pressure (psi)
1 226 85 15 221 107
2 215 111 16 205 119
3 233 79 17 223 103
4 245 73 18 225 91
5 244 79 19 232 90
6 251 68 20 226 86
7 204 113 21 226 98
8 227 90 22 226 104
9 243 85 23 246 81
10 217 95 24 229 94
11 222 110 25 228 96
12 210 110 26 224 91
13 208 106 27 254 62
14 218 96
14_045191 ch08qxd 81606 1141 PM Page 115
116 Part III Mastering Measuring
A What do you think Is it clear from the listed data whether a relationship exists between the twovariables Creating a scatter plot will help you answer this question
The first step is to recognize that each weekrsquos measurements form an x-y data pair Thatrsquos becauseeach measurement is linked in time For this example assign the production speed settings to thehorizontal axis and the compressed air pressure to the vertical axis
Begin your scatter plot by creating the horizontal and vertical axis framework and by plottingeach weekrsquos data pair like this for Week 1
Notice how the plotted point lines up with the 226 value on the horizontal axis and the 85 psivalue on the vertical axis Finish creating the scatter plot by repeating this process for all theweeksrsquo data pairs
Is there a pattern here Yes The plotted points follow a trend that slopes downward as you movefrom left to right This means yoursquove discovered a relationship between the production speed set-ting and the compressed air pressure variable As the production speed setting increases thecompressed air pressure goes down And as you decrease production speed air pressure goesup This is a negative relationship
Production Speed Setting
Com
pre
ssed
Air
Pre
ssu
re (
p)
200 210 220 230
130
110
90
70
50240 250 260
Production Speed Setting
Com
pre
ssed
Air
Pre
ssu
re (
p)
200 210 220 230
226 85
130
110
90
70
50240 250 260
14_045191 ch08qxd 81606 1141 PM Page 116
117Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
9 A company performed an experiment where glue samples were allowed to cure for different lengthsof time They then measured the force required to peel away the glue bond Create a scatter plot byusing the following data Make sure you have plenty of scrap paper
Cure Time Peel Force Cure Time Peel ForceSample (Days) (Lbs) Sample (Days) (Lbs)
1 524 80 6 516 74
2 493 72 7 542 83
3 491 70 8 532 77
4 452 67 9 539 78
5 463 66 10 479 70
Solve It
The strength of the relationship is determined by the slope of the general line formed by the plot-ted data The figure below draws in a line that roughly fits the pattern in the plotted points
For every 2 units of increase in production speed setting is a 20 psi decrease in the air pressureAnother way of saying this is that there is a 10 psi change in air pressure for every 10 change inthe production speed setting Knowing this fact about the relationship helps you forecast whatthe pressure will be for any setting of production speed and vice versa
Production Speed Setting
Com
pre
ssed
Air
Pre
ssu
re (
p)
200 210 220 230
20
20
130
110
90
70
50240 250 260
10 Describe the relationship if any between the cure time and the peel force variables from the scat-ter plot of Problem 9 of this chapter
Solve It
14_045191 ch08qxd 81606 1141 PM Page 117
12 Describe the relationship if any between Variable D and Variable C shown in the following scatterplot
Solve It
176
1900 2000 2100
Variable D
Vari
able
C
2200170
172
174
118 Part III Mastering Measuring
11 Describe the relationship if any between the Input X3 variable and the Output variable shown inthe following scatter plot
Solve It
160
132
90 130 170
Input X3
Ou
tpu
t
210
104
76
48
20
14_045191 ch08qxd 81606 1141 PM Page 118
119Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
Hindsight Is 2020 Using Process Behavior or Time Series Charts
Dot plots histograms box plots and scatter plots all neglect an important variable mdashtime To see how a characteristic changes over time you have to plot its measure-ments in the sequence or series in which they occurred When variation behavior islinked to time yoursquore then able to link process behavior and changes back to historicalevents or conditions mdash such as the changing of a work shift the gradual dulling of acutting tool or the steady jitter of the status quo
A time series or process behavior chart is made by simply plotting measured perform-ance on a vertical scale against a horizontal scale representing the progress of timeWhen a process is free of special causes or outside influences its behavior or meas-ured performance bounces randomly around a central flat level on the behaviorchart Most of the observed variation will be clustered close to this central level Andevery now and then some measurements will land farther away from the central levelThis variation will be completely random over time without patterns or trends
You interpret your behavior chart or time series plot by looking for signs of variationthat are out of the ordinary After yoursquove identified non-normal behavior you investi-gate what historical event (or special cause) could have caused such a change in performance
Drawing process behavior charts or time series charts isnrsquot difficult The part that istrickier however is interpreting these charts For that reason the example and prac-tice problems for this section will concentrate on interpretation rather than on themechanics of creating the charts
Q A plastic injection molded housing has a critical dimension between two bosses A series of meas-urements of the dimension was taken at five-minute intervals Theyrsquore plotted over time in the fol-lowing time series chart
Review this time series chart and indicate if there is any evidence of special causes affecting theperformance of the injection mold
235
245
17
168
174
176
178
172
255
305
315
325
335
345
355
405
415
425
435
445
455
505
515
525
Time
Len
gth
(m
m)
14_045191 ch08qxd 81606 1141 PM Page 119
14 Identify any non-normal behavior in the following time series chart of delivery time data
Solve It
150
Month
Del
iver
y
May
-03
132
134
136
138
140
142
144
146
148
Jun-
03Ju
l-03
Aug
-03
Sep
-03
Oct
-03
Nov
-03
Dec
-03
May
-04
Jun-
04Ju
l-04
Aug
-04
Sep
-04
Oct
-04
Nov
-04
Dec
-04
Jan-
04Fe
b-0
4M
ar-0
4A
pr-
04
May
-05
Jun-
05Ju
l-05
Aug
-05
Sep
-05
Jan-
05Fe
b-0
5M
ar-0
5A
pr-
05
120 Part III Mastering Measuring
13 A CEO is concerned about his companyrsquos recent monthly expenses After some temporary successin reducing expenses two months ago itrsquos now the second month in a row since then that expenseshave increased And the expenses in this latest month are 6 percent higher than the same month ayear ago The CEOrsquos plan is to enact a spending freeze on all departments to fix the problem If thatdoesnrsquot work he says ldquoHeads are going to rollrdquo To check the validity of the CEOrsquos plan you createa time history chart of the companyrsquos expenses over the previous 21 months as shown here
Based on the time series data do you agree or disagree with the CEOrsquos plan Why
Solve It
-21
$180000
Month
Exp
ense
s $160000
$140000
$120000
$100000-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
A Two things stand out from the normal variation of the boss dimension First at time 255 a singlepoint is observed much higher than the normal level of the rest of the variation mdash this is an out-lier Later starting at about 400 you can see an upward trend After the trend from about 430and on the central level of the dimension variation seems to stay at this higher level These out-of-the-ordinary observations mdash the outlier and the trend mdash indicate that something abnormal hasaffected the injection mold process
14_045191 ch08qxd 81606 1141 PM Page 120
121Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
Solutions to Exercises for Yields and Defects Problems
a Create a 10-bucket dot plot or histogram with the purchase order data shown in the originalproblem earlier in the chapter
Start by creating a horizontal line representing the dollar scale of the purchase order data Asshown in the following figure on the left end of the line place a tick mark for the smallest valuein the data set and on the right end place a tick mark for the largest value
Next divide the horizontal scale between the max and min into 10 equal buckets To find thewidth of the buckets divide the distance between the max and the min by ten as shown below
$ $ x x
101 614 07 797 54 81 65bucket width number buckets
MAX MIN=-
=-
=
Using this bucket width value calculate and draw in the remaining tick marks along the hori-zontal axis to create the 10 dot plot buckets
Now place a dot for each data point in its corresponding bucket Remember if a data point fallsright on a bucket division place it in the higher of the buckets Herersquos what the completed dotplot looks like
b Use your dot plot or histogram from Problem 1 of this chapter to describe the variation of thepurchase order data Whatrsquos the shape of the variation distribution Wherersquos the variation cen-tered How spread out is the variation Are there any outliers in the data
This dot plot tells you about the nature of the variation in the purchase order measurements inthe following ways
The shape of the variation distribution isnrsquot bell-shaped Most cycle time measurementsare clustered on the left or lower end of the histogram with a secondary grouping occur-ring at the far right or higher end This behavior may be evidence of a bi-modal distribu-tion (two separate distributions mixed on one chart) which may have occurred due tohigher spending in the last two months of the fiscal year
$879
19
$960
84
$10
425
0
$11
241
5
$12
058
1
$12
874
6
$13
691
1
$14
507
7
$15
324
2
Purchase Order Amount ($)
$797
54
$16
140
7
$879
19
$960
84
$10
425
0
$11
241
5
$12
058
1
$12
874
6
$13
691
1
$14
507
7
$15
324
2
Purchase Order Amount ($)
$797
54
$13
140
7
Purchase Order Amount ($)$7
975
4
$13
140
7
14_045191 ch08qxd 81606 1141 PM Page 121
122 Part III Mastering Measuring
The mode (the tallest peak in the dot plot) is located at the far left end of the dot plot inthe $79754 ndash $87919 data bucket and the mean (the pivot point on the horizontal axiswhere all the dots balance) seems to be around the value of about $112500 This largedifference between the mode and the mean tells you that the data is clearly not normalbut is skewed
The spread of the variation is easily seen as ranging from $161407 to $79754 or a valueof $81653
No outlying data points appear away from the rest of the variation distribution otherthan the possible bi-modal indication
c Draw a 12-bucket dot plot or histogram for the processing time measurements shown in theoriginal problem earlier in the chapter
Start by creating a horizontal line representing the minute scale of the processing time dataOn the left end place a tick mark for the smallest value in the data set On the right end placea tick mark for the largest value
Next divide the horizontal scale between the max and min into 12 equal buckets To find thewidth of the buckets divide the distance between the max and the min by 12 as shown here
x x
1211 30 5 20 0 508bucket width number buckets
MAX MIN=-
= - =
Using this bucket width value calculate and draw in the remaining tick marks along the hori-zontal axis to create the 12 dot plot buckets
Now place a dot for each data point in its corresponding bucket Herersquos what the completed dotplot looks like
d Use your dot plot or histogram from Problem 3 of this chapter to describe the variation of theprocessing time measurements Whatrsquos the shape of the variation distribution Wherersquos thevariation centered How spread out is the variation Are there any outliers in the data
571
622
673
723
774
825
876
927
978
Processing Time (minutes)
520
102
8
107
9
113
0
571
622
673
723
774
825
876
927
978
Processing Time (minutes)
520
102
8
107
9
113
0
Processing Time (minutes)
520
113
0
14_045191 ch08qxd 81606 1141 PM Page 122
The dot plot of Problem 3 tells you about the nature of the variation in the processing timemeasurements in the following ways
With the exception of the two isolated points on the right the shape of the variation dis-tribution is somewhat bell-shaped with most processing time measurements clusteredtogether in a normal shape
The mode is located near the left end of the dot plot in the 622 ndash 673 minute bucketBecause of the two extreme points on the right the mean seems to be a bit higher thanthe mode at around the 673 ndash 723 minute bucket This relatively small differencebetween the mode and the mean tells you that the data is nearly normal
The spread of the variation ranges from 1130 minutes to 520 minutes or a value of 610minutes
The two extreme data points on the right are outliers Ask yourself what caused thesetwo measurements to be so different from all the rest
e Create a box plot for the set of monthly sales figures shown in the original problem earlier inthe chapter
The first step is to rank order the data as shown here
Order Monthly Sales
1 $13168
2 $13245
3 $15705
4 $16151
5 $21576
6 $22154
7 $22409
8 $24036
9 $30221
10 $32689
11 $38267
12 $60231
With the data rank ordered you can easily pick off the xMIN and xMAX points For this monthlysales data xMIN = $13168 and xMAX = $60231
Finding the median is also easy Because you have an even number of measurements (12) themedian is the average value of the middle two points mdash $22281
To find Q1 and Q3 you have to calculate the average between the points that divide the four-point quarters For Q1 you calculate the average of the third and fourth ranked points For Q2you calculate the average of the ninth and tenth ranked points That leads to Q1 being $15928and Q3 being $31455
With the calculation of all these points prepared your next step is to create an axis to put yourbox plot on For the monthly sales dollar values you usually settle on a vertical axis Draw avertical line for the axis create tick marks along its length to indicate the scale and plot pointsfor each of the values mdash xMIN Q1 median Q3 and xMAX
123Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
14_045191 ch08qxd 81606 1141 PM Page 123
All you need to do now is draw the box between the Q1 and Q3 points and then draw thinwhiskers from the box edges out to xMIN and xMAX You complete the box plot by putting the linethrough the box at the point of its median like this
f Describe the variation of the monthly sales data from Problem 5 of this chapter Now supposethe monthly sales data is from the previous twelve months What amount of sales would youexpect for the next month following
The drawn box plot shows that the monthly sales variation is mostly near the lower end witha few monthsrsquo instances rising up to high values The median is toward the lower end of theobserved range of data All things staying the same yoursquod expect future monthly sales figuresto fit this same shape So taking a guess at what the next monthrsquos value will be yoursquod have tosay that it would be somewhere between about $16000 and $30000
g Three products mdash A B and C mdash are being compared for warranty returns Using the data givenin the original problem earlier in the chapter which shows the monthly number of warrantyreturns create box plots for all three products
Mon
thly
Sal
es (
$)
$60000
$50000
$40000
$30000
$20000
$10000
Mon
thly
Sal
es (
$)
$60000
$50000
$40000
times xMAX = $60231
times Q3 = $31455
times median = $22281
times Q1 = $15928
times xMIN = $13168
$30000
$20000
$10000
124 Part III Mastering Measuring
14_045191 ch08qxd 81606 1141 PM Page 124
125Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
The first step is to rank order all the warranty data
Order Product A Product B Product C
1 17 24 14
2 19 27 16
3 19 28 16
4 20 30 16
5 20 30 20
6 22 31 20
7 22 31 20
8 27 31 22
9 32 24
10 32 24
11 32 29
12 32 29
13 32
14 33
15 34
16 39
After reviewing each ordered set of data you can now find the critical values for creating boxplots Their values are shown in the table below
Product xMIN Q1 Median Q3 xMAX
A 17 18 20 22 27
B 24 30 315 32 39
C 14 16 20 24 29
Now use these values to construct the box plot for each product as shown below
War
ran
ty R
etu
rns
Product A Product B Product C
40
35
30
25
20
15
14_045191 ch08qxd 81606 1141 PM Page 125
h Using the box plots from Problem 7 of this chapter describe the differences and similaritiesbetween the warranty performance of the three products
Product A and Product C have nearly the same medians but their distributions are very differ-ent Product A has relatively little variation in its warranty performance whereas Product C hasa much wider spread Product B stands out from the others mdash its level of warranty returns issignificantly higher than either Product A or C The product that has consistently lower war-ranty returns is Product A
i A company performed an experiment where glue samples were allowed to cure for differentlengths of time They then measured the force required to peel away the glue bond Create ascatter plot by using the data from the original problem earlier in the chapter
The x-y data pairs to be plotted are the cure time and peel force for each sample Make the curetime the horizontal axis and the peel force the vertical axis and begin plotting the points Thecompleted plot should look something like this
j Describe the relationship if any between the cure time and the peel force variables from thescatter plot of Problem 9 of this chapter
When looking at the plot for Problem 9 itrsquos clear that the points arenrsquot just randomly scatteredInstead they roughly form a line That means there is a relationship between the two variablesIf you look more closely you see that an increase in cure time leads to an increase in peel forceAnd a decrease in cure time leads to a decrease in peel strength When the same directionalchanges happen in both variables that means itrsquos a positive relationship If you draw a roughline through the clustered dots in the plot as shown here you see more about the relationship
90
440 480 520
Cure Time (Days)
Pee
l Fo
rce
(lb
s)
56060
70
80
126 Part III Mastering Measuring
14_045191 ch08qxd 81606 1141 PM Page 126
From the slope of the line you can see that every tenth of a minute increase in cure time resultsin a 02-pound increase in peel strength and vice versa mdash every tenth of a minute decrease incure time leads to a 02-pound reduction in peel strength
By creating a scatter plot you can even go as far as forecasting what the peel strength will be ifyou know the cure time For example if you cure a specimen for exactly five days it will have acure strength of about 73 pounds
k Describe the relationship if any between the Input X3 variable and the Output variable shownin the scatter plot from the original problem earlier in the chapter
Something out of the ordinary is going on in the relationship between variables Input X3 andOutput The dots arenrsquot randomly scattered on the plot Note the great spread of variation onthe left side of the plot As Input X3 gets bigger though the amount of variation reduces whichmeans that some mechanism in the system must be affecting the amount of variation as InputX3 changes
l Describe the relationship if any between Variable D and Variable C shown in the scatter plotfrom the original problem earlier in the chapter
The scatter plot for this problem is a great example of a set of points that are randomly distrib-uted There arenrsquot any trends patterns or groupings in the plotted points which means that norelationship exists between the two variables mdash Variable D and Variable C
m A CEO is concerned about his companyrsquos recent monthly expenses After some temporary suc-cess in reducing expenses two months ago itrsquos now the second month in a row since then thatexpenses have increased And the expenses in this latest month are 6 percent higher than forthe same month a year ago The CEOrsquos plan is to enact a spending freeze on all departments tofix the problem If that doesnrsquot work he says ldquoHeads are going to rollrdquo To check the validity ofthe CEOrsquos plan you create a time series chart of the companyrsquos expenses over the previous 21months Based on the time series data shown in the original problem earlier in the chapter doyou agree or disagree with the CEOrsquos plan Why
90
440 480 520
Cure Time (Days)
Pee
l Fo
rce
(lb
s)
56060
70
80
127Chapter 8 A Picturersquos Worth 1000 Words Measuring with Charts and Graphs
14_045191 ch08qxd 81606 1141 PM Page 127
Reviewing raw numbers is difficult and often misleading Plotted or charted data on the otherhand immediately reveals information that would be very difficult to see otherwise In thisproblem the CEO is being misled If he looked at the stream of monthly expenditures plottedover the last several months he would quickly see that the monthly amount fluctuates ran-domly over time There is no evidence that the efforts to reduce expenses have had the desiredeffect and the increase observed in the last two months is easily within the typical variationexpected for this variable Therefore nothing out of the ordinary has occurred and any specialintervention would be premature and possibly counterproductive
n Identify any non-normal behavior in the time series chart of delivery time data (flip to the origi-nal problem earlier in the chapter to see the time series chart)
After reviewing the time series plot of the delivery time data what looks non-normal is thejump in the average level around October-November 2004 Something out of the ordinary musthave occurred to affect this type of permanent shift in the length of delivery times
128 Part III Mastering Measuring
14_045191 ch08qxd 81606 1141 PM Page 128
Chapter 9
Yield and Defects Calculating the Goodthe Bad and the Ugly
In This Chapter Setting meaningful and appropriate performance specifications
Calculating and interpreting Six Sigma yield metrics
Determining how many defects are being created and how often
Measuring performance is at the core of Six Sigma Fundamentally measuring perform-ance isnrsquot complicated mdash you just count how many times things go wrong compared
to how many times they go right
Yield and defect rate are the metrics you use in Six Sigma to communicate how often thingsgo wrong or on the flip side how often they go right To be effective in Six Sigma you needto be fluent in calculating and translating between these different yield and defect rate meas-urements This chapter explains how
Get Real Creating Realistic SpecificationsSpecifications represent the specific limit values that separate good performance from poorperformance mdash they define whatrsquos acceptable and whatrsquos not Specifications form the basis todetermine what process outputs or characteristics are either good or bad
Itrsquos critical that you donrsquot set your specifications arbitrarily If a specification is set too looselyyour performance may upset or dissatisfy your customer even though it meets the specifica-tion And unfortunately the excuse ldquoBut we met our specificationsrdquo wonrsquot appease yourunhappy customer On the other hand if a specification is set too tightly yoursquoll spend moretime and money than you should getting the characteristic performance always inside theoverly narrow goalpost
The mind-jogging acronym RUMBA helps you check the appropriateness of any specification
Reasonable Is the specification based on a realistic assessment of the customerrsquosactual needs Does the specification relate directly to the performance of the charac-teristic
Understandable Is the specification clearly stated and defined so that there is no argument about its interpretation
Measurable Can you measure the characteristicrsquos performance against the specifica-tion If not there will be a lot of debate between you and your customer as to whetherthe specification has been met or not
15_045191 ch09qxd 81606 1127 PM Page 129
Believable Have you bought into the specification setting Can you and your co-workers strive to meet the specification
Attainable or Achievable Can you reach the level and range of the specification
You need to review each specification to make sure it passes the RUMBA test If it fallsshort in any of the RUMBA categories you should begin developing a plan to bring therogue specification back under control
You can use the following handy checklist to make sure a specification passes theRUMBA test
RUMBA Specification Checklist
Write your specification here ____________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
R Is the specification realistic If not write what needs to be changed to make thespecification realistic ___________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
U Is the specification understandable If not write what needs to be changed tomake the specification understandable_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
M Is the specification measurable If not write what needs to be changed to makethe specification measurable ____________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
B Is the specification believable If not write what needs to be changed to makethe specification believable _____________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
A Is the specification attainable or achievable If not write what needs to bechanged to make the specification attainable or achievable_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
130 Part III Mastering Measuring
15_045191 ch09qxd 81606 1127 PM Page 130
Getting It Right the First Time CalculatingFirst Time Yield (FTY)
With an appropriate specification in place you can move on to calculating the yield per-formance of your process or characteristic And as you probably know the goal of SixSigma is to do things right the first time Instead of comparing the number of gooditems that eventually come out of a process to the total number of items started (whichcompletely misses the extra inspection rework and effort expended) first time yield(FTY) accurately measures how well your process works in its true form mdash without anymakeup FTY is the basic measure of a processrsquos real capability or ability to producegood items the first time
The formula for first time yield (FTY) is
FTY inin scrap rework
=- -
where in is the number of items started into the process scrap is the number of itemsthat end up unusable and rework is the number of items requiring additional effort tomeet the specification
131Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly
Q From the tire inflation process flow map in the following figure whatrsquos the first time yield of theprocess
A Even though 347 of the 352 tires that started through the inflation process eventually end upwithin specifications many of them have to be reworked or repeated before theyrsquore rightFrom the flow map you can see that 103 tires arenrsquot correctly inflated the first time through(98 reworked and 5 scrapped) leaving only 249 out of 352 that the process truly produces correctly the first time The FTY is
FTY inin scrap rework
352352 5 98
352249 0 707 70 7=
- -= - - = = =
Tire InflationProcess
Inspection
Items In
(352)
Items Out
(347)
Rework
Scrap
(98)
(103)
(5)
15_045191 ch09qxd 81606 1127 PM Page 131
132 Part III Mastering Measuring
1 A study of a hospitalrsquos blood test recordsreveals that out of 127 samples of drawnblood five were done incorrectly andended up being unusable Also for 31 ofthe samples the donors had to be con-tacted again to correct personal informa-tion Whatrsquos the first time yield of thisprocess
Solve It
2 A water heater production facility is show-ing a 97 percent yield at the final teststep in its assembly process As you lookdeeper however you find that this yieldcalculation only includes the number ofunits that must be scrapped at the end ofthe test In reality 62 percent of all unitsfail final inspection on their first attemptand must be corrected before they passWhatrsquos the true FTY of this productionsystem
Solve It
3 In a one-month study an airline lost four out of 9824 pieces of luggage Further 21 pieces of lug-gage were mishandled and were only connected with their passengers after a one- or two-daydelay Whatrsquos the FTY for the 9824 pieces of luggage
Solve It
Rolling Many into One Calculating RolledThroughput Yield (RTY)
A product or process rarely consists of only a single opportunity for success or failureSome like an automobile have hundreds of thousands of component specificationsthat must be met More precisely they all must be met simultaneously for the car tooperate properly Combining the yields for each of these opportunities is called rolledthroughput yield (RTY) Whether itrsquos a series of process steps or a collection of compo-nent parts in a product RTY tells you how well the entire system works together
The formula for rolled throughput yield (RTY) is
RTY FTY FTY FTY FTYii
n
n1
1 2 f= ==
where FTYi is the first time yield of each of the n product or process components fori = 1 to n
15_045191 ch09qxd 81606 1127 PM Page 132
133Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly
4 A cellphone keypad has 17 specifieddimensions and characteristics If each ofthese specifications is met with 99 percentaccuracy whatrsquos the chance of getting adefect-free keypad
Solve It
5 A hiring process has seven steps eachwith the following first time yields
FTY1 = 092 FTY5 = 098
FTY2 = 095 FTY6 = 089
FTY3 = 090 FTY7 = 087
FTY4 = 099
Whatrsquos the rolled throughput yield for thisprocess
Solve It
Q The following figure illustrates a purchase order process thatrsquos made up of five individualprocess steps Whatrsquos the RTY of this process
A For this purchase order process the rolled throughput yield is calculated as follows
RTY = FTY1 times FTY2 times FTY3 times FTY4 times FTY5
RTY = 075 times 095 times 085 times 095 times 090
RTY = 0518
An RTY of 0518 means that the chance of a purchase order going through the entire processcorrectly the first time with no rework or scrap is only 518 percent
Submitrequisition to
PurchasingDept
Fill outrequisition
Requisitionentered into
computersystem
Purchaseorder sent to
supplier
Confirmationsent to
requisitionoriginator
FTY2 = 095FTY1 = 075 FTY3 = 085 FTY4 = 095 FTY5 = 090
Like a chain only being as strong as its weakest link an RTY can never be greater than the lowestFTY within the system Use this mathematical fact to check your calculated answer
15_045191 ch09qxd 81606 1127 PM Page 133
ldquoHow Bad Is It Docrdquo CalculatingDefect Rates
The complementary measurement to yield is defects If your yield is 90 percent 10 per-cent of your units contain one or more defects Measuring defects and calculating therate or how often the defects occur is like looking at the flip side of the yield coin
A basic assessment of characteristic or process capability is to measure the totalnumber of defects that occur over a known number of units produced This number isthen transformed into a calculation of how often defects occur on a single unit whichis called defects per unit (DPU)
When you need to compare the defect rate of one product or process to the defect rateof a different product or process you first have to put the rates for each into compara-ble units mdash you want to compare apples to apples You can compare the defect ratesof different products or processes by counting the number of observed defects in theunits inspected and compare this count to the total number of opportunities for suc-cess or failure in the inspected product or process When you compare defect rates inthis way the metrics you use are defects per opportunity (DPO) and defects per millionopportunities (DPMO)
The following equations show you how to calculate DPU DPO and DPMO No order isimportant Which one or ones you use depends on your situation and what yoursquoretrying to communicate
DPU number of units inspectednumber of defects observed
DPO total number of opportunities inspectedtotal number of defects observed
DPMO DPO 1 000 000
=
=
=
134 Part III Mastering Measuring
6 A football coach is trying to improve his offensersquos ball possession Looking back at game filmshe finds that his offense has committed 12 turnovers through 305 plays from scrimmage giving theteam an overall probability of retaining possession at 96 percent for any single play Whatrsquos theprobability of the team retaining possession for 16 consecutive plays
Solve It
15_045191 ch09qxd 81606 1127 PM Page 134
135Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly
7 Whatrsquos the DPU of a product if you find fivedefects on 23 units
Solve It
8 A bottle-filling process has three opportu-nities for success or failure mdash the amountof product put into the bottle the correctassembly of the lid and the correct appli-cation of the bottle label Find the DPO ofthis process if it produces 7 defects after40 bottles have been produced
Solve It
Q You process 23 loan applications during a month and find that 11 defects occurred mdash suchthings as misspelled names missing prior residence information and incorrect loan amountsWhatrsquos the DPU for this process
A The DPU for your loan process is
DPU number of units inspectednumber of defects observed
2311 0 478= = =
This answer shows that you would expect to see defects in almost half the loan applicationsthat leave your desk
Q If there are 7 opportunities for success or failure on each loan application (from the previousexample) what is the DPO
A The DPO for this loan process is
DPO total number of opportunities inspectedtotal number of defects observed
23 711
16111 0 068
= = = =
This answer shows that for every 100 opportunities for success or failure you encounter in thisloan application the process will produce 6 or 7 defects
15_045191 ch09qxd 81606 1127 PM Page 135
Whatrsquos Missing Linking Yield to DefectsWhen you have an overall process with a relatively low defect rate mdash say a processthat produces units with a DPU less than 010 (or 10 percent) mdash you can mathemati-cally link the process defect rate to the overall process yield
RTY = endashDPU and
DPU = ndashln(RTY)
where e in the equation is a mathematical constant equal to 2718 and ln is the naturallogarithm function
On your scientific calculator or spreadsheet program yoursquoll find a function or key forraising e to any number (Look for the ex key on your calculator) Yoursquoll also find thesame for the ln function
136 Part III Mastering Measuring
9 A hospitalrsquos emergency room admittance process produces 17 defects per 500 opportunities Theradiology department produces 15 defects per 600 opportunities Which area of the hospital has ahigher defect rate How many defects should you expect after one million patients have gonethrough both of these hospital departments
Solve It
Q An experienced machine operator esti-mates that his rolled throughput yieldhovers around 95 percent What then isthe DPU for a typical unit produced bythis machine operator
A DPU = ndashln(095) = 00513
10 The manager of a product line with a meas-ured DPU of 001 wants you to estimate theRTY for his process What estimate do yougive
Solve It
11 If a specific model of automobile is foundto have an average of four defects whenpurchased new what would you expect therolled throughput yield (RTY) of its manu-facturing process to be`
Solve It
15_045191 ch09qxd 81606 1127 PM Page 136
137Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly
12 A hotel operator measured the first timeyields for each step of the guest checkoutprocess The rolled throughput yield forthe entire checkout process is 90 percentHow many defects should the operatorexpect the average guest to experience
Solve It
13 For an RTY of 064 whatrsquos the correspon-ding DPU
Solve It
15_045191 ch09qxd 81606 1127 PM Page 137
Solutions to Yield and Defects Problemsa A study of a hospitalrsquos blood test records reveals that out of 127 samples of drawn blood five
were done incorrectly and ended up being unusable Also for 31 of the samples the donors hadto be contacted again to correct personal information Whatrsquos the first time yield of this process
Recognizing that there are 5 scrapped and 31 worked samples from the 127 inspected you canuse the formula for FTY to calculate the correct answer
FTY inin scrap rework
127127 5 31
12791 0 717 71 7=
- -= - - = = =
b A water heater production facility is showing a 97 percent yield at the final test step in itsassembly process As you look deeper however you find that this yield calculation onlyincludes the number of units that must be scrapped at the end of the test In reality 62 percentof all units fail final inspection on their first attempt and must be corrected before they passWhatrsquos the true FTY of this production system
Because 97 percent of the water heaters eventually make it through the final test step thismeans that the other 3 percent end up scrapped as the problem description says Combiningthis scrap number with the provided rework number of 62 percent allows you to use the FTYformula on a per 100 (percent) basis
FTY inin scrap rework
100100 3 62
10035 0 35 35=
- -= - - = = =
c In a one-month study an airline lost four out of 9824 pieces of luggage Further 21 pieces ofluggage were mishandled and were only connected with their passengers after a one- or two-day delay Whatrsquos the FTY for the 9824 pieces of luggage
Looking at the one-month airline study data the total number of defects after the first try is 25 mdash4 completely lost pieces of luggage (scrap) and 21 delayed deliveries (rework) Applying the for-mula for FTY you have
FTY in
in scrap rework9 824
9 824 4 219 8249 799 0 9975 99 75=
- -=
- -= = =
d A cellphone keypad has 17 specified dimensions and characteristics If each of these specifica-tions is met with 99 percent accuracy whatrsquos the chance of getting a defect-free keypad
Rolled throughput yield (RTY) is calculated by simply multiplying all the yields together for thecharacteristics of a system So for this problem because each of the 17 characteristics runs at99 percent the RTY is 099 multiplied by itself 17 times
RTY = FTYn = 09917 = 0843 = 843
e A hiring process has seven steps each with the following first time yields
FTY1 = 092 FTY5 = 098
FTY2 = 095 FTY6 = 089
FTY3 = 090 FTY7 = 087
FTY4 = 099
Whatrsquos the rolled throughput yield for this process
138 Part III Mastering Measuring
15_045191 ch09qxd 81606 1127 PM Page 138
To find RTY you multiply all the FTYs
RTY = FTY1 times FTY2 times FTY3 times FTY4 times FTY5 times FTY6 times FTY7
RTY = 092 times 095 times 090 times 099 times 098 times 089 times 087
RTY = 0591 = 591
f A football coach is trying to improve his offensersquos ball possession Looking back at game filmshe finds that his offense has committed 12 turnovers through 305 plays from scrimmage givingthe team an overall probability of retaining possession at 96 percent for any single play Whatrsquosthe probability of the team retaining possession for 16 consecutive plays
Even though this problem deals with sports the math is no different If the probability of theoffense retaining possession on a single play is 096 or 96 percent the chance of retaining pos-session for 16 consecutive plays is 096 multiplied by itself 16 times
RTY = FTY16 = 09616 = 053 = 53
Even with a very high chance of retaining possession on a single play you can see why coachesget gray hair when trying to get their teams to have long sustained drives mdash the chances of aturnover quickly multiply
g Whatrsquos the DPU of a product if you find five defects on 23 units
DPU compares the number of defects found to the total number of units inspected
DPU 235 0 217= =
h A bottle-filling process has three opportunities for success or failure mdash the amount of productput into the bottle the correct assembly of the lid and the correct application of the bottlelabel Find the DPO of this process if it produces 7 defects after 40 bottles have been produced
DPO is based on the number of defects found per opportunity So you have to compare thedefects found to the total number of opportunities within the inspected units
rsquo DPO total number of opportunities inspectedtotal number of defects observed
40 37
1207 0 058
= = = =
i A hospitalrsquos emergency room admittance process produces 17 defects per 500 opportunitiesThe radiology department produces 15 defects per 600 opportunities Which area of the hospi-tal has a higher defect rate How many defects should you expect after one million patientshave gone through both of these hospital departments
If you use the DPO formula for each area of the hospital you get
DPO
DPO
50017 0 034
60015 0 025
ER
RAD
= =
= =
So the emergency room with its DPO of 0034 has a slightly higher defect rate
Putting these into defect per million opportunities (DPMO) units tells you how many defectswill occur when a million patients go through each of these hospital departments
DPMOER = DPOER times 1000000 = 0034 times 1000000 = 34000
DPMORAD = DPORAD times 1000000 = 0025 times 1000000 = 25000
139Chapter 9 Yield and Defects Calculating the Good the Bad and the Ugly
15_045191 ch09qxd 81606 1127 PM Page 139
So the total DPMO for the two departments together is
DPMOTOT = DPMOER + DPMORAD = 34000 + 25000 = 59000
j The manager of a product line with a measured DPU of 001 wants you to estimate the RTY forhis process What estimate do you give
If you know DPU you can use the formula to find RTY
RTY = endashDPU = endash001 = 099005
k If a specific model of automobile is found to have an average of four defects when purchased newwhat would you expect the RTY of its manufacturing process to be
If a car was found to have a DPU of four the RTY is
RTY = endashDPU = endash4 = 00183 = 183
For something as complex as a car with thousands of parts itrsquos not unusual to have a verylow RTY
l A hotel operator measured the first time yields for each step of the guest checkout processThe rolled throughput yield for the checkout process is 90 percent How many defects shouldthe operator expect the average guest to experience
If you know RTY you can use the formula to find DPU
DPU = ndashln(RTY) = ndashln(090) = 01054
m For an RTY of 064 whatrsquos the corresponding DPU
To find the corresponding DPU to an RTY of 064 use the following formula
DPU = ndashln(RTY) = ndashln(064) = 0446 = 446
140 Part III Mastering Measuring
15_045191 ch09qxd 81606 1127 PM Page 140
Part IVAssessing the Right
Approach to Your Analysis
16_045191 pt04qxd 81606 1141 PM Page 141
In this part
What underlies your data What are the root causesWhich are the critical inputs The answers to these
questions come from the skills of the Analyze phase of theDMAIC process This part exercises and builds your SixSigma analytical muscles
16_045191 pt04qxd 81606 1141 PM Page 142
Chapter 10
Mastering Measurement SystemAnalysis (MSA)
In This Chapter Completing a basic measurement system audit
Performing attribute measurement system analysis
Interpreting the results of your continuous variable measurement systems analysis
Measurement is the foundation of knowledge and subsequent improvement Measurementis the tool you use to verify that you have come to the right answer have corrected
the problem or have improved the situation Measurement is unavoidable because youcanrsquot observe anything without the filter of some kind of measurement system mdash your eyesyour brain your perception a ruler a stopwatch or a laser interferometer Everything comesto you through some kind of measurement system ldquolensrdquo Because measurement is so impor-tant you need to know whether your measurement system is giving you an accurate pictureof reality
As is always the case no measurement system is ever perfect Your job in Six Sigma is tostart by finding out how good your measurement system is You have to ask whether itrsquosgood enough to base decisions on and whether itrsquos consistent Then ask Is it accurate Howmuch of the error or variation you observe is a result of the measurement system itself Agood start at answering these basic questions is to perform a measurement system auditRead on to find out how
Your Basic Sanity Check AuditingMeasurement Systems
An audit is when you compare your measurements to a known correct standard In an audityour goal is to determine how often the recorded measurement hasnrsquot matched with what itshould be Here are some points to remember when performing a basic audit on a measure-ment system
Experience has shown that human inspection systems detect only about 80 percent ofactual defects Remember that eyewitness accounts often are incomplete or wrong Ifyou want to use human inspection you need to take precautions to try to make themaccurate enough for your needs
17_045191 ch10qxd 81606 1130 PM Page 143
If your measurement system is one that classifies problems or defects into cate-gories a Pareto chart (or bar chart) of the number of problems or defects placedinto each category shouldnrsquot be flat or even across the charted categoriesInstead about 80 percent of the total defects should come from a smaller subsetof the possible defect categories If the Pareto chart of your defects is flat thatmeans your measurement system probably isnrsquot accurately reflecting reality Alsoif a Pareto chart of classified defects looks extremely peaked with almost alldefects in a single category your measurement system doesnrsquot have enough dis-criminating power and needs to be addressed (Refer to Six Sigma for Dummiesfor background on creating and interpreting Pareto diagrams)
144 Part IV Assessing the Right Approach to Your Analysis
Q A stockroom manager uses a computer system to track inventory The computer system isupdated from production records meaning that when a product is counted in the computer systemas having been finished in production the items in its Bill of Materials are automatically subtractedfrom the computer inventory The stockroom manager wants to know how accurate this computermeasure of inventory really is So one day he went out into the stockroom and performed a verycareful physical count of all items Herersquos what he found for a selection of critical inventory items
Item Physical Count Computer Count
Polymer resin 20452 kg 21310 kg
Spindle motors 17721 units 17750 units
Fasteners 51897 units 52000 units
Adhesive 44 barrels 42 barrels
Label sets 18366 pairs 18370 pairs
Packaging 20204 units 20180 units
What is the accuracy of the computer inventory measurement system
A You can quickly calculate how close each computer system value is to the actual physical countnumber by calculating the amount each inventory item is ldquooffrdquo (the difference between the com-puter count and the physical count) and transforming the amount ldquooffrdquo into a percentage (divid-ing each amount ldquooffrdquo by the actual physical count of each item) This method is accomplishedby adding two columns to the table as seen below
Item Physical Count Computer Count Amount ldquoOffrdquo Percent ldquoOffrdquo
Polymer resin 20452 kg 21310 kg 858 kg 420
Spindle motors 17721 units 17750 units 29 units 016
Fasteners 51897 units 52000 units 103 units 020
Adhesive 44 barrels 42 barrels 2 barrels 455
Label sets 18366 pairs 18370 pairs 4 pairs 002
Packaging 20204 units 20180 units 24 units 012
The average percent that these critical items is ldquooffrdquo by is 154 percent This average tells youhow close your computer inventory is from being fully accurate In other words this measure-ment system is 9846 percent accurate
17_045191 ch10qxd 81606 1130 PM Page 144
145Chapter 10 Mastering Measurement System Analysis (MSA)
As a further check you can create a Pareto or bar chart to show how ldquooffrdquo each of the selectedinventory items is Herersquos what the chart looks like
By looking at the Pareto chart you can see that the accuracy of the adhesive and polymer resin ismuch different than the rest of the inventory items This fact tells you that the inventory measure-ment system is following the Pareto principle which states that about 80 percent of the problemis arising from about 20 percent of the items This ratio is exactly what yoursquod expect when a meas-urement system is operating correctly However if the bars of the Pareto chart had all been aboutthe same height yoursquod suspect something was fishy To improve this inventory measurementsystem yoursquod want to start with the way the stock levels for adhesive and polymer resin areaccounted for in the computer system
4
3
Adhesive Polymer Resin Fasteners Spindle Motors
Critical Inventory Items
Per
cen
t ldquoo
ffrdquo
in c
omp
ute
r sy
stem
Packaging Label Sets
0
2
1
5
1 You perform an audit of your companyrsquos employee information system Whenever a new employee ishired the hiring manager is required to gather specific background information for the employee suchas spousersquos name home address emergency contact and so on So you select a few random employ-ees and pull their filed background sheets You then have the employees review each of the 20 fieldsthat the hiring manager filled in to verify that theyrsquore all correct Herersquos what you found in your audit
Employee Errors in Background Info
Bob 3
John B 4
Ruth 0
David 4
Elizabeth 3
Stacy 4
Noah 2
Sam 0
Erin 2
John M 5
What is the average accuracy of the employee background information Get some scrap paper towork out your calculations
Solve It
17_045191 ch10qxd 81606 1130 PM Page 145
146 Part IV Assessing the Right Approach to Your Analysis
2 In the audit described in Problem 1 the following types and number of errors werefound
Error Type Number
Missing 15
Misspelled 5
Outdated 5
Incorrect 2
Create a Pareto chart of the error type datato see whether you can find evidence of adeeper problem with the system that cap-tures employee background information
Solve It
3 Using the scenario from Problems 1 and 2list some specific actions you can take toimprove the accuracy of the employeebackground information system
Solve It
Do We Agree Performing an AttributeMeasurement System Analysis
The purpose of some measurement systems is to categorize items by their attributes mdashto separate ldquogoodrdquo items from ldquobadrdquo ones sort samples into ldquobluerdquo ldquogreenrdquo and ldquocyanrdquogroups and assign invoices to ldquoengineeringrdquo ldquoproductionrdquo or ldquosalesrdquo departmentsThese types of measurement systems are called attribute measurement systems becausethey determine or measure one or more attributes of the item being inspected
The question is how repeatably and reliably can one of these systems determine thespecific attribute yoursquore looking for For example how repeatably and reliably doesyour attribute measurement system detect ldquobadrdquo disk drives from among all theldquogoodrdquo ones being completed in production To quantify how well an attribute meas-urement system is working you perform an attribute measurement system analysis
Here are the steps to complete an attribute measurement system analysis
1 Set aside 15 to 30 test samples of the item yoursquore measuring
Make sure these samples represent the full range of variation being encounteredand make sure approximately equal amounts of each possible attribute category(for example ldquopassrdquo or ldquofailrdquo or ldquobluerdquo ldquogreenrdquo or ldquocyanrdquo) are present within thesamples yoursquove set aside
2 Create a ldquomasterrdquo standard that designates each of the test samples into itstrue attribute category
17_045191 ch10qxd 81606 1130 PM Page 146
3 Select two or three typical inspectors and have them review the sample itemsjust as they normally would in the measurement system but in random orderRecord their attribute assessment for each item
4 Place the test samples in a new random order and have the inspectors repeattheir attribute assessments (Donrsquot reveal the new order to the inspectors)Record the repeated measurements
5 For each inspector go through the test sample items and calculate the per-centage of items where their first and second measurements agree
This percentage is the repeatability of that inspector To determine the repeata-bility of the overall attribute measurement system calculate the averagerepeatability for all three inspectors
6 Going through each of the sample items of the study calculate the percentageof times where all of the inspectorsrsquo attribute assessments agree for the firstand second measurements for each sample
This percentage is the reproducibility of the measurement system
7 You can also calculate the percent of the time all the inspectorsrsquo attribute assess-ments agree with each other and with the ldquomasterrdquo standard created in Step 2
This percentage which is referred to as the effectiveness of the measurementsystem tells you how consistently the measurement system correctly measuresthe true attribute itrsquos looking for
147Chapter 10 Mastering Measurement System Analysis (MSA)
Q Imagine yoursquore starting up a Six Sigma project It involves reducing the number of parts rejectedfrom a manufacturing process that makes shear pins As the pins are completed theyrsquorereviewed by an inspector and either passed or failed (and scrapped) Before you get too far intoyour project you decide to perform an attribute measurement system analysis to make sure thepass-fail assessments of the pins are reliable How would you go about doing this analysis
A You specially arrange 15 random shear pin parts mdash about half (8) known to be good and half (7)known to be bad Then you pick three inspectors for the study After randomizing the order ofthe parts and having each inspector measure the parts two different times you record theirpass-fail measurements A filled-out worksheet for your attribute measurement system analysisis shown in the next figure
The worksheet helps guide you through the steps of the calculations of the analysis Here are theactual calculation steps
bull To find the repeatability of the individual inspectors first count the number of times eachinspectorrsquos measurements agree between the two trials Then divide this by the number ofparts inspected and multiply by 100 to get the percent repeatability
bull To find the overall repeatability calculate the average of the three individual inspectorrepeatabilities
bull The overall reproducibility is calculated from a review of how many times all three inspectorsagree with each other
bull The overall effectiveness of the system is calculated from a review of how many times all threeinspectors agree and their agreement matches the designated ldquomasterrdquo standard
You can see by the worksheet that the overall repeatability of this measurement system is 844percent which means that if you repeated the measurements on the same set of pins yoursquod havean 844 percent chance of getting the same result mdash a marginally acceptable level But the over-all reproducibility or how well this system will do given a different set of pins or different set ofinspectors is only 60 percent This is not good enough for valid measurements As a result ofyour analysis your first project task has to be to improve the pass-fail measurement system forthe shear pins After you have valid measurement data you can proceed if necessary
17_045191 ch10qxd 81606 1130 PM Page 147
Item
ID
Tru
eA
ttri
bu
teldquoM
aste
r
1st
Ass
essm
ent
Ord
er
2nd
Ass
essm
ent
Ord
er
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
PASS
FAIL
FAIL
PASS
PASS
FAIL
PASS
FAIL
FAIL
PASS
FAIL
FAIL
PASS
PASS
FAIL
14 2 10 8 13 7 5 9 1 6 15 4 3 12 11
13 15 12 9 6 11 8 10 5 1 7 4 2 3 4
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
1st
Ass
essm
ent
2nd
Ass
essm
ent
PASS
FAIL
PASS
PASS
PASS
FAIL
PASS
FAIL
PASS
FAIL
FAIL
FAIL
PASS
PASS
FAIL
A1
Cou
nt o
f Agr
eem
ents
B1
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A1
B1times
100
PASS
FAIL
FAIL
PASS
PASS
FAIL
FAIL
FAIL
PASS
FAIL
FAIL
FAIL
PASS
PASS
FAIL
Y Y N Y Y Y N Y Y Y Y Y Y Y Y
Y Y N Y Y Y N Y N N Y Y Y Y Y
13 15
861
7
C1
Ind
ivid
ual
Rep
eata
bil
ity
867
11 15 733
Ind
ivid
ual
Eff
ecti
ven
ess
733
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
1st
Ass
essm
ent
2nd
Ass
essm
ent
PASS
FAIL
FAIL
PASS
PASS
PASS
PASS
FAIL
PASS
FAIL
FAIL
FAIL
PASS
PASS
FAIL
PASS
FAIL
PASS
PASS
PASS
PASS
FAIL
FAIL
PASS
FAIL
PASS
FAIL
FAIL
PASS
FAIL
Y Y N Y Y Y N Y Y Y N Y N Y Y
Y Y N Y Y N N Y N N N Y N Y Y
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
1st
Ass
essm
ent
2nd
Ass
essm
ent
PASS
FAIL
FAIL
PASS
PASS
FAIL
PASS
FAIL
PASS
PASS
FAIL
FAIL
PASS
PASS
FAIL
PASS
FAIL
FAIL
PASS
PASS
FAIL
PASS
FAIL
FAIL
PASS
FAIL
FAIL
PASS
PASS
FAIL
Y Y Y Y Y Y Y Y N Y Y Y Y Y Y
Y Y Y Y Y Y Y Y N Y Y Y Y Y Y
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
Y Y N Y Y N N Y Y N N Y N Y Y
Y Y N Y Y N N Y N N N Y N Y Y
A2
Cou
nt o
f Agr
eem
ents
B2
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A1
B1times
100
11 15 733
C2
Ind
ivid
ual
Rep
eata
bil
ity
733
7 15 467
Ind
ivid
ual
Eff
ecti
ven
ess
467
A3
Cou
nt o
f Agr
eem
ents
B3
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A1
B1times
100
14 15 933
C3
Ind
ivid
ual
Rep
eata
bil
ity
933
14 15 933
Ind
ivid
ual
Eff
ecti
ven
ess
933
Ove
rall
Rep
eata
bil
ity
(C
1+C
2+C
3)3
844
A4
Cou
nt o
f Agr
eem
ents
B4
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A4
B4times
100
9 15 300
C4
Ind
ivid
ual
Rep
eata
bil
ity
600
8 15 533
Ind
ivid
ual
Eff
ecti
ven
ess
533
148 Part IV Assessing the Right Approach to Your Analysis
17_045191 ch10qxd 81606 1130 PM Page 148
149Chapter 10 Mastering Measurement System Analysis (MSA)
Using the example from this section as a guideline perform an attribute measurement systemanalysis on a measurement system from your own work or better yet from your own Six Sigmaproject Use this template or print it from wwwdummiescomgosixsigmaworkbook
Item
ID
Tru
eA
ttri
bu
teldquoM
aste
r
1st
Ass
essm
ent
Ord
er
2nd
Ass
essm
ent
Ord
er
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
1st
Ass
essm
ent
2nd
Ass
essm
ent
A1
Cou
nt o
f Agr
eem
ents
B1
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A1
B1times
100
C1
Ind
ivid
ual
Rep
eata
bil
ity
Ind
ivid
ual
Eff
ecti
ven
ess
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
1st
Ass
essm
ent
2nd
Ass
essm
ent
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
1st
Ass
essm
ent
2nd
Ass
essm
ent
1st a
nd
2n
d
Ass
essm
ent
Agr
ee
(YN
)
1st a
nd
2n
d
Ass
essm
ent
Agr
ee w
ith
Mas
ter
(Y
N)
A2
Cou
nt o
f Agr
eem
ents
B2
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A1
B1times
100
C2
Ind
ivid
ual
Rep
eata
bil
ity
Ind
ivid
ual
Eff
ecti
ven
ess
A3
Cou
nt o
f Agr
eem
ents
B3
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A1
B1times
100
C3
Ind
ivid
ual
Rep
eata
bil
ity
Ind
ivid
ual
Eff
ecti
ven
ess
Ove
rall
Rep
eata
bil
ity
(C
1+C
2+C
3)3
A4
Cou
nt o
f Agr
eem
ents
B4
Num
ber
of I
tem
s In
spec
ted
Per
cent
Agr
eem
ent
A4
B4times
100
C4
Ind
ivid
ual
Rep
eata
bil
ity
Ind
ivid
ual
Eff
ecti
ven
ess
17_045191 ch10qxd 81606 1130 PM Page 149
Gauging Gages Analyzing ContinuousVariable Measurement Systems
Many measurement systems operate in the realm of continuous variable data To illus-trate this point think about thermometers rulers and stopwatches mdash all these areused to measure variables that vary continuously rather than just fall into attributebuckets such as ldquosmallrdquo ldquomediumrdquo or ldquolargerdquo For example the temperature in yourroom right now may be 712 degrees Thirty minutes later the temperature may be708 degrees A thermometer can capture these up and down changes But as with allmeasurement systems some of the observed change is due to variation from the meas-urement system itself
When dealing with continuous variable measurement systems the total variation youobserve is a combination of the actual variation of the parts or process yoursquore investi-gating and the variation from the measurement system itself Mathematically this canbe written as
σ σ σOBSERVED ACTUAL MEASUREMENTS2 2 2= +
Ideally you want the part of the observed variation stemming from your measurementsystem to be as small as possible That way the actual part or process variation wonrsquotbe clouded or drowned out by the measurement variation
When you study the ldquogoodnessrdquo of a continuous variable measurement system you doexperiments and calculations to quantify the observed variation the actual variationand the measurement system variation These studies usually involve two to threeinspectors and five to ten process outputs or characteristics to measure Each inspec-tor also measures each process output or characteristic two to three times
At this stage statistical analysis software such as Minitab or JMP is used to automati-cally calculate the observed actual and measurement system variations Interpretingthe results of your continuous variable measurement system analysis is straightfor-ward but is often glossed over in the explanations of your software tool After yoursoftware spits out values for σOBSERVED σACTUAL and σMEASUREMENTS you must clearly under-stand how to interpret them if you want to know whether yoursquore using an adequatemeasurement system You interpret these values by creating a ratio betweenσMEASRUEMENTS and σOBSERVED
Table 10-1 summarizes the ratio values you may get describes the situation and helpsyou interprets the results
150 Part IV Assessing the Right Approach to Your Analysis
17_045191 ch10qxd 81606 1130 PM Page 150
151Chapter 10 Mastering Measurement System Analysis (MSA)
Table 10-1 Measurement-to-Observed Variation Ratio Values and Interpretation
Calculated Variance Ratio Diagnosis Prescription
σσ
0 1OBSERVED
MEASUREMENTS This is an effective measure- Use the measurement ment system Contribution of system as it is now Look for the measurement system opportunities to simplify or itself to the overall observed make the measurement variation is small enough to system less expensive or enable good decisions from more efficientthe measurements
σσ
0 1 0 3OBSERVED
MEASUREMENTS This is a marginal measure- Use this system with caution ment system Contribution of and only if no better meas-the measurement system urement alternative existsitself to the overall observed Begin to improve the meas-variation is beginning to urement system by trainingcloud results You risk making operators standardizinga wrong decision due to measurement procedures the extra variation of the and investigating newmeasurement system measurement equipment
σσ
0 3OBSERVED
MEASUREMENTS $ This is an unacceptable This measurement system measurement system needs to be corrected Guessing is probably just before any valid information as precise Donrsquot base impor- can be derived from the tant decisions on information system Investigate causes compiled from a measurement of gross inconsistencysystem thatrsquos in this condition
Go through the example and practice exercises that follow to become proficient atinterpreting the results of a continuous variable measurement system analysis
Q You just performed a measurement system study on the temperature sensor used in the oven ofyour shrink-wrap process Here are the results that your computer software spit out
σ 0 0325OBSERVED2 =
σ 0 0007MEASUREMENTS2 =
What is your assessment of this situation Is the temperature measurement system effective
A The first thing to do with the output of your computer calculations is to create a ratio of the stan-dard deviation of the measurement system to the standard deviation of the total observed variation
σ
σσ
σ0 03250 0007
0 18030 0265 0 147
OBSERVED
MEASUREMENTS
OBSERVED
MEASUREMENTS
2
2
= = = =
After you achieve this number and look at Table 10-1 you can see that this system falls into themarginal category between 01 and 03 which means that the temperature sensor system in theshrink-wrap oven has some limitations but is still good enough to use if no other system is avail-able If more reliable measurements are necessary you need to make changes to the temperaturemeasurement system
17_045191 ch10qxd 81606 1130 PM Page 151
152 Part IV Assessing the Right Approach to Your Analysis
4 An analysis of a length measurementsystem mdash that is a ruler mdash produces thefollowing breakdown of the variation inthe system
σ 0 250OBSERVED2 =
σ 0 042MEASUREMENTS2 =
What is your diagnosis of the measurementsystem
Solve It
5 The results of your measurement systemanalysis on the method you use to measuresliding friction on a conveyor systemreveal that
σ 178OBSERVED2 =
σ 121ACTUAL2 =
Is this measurement system reliable
Solve It
17_045191 ch10qxd 81606 1130 PM Page 152
Solutions to MSA Problemsa You perform an audit of your companyrsquos employee information system Whenever a new
employee is hired the hiring manager is required to gather specific background information forthe employee such as spousersquos name home address emergency contact and so on So youselect a few random employees and pull their filed background sheets You then have theemployees review each of the 20 fields that the hiring manager filled in to verify theyrsquore all cor-rect Check the original question earlier in the chapter to see what you found in your audit
Each background form has 20 fields so the accuracy as a percentage of any form would be thenumber of correct fields (20 minus the number of errors) divided by the total number of fields(20) The table below shows the calculated accuracy percentage for each background formaudited
Employee Errors in Background Info Correct Fields Accuracy ( Correct)
Bob 3 17 85
John B 4 16 80
Ruth 0 20 100
David 4 16 80
Elizabeth 3 17 85
Stacy 4 16 80
Noah 2 18 90
Sam 0 20 100
Erin 2 18 90
John M 5 15 75
The average accuracy is just the average of all the individual accuracies which is 865 percent
b Create a Pareto chart of the error type data from Problem 1 to see whether you can find evi-dence of a deeper problem with the system that captures employee background information(see original question earlier in the chapter for more details)
The Pareto chart for the types of errors with their stated quantities would look like the follow-ing figure
15
Missing Misspelled Outdated Incorrect
Types of Errors
Nu
mb
er
0
10
5
20
153Chapter 10 Mastering Measurement System Analysis (MSA)
17_045191 ch10qxd 81606 1130 PM Page 153
You can see that the Pareto diagram looks just as it should mdash about 20 percent of the cate-gories accounting for about 80 percent of all the errors This tells you that the variation in themeasurement system is normal and that there is no cause for alarm
c Using the scenario from Problems 1 and 2 list some specific actions you can take to improvethe accuracy of the employee background information system
Herersquos a list of some immediate actions that can be taken to improve the employee backgroundinformation (you may have thought of others)
Have the new employee instead of the hiring manger fill out the form
Send a copy of each employeersquos background form to the employee annually for reviewDoing so gives the employee a chance to update addresses phone numbers or otherinformation that may be obsolete
Have the appropriate background information copied over from the same informationalready recorded on the official company job application
d An analysis of a length measurement system mdash that is a ruler mdash produces a breakdown of thevariation in the system (see the original question earlier in the chapter for details) What isyour diagnosis of the measurement system
Start by calculating the ratio of the standard deviation of the measurement system to the stan-dard deviation of the total observed variation
This ratio shows that the variation from the length measurements accounts for only a smallportion of the observed variation In fact this is better than the 010 rule of thumb You cantrust the measurements from this ruler and can go forward knowing that your work is basedon a sound foundation of data
e The results of your measurement system analysis of the method you use to measure slidingfriction on a conveyor system are revealed (see the original problem earlier in the chapter fordetails) Is the measurement system reliable
The way you can tell whether this is a reliable measurement system is to construct the ratio of thestandard deviation of the measurement system to the standard deviation of the entire system
The standard deviation for the measurements themselves isnrsquot given But you can calculate itjust by knowing the other two components
σ σ σ
σ σ σ 178 151 27OBSERVED ACTUAL MEASUREMENTS
MEASUREMENTS OBSERVED ACTUAL
2 2 2
2 2 2
= +
= - = - =
Now you can plug this value into the ratio formula
σ
σσ
σ17827
13 345 20 0 39
OBSERVED
MEASUREMENTS
OBSERVED
MEASUREMENTS
2
2
= = = =
This ratio value is higher than 03 which means that your measurement system isnrsquot capable ofreliably telling you what the sliding friction on the conveyor system really is Before you do anyfurther Six Sigma work you need to improve the measurement part of the system
154 Part IV Assessing the Right Approach to Your Analysis
17_045191 ch10qxd 81606 1130 PM Page 154
Chapter 11
Capability Matching Performance to Need
In This Chapter Discovering how to calculate sigma (Z) scores
Understanding the shift between short- and long-term performance
Using capability indices (CP CPK PP PPK) to apply an improvement plan
Capability is a measure of how well a product or process is able to meet a specificrequirement If PGA Tour fairways are 30 yards wide the capability of a golferrsquos tee
shot is his or her measured ability to stay within this 30-yard width Does the golf ball stayon the fairway all of the time Part of the time How capable is the golfer ldquoPoorrdquo ldquoprettygoodrdquo or ldquoTiger Woods goodrdquo are basically useless measures of capability in Six Sigmabecause theyrsquore too vague They offer no improvement traction In this chapter you prac-tice how to calculate the precise capability of a process or characteristic With a propercapability metric in hand you can compare the performance of one process to another youcan determine how many items will end up outside the required performance limits andyoursquoll have a basis for measuring performance improvement
Many of the formulas for capability rely on measures of the average and short- and long-termstandard deviation Flip to Chapter 7 to find out how to calculate these average and standarddeviation values
Calculating and Interpreting Sigma (Z) ScoresThe sigma (Z) score is one of the most basic ways to describe the capability of a process orcharacteristic (see Table 11-1) For example if you know the Z score you can quickly look upthe corresponding level of defect rate performance
Calculating a sigma score is straightforward A small Z is the number of short-term standarddeviations you can fit between the mean value of the processrsquos or characteristicrsquos perform-ance and its nearest specification limit (SL)
ZSL x
σ ST=
-
18_045191 ch11qxd 81606 1104 PM Page 155
Table 11-1 Sigma (Z ) Score Defects Per Million Opportunities (DPMO)
ZST DPMOLT
00 933193
05 841345
10 691462
15 500000
20 308538
25 158655
30 66807
35 22750
40 6210
45 1350
50 233
55 32
60 34
An example will help illustrate how to calculate and interpret a sigma (Z) score
156 Part IV Assessing the Right Approach to Your Analysis
Q A plastic housing part has a criticallength dimension of 3230 plusmn 010 mmYoursquove measured a short-term sampleof housings and calculated the followingabout the variation in this characteristic
x = 3234
σST = 0017
What is the sigma (Z) score for this criti-cal characteristic What is its expectedlong-term defect rate
A The sigma (Z) score is how many short-term standard deviations you can fitbetween the mean and the nearest specifi-cation limit In this example the measuredmean of 3234 mm is slightly closer to theupper specification limit of 3240 mm Sothe formula is
ZSL x
σ 0 01732 4 32 34
3 53ST
=-
=-
=
This characteristic is operating at a sigmalevel of 353 mdash not quite Six Sigma levelWhat is the defect rate You find this byfinding the calculated Z value and thenlooking up the corresponding DPMOvalue From Table 11-1 you can see thatthe Z entry for 350 has a DPMO of 22750This characteristic has a slightly higherZ value than that which means that itsperformance will be slightly better Soroughly estimating the characteristicrsquosDPMO will be about 21000
18_045191 ch11qxd 81606 1104 PM Page 156
157Chapter 11 Capability Matching Performance to Need
1 Your company has payment terms of net 30days on its accounts receivables Any pay-ment received later than 30 days is consid-ered late Yoursquove measured a sample ofpayment data and found the average pay-ment time to be 25 days with a standarddeviation of 2 days Calculate the sigma(Z) score and estimate the DPMO for thisprocess
Solve It
2 A fill-volume setting for a 1-liter bottle fillingprocess is set at 11 liters which provides asmall cushion against the bottle being filledtoo low A sample shows that the average fillvolume is actually 115 liters with a standarddeviation of 010 liters Calculate the sigma(Z) score and indicate about how many bot-tles will have a fill volume lower than the11-liter production limit
Solve It
3 Your preferred plastic injection moldertells you that when all things are carefullycontrolled he can operate his process to astandard deviation of 0001 inches for criti-cal dimensions In the plastic part yoursquoredesigning there is a screw boss that mustbe positioned 5010 inches from a datumfeature on the part If your engineeringdepartmentrsquos policy is that all dimensionsmust have a sigma (Z) value of 6 at whatplus or minus value do you set the specifi-cations for this dimension
Solve It
4 The reaction time of a certain chemicalprocess is on average 145 minutes witha standard deviation of 31frasl3 minutes Therequirement for the reaction to be com-pleted is between 135 and 150 minutesAnything shorter or longer than thisresults in a defect What change in defectrate will you get if you adjust the processaverage to be centered at 1371frasl2 minutes
Solve It
18_045191 ch11qxd 81606 1104 PM Page 157
Shift Happens Transforming Between Short- and Long-Term Performance
The calculation of a processrsquos or characteristicrsquos sigma (Z) score is a based off itsshort-term standard deviation Short-term standard deviations are easy to measureyou just go out and quickly capture a data sample from the process or characteristic
But in the sigma (Z) value look-up tables like in Table 11-1 the defect rate or yieldlisted is for the corresponding long-term performance For example a process witha short-term sigma (Z) value of 6 will have a short-term defect rate of 0001 DPMO Butas you can see from Table 11-1 it will have a long-term defect rate of 34 DPMO
Why all the mixing of short- and long-term values The reason is that early Six Sigmapractitioners found it easier and more practical to gather a short-term sample and cal-culate a short-term standard deviation and Z score but at the same time they wantedto convey the loss of capability that always occurs over the long-term To accomplishthis the early practitioners contrived the now-famous 15-sigma shift and built it rightinto all of their sigma (Z) score tables
orZ Z
Z Z
1 5
1 5LT ST
ST LT
= -
= +
The 15 value in the equations is called the 15-sigma shift An example shows youhow to use the idea of the 15-sigma shift to easily transform between the short- andlong-term domains
158 Part IV Assessing the Right Approach to Your Analysis
Q An engineering design manager has justbeen told that the most recent set of pro-totypes for a soon-to-be-launched producthas a defect rate of 0621 percent or 6210per million The manager turns to youand remarks ldquoThis is great At this ratewersquoll be able to launch full-scale produc-tion overseas rather than proving thingsout first hererdquo Do you agree with themanagerrsquos assessment
A Prototypes are always treated with spe-cial care Theyrsquore carefully crafted andassembled and tuned and maintainedAnd usually only a few are made Whatthis means is that prototypes provideshort-term data because they donrsquot cap-ture all the variation effects that occurover the long-term So the key point tokeep in mind in this example is that the6210 DPMO reported for the prototypesis a short-term measure of the productrsquosdefect rate
In the long-term however the sigma (Z)score for the product will degradeReferring to Table 11-1 a DPMO of 6210corresponds to a short-term sigma (Z)value of 4 In the long-term the Z score willbe 15 less than this or 25 This value cor-responds to the true long-term DPMO of158655 As real production for the productramps up specialized prototype toolingwill be replaced by multi-cavity productiontooling multiple suppliers will begin pro-ducing raw materials and full productionstaffs will be used for assembly All thesechanges lead to extra sources of variationthat the prototype never encounteredSo you can tell your manager that with adefect rate of 158655 per million the soon-to-be-launched product isnrsquot quite readyfor off-shore production
18_045191 ch11qxd 81606 1104 PM Page 158
159Chapter 11 Capability Matching Performance to Need
5 You calculate the short-term sigma (Z)score for a packaging process to be 512What will the long-term Z value be
Solve It
6 The long-term sigma (Z) score for a criticalcharacteristic is 25 What will the defectrate for this characteristic be in the long-term
Solve It
7 The short-term sigma (Z) value for cor-rectly filling out a loan application form is37 What is the short-term defect rate forthis application form
Solve It
8 When you measure one carrsquos fuel efficiencyyou find that the sigma (Z) score for itmeeting your companyrsquos minimum fuel effi-ciency requirement is 22 What will the Zscore and defect rate be for the companyrsquosentire fleet of cars
Solve It
18_045191 ch11qxd 81606 1104 PM Page 159
Calculating and InterpretingCapability Indices
Six Sigma practitioners use indices to communicate the capability of a process or char-acteristic These indices CP CPK PP PPK each compare the width of the processrsquos orcharacteristicrsquos specification requirement to its short- or long-term variation widthKnowing these index values you can quickly figure out how the process or character-istic is performing and you can compare the capabilities of different processes toeach other See Table 11-2 for an outline of the formulas and descriptions for eachcapability index
Table 11-2 Short- and Long-Term Capability IndicesIndex Name Formula Description
Short-term C USL LSL6P
ST=
-v
Compares the width of the capability index (Cp) specification to the short-term
width of the process
Adjusted short-term min σ σC USL x x LSL3 3PK
ST ST=
- -e o Compares the width of the
capability index (Cpk) specification to the short-term widthof the process AND accounts foroff-centering of the process fromthe specification
Long-term capability σP USL LSL6P
LT=
- Compares the width of the index (Pp) specification to the long-term width
of the process
Adjusted long-term min σ σP USL x x LSL3 3PK
LT LT=
- -e o Compares the width of the
capability index (Ppk) specification to the long-term widthof the process AND accounts foroff-centering of the process fromthe specification
Note ldquoUSLrdquo stands for Upper Specification Limit ldquoLSLrdquo stands for Lower Specification Limit
Prescribing an Improvement PlanHaving calculated each of the four capability indices for a process or characteristic youcan use these values to determine a plan for improvement Table 11-3 shows you how
Table 11-3 Prescriptive Capability Improvement PlanSymptom Diagnosis Prescription
CP ne CPK ne PP ne PPK Your process or characteristic Begin by eliminating special isnrsquot centered and you find causes Later focus on centering special causes are expanding the processthe long-term variation
160 Part IV Assessing the Right Approach to Your Analysis
18_045191 ch11qxd 81606 1105 PM Page 160
Symptom Diagnosis Prescription
CP = CPK Overall your process or As needed focus on reducing theand characteristic is centered long-term variation in your processPP = PPK within its specifications or characteristic while maintain-
ing on-center performance
CP = PP Your process or characteristic Focus on correcting the set point and suffers from a consistent of your process or characteristic CPK = PPK offset in its center location until itrsquos centered
CP = PPK Your process is operating at Continue to monitor the capability its entitlement level of variation of your process Redesign your
process to improve its entitlementlevel of performance
You can link the CPK back to the sigma (Z) score and the corresponding defect rate withthe formula
Z = 3CPK
161Chapter 11 Capability Matching Performance to Need
Q A progressive pizza chain has studied the preferences of its customers The findings are that eachsmall individual pizza must have at least 8 pepperonis and that having more than 12 pepperonisadds extra cost with no benefit The manager of the pizza chain wants to know how capable thechain is at meeting this requirement so over a period of time she counts the number of pepperonison small pizzas From her data this is what she finds
x =9
σST = 033
σLT = 050
Calculate CP CPK PP and PPK and describe the current situation at the pizza chain How many pizzaswill end up being outside the required range of 8 to 12 pepperonis Then based on the calculatedcapability indices devise an improvement strategy for the pizza chain
A Recognizing that the upper specification limit (USL) is 12 and the lower specification limit (LSL) is8 you can plug the average and standard deviation values from the managerrsquos measurements intothe formulas from Table 11-2 to calculate all the capability indices like this
min min min
min min min min
C USL LSL
C USL x x LSL
P USL LSL
P USL x x LSL
σ
σ σ
σ
σ σ
6 6 0 3312 8
24 2 00
3 3 3 0 3312 9
3 0 339 8
13
11 3 1 1 00
6 6 0 5012 8
34 1 33
3 3 3 0 512 9
3 0 59 8
1 53
1 51 2 0 67 0 67
PST
PKST ST
PLT
PKLT LT
= - = - = =
= - - = - - = = =
= - = - = =
= - - = - - = = =
$
$ $
$
$ $
e c c _
e c c _
o m m i
o m m i
The long-term defect rate for the current situation at the pizza chain can be found by multiplyingthe calculated CPK by 3 to get the sigma (Z) value and then looking up the corresponding defectrate in Table 11-1
Z = 3CPK = 3 sdot 100 = 300
18_045191 ch11qxd 81606 1105 PM Page 161
162 Part IV Assessing the Right Approach to Your Analysis
9 For a manufacturing process with an upperspecification limit of 40 mm a lower speci-fication limit of 375 mm and a short-termstandard deviation of 097 calculate the CP
Solve It
10 An accounting firm takes on average 39days with a short-term standard deviationof 3 days to prepare a clientrsquos taxes Theywant to have all clientsrsquo taxes preparedwithin 45 days Calculate the CPK for thisprocess at the accounting firm Also calcu-late the Z score and determine the percent-age of clients who will wait longer than45 days for their taxes to be completed
Solve It
At a Z of 3 Table 11-1 shows that the long-term defect rate will be 66807 per million Thatrsquos about67 percent of customers who will think that their pizza has either too many or too few pepperonis
How should the pizza chain improve its capability Because CP ne CPK ne PP ne PPK this situation fallsinto the first listed scenario in Table 11-3 The improvement step for this diagnosis is to eliminatethe special out-of-the-ordinary causes of variation By doing so the short-term and long-termstandard deviations will be equal making CP = PP and CPK = PPK The improvement plan for this diag-nosis is to center the process variation between the 8 and 12 specification limits With this accom-plished CP = CPK = PP = PPK and the pizza chain is operating at its entitlement level
18_045191 ch11qxd 81606 1105 PM Page 162
163Chapter 11 Capability Matching Performance to Need
11 What is going on in a process when the cal-culated CP is the same as the CPK
Solve It
12 Yoursquove performed a lot of measurementson a new adhesive application processHere are the statistics yoursquove determinedfrom the process
x =1053
σST = 512
σLT = 528
To be successful the result of this adhe-sive process must be between 90 and 130Calculate the CP for this process
Solve It
13 Calculate the CPK for the scenario describedin Problem 12
Solve It
14 Calculate the PP for the scenario describedin Problem 12
Solve It
18_045191 ch11qxd 81606 1105 PM Page 163
164 Part IV Assessing the Right Approach to Your Analysis
15 Calculate the PPK for the scenario describedin Problem 12
Solve It
16 Whatrsquos the defect rate for the scenariodescribed in Problem 12
Solve It
17 Outline an improvement strategy for the scenario described in Problem 12
Solve It
18_045191 ch11qxd 81606 1105 PM Page 164
Solutions to Capability Problemsa Your company has payment terms of net 30 days on its accounts receivables Any payment
received later than 30 days is considered late Yoursquove measured a sample of payment data andfound the average payment time to be 25 days with a standard deviation of 2 days Calculatethe sigma (Z) score and estimate the DPMO for this process
The specification limit in this problem is the 30-day limit on accounts receivables Knowing theaverage or mean (25 days) and the standard deviation (2 days) you can plug these into the for-mula to calculate the sigma (Z) score
ZSL x
σ 230 25
25 2 5
ST=
-=
-= =
Referring to Table 11-1 the DPMO corresponding to this Z score is 158655 Thatrsquos how manyaccounts out of a million that will end up taking longer than the 30-day limit
b A fill-volume setting for a 1-liter bottle filling process is set at 11 liters which provides a smallcushion against the bottle being filled too low A sample shows that the average fill volume isactually 115 liters with a standard deviation of 010 liters Calculate the sigma (Z) score andindicate about how many bottles will have a fill volume lower than the 11-liter production limit
Knowing the 11-liter specification level the average (115 liters) and the standard deviation(01 liters) you can plug these values into the formula to calculate the Z value
Z
SL xσ 0 1
1 1 1 150 1
0 05 0 5ST
=-
=-
= =
This is a low Z value From Table 11-1 you can see that the defect rate for this Z value is veryhigh 841345 defects per million opportunities
c Your preferred plastic injection molder tells you that when all things are carefully controlledhe can operate his process to a standard deviation of 0001 inches for critical dimensions Inthe plastic part yoursquore designing there is a screw boss that must be positioned 5010 inchesfrom a datum feature on the part If your engineering departmentrsquos policy is that all dimensionsmust have a sigma (Z) value of 6 at what plus or minus value do you set the specifications forthis dimension
In this problem yoursquore not solving for Z Instead yoursquore solving for the plus or minus tolerance(plusmnTol) that needs to be specified around the target design nominal of 5010 inches The firstthing then is to use some basic algebra to rearrange the formula into a form that will calculatethe plusmnTol value
ZSL x Tol
Tol Z
σ σσ
ST ST
ST
=-
=
=
Now plugging the provided values into the rearranged equation you get
plusmnTol = ZσST = 6 sdot 0001 = 0006
With the plus or minus tolerance value being 0006 inches the specification limits for the designwill have to be between 5016 and 5004 inches in order to meet your 6 sigma design goal
d The reaction time of a certain chemical process is on average 145 minutes with a standarddeviation of 31frasl3 minutes The requirement for the reaction to be completed is between 135 and150 minutes Anything shorter or longer than this results in a defect What change in defect ratewill you get if you adjust the process average to be centered at 1371frasl2 minutes
165Chapter 11 Capability Matching Performance to Need
18_045191 ch11qxd 81606 1105 PM Page 165
The closest specification limit to the process average is 150 minutes So plug this valuetogether with the stated process mean and standard deviation into the formula for the sigma(Z) score
ZSL x
σ 3 333150 145
3 3335 1 5
ST=
-=
-= =
From Table 11-1 the defect rate at this Z level is 500000 DPMO
Now calculate the Z value if the process average were shifted to the centered value of 1375minutes
Z
SL xσ 3 333
150 137 53 33312 5 3 75
ST=
-=
-= =
Using Table 11-1 the defect rate for this Z value between 35 and 40 can be estimated to beabout 13000 to 15000 What an improvement just centering the process made
e You calculate the short-term sigma (Z) score for a packaging process to be 512 What will thelong-term Z value be
Z Z 1 5 5 12 1 5 3 62LT ST= - = - =
f The long-term sigma (Z) score for a critical characteristic is 25 What will the defect rate forthis characteristic be in the long-term
ZST = ZLT + 15 = 25 + 15 = 40
Table 11-1 lists long-term DPMOs for short-term Z scores The long-term DPMO associated withthe short-term Z score of 40 is 6210
g The short-term sigma (Z) value for correctly filling out a loan application form is 37 What isthe short-term defect rate for this application form
Because Table 11-1 gives long-term defect rates for short-term sigma (Z) scores you have toadjust the Z value you look up by 15 By looking at the table you can see that the long-termdefect rate for the short-term sigma (Z) value of 37 is about 14000 DPMO The short-termdefect rate will be found at a sigma (Z) value of 15 sigma higher than this or 52 At a Z of 52Table 11-1 shows that the defect rate is about 100 which is the short-term defect rate for theshort-term sigma (Z) value of 37
h When you measure one carrsquos fuel efficiency you find that the sigma (Z) score for it meetingyour companyrsquos minimum fuel efficiency requirement is 22 What will the Z score and defectrate be for the companyrsquos entire fleet of cars
Measuring one car provides a short-term estimate of the sigma (Z) value of the companyrsquosentire fleet The long-term Z for the fleet will be 15 sigmas less than this which in this case is avalue of 07 But in Table 11-1 you can only look up the short-term Z value to find the long-termdefect rate In this case the long-term defect rate corresponds to the look-up value of 22 inTable 11-1 or about 250000 DPMO
Estimating a value between two entries in a table is called interpolating For example a Z valueof 22 falls between 20 and 25 in Table 11-1 but not exactly in between 22 is a little closer to20 than it is to 25 In fact the value of 22 is located exactly two-fifths of the way from 20 to25 So the corresponding DPMO value for 22 will be two-fifths of the way from 308538 to158655 You can solve for this value mathematically with the following expression
308 538 52 308 538 158 655 248 585 250 000- - = =_ i
166 Part IV Assessing the Right Approach to Your Analysis
18_045191 ch11qxd 81606 1105 PM Page 166
i For a manufacturing process with an upper specification limit of 40 mm a lower specificationlimit of 375 mm and a short-term standard deviation of 097 calculate the CP
C USL LSL
σ6 0 9740 0 37 5 0 43P
ST
= - = - =
j An accounting firm takes on average 39 days with a short-term standard deviation of 3 days toprepare a clientrsquos taxes They want to have all clientsrsquo taxes prepared within 45 days Calculatethe CPK for this process at the accounting firm Also calculate the Z score and determine thepercentage of clients who will wait longer than 45 days for their taxes to be completed
First calculate the adjusted short-term capability index CPK In this exercise you only have anupper specification limit (USL) so the CPK is
C USL xσ3 3 3
45 3996 0 67PK
ST
= - = - = =$
The Z for this scenario is calculated from
Z C3 3 0 67 2 00PK= = =$Now looking up this Z value in Table 11-1 you get a DPMO of 308538 or 308 percent So in thelong-run nearly one in three clients will have their taxes take longer than the 45-day goal of thecompany
k What is going on in a process when the calculated CP is the same as the CPK
When the calculated CP is the same as the calculated CPK the process or product characteristicis centered between the specification limits such that no ldquodiscountingrdquo of the capability indexoccurs
l Yoursquove performed a lot of measurements on a new adhesive application process Here are thestatistics yoursquove determined from the process
x =1053
σST = 512
σLT = 528
To be successful the result of this adhesive process must be between 90 and 130 Calculate theCP for this process
The CP is found from the formula
C USL LSLσ6 6 5 12
130 9030 72
40 1 30PST
= - = - = =$
m Calculate the CPK for the scenario described in Problem 12
min min
min min
C USL x x LSLσ σ3 3 3 5 12
130 105 33 5 12
105 3 90
15 3624 7
15 3615 3 1 61 1 00 1 00
PKST ST
= - - = - - =
= =
$ $e c
c _
o m
m i
n Calculate the PP for the scenario described in Problem 12
P USL LSLσ6 6 5 28
130 9031 68
40 1 26PLT
= - = - = =$
167Chapter 11 Capability Matching Performance to Need
18_045191 ch11qxd 81606 1105 PM Page 167
o Calculate the PPK for the scenario described in Problem 12
min min
min min
P USL x x LSLσ σ3 3 3 5 28
130 105 33 5 28
105 3 90
15 8424 7
15 8415 3 1 56 0 97 0 97
PKLT LT
= - - = - - =
= =
$ $e c
c _
o m
m i
p Whatrsquos the defect rate for the scenario described in Problem 12
The defect rate is calculated by finding the sigma (Z) value and then looking up the correspon-ding DPMO value in Table 11-1
Z C3 3 1 00 3 00PK= = =$Looking up this value in Table 11-1 gives you the DPMO value of 66807
q Outline an improvement strategy for the scenario described in Problem 12
Reviewing the answers from Exercises 12 through 16 you can see that both of the following aretrue
CP asymp PP
CPK asymp PPK
Fitting this into the various scenarios of Table 11-3 you can see that the prescription to thisdiagnosis is to first work on centering the process average which needs to be set as closelyas possible to the middle of the specification at a value of 110
Subsequently if more improvement is still needed you would begin to eliminate commoncauses which would reduce the short- and long-term standard deviations of the process
168 Part IV Assessing the Right Approach to Your Analysis
18_045191 ch11qxd 81606 1105 PM Page 168
Chapter 12
Narrowing Your Inputs with Confidence
In This Chapter Quantifying with confidence intervals for means
Using confidence intervals for standard deviations
Understanding how to use confidence intervals for proportions
What the wheel is to transportation sampling is to statistics Statistical samplingallows you to understand an entire population thatrsquos too large to measure individu-
ally by inspecting only a few representative samples of the population For example whatrsquosthe average height of European males If answering that question required you to locate andmeasure every man on the continent yoursquod never finish However the invention of statisti-cal sampling allows you to find the answer by measuring only a small representative sampleof the population (Next time you meet a statistician be sure to stop and thank him for hiscontribution to humanity)
Therersquos one catch to using samples though When you work with a sample instead of theentire population you may discover that your sample doesnrsquot exactly represent the popula-tion In fact the smaller your sample the greater this potential for error becomes (Can youhonestly say that the average of Pierrersquos and Hansrsquo heights accurately represents the averageheight of all men in Europe)
Statisticians have studied sampling to an almost sickening extreme They use the Greek sym-bols such as micro and σ to represent the exact population parameters and the Roman symbolssuch as x and s to represent the calculated parameters of a sample What theyrsquove found isthat each time you take a different sample from the same population and calculate thesamplersquos mean or its standard deviation mdash or any other statistical measure mdash you end upwith a slightly different calculation result When you collect the repeated sample calculationstogether they form whatrsquos called a sampling distribution And statisticians know exactly howwide this variation will be depending on how many data points are in your sample
A confidence interval quantifies the potential variation around your calculated metrics suchas the mean or the standard deviation If for example yoursquore basing your mean calculationoff of a sample rather than off of the entire population (which is almost always the case) aconfidence interval allows you to say ldquoWith 95 percent confidence I know the average heightof the European male population is between 175 and 178 metersrdquo
Because in the real world you almost always work with calculated values from samples insteadof entire populations itrsquos important that you know how to quantify the potential error in yourmeasurements and how this quantification affects your decisions This chapter provides exam-ples and practice problems that help you become an expert in creating confidence intervalsaround your calculations
19_045191 ch12qxd 81606 1128 PM Page 169
Creating Confidence Intervals for MeansWhenever you use the calculated average of a sample (x ) to infer what the average ofthe entire population (micro) is you have a potential for error Confidence intervals formeans allow you to quantify how much your inference may be off based on thenumber of data points in your sample
When your sample has 30 or more data points the confidence interval for the calcu-lated mean is
x Znsmicro =
where
micro is the true population mean
x is the calculated sample mean
F is the sigma value corresponding to the desired level of confidence you wantto have
s is the sample standard deviation
n is the number of data points in your sample
On the other hand when your sample has fewer than 30 data points the confidenceinterval for the calculated mean must instead be based on an adjusted t value whichdepends on the size of your sample Herersquos the equation
x tnsmicro =
Table 12-1 provides t and F values for various levels of desired confidence
Table 12-1 t and F Values for Mean Confidence LevelsConfidence t (n = 2) t (n = 5) t (n = 10) t (n = 25) F (n ge 30)
68 1837 1142 1059 1021 1
95 13968 2869 2320 2110 2
997 235811 6620 4094 3345 3
170 Part IV Assessing the Right Approach to Your Analysis
19_045191 ch12qxd 81606 1128 PM Page 170
171Chapter 12 Narrowing Your Inputs with Confidence
Q Yoursquove taken 32 parts at random from ashipment of 10000 switches The platingthickness measurements for the 32 partsare given below With 95 percent confi-dence whatrsquos the mean plating thicknessfor the entire 10000-part shipment
0068 0075 0067 0069
0064 0065 0068 0067
0071 0067 0071 0080
0076 0062 0068 0074
0076 0061 0068 0082
0079 0065 0068 0067
0059 0066 0077 0078
0069 0059 0070 0062
A The first thing you must recognize is thatyour sample is large enough (greater than30 data points) that you can use the F for-mula for calculating the confidence inter-val for the mean Reviewing that formulayou need to know values for the samplemean (x ) F for a confidence level of 95percent the sample standard deviation(s) and the number of data points in yoursample (n) x and s can be calculatedquickly from the 32 measurements(refer to Chapter 7 if necessary) FromTable 12-1 the F value for 95 percent confidence is 2 Now just plug thesevalues into the formula
x Znsmicro 0 069 2
320 006
0 069 0 002 0 067 0 071
= = =
= 7 A
Even though the calculated mean for thesample is exactly 0069 you havenrsquot meas-ured every switch in the 10000-part pop-ulation which means that you donrsquot haveexact certainty in estimating the popula-tion mean What you can say with 95 per-cent confidence though is that the truepopulation mean lies between 0067 and0071
Q Suppose now that yoursquove received a second10000-part shipment of the same type ofswitches as in the last example problemThis time you take a 25-piece sample andfind the sample mean (x) to be 0074 andthe sample standard deviation (s) to be0005 With a 95 percent level of confidencecan you tell whether a difference existsbetween the average plating thickness forthe two 10000-part shipments
A For an n = 25 sample you have to usethe t formula to calculate the confidenceinterval for the population mean FromTable 12-1 the t value for a 25-pointsample at 95 percent confidence is 211So you can figure the confidence intervalfor the mean as follows
x tnsmicro 0 074 2 110
250 005
0 074 0 002 0 072 0 076
= =
= = 7 A
Comparing the confidence intervals forthe two shipments you can see that theydonrsquot overlap mdash the predicted range forthe first shipment is completely differentfrom the predicted range for the secondshipment This indicates that with 95 percent confidence you can say that adifference does exist between the averageplating thickness of the two shipmentswith the second shipment being 0004thicker on average than the first
If the two confidence intervals in theexample had overlapped you would havehad to conclude that no differenceexisted between the averages of the twoshipments Overlap in the confidenceintervals is what you look for when com-paring means
19_045191 ch12qxd 81606 1128 PM Page 171
172 Part IV Assessing the Right Approach to Your Analysis
1 You have received material samples fromfour different suppliers mdash A B C and DThe compiled data on the density of eachsupplierrsquos samples is summarized below
Supplier Sample Sample Sample StdSize (n) Mean ( x) Dev (s)
A 60 107 0950
B 10 120 0048
C 25 98 0220
D 34 100 0300
Calculate the 997 percent confidence inter-val for each population mean and identifyany differences in average density betweenthe suppliers
Solve It
2 Yoursquove developed two alternative curingprocesses for an adhesive Now you needto determine which one results in a highershear strength for the adhesive Knowingthat yoursquoll conduct 25 tests using eachprocess and estimating that the standarddeviation of both processes will be 12pounds how large must the difference bebetween the sample means of each of the25 tests before you declare with 95 percentconfidence that one process is better thanthe other
Solve It
3 Calculate the 997 percent confidence inter-val for the population mean from the fol-lowing sample data set
476 373 699 468
598
Solve It
4 You receive an e-mail from a colleague stat-ing that Process A is better than Process Bbecause the average cycle time of ProcessA is better than the average cycle time ofProcess B What other information do youneed to know to verify your colleaguersquosclaim
Solve It
19_045191 ch12qxd 81606 1128 PM Page 172
Calculating Confidence Intervalsfor Standard Deviations
Just as with the mean of a sample the standard deviation of a sample (s) doesnrsquotexactly tell you what the true population standard deviation (σ) is But you can bindyour calculation with a confidence interval surrounding the true value And just aswith confidence intervals for means the more data points you have in your samplethe tighter your confidence interval for the population standard deviation will be
Confidence intervals for standard deviations are based on the χ2distribution
n s n s
σ1 1
LOWER UPPER2
2
2
2
=- -
| |^ ^h h
R
T
SSS
V
X
WWW
Table 12-2 provides χ2 values for common sample sizes and confidence levels
Table 12-2 χ2 Values for Standard Deviation Confidence Intervals
Confidence n = 2 n = 5 n = 10 n = 25
68 1987 6599 13088 308330040 1416 4919 17169
95 5187 11365 19301 397490001 0460 2628 12225
997 10273 17800 27093 501630000 0106 1241 8382
Note The first value listed in each table cell is χ2LOWER The second value listed in each cell is χ2
UPPER
When comparing two sample standard deviations you can determine which popula-tion standard deviation is larger by creating a confidence interval around the ratio oftheir corresponding sample variances If the confidence interval includes the value of1 no detectible difference exists between the population standard deviations But ifthe range of the confidence interval ends up being greater than 1 the population stan-dard deviation from the numerator of your ratio is greater than the population stan-dard deviation from the denominator And the reverse is also true If the range of theconfidence interval ends up being less than 1 the population standard deviation fromthe numerator of your ratio is smaller than the population standard deviation from thedenominator
Confidence intervals for the ratio of two variances are based on the F distribution
F c n r n ss
F c n r nss
σσ 1
2212
2 1 2212
1 22212
== =
= =_
_i
i
R
T
SSS
V
X
WWW
Table 12-3 provides F distribution values for common sample sizes at a confidencelevel of 95 percent
173Chapter 12 Narrowing Your Inputs with Confidence
19_045191 ch12qxd 81606 1128 PM Page 173
Table 12-3 F Distribution Values for 95 Confidence Intervals with Various Sample Sizes
c = 2 c = 5 c = 10 c = 25
r = 2 161446 224583 240543 249052
r = 5 7709 6388 5999 5774
r = 10 5117 3633 3179 2900
r = 25 4260 2776 2300 1984
When yoursquore given an F value to look up it will be in the form of F(c r) which helpsyou determine which column and row to look in For example the first value given forlooking up F mdash the ldquocrdquo in F(c r) mdash tells you which column to look in The second valuegiven mdash the ldquorrdquo in F(c r) mdash tells you which row to look in So F(10 25) would be in thecolumn corresponding to c = 10 and the row corresponding to r = 25 giving a valueof 2300
174 Part IV Assessing the Right Approach to Your Analysis
Q You have a 10-point sample of purchase order data from a longtime supplier to your companyACME Engineering Services Ltd Herersquos the data
$2930946 $3293589 $2497775 $3251855
$2706133 $3275621 $2946138 $2750059
$3056179 $3398621
Calculate the 95 percent confidence interval for the population standard deviation of this data
A Reviewing the formula for the confidence interval for a standard deviation you need to knowthe sample standard deviation (s) the size of the sample (n) and the correct χ2
LOWER and χ2UPPER
values corresponding to your sample size and to 95 percent confidence The calculated samplestandard deviation (refer to Chapter 7 if necessary) is $297253 You need to look up the χ2
values from Table 12-2 For 95 percent confidence and a sample size of n = 10 the values areχ2
LOWER = 19301 and χ2UPPER = 2628 Now just plug these values into the formula as follows
$
$
$ $
n s n sσ
1 119 301
10 1 2 972 532 628
10 1 2 972 53
2 029 82 5 500 91
LOWER UPPER2
2
2
2 2 2
=- -
=- -
=| |
^ ^ ^ ^h h h hR
T
SSS
R
T
SSS
7
V
X
WWW
V
X
WWW
A
So from your 10-point sample you can say with 95 percent confidence that the true populationstandard deviation lies somewhere between $202982 and $550091
Donrsquot be surprised if your calculated confidence interval for a standard deviation looks very wideUnless you have a lot of data points in your sample your confidence interval will be wide because esti-mates of standard deviations are always less accurate than estimates of means
19_045191 ch12qxd 81606 1128 PM Page 174
175Chapter 12 Narrowing Your Inputs with Confidence
5 Calculate a 997 percent confidence inter-val for the population standard deviationof the following cycle time data
202 153 137 169
181 180 165 164
170 177 151 174
183 182 184 209
148 172 182 156
142 169 173 162
215
Solve It
6 Calculate the 95 percent confidence inter-val for the population standard deviationof the following cycle time data
1691 1694 1712 1694
1696 1713 1697 1693
1700 1693
Solve It
Q Now in addition to the supplier from the previous example suppose that you have a second sup-plier Jones Brothers LLC with a standard deviation from a 25-point sample of their historicalpurchase orders of $3500 With 95 percent confidence is there a difference in the variation ofthese two suppliers
A To compare the width of the populations of these two suppliers you create a confidence intervalfor the ratio of their variances You need to know the sample standard deviations for each sup-plier (s1 and s2) the size of the sample for each supplier (n1 and n2) and the F statistics followingthe pattern in the formula for the confidence interval After you have all of the required numbersyou can plug them into the formula like this
$ $
$ $
F c n r n ss
F c n r nss
F c r ss
F c rss
σσ 1
25 101 10 25
2 9001
3 500 002 972 53 2 300
3 500 002 972 53 0 249 1 659
2212
2 1 2212
1 22212
2212
2212
2
2
2
2
== =
= = == =
= = =
=
__
__
ii
ii
R
T
SSS
R
T
SSS
= 7
V
X
WWW
V
X
WWW
G A
Reviewing the calculated limits of the confidence interval you can say that with a confidence of95 percent that the ratio includes the value of 1 So you know that the width of the variation ofthe first supplier is no different than the width of the second supplier
19_045191 ch12qxd 81606 1128 PM Page 175
176 Part IV Assessing the Right Approach to Your Analysis
7 Create a 95 percent confidence interval forthe ratio of the variance from Problem 5 tothe variance from Problem 6
Solve It
8 At a 95 percent confidence level is there adifference in the population variances ofthe data in Problems 5 and 6
Solve It
Four Out of Five Recommend UsingConfidence Intervals for Proportions
Sometimes the data yoursquove collected creates a proportion such as 8 ldquogoodrdquo disk drivesout of 10 disk drives inspected The group of 10 disk drives inspected in this exampleis a sample of the larger population of total disk drive items produced So you cancreate a confidence interval You can also create confidence intervals around the dif-ference between two proportions
If y is the number of items identified out of the total number inspected (n) the propor-tion is written mathematically as
ny
The confidence interval around the true population proportion (p) is written as
p n
yZ n
y n y n1=
-_ _i i
To compute p all you need to know is y (the number of items identified out of yoursample) n (the actual number of items in your sample) and F (the sigma value corre-sponding to the level of confidence you want your interval to have)
The confidence interval for the difference between two proportions is written as
ny
ny
Z ny n y n
ny n y n1 1
1
1
2
2
1
1 1 1 1
2
2 2 2 2-
-+
-_ _ _ _i i i i
In reality proportions can never be less than zero or greater than one So if the calcu-lated confidence interval for your proportion exceeds these natural limits just adjustthe confidence interval to the natural limit
19_045191 ch12qxd 81606 1128 PM Page 176
177Chapter 12 Narrowing Your Inputs with Confidence
Q You do a quick survey of five of your department colleagues as to whether they like the neworganization structure that your company has just announced You find that three out of five ofthe colleagues think that the new organization structure will work better Create a F = 164 (90 per-cent) confidence interval for the true proportion of the companyrsquos entire workforce populationthinking the same way
A Recognizing that y = 3 n = 5 and F = 164 you plug these values into the equation
p n
yZ n
y n y n153 1 64 5
3 5 1 3 50 6 0 36 =
-=
-=
_ _ _ _i i i i
Q A survey of a different department at the same company explained in the previous exampleresults in 7 colleagues out of 8 sampled thinking that the new organization structure will be betterWith 90 percent confidence (F = 164) is there really any difference between the feelings of thesetwo departments
A To find out just plug the values into the confidence interval equation for the difference of two proportions
p p ny
ny
Z ny n y n
ny n y n1 1
53
87 1 64 5
3 5 1 3 58
7 8 1 7 80 275 0 407
1 21
1
2
2
1
1 1 1 1
2
2 2 2 2
- = --
+-
=
--
+-
= -
_ _ _ _
_ _ _ _
i i i i
i i i i
Because this confidence interval spans the value of zero with 90 percent confidence you can saythat even though the sample proportions look different no difference exists between the true pop-ulation proportions
9 You review 17 purchase order forms fromthe previous year Out of the 17 forms 12have no mistakes Calculate the 95 percentconfidence interval for the true populationproportion of correct purchase orders forthe previous year
Solve It
10 Yoursquove just received a shipment of 2000shock sensors Your plan is to place thesesensors into some of your own shipmentsto see whether theyrsquore exceeding the 10gshock limit your customers require of youThe shock sensors work like this Capsulesinside the sensors break when they experi-ence any shock above 10grsquos which causesa viewing tube to fill with dye So you take12 shock sensors from your shipment andgive them a known shock of 15grsquos Eleven ofthe shock sensors properly break whichindicates that the 10g shock level wasexceeded Calculate the 95 percent confi-dence interval for the proportion of theentire shipment that will work properly
Solve It
19_045191 ch12qxd 81606 1128 PM Page 177
178 Part IV Assessing the Right Approach to Your Analysis
11 Your marketing team has conducted twofocus group sessions to evaluate a newproduct design The first focus group con-sisted of teenagers and resulted in four outof five of the teens liking the new productThe second group was made up of parentsof young children Only one out of fivepeople in this group said they liked thesame new product as the teens With 90percent confidence (F = 164) can you saythat there is a difference between the viewsof these two populations
Solve It
12 Four out of five and 40 out of 50 are thesame proportion So why should you treatthese proportions differently
Solve It
19_045191 ch12qxd 81606 1128 PM Page 178
Solutions to Narrowing Inputs Problemsa You have received material samples from four different suppliers mdash A B C and D The com-
piled data on the density of each supplierrsquos samples is summarized below
Supplier Sample Size (n) Sample Mean ( x ) Sample Std Dev (s)
A 60 107 0950
B 10 120 0048
C 25 98 0220
D 34 100 0300
Calculate the 997 percent confidence interval for each population mean and identify any differ-ences in average density between the suppliers
Start with Supplier A Its sample has 60 data points so itrsquos large enough that you can use the Fformula for the confidence interval for the mean All the values needed for the calculation aregiven in the problem statement Plug the numbers into the formula like this
x Znsmicro 10 7 3
600 501 10 7 0 37 10 33 11 07A A
A
A = = = = 7 A
Suppliers B and C both have too few samples to use the F formula Instead you have to use thet formula With a 997 percent confidence level you have to look up the t value for a sample sizeof 10 for Supplier B and a sample size of 25 for Supplier C Here are the calculations
x tns
x tns
micro
micro
12 0 4 09410
0 048 12 0 0 06 11 94 12 06
9 8 3 34525
0 22 9 8 0 15 9 65 9 95
B B n
B
B
C C n
C
C
0 997 10
0 997 25
= = = =
= = = =
=
=
7
7
A
A
Supplier D has enough points in its sample to use the F formula Here are the calculations
x Znsmicro 10 0 3
340 300 10 0 0 15 9 85 10 15D D
D
D = = = = 7 A
Compare these confidence intervals graphically by lining up each of the confidence intervalson a chart as in the following figure
A
900
1000
1100
1200
1300
B C D
Supplier
Den
sity
179Chapter 12 Narrowing Your Inputs with Confidence
19_045191 ch12qxd 81606 1129 PM Page 179
Whenever confidence intervals overlap you can say that no significant difference existsbetween the populations Suppliers C and D for example overlap meaning theyrsquore the sameSuppliers A and B on the other hand have obvious gaps between them So theyrsquore differentfrom each other and from Suppliers C and D
b Yoursquove developed two alternative curing processes for an adhesive Now you need to determinewhich one results in a higher shear strength for the adhesive Knowing that yoursquoll conduct 25tests using each process and estimating that the standard deviation of both processes will be12 pounds how large must the difference be between the sample means of each of the 25 testsbefore you declare with 95 percent confidence that one process is better than the other
The average performance of the adhesives will be statistically different if their confidence inter-vals donrsquot overlap With the same standard deviation the same number of samples and thesame t value the minimum difference between the means would have to be as follows
tns2 2 2 110
251 2 1 013 n0 997 25 = == $
If the sample averages of the adhesives are different by 1013 or more the gap between the confidence intervals will verify that the populations are truly different
c Calculate the 997 percent confidence interval for the population mean from the followingsample data set
476 373 699 468
598
With a small sample size of only five data points simply follow the t formula for calculatingthe confidence interval for the population mean First calculate the sample mean and standarddeviation and then plug them into the formula like this
x tnsmicro 5 23 6 620
51 271 5 23 3 76 1 46 8 99 n0 997 5 = = = == 7 A
d You receive an e-mail from a colleague stating that Process A is better than Process B becausethe average cycle time of Process A is better than the average cycle time of Process B Whatother information do you need to know to verify your colleaguersquos claim
A sample average without a confidence interval doesnrsquot provide enough information When youcompare the processes without creating a confidence interval your conclusions may end upbeing completely wrong The information you need to verify your colleaguersquos claim is the stan-dard deviation sample size and required confidence level for the decision being made Withthis information the confidence interval can be created and the processes validly compared
e Calculate a 997 percent confidence interval for the population standard deviation of the follow-ing cycle time data
202 153 137 169
181 180 165 164
170 177 151 174
183 182 184 209
148 172 182 156
142 169 173 162
215
180 Part IV Assessing the Right Approach to Your Analysis
19_045191 ch12qxd 81606 1129 PM Page 180
The χ2 distribution formula is used to calculate the confidence interval around the standarddeviation Just plug the values into the formula like this
n s n s
σ1 1
50 16325 1 1 912
8 38225 1 1 912
1 323 3 236LOWER UPPER2
2
2
2 2 2
=- -
=- -
=| |
$ $^ ^ ^ ^h h h hR
T
SSS
R
T
SSS
7
V
X
WWW
V
X
WWW
A
With this confidence interval you know with 997 percent confidence that the true populationstandard deviation lies between these values
f Calculate the 95 percent confidence interval for the population standard deviation of the fol-lowing cycle time data
1691 1694 1712 1694
1696 1713 1697 1693
1700 1693
The χ2 distribution formula is used to calculate the confidence interval around the standarddeviation Just plug the values into the formula as follows
n s n s
σ1 1
19 30110 1 0 080
2 62810 1 0 080
0 054 0 147LOWER UPPER2
2
2
2 2 2
=- -
=- -
=| |
$ $^ ^ ^ ^h h h hR
T
SSS
R
T
SSS
7
V
X
WWW
V
X
WWW
A
With this confidence interval you know with 95 percent confidence that the true populationstandard deviation lies between these values
g Create a 95 percent confidence interval for the ratio of the variance from Problem 5 to the vari-ance from Problem 6
A confidence interval for the ratio of variances is based on the F distribution You need to knowthe variances for each of the two samples and you need to know the correct F value based onthe sizes of the two samples
For this problem the sample size from Problem 5 is n1 = 25 and the sample size from Problem 6is n2 = 10 Using the previously calculated values for the sample standard deviations fromProblems 5 and 6 you can plug these values into the F value equation like this
F c n r n ss
F c n r nss
σσ 1
2 3001
0 0801 912 2 900
0 0801 912 250 6 1 671 7
2212
2 1 2212
1 22212
2
2
2
2
== =
= = = =_
_i
i
R
T
SSS
lt 7
V
X
WWW
F A
h At a 95 percent confidence level is there a difference in the population variances of the data inProblems 5 and 6
The ratio of these two variances and the surrounding confidence interval show that the widthof the variation from Problem 5 is much larger than the width of the variation in Problem 6
If the two variations were the same the confidence interval of their ratio would have includedthe value of 1
i You review 17 purchase order forms from the previous year Out of the 17 forms 12 have nomistakes Calculate the 95 percent confidence interval for the true population proportion ofcorrect purchase orders for the previous year
To create this confidence interval for the true population proportion you just need to plug thevalues into the formula Remember that a confidence level of 95 percent corresponds to a Fvalue of 2 Here are the calculations
p n
yZ n
y n y n11712 2 17
12 17 1 12 170 71 0 22 0 48 0 93 =
-=
-= =
_ _ _ _i i i i7 A
181Chapter 12 Narrowing Your Inputs with Confidence
19_045191 ch12qxd 81606 1129 PM Page 181
j Yoursquove just received a shipment of 2000 shock sensors Your plan is to place these sensors intosome of your own shipments to see whether theyrsquore exceeding the 10g shock limit your cus-tomers require of you The shock sensors work like this Capsules inside the sensors breakwhen they experience any shock above 10grsquos which causes a viewing tube to fill with dye Soyou take 12 shock sensors from your shipment and give them a known shock of 15grsquos Elevenof the shock sensors properly break which indicates that the 10g shock level was exceededCalculate the 95 percent confidence interval for the proportion of the entire shipment that willwork properly
In this exercise the total number inspected (n) is equal to 12 and the number of successes (y) isequal to 11 So the 95 percent (F = 2) confidence interval for the true proportion is as follows
p n
yZ n
y n y n11211 2 12
11 12 1 11 120 92 0 16 0 76 1 00 =
-=
-= =
_ _ _ _i i i i7 A
Notice that on the high end of the confidence interval the value is capped at the theoreticallimit for proportions at 100
k Your marketing team has conducted two focus group sessions to evaluate a new productdesign The first focus group consisted of teenagers and resulted in four out of five of the teensliking the new product The second group was made up of parents of young children Only oneout of five people in this group said that they liked the same new product as the teens With 90percent confidence (F = 164) can you say that there is a difference between the views of thesetwo populations
This exercise requires you to use the formula for calculating the confidence interval around thedifference of two proportions Itrsquos just a matter of plugging in the right values for the first andsecond proportions Herersquos the formula
p p ny
ny
Z ny n y n
ny n y n1 1
54
51 1 64 5
4 5 1 4 55
1 5 1 1 50 60 0 41 0 19 1 00
1 21
1
2
2
1
1 1 1 1
2
2 2 2 2
- = --
+-
=
--
+-
= =
_ _ _ _
_ _ _ _
i i i i
i i i i7 A
Because the calculated confidence interval for the difference is positive you know that there isin fact a difference between the teenager and the young parent populations (Not that youneeded a statistical study to tell you something that obvious)
l Four out of five and 40 out of 50 are the same proportion So why should you treat these pro-portions differently
The answer to this problem is all in the size of the samples Four out of five produces a muchless confident measure of the true proportion than 40 out of 50 does
182 Part IV Assessing the Right Approach to Your Analysis
19_045191 ch12qxd 81606 1129 PM Page 182
Part VImproving and Controlling
20_045191 pt05qxd 81606 1142 PM Page 183
In this part
The final stages of DMAIC are Improve and Control Skillsfor synthesizing improvements can be mastered by
anyone through the exercises and worksheets provided inthis part And finally you learn Control mdash you becomeproficient in the skills of maintaining and sustaining theimprovements yoursquove implemented
20_045191 pt05qxd 81606 1142 PM Page 184
Chapter 13
Quantifying Variable RelationshipsIn This Chapter Calculating how much correlation there is between variables
Determining equations with curve fitting
Making sure fitted lines are valid
After yoursquove defined your most important process or product outputs (Ys) and thoseprocess or product inputs most likely to affect them (Xs) you need to understand and
quantify the relationship between them The first step is to identify which Xs are correlatedto Y You can get hints about correlations by looking at scatter plots discussed in Chapter 8But to be sure you need to quantify the correlation and test for statistical significance Youcan also develop an actual line equation mathematically relating X to Y This chapter coversthese topics
Quantifying Correlation between VariablesCorrelation is a frequent and popular subject of the news ldquoHigh income linked to schoolingoutcomesrdquo ldquoLow-fat diets correlated with reduced heart diseaserdquo and ldquoSodas contribute toobesityrdquo Almost every day a headline reports on a new study linking two variables togetherin a cause-effect relationship
Correlation is all about quantifying the strength of the relationship between two variablesYou want to know whether itrsquos a loose relationship or whether itrsquos so strong that by knowingone variable you can predict the other confidently
To quantify the linear relationship between two variables (x and y) you use the following for-mula to calculate their correlation coefficient (r) The equation may seem complicated but itrsquosactually quite simple when you break it down
r nx x y y
σ σ11
x
i
y
i
i
n
1=
-- -
=
d dn n
where n is the number of data pair measurements
xi and yi are the individual x-variable and y-variable measurements taken at the same time orwithin the same subject to create a data pair
x and y are the averages of the x- and y-variable measurements respectively
σx and σy are the standard deviations of the x- and y-variable measurements respectively and
Σ is the Greek letter telling you to sum up all the x x y y
σ σx
i
y
i- -d dn n terms from 1 to n
21_045191 ch13qxd 81606 1136 PM Page 185
The calculated correlation coefficient value has some interesting properties
The calculated r will always be between ndash1 and 1
The sign of r tells you the direction of the relationship between the variables
If r is positive then when one variable increases in value the other variable willalso increase The opposite is also true If one variable decreases the other vari-able will also decrease This is called a positive correlation
If r is negative then when one variable increases in value the other variable willdecrease and vice versa This is called a negative correlation
The absolute value of r tells you how strong the relationship between the vari-ables is
The closer r gets to its theoretical limits of ndash1 or 1 the stronger the correlationis An r equal to ndash1 or 1 indicates a perfect linear relationship with all the x-ypoints lying exactly on a straight line
An r close to 0 indicates an absence of a linear fit or correlation to the data
186 Part V Improving and Controlling
Q An HR manager has just finished a Six Sigma training course Shersquos been working on understand-ing the perception among new hires that the hiring process the company uses ldquostinksrdquo Over thelast year the HR manager has had all new hires complete a survey rating their hiring experienceA score of 20 on the survey means that a new hire thought the hiring experience was wonderfulwhereas a survey score of 0 means that the new hire thought the process was awful On a hunchthe HR manager has created data points for each employee hired over the last year combininghow many days it took the person to be hired with the personrsquos new-hire survey score The tabu-lated and plotted forms of the following data pairs are shown in the accompanying figure
Number of days to hire Hiring process satisfaction
38 9
32 15
41 13
48 12
47 7
50 4
27 15
39 9
47 7
33 16
36 7
30 11
21_045191 ch13qxd 81606 1136 PM Page 186
187Chapter 13 Quantifying Variable Relationships
Table 13-1 Table Used to Calculate the Correlation Coefficient for the Example Problem
xi yi σx x
x
i -σ
y yy
i -σ σ
x x y yx
i
y
i- -d dn n
38 9 ndash013 ndash037 005
32 15 ndash091 119 ndash108
41 13 026 067 017
48 12 117 041 048
47 7 104 ndash089 ndash092
50 4 143 ndash167 ndash238
27 15 ndash156 119 ndash185
39 9 000 ndash037 000
47 7 104 ndash089 ndash092
33 16 ndash078 145 ndash113
36 7 ndash039 ndash089 035
30 11 ndash117 015 ndash018
Averages ( x y ) 3900 1042 σ σx x y y
x y
i
i
2
1
12 - -
=
d dn n ndash741
Standard 771 385 σ σr nx x y y
11
x
i
i y
i
1
12
=-
- -
=
d dn n ndash067Deviations (σx σy)
Do you find a correlation between the number of days from when the employees submitted theirapplications to the start date of their employment and their perceived satisfaction with the hiringprocess
A Calculating the correlation coefficient (r) between the two variables will quantify the strength oftheir relationship See Table 13-1
250
20
15
10
5
30 35 40 45 50 55
Time from Receiving Application to Employment Start Date (Days)
Em
plo
yeersquo
s H
irin
g Sa
tisf
acti
on S
urv
ey
21_045191 ch13qxd 81606 1137 PM Page 187
1 An improvement project leader is trying tounderstand whether the length of time anassembled top cover is cured in an ovenrelates to the final strength of the adhesivebond of its assembly The leader has col-lected data on some assemblies mdash howlong they were allowed to cure in the ovenand how strong each adhesive bond endedup being after curing Here are the datapairs for each assembly
Cure Time Bond Strength (minutes) (grams)
24 150
23 140
21 150
17 109
26 143
22 146
22 124
22 141
20 127
22 122
15 88
21 128
17 100
10 82
20 127
Quantify the correlation and describe therelationship between these two variablesGet your scrap paper ready
Solve It
2 You suspect that the resistance of a certaincircuit board component is leading to prob-lems with the output current You grab 10circuit boards and measure the resistance ofthis component and then record their corre-sponding output voltage Here is your data
Resistance (KΩ) Output Current (A)
506 93
505 89
501 90
511 96
522 90
509 96
517 90
503 94
518 94
499 91
Is the resistance of the component corre-lated with the output current of the circuit
Solve It
An r of ndash067 indicates a medium level of correlation between the time it takes to hire the employeesand their satisfaction with the hiring process Also because this value is negative you know that if thehiring time increases the employeersquos satisfaction is likely to decrease So to improve satisfaction theHR manager should work to shorten the hiring time
188 Part V Improving and Controlling
21_045191 ch13qxd 81606 1137 PM Page 188
189Chapter 13 Quantifying Variable Relationships
3 A production manager believes that thequality of machined parts defined by thenumber of defects found out of 100 partsinspected is related to the hours of train-ing the machine operator has had Here aredata pairs for 18 machine operators
Training (Hours) Defects (Per 100)
81 5
84 3
76 3
82 7
86 9
89 6
96 3
75 6
83 7
77 7
95 1
97 1
79 7
79 5
83 7
97 5
99 7
85 11
Calculate the correlation between these twovariables and describe their relationship
Solve It
4 A news story reports that reading compre-hension has been found to be correlated toa personrsquos height Whatrsquos your response tothis reported cause-effect relationship
Solve It
21_045191 ch13qxd 81606 1137 PM Page 189
190 Part V Improving and Controlling
Fitting Lines to Variable RelationshipsCurve fitting goes a step beyond correlation In curve fitting you actually determine theequation for the curve that best fits your x-y data Armed with this information youcan quantitatively predict what effect one variable will have on another
The most basic curve to fit to your data is a straight line The equation for a line fittingyour data can be written as y
= β0 + β1x + ε where β0 defines the intercept where the linecrosses the y axis and β1 defines the slope of the line ε represents all the variation thatisnrsquot due to the linear relationship (meaning what the fitted line doesnrsquot capture mdash flipto the next section for more about ε) You calculate β0 and β1 from these equations
x x
x x y
y x
β
β β
ii
n
i ii
n
1 2
1
1
0 1
=-
-
= -
=
=
_
_
i
i
where xi and yi are the paired data points and x and y are the calculated averages forall the x data points and all the y data points respectively
Q The HR manager from the example problem in the previous section now wants to fit a line to thehiring time ndash satisfaction data Find the equation for this line
A A calculation table like Table 13-2 helps in computing the values for β0 and β1
Table 13-2 Table Used To Calculate β0 and β1 For Example Problem
xi yi xi ndash x (xi ndash x )yi (xi ndash x ) 2
38 9 ndash1 ndash9 1
32 15 ndash7 ndash105 49
41 13 2 26 4
48 12 9 108 81
47 7 8 56 64
50 4 11 44 121
27 15 ndash12 ndash180 144
39 9 0 0 0
47 7 8 56 64
33 16 ndash6 ndash96 36
36 7 ndash3 ndash21 9
30 11 ndash9 ndash99 81
Averages (x y ) 3900 1042 Σ ndash220 654
β1 ndash034
β0 2354
21_045191 ch13qxd 81606 1137 PM Page 190
191Chapter 13 Quantifying Variable Relationships
You start the calculation table by computing the mean or average for both the x and y data After themean and average are calculated they feed into the calculation of the other columns At the endyou sum up the (xi ndash x)yi and (x ndash x)2 columns As a final step you divide the sum of the (xi ndash x)yi
column by the sum of the (x ndash x)2 column This calculation gives you the β1 term Knowing β1 youuse the formula β0 = y ndash β1 x to solve for β0
By plugging in the values you calculated you find that the equation for the best-fit line throughthe data is y = 2354 ndash 034x
You can plot this line through the set of points to see how it fits as in the following figure As youcan see by the figure the line does in fact fit
250
20
15
10
5
30 35 40 45 50
y = -034x + 2354
55
Time from Receiving Application to Employment Start Date (Days)
Em
plo
yeersquo
s H
irin
g Sa
tisf
acti
on S
urv
ey
5 Calculate the equation for a line fitting the cure time ndash bond strength data inProblem 1 Plot the line on a graph alongwith the original data Make sure you haveextra paper on hand
Solve It
6 Using the line equation you created inProblem 5 calculate the expected bondstrength for an assembly that had a curetime of 17 minutes
Solve It
21_045191 ch13qxd 81606 1137 PM Page 191
7 Calculate the equation for the line fittingthe training hours ndash part quality data inProblem 3 Plot the line on a graph alongwith the original data Keep that scrappaper handy
Solve It
8 Using the line equation you created inProblem 7 calculate the expected defectsper 100 parts for an operator with 9 hoursof training
Solve It
192 Part V Improving and Controlling
Assessing the Adequacy of a Fitted LineFitting a line to your data isnrsquot always statistically valid Sometimes no significant rela-tionship exists and sometimes a line isnrsquot the right type of curve to fit your data Soeach time you fit a line to your data you need to check to make sure that the result isstatistically valid
You need to remember several things about checking the adequacy of your fitted linemodel
Residuals are the errors between your actual data and the prediction of yourfitted line model Each point in your data set has a residual (ei) which can bewritten mathematically as
ei = yi ndash y
i
where yi are the actual observed data values and y
i are the predicted valuesfrom your fitted line equation
To determine whether your fitted line equation is statistically valid you need toplot the residuals in several different ways to make sure they appear normallydistributed which is the criteria for your line equation to be statistically validUsing your residuals create the following plots
bull A scatter plot of the residuals (ei) versus the predicted values ( y
i) fromyour line equation
bull A scatter plot of the residuals (ei) versus the observed xi data
bull If you collected your observed xi yi data sequentially over time a run orderchart of each of the residuals (ei) versus their preceding residuals (eindash1)
bull Additional scatter plots of the residuals (ei) versus other x variables thatyou didnrsquot include in your equation
21_045191 ch13qxd 81606 1137 PM Page 192
Calculate the coefficient of determination (R2) for your fitted line using this formula
Ry y
y y
ii
n
ii
n
2
2
1
2
1=-
-
=
=
_
b
i
l
R2 should generally be greater than 08 for a strong linear fit Lesser values of R2
should be used with caution for predictions
Calculate the F statistic with the following equation to test the statistical signifi-cance of the fitted line
n y y y y
y yF n
21
1 2i i
i
n
i
n
ii
n
2 2
11
2
1 $
-- - -
--
==
=
_ b
b
_
i l
l
i
= G
If the left side of this equation is greater than the right side F statistic from thelook-up table in Chapter 12 (Table 12-3) you can say that the line you fitted toyour data is statistically significant with 95 percent confidence
ε represents a random normal distribution centered at a value of zero This valueis an inherent part of your predictive linear equation An estimate of the standarddeviation of the ε distribution can be calculated using the following equation
n y y y yσ 21
i ii
n
i
n
ε
22
11=
-- - -
==
_ ci m
R
T
SSS
V
X
WWW
The estimate of the standard deviation of the ε distribution comes in handy when youwant to mimic what might happen in reality You use your derived linear model to pre-dict the average or expected performance of the output y
and then add to it a randomnumber generated from the ε distribution mdash with a mean of zero and standard devia-tion equal to σ ε In this way you can simulate what would happen if your process orcharacteristic were repeated over and over again
193Chapter 13 Quantifying Variable Relationships
Q The HR manager from the previous examples in this chapter wants to determine whether the lineequation she has fitted to her hiring time ndash satisfaction data is statistically valid Perform all thechecks to make sure her fitted line is valid
A In setting up to check the validity of a fitted line use a table to lay out all of the values and calcu-lations See Table 13-3 which was created for this hiring example
Table 13-3 Table for Calculating Standard Deviation of the Unexplained Variation in Your Line Model
xi yi yi
ei (yi
ndash y ) 2 (yi ndash y ) 2
38 9 1075 ndash175 011 201
32 15 1277 223 554 2101
41 13 974 326 045 667
48 12 739 461 917 251
47 7 773 ndash073 724 1167
(continued)
21_045191 ch13qxd 81606 1137 PM Page 193
194 Part V Improving and Controlling
The y
i values used in the above calculation table are found by plugging the xi values into the fittedline equation
Now plot the residuals (ei) versus the predicted fit values ( y
i) In this plot yoursquore checking tomake sure the residuals look normally distributed around a center of zero Take a look at the fol-lowing figure and determine whether the residuals look normally distributed
The residuals indeed look normally distributed Therersquos no clumping or patterns in the plotteddata mdash it all looks randomly scattered So far so good Now check the plot of the residuals versusthe xi data values as in the following figure
600 800 1000 1200 1400-600
600
400
200
000
-200
-400
1600
yi
e i
Table 13-3 (continued)xi yi yi
ei (yi
ndash y ) 2 (yi ndash y ) 2
50 4 672 ndash272 1369 4117
27 15 1445 055 1629 2101
39 9 1042 ndash142 000 201
47 7 773 ndash073 724 1167
33 16 1244 356 407 3117
36 7 1143 ndash443 102 1167
30 11 1344 ndash244 917 034
Averages (x y ) 3900 1042 Σ 7401 16292
21_045191 ch13qxd 81606 1137 PM Page 194
195Chapter 13 Quantifying Variable Relationships
Again these residuals all look normal and random with no clumps trends or patterns Becausethis example includes no other x variables and because the data measurements werenrsquot takensequentially you donrsquot need to check any other residual plots
Next calculate the coefficient of determination R2 for the fitted line You calculate R2 by dividingthe sum of the ( y
i ndash y)2 column by the sum of the (yi ndash y)2 column located at the right side of thecalculation table The result looks like this
R
y y
y y
162 9274 01 0 45
ii
n
ii
n
2
2
1
2
1=-
-= =
=
=
_
b
i
l
An R2 of 045 means that only 45 percent of the observed variation of the data is explained by thefitted line That percentage is not so hot If you used the equation of the fitted line to predictemployee satisfaction with the hiring process most of the time you wouldnrsquot be very close
A final test of validity can be done by calculating the F statistic with the following formula
n y y y y
y yF F n
21
12 21 162 92 74 01
74 01 8 32 4 96 1 10 1 2i i
i
n
i
n
ii
n
2 2
11
2
1 $
-- - -
-=
--
= = = -
==
=
_ b
b
_ _
i l
l
i i
= 6G
The story here is that because your calculated F statistic is greater than the look-up test statisticof F(110) from Table 12-3 (flip to Chapter 12) a weak linear relationship exists that is statisticallyvalid for your fitted line However the line only accounts for 45 percent of the total variation sothere may be errors or other factors that weaken the relationship This low percentage is a signalto investigate further before making predictions
A final piece of information you can glean from the fitted line is an estimate of the standard devia-tion of ε which is the portion of the line equation thatrsquos unaccounted-for variation You just plugthe values from the calculation table into the formula like this
n y y y yσ 21
12 21 162 92 74 01 2 98i i
i
n
i
n
ε
2 2
11=
-- - - =
-- =
==
_ bi l= 6G
So the full equation for the fitted line ends up looking like this
y = 2354 ndash 034x + N(0 298)
Overall you can say that the HR manager definitely has a statistically significant fitted line for thehiring data But she has to consider that fitted line only explains 45 percent of the observed varia-tion The rest is unaccounted-for random variation with a standard variation of 298
20 25 30 35 40 45 50-600
600
400
200
000
-200
-400
55
xi
e i
21_045191 ch13qxd 81606 1137 PM Page 195
196 Part V Improving and Controlling
9 Assess the adequacy of the line equationfrom the cure time ndash bond strength data inProblems 1 and 5
Solve It
10 Investigate the strength of the fitted line forthe training hours ndash product quality data inProblems 3 and 7
Solve It
21_045191 ch13qxd 81606 1137 PM Page 196
Solutions to Quantifying VariableRelationships Problems
a An improvement project leader is trying to understand whether the length of time an assem-bled top cover is cured in an oven relates to the final strength of the adhesive bond of itsassembly The leader has collected data on some assemblies mdash how long they were allowedto cure in the oven and how strong each adhesive bond ended up being after curing Here arethe data pairs for each assembly
Cure Time (minutes) Bond Strength (grams)
24 150
23 140
21 150
17 109
26 143
22 146
22 124
22 141
20 127
22 122
15 88
21 128
17 100
10 82
20 127
Quantify the correlation and describe the relationship between these two variables
To find the correlation coefficient you create a table that guides you through the calculationsSee Table 13-4 to check your own table
Table 13-4 Table Used for Calculating Correlation Coefficient
xi yi σx x
x
i -σ
y yy
i -σ σ
x x y yx
i
y
i- -d dn n
24 150 097 114 111
23 140 072 068 049
21 150 022 114 025
17 109 ndash079 ndash074 058
26 143 147 082 121
22 146 047 096 045
22 124 047 ndash005 ndash002
(continued)
197Chapter 13 Quantifying Variable Relationships
21_045191 ch13qxd 81606 1137 PM Page 197
Table 13-4 (continued)
xi yi σx x
x
i -σ
y yy
i -σ σ
x x y yx
i
y
i- -d dn n
22 141 047 073 034
20 127 ndash003 009 000
22 122 047 ndash014 ndash007
15 88 ndash129 ndash171 220
21 128 022 013 003
17 100 ndash079 ndash116 091
10 82 ndash255 ndash198 505
20 127 ndash003 009 000
Averages 2013 12513 σ σx x y y
x
i
i y
i
1
12 - -
=
d dn n 1253(x y )
Standard 398 2175 σ σr nx x y y
11
x
i
y
i
i 1
12
=-
- -
=
d dn n 090Deviations (σx σy)
After going through all of the calculations you should determine that the correlation coefficientr is 090 This value indicates that a strong positive correlation between the two variablesexists When cure time goes up bond strength goes up too When cure time goes down bondstrength also goes down
b You suspect that the resistance of a certain circuit board component is leading to problemswith the output current You grab 10 circuit boards and measure the resistance of this compo-nent and then record their corresponding output voltage Here is your data
Resistance (KΩ) Ouput Current (A)
506 93
505 89
501 90
511 96
522 90
509 96
517 90
503 94
518 94
499 91
Is the resistance of the component correlated with the output current of the circuit
To calculate r create a calculation table and compare it to Table 13-5
198 Part V Improving and Controlling
21_045191 ch13qxd 81606 1137 PM Page 198
Table 13-5 Table Used to Calculate the Correlation Coefficient
xi yi σx x
x
i -σ
y yy
i -σ σ
x x y yx
i
y
i- -d dn n
506 93 ndash040 027 ndash011
505 89 ndash053 ndash126 066
501 90 ndash104 ndash088 091
511 96 024 141 034
522 90 166 ndash088 ndash145
509 96 ndash001 141 ndash002
517 90 102 ndash088 ndash089
503 94 ndash079 065 ndash051
518 94 115 065 074
499 91 ndash130 ndash049 064
Averages 509 9230 σ σx x y y
x
i
y
i
i 1
12 - -
=
d dn n 033(x y )
Standard 008 263 σ σr nx x y y
11
x
i
y
i
i 1
12
=-
- -
=
d dn n 004Deviations (σx σy)
As you can see by Table 13-5 the correlation coefficient r for this set of data is 004 which isnrsquota good correlation between the variables
The next step would be to look for other stronger correlations test for statistical significanceor investigate higher-order relationships
c A production manager believes that the quality of machined parts defined by the number ofdefects found out of 100 parts inspected is related to the hours of training the machine opera-tor has had Here are data pairs for 18 machine operators
Training (Hours) Defects (Per 100) Training (Hours) Defects (Per 100)
81 5 77 7
84 3 95 1
76 3 97 1
82 7 79 7
86 9 79 5
89 6 83 7
96 3 97 5
75 6 99 7
83 7 85 11
Calculate the correlation between these two variables and describe their relationship
Use a table like Table 13-6 to calculate all the intermediate steps in the calculation
199Chapter 13 Quantifying Variable Relationships
21_045191 ch13qxd 81606 1137 PM Page 199
Table 13-6 Table for Calculating the Correlation Coefficient
xi yi σx x
x
i -σ
y yy
i -σ σ
x x y yx
i
y
i- -d dn n
81 5 ndash030 ndash029 009
84 3 017 ndash126 ndash022
76 3 ndash108 ndash126 136
82 7 ndash014 068 ndash010
86 9 048 165 080
89 6 095 019 018
96 3 205 ndash126 ndash258
75 6 ndash123 019 ndash024
83 7 002 068 001
77 7 ndash092 068 ndash062
95 1 189 ndash223 ndash421
97 1 220 ndash223 ndash490
79 7 ndash061 068 ndash041
79 5 ndash061 ndash029 018
83 7 002 068 001
97 5 220 ndash029 ndash064
99 7 251 068 170
85 11 033 261 086
Averages 829 560 σ σx x y y
x
i
y
i
i 1
12 - -
=
d dn n ndash132(x y )
Standard 064 207 σ σr nx x y y
11
x
i
y
i
i 1
12
=-
- -
=
d dn n ndash015Deviations (σx σy)
In this example the correlation coefficient r ends up being ndash015 Here the correlation betweenthe two variables is weak The correlation is also negative which means that when the x valueincreases the y values decrease and vice versa
d A news story reports that reading comprehension has been found to be correlated to apersonrsquos height Whatrsquos your response to this reported cause-effect relationship
In this practice problem you can see that someone has misunderstood the tool of correlationEven though a correlation exists numerically between a personrsquos height and his or her readingcomprehension itrsquos probably due to the fact that in general older taller individuals naturallyhave learned to read better than those people who are younger and shorter mdash height isnrsquot thecausing factor but instead is just an indicating factor
Correlation doesnrsquot automatically indicate cause
200 Part V Improving and Controlling
21_045191 ch13qxd 81606 1137 PM Page 200
e Calculate the equation for a line fitting the cure time ndash bond strength data in Problem 1 Plot theline on a graph along with the original data
To help in computing the values for β0 and β1 you should have created a calculation tableCompare your table to Table 13-7
Table 13-7 Table of Intermediate Values Used in Calculating β0 and β1
xi yi xi ndash x (xi ndash x)yi (xi ndash x ) 2
24 150 387 58000 1495
23 140 287 40133 822
21 150 087 13000 075
17 109 ndash313 ndash34153 982
26 143 587 83893 3442
22 146 187 27253 348
22 124 187 23147 348
22 141 187 26320 348
20 127 ndash013 ndash1693 002
22 122 187 22773 348
15 88 ndash513 ndash45173 2635
21 128 087 11093 075
17 100 ndash313 ndash31333 982
10 82 ndash1013 ndash83093 10268
20 127 ndash013 ndash1693 002
Averages (x y ) 2013 12513 Σ 108473 22173
From the calculation table you can plug the intermediate sums into the formulas for β0 and β1
x x
x x y
y x
β
β β
221 731084 73 4 89
125 13 4 89 20 13 26 64
andi
i
n
i ii
n
i
1 2
1
1
0
=-
-= =
= - = - =
=
=
$
_
_
i
i
The equation for the fitted line is
y
= β0 + β1x = 2664 + 489x
The next step is to plot the original data and add this line to the plot like in the followingfigure
201Chapter 13 Quantifying Variable Relationships
21_045191 ch13qxd 81606 1137 PM Page 201
You can see that the calculated line equation does indeed fit the data
f Using the line equation you created in Problem 5 calculate the expected bond strength for anassembly that had a cure time of 17 minutes
To estimate the bond strength for a 17-minute cure time you simply plug the value of 17 min-utes into the derived equation and calculate what the resulting bond strength output is Herersquosthe calculation
y
= 2664 + 489x = 2684 + 489 times 17 = 1098
g Calculate the equation for the line fitting the training hours ndash part quality data in Problem 3Plot the line on a graph along with the original data
A calculation table helps in computing the values for β0 and β1 See Table 13-8 to check yourown table
Table 13-8 Table of Intermediate Values Used in Calculating β0 and β1
xi yi xi ndash x (xi ndash x )yi (xi ndash x ) 2
81 5 ndash047 ndash236 022
84 3 ndash017 ndash052 003
76 3 ndash097 ndash292 095
82 7 ndash037 ndash261 014
86 9 003 025 000
89 6 033 197 011
96 3 103 308 106
75 6 ndash107 ndash643 115
83 7 ndash027 ndash191 007
77 7 ndash087 ndash611 076
95 1 093 093 086
5 10 15 20 25
y = 489x + 2664
70
160
150
140
130
120
110
100
90
80
30
Cure Time (minutes)
Bon
d S
tren
gth
(gr
ade)
202 Part V Improving and Controlling
21_045191 ch13qxd 81606 1137 PM Page 202
xi yi xi ndash x (xi ndash x )yi (xi ndash x ) 2
97 1 113 113 127
79 7 ndash067 ndash471 045
79 5 ndash067 ndash336 045
83 7 ndash027 ndash191 007
97 5 113 564 127
99 7 133 929 176
85 11 ndash007 ndash079 001
Averages (x y ) 829 560 Σ ndash1132 1064
From the calculation table you can plug the intermediate sums into the formulas for β0 and β1Here are the calculations
x x
x x y
y x
β
β β
10 6411 32 1 06
5 60 1 06 8 29 14 68
andi
i
n
i ii
n
1 2
1
1
0 1
=-
-= - = -
= - = + =
=
=
$
_
_
i
i
The equation for the fitted line is
y
= β0 + β1x = 1468 ndash 106x
The next step is to plot the original data and add this line to the plot like in the next figure
h Using the line equation you created in Problem 7 calculate the expected defects per 100 partsfor an operator with 9 hours of training
To estimate what the defects per 100 parts will be for an operator with 9 hours of training yousimply plug the input of 9 hours into the equation for the line like this
y
= 1468 ndash 106x = 1468 ndash 106 times 9 = 51
70 75 80 85 95
y = -106x + 1468
0
12
10
8
6
4
2
100
Training (Hours)
Def
ects
Per
100
Par
ts
90
203Chapter 13 Quantifying Variable Relationships
21_045191 ch13qxd 81606 1137 PM Page 203
i Assess the adequacy of the line equation from the cure time ndash bond strength data in Problems 1and 5
In setting up to check the validity of a fitted line itrsquos good to use a table to lay out all the valuesand calculations See Table 13-9 to check your own table
Table 13-9 Table for Calculating Residualsxi yi y
i ei (y
ndash y ) 2 (yi ndash y ) 2
24 150 14405 595 35781 61835
23 140 13916 084 19667 22102
21 150 12937 2063 1798 61835
17 109 10980 ndash080 23496 26028
26 143 15383 ndash1083 82370 31922
22 146 13427 1173 8339 43542
22 124 13427 ndash1027 8339 128
22 141 13427 673 8339 25175
20 127 12448 252 043 348
22 122 13427 ndash1227 8339 982
15 88 10002 ndash1202 63064 137888
21 128 12937 ndash137 1798 822
17 100 10980 ndash980 23496 63168
10 82 7556 644 245747 186048
20 127 12448 252 043 348
Averages (x y ) 2013 12513 Σ 530658 662173
Now plot the residuals (ei) column of the calculation table versus the predicted fit values ( y
i)column (see the next figure) In this plot yoursquore checking to make sure the residuals look nor-mally distributed around a center of zero
-2500
2500
2000
1500
1000
500
000
-500
-1000
-1500
-2000
yi
e i
60 80 100 120 140 160
204 Part V Improving and Controlling
21_045191 ch13qxd 81606 1137 PM Page 204
The scattering of the residuals versus the predicted values looks randomly distributed Noproblems here Now plot the residuals (ei) column of the calculation table versus the observedx values (xi) (see the next figure)
Still after studying the plot in the figure everything appears normally and randomly distrib-uted The residuals donrsquot point to any problems with the fitted line equation
Now calculate the coefficient of determination (R2) for the fitted line by dividing the sum of the( y
i ndash y)2 column by the sum of the (yi ndash y)2 column which is located at the right side of the cal-culation table The calculations look like this
Ry y
y y
6621 735306 58 0 80
ii
n
ii
n
2
2
1
2
1=-
-= =
=
=
_
b
i
l
An R2 of 080 means that 80 percent of the observed variation of the data is explained by thefitted line Thatrsquos just enough to have confidence in your predictions from the line If you usethe equation of the fitted line to predict employee satisfaction with the hiring process yoursquollbe fairly close to the right answer
A final test of validity can be done by calculating the F statistic Use the following formula
n y y y y
y y
F F n
21
15 21 6621 73 5306 58
5306 58
52 45 4 67 1 13 1 2
ii
n
ii
n
ii
n
2
1
2
1
2
1
$
-- - -
-=
--
=
= = -
= =
=
_ b
b
_ _
i l
l
i i
= 6G
Your calculated F statistic is greater than the look-up test statistic of F(113) from Table 12-3(see Chapter 12) so your fitted line is also statistically valid
A final piece of information you can glean from the fitted line is an estimate of the standarddeviation of ε which is the portion of the line equation thatrsquos unaccounted-for variation Youjust plug the values from the calculation table into the formula like this
n y y y yσ 21
15 21 6621 73 5306 58 10 06i i
i
n
i
n
ε
2 2
11=
-- - - =
-- =
==
_ bi l= 6G
So the full equation for the fitted line ends up being
y = 2644 + 489x + N(0 1006)
-2500
2500
2000
1500
1000
500
000
-500
-1000
-1500
-2000
xi
e i
5 10 15 20 25 30
205Chapter 13 Quantifying Variable Relationships
21_045191 ch13qxd 81606 1137 PM Page 205
206 Part V Improving and Controlling
Overall you can say that you definitely have a statistically significant fitted line for the curetime ndash bond strength data and you know that the fitted line explains 80 percent of the observedvariation The rest is unaccounted-for random variation with a standard variation of 1006
j Investigate the strength of the fitted line for the training hours ndash product quality data inProblems 3 and 7
Use a table to lay out all the values and calculations you need to assess the adequacy of thefitted line See Table 13-10 to check your own table
Table 13-10 Table of Values Used to Calculate Residualsxi yi y
i ei (y
i ndash y ) 2 (yi ndash y ) 2
17 100 10980 ndash980 23496 63168
81 5 606 ndash106 025 031
84 3 574 ndash274 003 653
76 3 659 ndash359 107 653
82 7 595 105 016 209
86 9 553 347 000 1186
89 6 521 079 012 020
96 3 446 ndash146 120 653
75 6 670 ndash070 130 020
83 7 585 115 008 209
77 7 648 052 086 209
95 1 457 ndash357 098 2075
97 1 436 ndash336 144 2075
79 7 627 073 051 209
79 5 627 ndash127 051 031
83 7 585 115 008 209
97 5 436 064 144 031
99 7 414 286 200 209
85 11 563 537 001 2964
Averages (x y ) 829 560 Σ 1205 11644
Now plot the residuals (ei) column of the calculation table versus the predicted fit values ( y
i)column In this plot yoursquore checking to make sure the residuals look normally distributedaround a center of zero (see the following figure)
21_045191 ch13qxd 81606 1137 PM Page 206
These residuals look normally scattered with no patterns or trends Now plot the observed xvalues (xi) versus the residuals (see the next figure)
After studying this figure you can see that no apparent problems exist in this residual plotNow calculate the coefficient of determination (R2) for the fitted line by dividing the sum of thecolumn ( y
i ndash y)2 by the sum of the (yi ndash y)2 column which is located at the right side of the cal-culation table The calculations look like this
R
y y
y y
116 4412 05 0 10
ii
n
ii
n
2
2
1
2
1=-
-= =
=
=
_
b
i
l
An R2 of 010 means that only 10 percent of the observed variation of the data is explained bythe fitted line Almost all of the variation is unaccounted for by the fitted line Even though themath works to provide you an equation for a line fitting the data the fitted line really serves nopractical purpose
70 75 80 85 95-600
600
400
200
000
-200
-400
100
xi
e i
90 105
40 45 50 55 65-600
600
400
200
000
-200
-400
70
yi
e i
60
207Chapter 13 Quantifying Variable Relationships
21_045191 ch13qxd 81606 1137 PM Page 207
As your last test of validity you have to calculate the F statistic by using the following formula
n y y y y
y y
F F n
21
18 21 116 44 12 05
12 05
1 85 4 49 1 16 1 2
i ii
n
i
n
ii
n
2 2
11
2
1
$
-- - -
-=
--
=
= = -
==
=
_ b
b
_ _
i l
l
i i
= 6G
Your calculated F statistic is much less than the look-up test statistic of F(116) from Table 12-3(see Chapter 12) so there is no statistical basis for your fitted line
With this poor of a fit you have no reason to perform any other checks mdash your fitted linesimply doesnrsquot provide any value here
208 Part V Improving and Controlling
21_045191 ch13qxd 81606 1137 PM Page 208
Chapter 14
Planning and Conducting 2k
Factorial ExperimentsIn This Chapter Setting up and implementing 2k factorial experiments
Blocking and randomizing variables
Calculating effects
Eliminating insignificant effects
Forming Y = f(X) equations
Chapter 13 shows how Xs are related to the critical Ys Now you have to determine whatcauses these effects The questions you have to ask yourself are ldquoWhat can I adjust or
change to improve my Ysrdquo and ldquoWhatrsquos the ideal adjustment to makerdquo
The answer to all of these questions is simply You have to experiment Experimenting is at theheart of Six Sigma Wersquore not talking about trial-and-error or knee-jerk experiments mdash wersquoretalking about thoughtfully planned and designed experiments Design of experiments (DOE) is a topic large enough to fill an entire For Dummies book by itself To give you immediate profi-ciency however this chapter gives you practice in planning conducting and analyzing themost common type of experiment used in Six Sigma mdash the 2k factorial 2k factorial experimentscan be adapted to provide screening characterization or optimization information
Donrsquot Be a Frankenstein Planning ExperimentsEvery experiment in Six Sigma targets a better understanding of the Y = f(X) relationshipbetween the input Xs and the output Ys of the process or system being improved Betterunderstanding from experimentation includes
Knowing which input Xs have a significant effect on the output Y and knowing which Xsare insignificant
Formulating and quantifying the mathematical relationship between the significant Xsand the output Y
Discovering where to set the values of the significant Xs so that they combine to pro-duce the optimal output value of Y
Fight the urge to rush into an experiment You can never spend too much time planning anddesigning your experiment In fact shifting some time up front to the planning and designphase of experimentation almost always saves time later and it ensures that the resourcesyou expend provide viable information
22_045191 ch14qxd 81606 1138 PM Page 209
When planning your 2k factorial experiment take the following items into consideration
2k factorial experiments are best suited to studying two to five input variablesSo first narrow your investigation down to this smaller subset of likely suspectsTypically these are factors that have already been found to have a statisticallysignificant impact on the output variable of interest Your selected variables orfactors as theyrsquore often called can be either continuous or attribute variables(In case yoursquore wondering the ldquokrdquo in 2k represents the number of factors in yourexperiment)
Select exactly two experimental levels for each input variable mdash one ldquohighrdquo andone ldquolowrdquo mdash which span the range you want to investigate for each variable(The ldquo2rdquo in 2k represents the number of levels for each factor in your experiment)
Your factorial experiment will have 2k experimental runs For example if youhave three factors you will have 23 = 2 times 2 times 2 = 8 experimental runs Each runrepresents a unique setting of the factors at their high and low levels
Create a coded experiment design matrix that captures each of the 2k uniqueexperiment settings by creating a column for each of the factors and a row foreach of the 2k runs Using ndash1rsquos as codes for the ldquolowrdquo variable settings and +1rsquosas codes for the ldquohighrdquo settings start with the left-most factor column Fill in thiscolumnrsquos cells with alternating ndash1rsquos and +1rsquos With the left-most column filled inmove on to the next column to the right and repeat the process mdash but this timefill the column in with alternating pairs of ndash1rsquos and +1rsquos Fill in the next column tothe right with alternating quadruplets of ndash1rsquos and +1rsquos and so on repeating thisprocess from left to right until in the right-most column you have the first halfof the runs marked as ndash1rsquos and the bottom half as +1rsquos This table of patterned+1rsquos and ndash1rsquos is called the coded design matrix and represents the experimentalsettings for each of the 2k experimental runs
210 Part V Improving and Controlling
Q An engineer at a food processing plant is getting ready to conduct an experiment With this exper-iment she wants to understand how the input variables of the caustic cleaning process reducethe pressure differential across the plantrsquos filter system The input variables are
bull The temperature of the caustic cleaning solution In the past solution temperatures rangingfrom 65 to 85 degrees Celsius have been used successfully
bull The time the caustic cleaning solution is allowed to contact and flow through the filter sur-faces The duration of the cleaning is sometimes as low as 10 minutes and sometimes as highas 45 minutes
bull The concentration of the acid in the caustic cleaning solution Records show that concentra-tions of 4 to 8 percent have been used
Help the engineer select the levels for the experiment factors and create a coded design matrixfor a 2k factorial experiment
A Three factors will be included in the engineerrsquos experiment A good practice is to select the highand low experimental levels for each factor by reviewing what operating ranges have been used inthe past So going through each factor
bull X1 Temperature The high level could be set at 85 degrees Celsius and the low level could beset at 65 degrees Celsius
bull X2 Time The high level could be set at 45 minutes and the low level could be set at 10 minutes
bull X3 Concentration The high level could be set at 8 percent and the low level could be set at 4percent
22_045191 ch14qxd 81606 1138 PM Page 210
Now you have to put these factors and their levels into a coded design matrix This experimentwhich has three factors will have 23 = 8 unique runs You lay out the coded design matrix by creat-ing a table with a row for each run (eight in this case) and columns for each of the factors In theTemperature column you fill in the values for each row with alternating ndash1rsquos and +1rsquos In nextcolumn to the right you fill in alternating pairs of ndash1rsquos and +1rsquos Finally in the right-most columnyou fill in alternating quadruplets of ndash1rsquos and +1rsquos When completed your table should look likethis
Run X1 Temperature X2 Time X3 Concentration
1 ndash1 ndash1 ndash1
2 +1 ndash1 ndash1
3 ndash1 +1 ndash1
4 +1 +1 ndash1
5 ndash1 ndash1 +1
6 +1 ndash1 +1
7 ndash1 +1 +1
8 +1 +1 +1
This coded design matrix shows you what the factor settings will be for each run in your experi-ment For example run Number 6 will have a temperature at its +1 ldquohighrdquo setting (85 degreesCelsius) time at its ndash1 ldquolowrdquo setting (10 minutes) and concentration at its +1 ldquohighrdquo setting (8 percent)
211Chapter 14 Planning and Conducting 2k Factorial Experiments
1 How many unique runs will a 2k factorialexperiment that investigates four factorshave
Solve It
2 Create a coded design matrix for a 22 facto-rial experiment
Solve It
22_045191 ch14qxd 81606 1138 PM Page 211
212 Part V Improving and Controlling
3 Create a coded design matrix for a 2k facto-rial experiment thatrsquos investigating five factors Make sure you have plenty ofscrap paper to use
Solve It
4 A project leader wants to set up an experi-ment to explore two production variablesproduction line and shift The companyhas two duplicate production lines mdash LineA and Line B And it has two shifts mdash dayshift and afternoon shift Create a codeddesign matrix for this 22 factorial experi-ment
Solve It
Managing Those Pesky Nuisance VariablesIn almost every experiment you have variables that can affect the Y outputs that arenrsquotexplicitly included in your experiment design plan For example an experiment studying theeffect of depth of cut and cutting speed on a machined surface finish doesnrsquot include thevariable of the machinist Whether the machining operation is performed by Bob or Hankmay affect the results of the surface finish The experiment also doesnrsquot include cutting toolsharpness Meaning that as the cutting tool dulls during the experiment runs it may beginto have an impact on your results In a well-designed experiment these influential factorsare addressed and managed so that the results of the experiment remain valid
A simple catchy phrase can help you remember how to manage potential nuisance vari-ables that arenrsquot included in your experiment Block what you can and randomize againstwhat you canrsquot block
The following descriptions show you how to apply this catch phrase
Blocking When you know the source of a potential nuisance variable you can pur-posely remove its influence completely or include its effect evenly through all of yourexperimental runs By blocking you guarantee that you wonrsquot have an unfair bias ononly a portion of your experimental settings
Randomizing To inoculate your experiment against the detrimental effects ofunknown nuisance variables you need to randomize all the variables that arenrsquotdirectly part of your experiment You randomize such things as the order of the exper-imental runs the materials being used the operators performing the work the time ofday the experiment runs are made and so on Randomizing spreads out the otherwiseconcentrated or confounding potential for unknown nuisance effects evenly and fairlyover all of the experimental runs and preserves the accuracy of your results
22_045191 ch14qxd 81606 1138 PM Page 212
213Chapter 14 Planning and Conducting 2k Factorial Experiments
5 Using the food plant caustic filter cleaningexample from the ldquoDonrsquot Be a FrankensteinPlanning Experimentsrdquo section earlier in this chapter determine some of the non-experiment factors that you think may exert unwanted influence on theexperiment
Solve It
6 For each of the non-experiment factors youidentify in Problem 5 explain how youmight manage its effect on the experiment
Solve It
Q An improvement leader is planning anexperiment to explore the effect of tem-perature and pressure on the quality ofmolded parts The leader has determinedseveral variables he knows will have aneffect on his results unless he does some-thing about them These variables includethe difference between the mold pressestypically used for the process (three dif-ferent presses are typically used) and thetime allowed for the machine to warm upto operating conditions He also suspectsthat other variables will influence hisexperiment but he isnrsquot sure what thosevariables might be Give some sugges-tions for what the improvement leadermight do to manage the effects of theseextraneous variables
A For the known nuisance variables of moldpress and warm-up time the improve-ment leader should do something toblock their effects The best way to blockthe extra variation from the three differ-ent mold presses is to conduct the experi-ment on just one of the presses By usingonly one press the typical variation thatrsquosobserved from press to press will be elim-inated from the experiment
To block the effect of the warm-up timethe experimental runs could be conductedrapidly at approximately the same point inthe warm-up cycle Or if each run requiresa lot of time a different way to block thewarm-up time variable would be to con-duct one run each day at precisely thesame time in the warm-up cycle
The most basic thing to do about the lurk-ing unknown nuisance variables is to ran-domize the order of the experiment runsso that all of the runs will have an equallikelihood of experiencing the unknowninfluences
22_045191 ch14qxd 81606 1138 PM Page 213
7 What should you always do to mitigate theeffect of unknown variable influences
Solve It
8 An investigator of car quality decides totest four newly manufactured cars Whatare some of the blocking and randomizingstrategies the investigator could use tomake sure his selection of newly manufac-tured cars is valid
Solve It
Calculating Main EffectsMain effects are the quantitative influences each single experiment factor (X) has indi-vidually on the output response (Y) Each factor in your experiment has a main effect
To explore and quantify the main effect of each factor in your experiment follow thesesteps
1 Create a main effects plot for each factor
You create this plot by plotting the line between the point representing the aver-age of the responses with the factor at its high level and the point representingthe average of the responses with the factor at its low level
The steeper the line the stronger the effect
2 Quantify the main effect for each factor Ei using this formula
E c y2
1i k i j j
j1
1
2k
= -=
214 Part V Improving and Controlling
22_045191 ch14qxd 81606 1138 PM Page 214
215Chapter 14 Planning and Conducting 2k Factorial Experiments
Q The engineer from the food plant caustic filter cleaning example described earlier in the chapterran the experiment and recorded the following results
X1 Temperature X2 Time X3 Concentration Y Pressure Differential Run Order Coded Value c1 Coded Value c2 Coded Value c3 Reduction ()
1 3 ndash1 ndash1 ndash1 608
2 7 +1 ndash1 ndash1 850
3 4 ndash1 +1 ndash1 686
4 2 +1 +1 ndash1 906
5 8 ndash1 ndash1 +1 678
6 6 +1 ndash1 +1 782
7 1 ndash1 +1 +1 753
8 5 +1 +1 +1 866
Plot quantify and compare the main effects for each factor
A The main effect for the X1 temperature factor can be seen graphically by plotting the line betweenthe point representing the average of the responses with the factor at its ldquohighrdquo level and thepoint representing the average of the responses with the factor at its ldquolowrdquo level The responsesfor this factor at its ldquohighrdquo level are those runs where the coded value for temperature c1 is at +1These runs include 2 4 6 and 8 The average response at the ldquohighrdquo level for this factor is calcu-lated like this
y y y y
4 485 0 90 6 78 2 86 6
4340 4 85 12 4 6 8+ + +
= + + + = =
The responses for the X1 temperature factor at its low level are those runs with the coded value c1
at a value of ndash1 These runs include 1 3 5 and 7 The average response at the ldquolowrdquo level for thisfactor is calculated like this
y y y y
4 460 8 68 6 67 8 75 3
4272 5 68 11 3 5 7+ + +
= + + + = =
Plotting these two points and drawing a straight line between them using standard statistical soft-ware shows you a graph of the X1 main effect like in the following figure
-1
70
85
X1 Temperature
Main Effects Plot for X1 Temperature
Mea
n o
f Y
75
80
1
22_045191 ch14qxd 81606 1138 PM Page 215
The main effect plot for the X1 temperature factor shows a positive slope which means that as thetemperature of the caustic cleaning solution is increased the percent reduction in pressure alsoincreases
The main effect plot for the X2 time factor is found in a similar way by plotting the line betweenthe point of the average response from the runs with X2 at its ldquohighrdquo level and the point of theaverage response from the runs with X2 at its ldquolowrdquo level Referring to the coded design matrix created at the beginning of this example the runs with X2 at its ldquohighrdquo level are those where c2 isequal to +1 These runs include 3 4 7 and 8 The calculation for the average response at theldquohighrdquo level is
y y y y
4 468 6 90 6 75 3 86 6
4321 1 80 33 4 7 8+ + +
= + + + = =
The runs with X2 at its ldquolowrdquo level are those where c2 is set at ndash1 These runs include 1 2 5 and 6The calculation for the average response at the ldquolowrdquo level is
y y y y
4 460 8 85 0 67 8 78 2
4291 8 73 01 2 5 6+ + +
= + + + = =
Plotting these two points and drawing a straight line between them gives you a graph of the X2
main effect The graph ends up looking something like the next figure
Like with X1 the main effect plot for the X2 time factor shows a positive slope which means thatas the exposure time of the caustic cleaning solution is increased the percent reduction in pres-sure also increases
To create the main effect plot for the X3 concentration factor you go through the same processThe point with X3 at its ldquohighrdquo level is the average of the runs where c3 is set at +1 These runsinclude 5 6 7 and 8 So the calculation for the average response from these runs is
y y y y
4 467 8 78 2 75 3 86 6
4307 9 77 05 6 7 8+ + +
= + + + = =
The point with X3 at its ldquolowrdquo level is the average of the runs where c3 is set at ndash1 These runsinclude 1 2 3 and 4 The calculation for this average is
y y y y
4 460 8 85 0 68 6 90 6
4305 0 76 31 2 3 4+ + +
= + + + = =
Plotting these two points gives you a graph of the X3 main effect which looks like the one in thefollowing figure
-1
72
73
74
75
76
77
78
79
80
81
X2 Time
Main Effects Plot for X2 Temperature
Mea
n o
f Y
1
216 Part V Improving and Controlling
22_045191 ch14qxd 81606 1138 PM Page 216
217Chapter 14 Planning and Conducting 2k Factorial Experiments
Like with the other two main effects the main effect for the X3 concentration factor is also posi-tive which means that the higher the concentration of the caustic solution the greater the pres-sure reduction
A very important step in reviewing the main effects plots is to place them side by side all on thesame vertical scale which allows you to compare the relative slopes of the lines visually Checkout the next figure to see what your plots for this example should look like
After examining the figure you can immediately see that X1 has a much stronger main effect thaneither X2 or X3 In fact even though you can calculate a number for the X3 main effect by compari-son to the others it really is nearly flat
A quantitative value Ei can be calculated for each main effect This value is simply the differencebetween the high-setting average and the low-setting average for the factor yoursquore calculating Butthe standard formula using the coded design matrix values produces this exact value in everycase You just plug the c values in from the coded design matrix and plug them into the formulaFor the X1 temperature factor the quantitative main effect E1 is calculated like this
70
85
Main Effects Plots
Mea
n o
f Y
75
-1 1
X1 X2 X3
-1 1 -1 1
80
-1
762
770
769
768
767
766
765
764
763
X3 Time
Main Effects Plot for X3 Concentration
Mea
n o
f Y
1
22_045191 ch14qxd 81606 1138 PM Page 217
9 An engineer has set up a 22 factorial experiment to study the effects of heat treatment tempera-ture on the yield strength of a material He recorded the following results
E c y c y c y c y
E
E
E
21
21
21 1 60 8 1 85 0 1 68 6 1 90 6 1 67 8 1 78 2 1 75 3 1 86 6
41 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 67 8 16 9
k j jj
k1 1 11
2
1 1 1 1 1 2 2 1 8 8
1 3 1
1
1
k
f= = + + +
= - + + + - + + + - + + + - + +
= - + - + - + - +
= =
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h
8
8
6
6
B
B
The same formula can be used to quantify the main effect for the second and third experiment fac-tors E2 and E3 Here are the calculations for the quantitative main effect E2
E c Y c Y c Y c Y
E
E
E
21
21
21 1 60 8 1 85 0 1 68 6 1 90 6 1 67 8 1 78 2 1 75 3 1 86 6
41 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 29 3 7 3
k j jj
k2 1 21
2
1 2 1 1 2 2 2 2 8 8
2 3 1
2
2
k
f= = + + +
= - + - + + + + + - + - + + + +
= - - + + - - + +
= =
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h
8
8
6
6
B
B
And finally here are the calculations for E3
E c y c y c y c y
E
E
E
21
21
21 1 60 8 1 85 0 1 68 6 1 90 6 1 67 8 1 78 2 1 75 3 1 86 6
41 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 2 8 0 7
k j jj
k3 1 31
2
1 3 1 1 3 2 2 3 8 8
3 3 1
3
3
k
f= = + + +
= - + - + - + - + + + + + + + +
= - - - - + + + +
= =
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^h h h h h h h h
8
8
6
6
B
B
Sure enough the math works and the formula quantifies the main effects from the plots perfectly
218 Part V Improving and Controlling
Run Order X1 Supplier X2 Heat Treatment Y YieldTemperature Strength (ksi)
Coded Value c1 Coded Value c2
1 2 ndash1 ndash1 27800
2 4 +1 ndash1 28740
3 1 ndash1 +1 32150
4 3 +1 +1 31240
Plot and calculate the main effect for the X1 supplier factor
Solve It
22_045191 ch14qxd 81606 1138 PM Page 218
219Chapter 14 Planning and Conducting 2k Factorial Experiments
10 Using the information from the experimentin Problem 9 plot and calculate the maineffect for the X2 heat treatment tempera-ture factor
Solve It
11 Using the information from Problems 9 and10 create a side-by-side main effects plotfor X1 and X2 and compare the relativestrengths of the two
Solve It
Calculating Interaction EffectsSometimes a variable has an effect on an outcome when it combines and interactswith another variable For example think of baking a cake A delicious-tasting out-come (the Y) is a function of several input Xs such as ldquoamount of flourrdquo ldquonumber ofeggsrdquo ldquooven temperaturerdquo ldquobaking timerdquo and so on Obviously the right value for thevariable of ldquobaking timerdquo depends on the setting for ldquooven temperaturerdquo How hot theoven is and how long you leave the cake in the oven are two input variables that inter-act with each other
A properly designed and conducted 2k factorial experiment allows you to identify andquantify all interaction effects among your experimental factors Yoursquoll have a potentialinteraction effect for each possible factor combination
To explore and quantify the interaction effect of each factor combination in yourexperiment follow these steps
1 Create a two-way interaction effect plot for each two-factor combination bycreating a plot with the following two lines
bull The first line is drawn between the point created by the average responseof the runs where the first-listed factor is at its ldquohighrdquo level and the second-listed factor is also at its ldquohighrdquo level and the point created by the averageresponse of the runs where the first-listed factor is still at its ldquohighrdquo leveland the second-listed factor is at its ldquolowrdquo level
bull The second line is drawn between the point created by the average responseof the runs where the first-listed factor is at its ldquolowrdquo level and the second-listed factor is at its ldquohighrdquo level and the point created by the averageresponse of the runs where the first-listed factor is at its ldquolowrdquo level and thesecond-listed factor is also at its ldquolowrdquo level
22_045191 ch14qxd 81606 1138 PM Page 219
220 Part V Improving and Controlling
The greater the difference in the slope of the two lines the stronger the interac-tion effect between the two factors
2 Quantify the interaction effect for any factor combination Eabz using the fol-lowing formula
E c c c y2
1 ab z k a j b j z j
jj1
1
2k
f=g -=
` j
Q Using the information from the food plant caustic filter cleaning example described earlier in thechapter plot the two-way interaction effects and quantify and compare all of the interactioneffects for each factor
A In this 23 factorial experiment you have the following three two-way interaction effects
bull E12 is the interaction effect between X1 Temperature and X2 Time
bull E13 is the interaction effect between X1 Temperature and X3 Concentration
bull E23 is the interaction effect between X2 Time and X3 Concentration
The interaction effect between X1 Temperature and X2 Time can be seen graphically by creating aplot with the following two lines
bull The first line on the plot is drawn between the point created by the average response of theruns where the X1 temperature factor is at its ldquohighrdquo level and the X2 time factor is also at itsldquohighrdquo level and the point created by the average response of the runs where the X1 tempera-ture factor is still at its ldquohighrdquo level and the X2 time factor is at its ldquolowrdquo level
bull The second line is drawn between the point created by the average response of the runs wherethe X1 temperature factor is at its ldquolowrdquo level and the X2 time factor is at its ldquohighrdquo level and thepoint created by the average response of the runs where the X1 temperature factor is at itsldquolowrdquo level and the X2 time factor is also at its ldquolowrdquo level
Start by calculating the first point for the first line Looking back at the coded design matrix theruns where c1 is at its coded ldquohighrdquo level of +1 and c2 is also at its coded ldquohighrdquo level of +1 are 4and 8 So the average response of these runs is
y y
2 290 6 86 6
2177 2 88 64 8+
= + = +
The second point for the first line is the average of the runs where the c1 is still at its ldquohighrdquo levelof +1 and c2 is at its ldquolowrdquo level of ndash1 After referring to the coded design matrix yoursquoll see thatruns 2 and 6 are at these levels The average response of these runs is
y y
2 285 0 78 2
2163 2 81 62 6+
= + = =
Now calculate the first point for the second line The runs where c1 is at its coded ldquolowrdquo level of ndash1and c2 is at its coded ldquohighrdquo level of +1 are 3 and 7 The average response of these runs is
y y
2 268 6 75 3
2143 9 72 03 7+
= + = =
Next calculate the second point for the second line The runs where c1 is at its coded ldquolowrdquo levelof ndash1 and c2 is also at its coded ldquolowrdquo level of ndash1 are 1 and 5 The average response of these runs is
y y
2 260 8 67 8
2128 6 64 31 5+
= + = =
22_045191 ch14qxd 81606 1139 PM Page 220
221Chapter 14 Planning and Conducting 2k Factorial Experiments
Now plot the two lines formed by these two pairs of calculated points as shown in the next figureYou have successfully created the two-way interaction effect plot for X1 Temperature and X2 Time
You can see that the slopes of these two lines are graphically identical mdash the lines are nearly parallel mdash which means that no interaction exists between the X1 temperature and X2 time factors
Follow the same process to create two-way interaction plots for the X1 Temperature ndash X3Concentration and the X2 Time ndash X3 Concentration interactions Lay out all of the two-way inter-action plots for review and inspection by plotting them on the same vertical scale and placingthem in a matrix pattern like in the following figure
When looking at all of the two-way interactions you see that only the plot for the X1 Temperaturendash X3 Concentration and the X2 Time ndash X3 Concentration interaction shows lines with significantlydifferent slopes This indicates an interaction between these two variables
-1 1Interaction Plots
X2
X1
X3
X1-11
X2-11
-1 190
80
70
90
80
70
90
Interaction Plot X1 and X2
X2
Mea
n85
80
75
70
65
-1 1
X1-11
22_045191 ch14qxd 81606 1139 PM Page 221
You quantify the interaction effect values by plugging the coded design matrix values and theexperiment run responses into the following formula
E c c c y2
1 ab z k a j b j z j
jj1
1
2k
f=g -=
` j
So the quantified value for the interaction effect between X1 Temperature and X2 Time is as follows
E c c y c c y c c y c c y
E
E
E
21
21
21 1 1 60 8 1 1 85 0 1 1 68 66 1 1 90 6
1 1 67 8 1 1 78 2 1 1 75 3 1 1 86 6
21 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 1 3 0 3
k j j jj
k12 1 1 21
2
1 1 1 2 1 1 1 2 2 2 2 1 8 2 8 8
12 3 1
12 2
12
k
f= = + + +
=- - + + - + - + + + +
- - + + - + - + + + +
= - - + + - - +
= - = -
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^
^ ^ ^ ^ ^ ^ ^ ^
h h h h h h h h
h h h h h h h h
R
T
SSS
8
6
6
V
X
WWW
B
The formula and calculations for the X1 Temperature ndash X3 Concentration interaction E13 look likethis
E c c y c c y c c y c c y
E
E
E
21
21
21 1 1 60 8 1 1 85 0 1 1 68 6 1 1 90 6
1 1 67 8 1 1 78 2 1 1 75 3 1 1 86 6
21 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 24 5 6 1
k j j jj
k13 1 1 31
2
1 1 1 3 1 1 1 2 3 2 2 1 8 3 8 8
13 3 1
13 2
13
k
f= = + + +
=- - + + - + - - + + - +
- + + + + + - + + + +
= - + - - + - +
= - = -
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^
^ ^ ^ ^ ^ ^ ^ ^
h h h h h h h h
h h h h h h h h
R
T
SSS
8
6
6
V
X
WWW
B
The formula and calculations for the X2 Time ndash X3 Concentration interaction E23 look like this
E c c y c c y c c y c c y
E
E
E
21
21
21 1 1 60 8 1 1 85 0 1 1 68 6 1 1 90 6
1 1 67 8 1 1 78 2 1 1 75 3 1 1 86 6
21 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 2 5 0 6
k j j jj
k23 1 2 31
2
1 2 1 3 1 1 2 2 3 2 2 2 8 3 8 8
23 3 1
23 2
23
k
f= = + + +
=- - + - - + + - + + - +
- + + - + + + + + + +
= + - - - - + +
= =
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^
^ ^ ^ ^ ^ ^ ^ ^
h h h h h h h h
h h h h h h h h
R
T
SSS
8
6
6
V
X
WWW
B
For the one three-way interaction of X1-X2-X3 E123 the formula and calculations look like this
E c c c y c c c y c c c y c c c y
E
E
E
21
21
21 1 1 1 60 8 1 1 1 85 0 1 1 1 68 6 1 1 1 90 6
1 1 1 67 8 1 1 1 78 2 1 1 1 75 3 1 1 1 86 6
21 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6
41 3 1 0 8
k j j j jj
k123 1 1 2 31
2
1 1 1 2 1 3 1 1 1 2 2 2 3 2 2 1 8 2 8 3 8 8
123 3 1
123 2
123
k
f= = + + +
=- - - + + - - + - + - + + + - +
- - + + + - + + - + + + + + +
= - + + - + - - +
= =
-=
-
-
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
h h h h h h h h h h h h
h h h h h h h h h h h h
R
T
SSS
8
6
6
V
X
WWW
B
222 Part V Improving and Controlling
22_045191 ch14qxd 81606 1139 PM Page 222
223Chapter 14 Planning and Conducting 2k Factorial Experiments
12 List all of the possible interaction effectsfor a 24 factorial experiment
Solve It
13 Create a two-way interaction plot for the 22
factorial experiment from Problem 9 Afteryou create the plot calculate the E12 valuefor this interaction effect
Solve It
14 For the two-way plots in the accompanying figure which ones display evidence of an interactioneffect
Solve It
150 200Interaction Plots
X2
X1
X3
X12050
X2150200
A B
48
46
44
48
46
44
22_045191 ch14qxd 81606 1139 PM Page 223
224 Part V Improving and Controlling
Determining Which Effects Are SignificantUnfortunately just because you can calculate the main effects and the interactioneffects for an experiment doesnrsquot mean that all of the values are statistically signifi-cant The Pareto Principle (see Chapter 6) indicates that only a relatively small minor-ity of all the possible effects will explain the majority of the changes in the responseAfter calculating all the possible main and interaction effects you have to test them tosee which few you should keep and which you should discard
To find out which calculated effects are significant you plot the calculated effectsagainst their normal Z scores If the effects are insignificant the plotted points will fallinto the shape of a line Any effect in which a corresponding point falls far off the linehowever is significant
You create the plotted points by following these steps
1 Rank order the calculated estimates from lowest to highest
2 Calculate the probability for each of the ranked estimates using this formula
P i2 1
0 5i k=
--
3 Look up the normal or Zi score corresponding to each calculated Pi
You can look up the Zi score in the Standard Normal Distribution Table atwwwdummiescomgosixsigmaworkbook
4 Plot each calculated effect versus its Zi
5 Identify the significant effects from those that are far off the line created bythe plotted points
Q Using the food plant caustic filter cleaning example from earlier in the chapter determine whichcalculated effects are significant by creating a normal probability plot of the effects
A The first step is to list all of the calculated effects in rank order and assign them their rank order(i) like this
Effect Calculated Value Rank(i)i
E13 ndash61 1
E12 ndash03 2
E23 06 3
E3 07 4
E123 08 5
E2 73 6
E1 169 7
22_045191 ch14qxd 81606 1139 PM Page 224
225Chapter 14 Planning and Conducting 2k Factorial Experiments
The next step is to calculate the probability for each ranked effect using this formula
P i2 1
0 5i k=
--
As an example P1 for the lowest rank ordered effect is calculated like this
P2 1
1 0 57
0 5 0 07141 3=-
- = =
You repeat this calculation for the Pi for each ranked effect The Pi column on the table of valuescan now be updated
Effect Calculated Value Rank(i) Pi
E13 ndash61 1 00714
E12 ndash03 2 02143
E23 06 3 03571
E3 07 4 05000
E123 08 5 06429
E2 73 6 07857
E1 169 7 09286
You can find similar Standard Normal Distribution tables online or in virtually every statistics andSix Sigma book Maybe thatrsquos why theyrsquore called ldquostandardrdquo mdash because theyrsquore a standard way forauthors to fill up the pages of their statistics books The final value to find for the ranked effect isits normal score or the Z value which you can find in any Standard Normal Distribution table Youcan find one at this bookrsquos bonus material Web page wwwdummiescomgosixsigmaworkbook
For E13 the calculated Pi is 00714 This value is a probability value in the half of the StandardNormal Distribution Table thatrsquos not listed You have to first figure its symmetrical probabilityHere are the calculations
1 ndash 00714 ndash 09286
After looking up the probability of 09286 in a Standard Normal Distribution Table you find thatthe Z value corresponding to this probability is 146 and 147 So a value of 1465 is probablyclose But donrsquot forget that because yoursquove used symmetry to look up this Z value you have touse a negative Z value (ndash1465)
By repeating this same look-up process you can find Z values for each of the rank ordered effectsYour table of values should now look like this
Effect Calculated Value Rank(i) Pi Zi
E13 ndash61 1 00714 ndash1465
E12 ndash03 2 02143 ndash0792
E23 06 3 03571 ndash0366
E3 07 4 05000 0
E123 08 5 06429 0366
E2 73 6 07857 0792
E1 169 7 09286 1657
With these values all calculated yoursquore ready to draw a normal probability plot to test the signifi-cance of the effects You simply plot the points created by each effect and its corresponding Zvalue like in the next figure
22_045191 ch14qxd 81606 1139 PM Page 225
The points created by effects E1 E2 and E13 are far off the line that was created by the rest of theeffects These off-line points signal that these effects arenrsquot just random chance but are signifi-cant You now have evidence that allows you to dismiss the other calculated effects
2
Normal Probability Plot of the Effects
Effect
Z S
core
1
0
E13
E2
E1
-5
-1
-2
0 5 10 15 20
Effect TypeNot SignificantSignificant
226 Part V Improving and Controlling
15 In the normal probability plot for theeffects in the accompanying figure circlewhich effects appear significant
Solve It
2
Normal Probability Plot of the Effects
Effect
Z S
core
1
0
-5
-1
-20 5 10 15 20
16 Plot the normal probabilities for the 22 fac-torial experiment in Problem 9 Indicatewhich effects are significant
Solve It
22_045191 ch14qxd 81606 1139 PM Page 226
227Chapter 14 Planning and Conducting 2k Factorial Experiments
The Ultimate Power Trip Forming Y = f(X) Equations
2k factorial experiments not only reveal which factors affect the output Y but they also allowyou to understand the form of the Y = f(X) equation for the system or process yoursquore improv-ing At the start a 2k experiment investigates the possibility of all main and interactioneffects being significant Later you whittle this list down to just the significant effects Whatyou do next is create a Y = f(X) equation for your system which allows you to predict futureoutputs from known inputs
The equation has a constant term (β0) and also potential terms for each main and interactioneffect (Xs) These terms each have a corresponding coefficient labeled β
For a three-factor system the general form of the equation with all its potential terms lookslike this
Y = β0 + β1X1 + β2X2 +β3X3 + β12X1X2 + β13X1X3 + β23X2X3 + β123X1X2X3
For a two-factor system the general form of the equation with all its potential terms lookslike this
Y = β0 + β1X1 + β2X2 + β12X1X2
To create the specific equation for your process or system you start with its general formand then remove all but the constant term β0 and the terms corresponding to the effects thatwere found to be significant You calculate the β coefficients from these equations
β0 = y
Eβ 21
ab z ab z=g g
Q Create the specific Y = f(X) equation for the food plant caustic filter cleaning example in thischapter
A The experiment for this example is a 23 factorial So the general form of the Y = f(X) equationwith all its potential terms will look like this
Y = β0 + β1X1 + β2X2 +β3X3 + β12X1X2 + β13X1X3 + β23X2X3 + β123X1X2X3
However only the E1 E2 and E13 effects were found to be significant which means that you canreduce the form of the equation so that it looks like this
Y = β0 + β1X1 + β2X2 +β13X1X3
To find the specific values for the βs in the reduced equation simply use the formulas
y y
E
E
E
β
β
β
β
21
21 60 8 85 0 68 6 90 6 67 8 78 2 75 3 86 6 8
1 612 9 76 6
21
21 16 9 8 5
21
21 7 3 3 7
21
21 6 1 3 1
k ii
01
2
3
1 1
2 2
13 13
k
= = = + + + + + + + = =
= = =
= = =
= = - = -
=
^
^
^
h
h
h
6 6
Plugging these β values back into the reduced Y = f(X) equation you get this equation
Y = 766 + 85X1 + 37X2 ndash 31X1X3
22_045191 ch14qxd 81606 1139 PM Page 227
228 Part V Improving and Controlling
17 Write out the general form of the Y = f(X)equation with all of its potential terms fora 24 factorial experiment
Solve It
18 For the experiment in Problem 9 write outthe general form of the equation includingall of its potential terms
Solve It
19 For the experiment in Problem 9 write out the specific form of the equation using only the signifi-cant terms
Solve It
22_045191 ch14qxd 81606 1139 PM Page 228
Solutions to 2k Factorial ExperimentProblems
a How many unique runs will a 2k factorial experiment that investigates four factors have
For 2k factorial experiments k represents the number of factors The number of runs is equal to2k or in this case 24 = 16 runs
b Create a coded design matrix for a 22 factorial experiment
For a 22 factorial experiment yoursquoll have two levels for each of the two factors mdash a ldquohighrdquo leveland a ldquolowrdquo level mdash and yoursquoll have 22 = 4 runs Each run represents a unique combination ofthe levels of the two factors Using the alternating pattern of +1rsquos to represent the ldquohighrdquo factorlevels and ndash1rsquos to represent the ldquolowrdquo factor levels shown in this pattern you create a list of thefour unique experiment runs This is what your coded design matrix for the experiment shouldlook like
Run X1 X2
1 ndash1 ndash1
2 +1 ndash1
3 ndash1 +1
4 +1 +1
c Create a coded design matrix for a 2k factorial experiment thatrsquos investigating five factors
A 25 factorial experiment will have 32 unique runs You use the alternating ndash1 +1 pattern shownin this chapter to create the coded design matrix It should look like this
Run X1 X2 X3 X4 X5
1 ndash1 ndash1 ndash1 ndash1 ndash1
2 +1 ndash1 ndash1 ndash1 ndash1
3 ndash1 +1 ndash1 ndash1 ndash1
4 +1 +1 ndash1 ndash1 ndash1
5 ndash1 ndash1 +1 ndash1 ndash1
6 +1 ndash1 +1 ndash1 ndash1
7 ndash1 +1 +1 ndash1 ndash1
8 +1 +1 +1 ndash1 ndash1
9 ndash1 ndash1 ndash1 +1 ndash1
10 +1 ndash1 ndash1 +1 ndash1
11 -1 +1 ndash1 +1 ndash1
12 +1 +1 ndash1 +1 ndash1
13 ndash1 ndash1 +1 +1 ndash1
14 +1 ndash1 +1 +1 ndash1
15 ndash1 +1 +1 +1 ndash1
16 +1 +1 +1 +1 ndash1
229Chapter 14 Planning and Conducting 2k Factorial Experiments
22_045191 ch14qxd 81606 1139 PM Page 229
17 ndash1 ndash1 ndash1 ndash1 +1
18 +1 ndash1 ndash1 ndash1 +1
19 ndash1 +1 ndash1 ndash1 +1
20 +1 +1 ndash1 ndash1 +1
21 ndash1 ndash1 +1 ndash1 +1
22 +1 ndash1 +1 ndash1 +1
23 ndash1 +1 +1 ndash1 +1
24 +1 +1 +1 ndash1 +1
25 ndash1 ndash1 ndash1 +1 +1
26 +1 ndash1 ndash1 +1 +1
27 ndash1 +1 ndash1 +1 +1
28 +1 +1 ndash1 +1 +1
29 ndash1 ndash1 +1 +1 +1
30 +1 ndash1 +1 +1 +1
31 ndash1 +1 +1 +1 +1
32 +1 +1 +1 +1 +1
d A project leader wants to set up an experiment to explore two production variables productionline and shift The company has two duplicate production lines mdash Line A and Line B And it hastwo shifts mdash day shift and afternoon shift Create a coded design matrix for this 22 factorialexperiment
With two factors and two levels for each factor yoursquoll have 22 = 4 experiment runs For the X1production line factor you have two levels Line A represented by ndash1 and Line B represented by+1 For the X2 shift factor you have two levels day shift represented by ndash1 and afternoon shiftrepresented by +1 The coded design matrix should look like this
Run X1 Line X2 Shift
1 ndash1 ndash1
2 +1 ndash1
3 ndash1 +1
4 +1 +1
e Using the food plant caustic filter cleaning example from the ldquoDonrsquot Be a Frankenstein PlanningExperimentsrdquo section earlier in this chapter determine some of the non-experiment factors thatyou think may exert unwanted influence on the experiment
The factors included in the experiment are temperature time and concentration Other influ-encing factors include
Flow rate of the caustic solution through the filter
Pressure differential measurement system being used
Length of time since the last caustic cleaning of the filters
Shelf life of caustic chemicals
230 Part V Improving and Controlling
22_045191 ch14qxd 81606 1139 PM Page 230
Your list may include some of the many other variables that influence the system Rememberto consult your list of Xs that have previously been identified as correlated when constructingyour experimental definition
f For each of the non-experiment factors you identify in Problem 5 explain how you mightmanage its effect on the experiment
Below for each of the identified influencing factors are suggestions on how you might managetheir effect on the experiment
Flow rate of the caustic solution through the filter Block the effect of this variable by setting and controlling the flow rate to a single value for all experiment runs
Pressure differential measurement system being used Randomize the order of theexperiment runs to spread out possible variation from the pressure differential measure-ment system
Length of time since the last caustic cleaning of the filters Block the effect of this vari-able by conducting each experiment run after a set length of time since the last cleaning
Shelf life of caustic chemicals Randomize which experiment runs get which causticchemical dates
g What should you always do to mitigate the effect of unknown variable influences
Always randomize the order of the runs of every experiment you perform
Randomizing each experiment may seem trivial and may seem like a lot of extra work but inthe end it helps your experiment stay free of bias from outside influences It always pays to randomize
h An investigator of car quality decides to test four newly manufactured cars What are some ofthe blocking and randomizing strategies the investigator could use to make sure his selectionof newly manufactured cars is valid
Imagine if one of the four selected cars was manufactured on a Monday but all the others weremanufactured on Tuesday Wednesday or Thursday The possible lower quality associated witha Monday-built car (due to the weekend layoff) would have an effect on the new car qualityinvestigations To correct this you could block for this factor by selecting study cars all comingfrom the same day of production
i An engineer has set up a 22 factorial experiment to study the effects of supplier and heat treat-ment temperature on the yield strength of a material See the original question earlier in thischapter for his results Plot and calculate the main effect for the X1 supplier factor
The plot for the main effect for X1 is found by drawing a line between the average of the runswith X1 at its ldquohighrdquo (+1) value and the average of the runs with X1 at its ldquolowrdquo (ndash1) valueLooking back at the coded design matrix for this example the average for the ldquohighrdquo runs is
y y
2 228 740 31 240
259 580 29 9902 4+
=+
= =
The average for the ldquolowrdquo runs is
y y
2 227 800 32 150 29 9751 3+
=+
=
Plotting these two points creates the plot for the X1 main effect (next figure)
231Chapter 14 Planning and Conducting 2k Factorial Experiments
22_045191 ch14qxd 81606 1139 PM Page 231