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1、Day 1 AgendavWelcome and IntroductionsvCourse StructureMeeting Guidelines/Course Agenda/Report Out CriteriavGroup ExpectationsvIntroduction to Six Sigma ApplicationsvRed Bead ExperimentvIntroduction to Probability DistributionsvCommon Probability Distributions and Their UsesvCorrelation AnalysisDay
2、2 AgendavTeam Report Outs on Day 1 MaterialvCentral Limit TheoremvProcess CapabilityvMulti-Vari AnalysisvSample Size ConsiderationsDay 3 AgendavTeam Report Outs on Day 2 MaterialvConfidence IntervalsvControl ChartsvHypothesis TestingvANOVA (Analysis of Variation)vContingency TablesDay 4 AgendavTeam
3、Report Outs on Practicum ApplicationvDesign of ExperimentsvWrap Up - Positives and DeltasClass GuidelinesvQ&A as we govBreaks HourlyvHomework ReadingsAs assigned in SyllabusvGradingClass Preparation30%Team Classroom Exercises30%Team Presentations40%v10 Minute Daily Presentation (Day 2 and 3) on Appl
4、ication of previous days workv20 minute final Practicum application (Last day)vCopy on Floppy as well as hard copyvPowerpoint preferredvRotate PresentersvQ&A from the classINTRODUCTION TO SIX SIGMA APPLICATIONSLearning ObjectivesvHave a broad understanding of statistical concepts and tools.vUndersta
5、nd how statistical concepts can be used to improve business processes.vUnderstand the relationship between the curriculum and the four step six sigma problem solving process (Measure, Analyze, Improve and Control).What is Six Sigma?A PhilosophyA Quality LevelA Structured Problem-Solving ApproachA Pr
6、ogramSCustomer Critical To Quality (CTQ) CriteriaSBreakthrough ImprovementsSFact-driven, Measurement-based, Statistically Analyzed PrioritizationSControlling the Input & Process Variations Yields a Predictable ProductS6 = 3.4 Defects per Million OpportunitiesSPhased Project:Measure, Analyze, Improve
7、, ControlSDedicated, Trained BlackBeltsSPrioritized ProjectsSTeams - Process Participants & OwnersPOSITIONING SIX SIGMA THE FRUIT OF SIX SIGMAGround FruitGround FruitLogic and IntuitionLow Hanging FruitLow Hanging FruitSeven Basic ToolsBulk of FruitBulk of FruitProcess Characterization and Optimizat
8、ionProcess EntitlementSweet FruitSweet Fruit Design for ManufacturabilityUNLOCKING THE HIDDEN FACTORYVALUE STREAM TO THE CUSTOMERPROCESSES WHICH PROVIDE PRODUCT VALUE IN THE CUSTOMERS EYESFEATURES OR CHARACTERISTICS THE CUSTOMER WOULD PAY FOR.WASTE DUE TO INCAPABLE PROCESSESWASTE SCATTERED THROUGHOU
9、T THE VALUE STREAMEXCESS INVENTORYREWORKWAIT TIMEEXCESS HANDLINGEXCESS TRAVEL DISTANCESTEST AND INSPECTIONWaste is a significant cost driver and has a major impact on the bottom line.Common Six Sigma Project AreasvManufacturing Defect ReductionvCycle Time ReductionvCost ReductionvInventory Reduction
10、vProduct Development and IntroductionvLabor ReductionvIncreased Utilization of ResourcesvProduct Sales ImprovementvCapacity ImprovementsvDelivery ImprovementsThe Focus of Six Sigma.Y = f(x)All critical characteristics (Y) are driven by factors (x) which are “upstream” from the results.Attempting to
11、manage results (Y) only causes increased costs due to rework, test and inspectionUnderstanding and controlling the causative factors (x) is the real key to high quality at low cost.INSPECTION EXERCISEThe necessity of training farm hands for first class farms in the fatherly handling of farm livestoc
12、k is foremost in the minds of farm owners. Since the forefathers of the farm owners trained the farm hands for first class farms in the fatherly handling of farm livestock, the farm owners feel they should carry on with the family tradition of training farm hands of first class farms in the fatherly
13、 handling of farm livestock because they believe it is the basis of good fundamental farm management.How many fs can you identify in 1 minute of inspection.INSPECTION EXERCISEThe necessity of* training f*arm hands f*or f*irst class f*arms in the f*atherly handling of* f*arm livestock is f*oremost in
14、 the minds of* f*arm owners. Since the f*oref*athers of* the f*arm owners trained the f*arm hands f*or f*irst class f*arms in the f*atherly handling of* f*arm livestock, the f*arm owners f*eel they should carry on with the f*amily tradition of* training f*arm hands of* f*irst class f*arms in the f*a
15、therly handling of* f*arm livestock because they believe it is the basis of* good f*undamental f*arm management.How many fs can you identify in 1 minute of inspection.36 total are available.SIX SIGMA COMPARISONSix Sigma Traditional “SIX SIGMA TAKES US FROM FIXING PRODUCTS SO THEY ARE EXCELLENT, TO F
16、IXING PROCESSES SO THEY PRODUCE EXCELLENT PRODUCTS” Dr. George Sarney, President, Siebe Control SystemsIMPROVEMENT ROADMAPBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlDefine the problem and verify the primary and secondary measu
17、rement systems.Identify the few factors which are directly influencing the problem.Determine values for the few contributing factors which resolve the problem.Determine long term control measures which will ensure that the contributing factors remain controlled.ObjectiveMeasurements are critical.If
18、we cant accurately measure something, we really dont know much about it.If we dont know much about it, we cant control it.If we cant control it, we are at the mercy of chance.WHY STATISTICS?THE ROLE OF STATISTICS IN SIX SIGMA.vWE DONT KNOW WHAT WE DONT KNOWIF WE DONT HAVE DATA, WE DONT KNOWIF WE DON
19、T KNOW, WE CAN NOT ACTIF WE CAN NOT ACT, THE RISK IS HIGHIF WE DO KNOW AND ACT, THE RISK IS MANAGEDIF WE DO KNOW AND DO NOT ACT, WE DESERVE THE LOSS. DR. Mikel J. HarryvTO GET DATA WE MUST MEASUREvDATA MUST BE CONVERTED TO INFORMATIONvINFORMATION IS DERIVED FROM DATA THROUGH STATISTICSWHY STATISTICS
20、?THE ROLE OF STATISTICS IN SIX SIGMA.vIgnorance is not bliss, it is the food of failure and the breeding ground for loss. DR. Mikel J. HarryYears ago a statistician might have claimed that statistics dealt with the processing of data. Todays statistician will be more likely to say that statistics is
21、 concerned with decision making in the face of uncertainty. BartlettmSales ReceiptsmOn Time DeliverymProcess CapacitymOrder Fulfillment TimemReduction of WastemProduct Development TimemProcess YieldsmScrap ReductionmInventory ReductionmFloor Space UtilizationWHAT DOES IT MEAN?Random Chance or Certai
22、nty.Which would you choose.?Learning ObjectivesvHave a broad understanding of statistical concepts and tools.vUnderstand how statistical concepts can be used to improve business processes.vUnderstand the relationship between the curriculum and the four step six sigma problem solving process (Measure
23、, Analyze, Improve and Control).RED BEAD EXPERIMENTLearning ObjectivesvHave an understanding of the difference between random variation and a statistically significant event.vUnderstand the difference between attempting to manage an outcome (Y) as opposed to managing upstream effects (xs).vUnderstan
24、d how the concept of statistical significance can be used to improve business processes.WELCOME TO THE WHITE BEAD FACTORYHIRING NEEDSBEADS ARE OUR BUSINESSBEADS ARE OUR BUSINESSPRODUCTION SUPERVISOR4 PRODUCTION WORKERS2 INSPECTORS1 INSPECTION SUPERVISOR1 TALLY KEEPERSTANDING ORDERSvFollow the proces
25、s exactly.vDo not improvise or vary from the documented process.vYour performance will be based solely on your ability to produce white beads.vNo questions will be allowed after the initial training period.vYour defect quota is no more than 5 off color beads allowed per paddle.WHITE BEAD MANUFACTURI
26、NG PROCESS PROCEDURESvThe operator will take the bead paddle in the right hand.vInsert the bead paddle at a 45 degree angle into the bead bowl.vAgitate the bead paddle gently in the bead bowl until all spaces are filled.vGently withdraw the bead paddle from the bowl at a 45 degree angle and allow th
27、e free beads to run off.vWithout touching the beads, show the paddle to inspector #1 and wait until the off color beads are tallied.vMove to inspector #2 and wait until the off color beads are tallied.vInspector #1 and #2 show their tallies to the inspection supervisor. If they agree, the inspection
28、 supervisor announces the count and the tally keeper will record the result. If they do not agree, the inspection supervisor will direct the inspectors to recount the paddle.vWhen the count is complete, the operator will return all the beads to the bowl and hand the paddle to the next operator.INCEN
29、TIVE PROGRAMvLow bead counts will be rewarded with a bonus.vHigh bead counts will be punished with a reprimand.vYour performance will be based solely on your ability to produce white beads.vYour defect quota is no more than 7 off color beads allowed per paddle.PLANT RESTRUCTUREvDefect counts remain
30、too high for the plant to be profitable.vThe two best performing production workers will be retained and the two worst performing production workers will be laid off.vYour performance will be based solely on your ability to produce white beads.vYour defect quota is no more than 10 off color beads al
31、lowed per paddle.OBSERVATIONS.WHAT OBSERVATIONS DID YOU MAKE ABOUT THIS PROCESS.?The Focus of Six Sigma.Y = f(x)All critical characteristics (Y) are driven by factors (x) which are “downstream” from the results.Attempting to manage results (Y) only causes increased costs due to rework, test and insp
32、ectionUnderstanding and controlling the causative factors (x) is the real key to high quality at low cost.Learning ObjectivesvHave an understanding of the difference between random variation and a statistically significant event.vUnderstand the difference between attempting to manage an outcome (Y)
33、as opposed to managing upstream effects (xs).vUnderstand how the concept of statistical significance can be used to improve business processes.INTRODUCTION TO PROBABILITY DISTRIBUTIONSLearning ObjectivesvHave a broad understanding of what probability distributions are and why they are important.vUnd
34、erstand the role that probability distributions play in determining whether an event is a random occurrence or significantly different.vUnderstand the common measures used to characterize a population central tendency and dispersion.vUnderstand the concept of Shift & Drift.vUnderstand the concept of
35、 significance testing.Why do we Care?An understanding of Probability Distributions is necessary to: Understand the concept and use of statistical tools.Understand the significance of random variation in everyday measures.Understand the impact of significance on the successful resolution of a project
36、. IMPROVEMENT ROADMAPUses of Probability DistributionsBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlEstablish baseline data characteristics.Project UsesIdentify and isolate sources of variation.Use the concept of shift & drift to
37、 establish project expectations.Demonstrate before and after results are not random chance.Focus on understanding the conceptsVisualize the conceptDont get lost in the math.KEYS TO SUCCESSMeasurements are critical.If we cant accurately measure something, we really dont know much about it.If we dont
38、know much about it, we cant control it.If we cant control it, we are at the mercy of chance.Types of MeasuresvMeasures where the metric is composed of a classification in one of two (or more) categories is called Attribute data. This data is usually presented as a “count” or “percent”.Good/BadYes/No
39、Hit/Miss etc.vMeasures where the metric consists of a number which indicates a precise value is called Variable data.TimeMiles/HrCOIN TOSS EXAMPLEvTake a coin from your pocket and toss it 200 times.vKeep track of the number of times the coin falls as “heads”.vWhen complete, the instructor will ask y
40、ou for your “head” count.COIN TOSS EXAMPLE1301201101009080701000050000Cumulative FrequencyResults from 10,000 people doing a coin toss 200 times.Cumulative Count1301201101009080706005004003002001000Head CountFrequencyResults from 10,000 people doing a coin toss 200 times.Count Frequency1301201101009
41、08070100500Head CountCumulative PercentResults from 10,000 people doing a coin toss 200 times.Cumulative PercentCumulative FrequencyCumulative PercentCumulative count is simply the total frequency count accumulated as you move from left to right until we account for the total population of 10,000 pe
42、ople.Since we know how many people were in this population (ie 10,000), we can divide each of the cumulative counts by 10,000 to give us a curve with the cumulative percent of population.COIN TOSS PROBABILITY EXAMPLE130120110100908070100500Cumulative PercentResults from 10,000 people doing a coin to
43、ss 200 timesCumulative PercentThis means that we can now predict the change that certain values can occur based on these percentages.Note here that 50% of the values are less than our expected value of 100.This means that in a future experiment set up the same way, we would expect 50% of the values
44、to be less than 100. COIN TOSS EXAMPLE1301201101009080706005004003002001000Head CountFrequencyResults from 10,000 people doing a coin toss 200 times.Count Frequency130120110100908070100500Head CountCumulative PercentResults from 10,000 people doing a coin toss 200 times.Cumulative PercentWe can now
45、equate a probability to the occurrence of specific values or groups of values.For example, we can see that the occurrence of a “Head count” of less than 74 or greater than 124 out of 200 tosses is so rare that a single occurrence was not registered out of 10,000 tries.On the other hand, we can see t
46、hat the chance of getting a count near (or at) 100 is much higher. With the data that we now have, we can actually predict each of these values.COIN TOSS PROBABILITY DISTRIBUTION-6-5-4-3-2-10123456NUMBER OF HEADSPROCESS CENTERED ON EXPECTED VALUE SIGMA ( ) IS A MEASURE OF “SCATTER” FROM THE EXPECTED
47、 VALUE THAT CAN BE USED TO CALCULATE A PROBABILITY OF OCCURRENCESIGMA VALUE (Z)CUM % OF POPULATION586572798693100107114121128135142.003.1352.27515.8750.084.197.799.8699.9971301201101009080706005004003002001000FrequencyIf we know where we are in the population we can equate that to a probability valu
48、e. This is the purpose of the sigma value (normal data).% of population = probability of occurrencemCommon Occurrence mRare EventWHAT DOES IT MEAN?What are the chances that this “just happened” If they are small, chances are that an external influence is at work that can be used to our benefit.Proba
49、bility and Statistics “the odds of Colorado University winning the national title are 3 to 1” “Drew Bledsoes pass completion percentage for the last 6 games is .58% versus .78% for the first 5 games” “The Senator will win the election with 54% of the popular vote with a margin of +/- 3%” Probability
50、 and Statistics influence our lives daily Statistics is the universal lanuage for science Statistics is the art of collecting, classifying, presenting, interpreting and analyzing numerical data, as well as making conclusions about the system from which the data was obtained.Population Vs. Sample (Ce
51、rtainty Vs. Uncertainty) Population Vs. Sample (Certainty Vs. Uncertainty) A sample is just a subset of all possible valuespopulationsample Since the sample does not contain all the possible values, there is some uncertainty about the population. Hence any Hence any statistics, such as mean and stan
52、dard deviation, are just statistics, such as mean and standard deviation, are just estimatesestimates of the true population parameters. of the true population parameters.Descriptive StatisticsDescriptive Statistics is the branch of statistics whichmost people are familiar. It characterizes and summ
53、arizesthe most prominent features of a given set of data (means, medians, standard deviations, percentiles, graphs, tables and charts. Descriptive Statistics describe the elements ofa population as a whole or to describe data that representjust a sample of elements from the entire populationInferent
54、ial Statistics Inferential StatisticsInferential Statistics is the branch of statistics that deals withdrawing conclusions about a population based on informationobtained from a sample drawn from that population.While descriptive statistics has been taught for centuries, inferential statistics is a
55、relatively new phenomenon havingits roots in the 20th century.We “infer” something about a population when only informationfrom a sample is known.Probability is the link betweenDescriptive and Inferential StatisticsWHAT DOES IT MEAN?-6-5-4-3-2-10123456NUMBER OF HEADS SIGMA VALUE (Z)CUM % OF POPULATI
56、ON586572798693100107114121128135142.003.1352.27515.8750.084.197.799.8699.9971301201101009080706005004003002001000FrequencyAnd the first 50 trials showed “Head Counts” greater than 130?WHAT IF WE MADE A CHANGE TO THE PROCESS?Chances are very good that the process distribution has changed. In fact, th
57、ere is a probability greater than 99.999% that it has changed. USES OF PROBABILITY DISTRIBUTIONSCritical ValueCritical ValueCommon OccurrenceRare OccurrenceRare OccurrencePrimarily these distributions are used to test for significant differences in data sets. To be classified as significant, the act
58、ual measured value must exceed a critical value. The critical value is tabular value determined by the probability distribution and the risk of error. This risk of error is called a risk and indicates the probability of this value occurring naturally. So, an a risk of .05 (5%) means that this critic
59、al value will be exceeded by a random occurrence less than 5% of the time.SO WHAT MAKES A DISTRIBUTION UNIQUE? CENTRAL TENDENCY Where a population is located. DISPERSIONHow wide a population is spread. DISTRIBUTION FUNCTIONThe mathematical formula that best describes the data (we will cover this in
60、detail in the next module). COIN TOSS CENTRAL TENDENCY1 301 201 1 01 009 08 07 06 0 05 0 04 0 03 0 02 0 01 0 00Number of occurrencesWhat are some of the ways that we can easily indicate the centering characteristic of the population? Three measures have historically been used; the mean, the median a
61、nd the mode. WHAT IS THE MEAN?ORDERED DATA SET-5-3-1-10000013-6-5-4-3-2-101234564The mean has already been used in several earlier modules and is the most common measure of central tendency for a population. The mean is simply the average value of the data.n=12xi= -2meanxxni=-= -21217.MeanWHAT IS TH
62、E MEDIAN?ORDERED DATA SET-5-3-1-10000013-6-5-4-3-2-101234564If we rank order (descending or ascending) the data set for this distribution we could represent central tendency by the order of the data points.If we find the value half way (50%) through the data points, we have another way of representi
63、ng central tendency. This is called the median value.Median ValueMedian50% of data pointsWHAT IS THE MODE?ORDERED DATA SET-5-3-1-10000013-6-5-4-3-2-101234564If we rank order (descending or ascending) the data set for this distribution we find several ways we can represent central tendency.We find th
64、at a single value occurs more often than any other. Since we know that there is a higher chance of this occurrence in the middle of the distribution, we can use this feature as an indicator of central tendency. This is called the mode.ModeModeMEASURES OF CENTRAL TENDENCY, SUMMARYMEAN ( )(Otherwise k
65、nown as the average)XXni=-=21217.XORDERED DATA SET-5-3-1-10000013-6 -5 -4 -3 -2 -101234564ORDERED DATA SET-5-3-1-10000013-6 -5 -4 -3 -2 -101234564ORDERED DATA SET-5-3-1-10000013-6 -5 -4 -3 -2 -101234564MEDIAN (50 percentile data point)Here the median value falls between two zero values and therefore
66、 is zero. If the values were say 2 and 3 instead, the median would be 2.5. MODE (Most common value in the data set)The mode in this case is 0 with 5 occurrences within this data.Mediann=12n/2=6n/2=6Mode = 0Mode = 0SO WHATS THE REAL DIFFERENCE?MEANThe mean is the most consistently accurate measure of
67、 central tendency, but is more difficult to calculate than the other measures. MEDIAN AND MODEThe median and mode are both very easy to determine. Thats the good news.The bad news is that both are more susceptible to bias than the mean. SO WHATS THE BOTTOM LINE?MEANUse on all occasions unless a circ
68、umstance prohibits its use. MEDIAN AND MODEOnly use if you cannot use mean.COIN TOSS POPULATION DISPERSION1 301 201 1 01 009 08 07 06 0 05 0 04 0 03 0 02 0 01 0 00Number of occurrencesWhat are some of the ways that we can easily indicate the dispersion (spread) characteristic of the population? Thre
69、e measures have historically been used; the range, the standard deviation and the variance. WHAT IS THE RANGE?ORDERED DATA SET-5-3-1-10000013-6-5-4-3-2-101234564The range is a very common metric which is easily determined from any ordered sample. To calculate the range simply subtract the minimum va
70、lue in the sample from the maximum value.RangeRangeMaxMinRangexxMAXMIN=-=- -=459()WHAT IS THE VARIANCE/STANDARD DEVIATION?The variance (s2) is a very robust metric which requires a fair amount of work to determine. The standard deviation(s) is the square root of the variance and is the most commonly
71、 used measure of dispersion for larger sample sizes.()sXXni221616712156=-=-=.DATA SET-5-3-1-10000013-6 -5 -4 -3 -2 -101234564XXni=-=212-.17XXi-5-(-.17)=-4.83-3-(-.17)=-2.83-1-(-.17)=-.83-1-(-.17)=-.830-(-.17)=.170-(-.17)=.170-(-.17)=.170-(-.17)=.170-(-.17)=.171-(-.17)=1.173-(-.17)=3.174-(-.17)=4.17(
72、-4.83)2=23.32(-2.83)2=8.01(-.83)2=.69(-.83)2=.69(.17)2=.03(.17)2=.03 (.17)2=.03(.17)2=.03(.17)2=.03(1.17)2=1.37(3.17)2=10.05(4.17)2=17.39 61.67MEASURES OF DISPERSIONRANGE (R)(The maximum data value minus the minimum)ORDERED DATA SET-5-3-1-10000013-6 -5 -4 -3 -2 -101234564ORDERED DATA SET-5-3-1-10000
73、013-6 -5 -4 -3 -2 -101234564VARIANCE (s2) (Squared deviations around the center point)STANDARD DEVIATION (s) (Absolute deviation around the center point)Min=-5RXX=-=- -=maxmin()4610Max=4DATA SET-5-3-1-10000013-6 -5 -4 -3 -2 -101234564XXni=-=212-.17()sXXni221616712156=-=-=.XXi-5-(-.17)=-4.83-3-(-.17)
74、=-2.83-1-(-.17)=-.83-1-(-.17)=-.830-(-.17)=.170-(-.17)=.170-(-.17)=.170-(-.17)=.170-(-.17)=.171-(-.17)=1.173-(-.17)=3.174-(-.17)=4.17(-4.83)2=23.32(-2.83)2=8.01(-.83)2=.69(-.83)2=.69(.17)2=.03(.17)2=.03 (.17)2=.03(.17)2=.03(.17)2=.03(1.17)2=1.37(3.17)2=10.05(4.17)2=17.39 61.67ss=2562 37.SAMPLE MEAN
75、AND VARIANCE EXAMPLE$m=XNXis()$221=-2sn-XXiXi1015 121410 91112 101212345678910S SXXi- -XXi( () )2- -XXi2sSO WHATS THE REAL DIFFERENCE?VARIANCE/ STANDARD DEVIATIONThe standard deviation is the most consistently accurate measure of central tendency for a single population. The variance has the added b
76、enefit of being additive over multiple populations. Both are difficult and time consuming to calculate.RANGEThe range is very easy to determine. Thats the good news.The bad news is that it is very susceptible to bias. SO WHATS THE BOTTOM LINE?VARIANCE/ STANDARD DEVIATIONBest used when you have enoug
77、h samples (10).RANGEGood for small samples (10 or less). SO WHAT IS THIS SHIFT & DRIFT STUFF.The project is progressing well and you wrap it up. 6 months later you are surprised to find that the population has taken a shift.-12-10-8-6-4-2024681012 USLLSLSO WHAT HAPPENED?All of our work was focused i
78、n a narrow time frame. Over time, other long term influences come and go which move the population and change some of its characteristics. This is called shift and drift.TimeHistorically, this shift and drift primarily impacts the position of the mean and shifts it 1.5 from its original position.Ori
79、ginal StudyVARIATION FAMILIESVariation is present upon repeat measurements within the same sample. Variation is present upon measurements of different samples collected within a short time frame.Variation is present upon measurements collected with a significant amount of time between samples.Source
80、s of VariationWithin Individual SamplePiece to PieceTime to TimeSO WHAT DOES IT MEAN?To compensate for these long term variations, we must consider two sets of metrics. Short term metrics are those which typically are associated with our work. Long term metrics take the short term metric data and de
81、grade it by an average of 1.5s.IMPACT OF 1.5 SHIFT AND DRIFTZPPMSTCpkPPMLT (+1.5 )0.0500,0000.0933,1930.1460,1720.0919,2430.2420,7400.1903,1990.3382,0890.1884,9300.4344,5780.1864,3340.5308,5380.2841,3450.6274,2530.2815,9400.7241,9640.2788,1450.8211,8550.3758,0360.9184,0600.3725,7471.0158,6550.3691,4
82、621.1135,6660.4655,4221.2115,0700.4617,9111.396,8010.4579,2601.480,7570.5539,8281.566,8070.5500,0001.654,7990.5460,1721.744,5650.6420,740Here, you can see that the impact of this concept is potentially very significant. In the short term, we have driven the defect rate down to 54,800 ppm and can exp
83、ect to see occasional long term ppm to be as bad as 460,000 ppm. SHIFT AND DRIFT EXERCISEWe have just completed a project and have presented the following short term metrics:Zst=3.5PPMst=233Cpkst=1.2Calculate the long term values for each of these metrics.Learning ObjectivesvHave a broad understandi
84、ng of what probability distributions are and why they are important.vUnderstand the role that probability distributions play in determining whether an event is a random occurrence or significantly different.vUnderstand the common measures used to characterize a population central tendency and disper
85、sion.vUnderstand the concept of Shift & Drift.vUnderstand the concept of significance testing.COMMON PROBABILITY DISTRIBUTIONS AND THEIR USESLearning ObjectivesvHave a broad understanding of how probability distributions are used in improvement projects.vReview the origin and use of common probabili
86、ty distributions.Why do we Care?Probability distributions are necessary to: determine whether an event is significant or due to random chance.predict the probability of specific performance given historical characteristics.IMPROVEMENT ROADMAPUses of Probability DistributionsBreakthroughStrategyChara
87、cterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlBaselining Processes Verifying ImprovementsCommon UsesFocus on understanding the use of the distributionsPractice with examples wherever possibleFocus on the use and context of the toolKEYS TO SUCCESSData poi
88、nts vary, but as the data accumulates, it forms a distribution which occurs naturally.LocationSpreadShapeDistributions can vary in:PROBABILITY DISTRIBUTIONS, WHERE DO THEY COME FROM?-4-3-2-10123401234567-4-3-2-101234012345670123401234567Original PopulationSubgroup AverageSubgroup Variance (s2)Contin
89、uous DistributionNormal Distributionc2 DistributionCOMMON PROBABILITY DISTRIBUTIONSTHE LANGUAGE OF MATHPopulation and Sample SymbologyValuePopulationSampleMean m mVariance 2s2Standard Deviation sProcess CapabilityCpBinomial MeanxPPCpTHREE PROBABILITY DISTRIBUTIONStXsnCALC=-mSignificantttCALCCRIT=Sig
90、nificantFFCALCCRIT=Fsscalc=1222()ca,dfeaefff22=-SignificantCALCCRIT=cc22Note that in each case, a limit has been established to determine what is random chance verses significant difference. This point is called the critical value. If the calculated value exceeds this critical value, there is very l
91、ow probability (P30)F Stat (n5)Z Stat (n30)Z Stat (p)t Stat (n30) t Stat (n30)Z Stat (p)t Stat (n30) t Stat (n30) c2 Stat (n5)Z Stat (m,n30)Z Stat (p)t Stat (m,n30) c2 Stat (s,norders) we look for $0 on this new distribution. Any occurrence to the right of this point will represent shipments orders.
92、 So, we need to calculate the percent of the curve that exists to the right of $0.$7000$0Shipments ordersX = $53,000 in orders/weeks = $8,000X = $60,000 shipped/weeks = $5,000OrdersShipmentsTRANSACTIONAL EXAMPLETRANSACTIONAL EXAMPLE, CONTINUEDXXXshipmentsordersshipmentsorders-=-=-=$60,$53,$7,0000000
93、00()()sssshipmentsordersshipmentsorders-=+=+=222250008000$9434To calculate the percent of the curve to the right of $0 we need to convert the difference between the $0 point and $7000 into sigma intervals. Since we know every $9434 interval from the mean is one sigma, we can calculate this position
94、as follows:$7000$0Shipments ordersm074-=-=Xss$0$7000$9434.Look up .74s in the normal table and you will find .77. Therefore, the answer to the original question is that 77% of the time, shipments will exceed orders.Now, as a classroom exercise, what percent of the time will shipments exceed orders b
95、y $10,000?MANUFACTURING EXAMPLEv2 Blocks are being assembled end to end and significant variation has been found in the overall assembly length.vThe blocks have the following dimensions:vDetermine the overall assembly length and standard deviation.X1 = 4.00 inchess1 = .03 inchesX2 = 3.00 inchess2 =
96、.04 inchesLearning ObjectivesvHave a broad understanding of how probability distributions are used in improvement projects.vReview the origin and use of common probability distributions.CORRELATION ANALYSISLearning ObjectivesvUnderstand how correlation can be used to demonstrate a relationship betwe
97、en two factors.vKnow how to perform a correlation analysis and calculate the coefficient of linear correlation (r).vUnderstand how a correlation analysis can be used in an improvement project.Why do we Care?Correlation Analysis is necessary to: show a relationship between two variables. This also se
98、ts the stage for potential cause and effect.IMPROVEMENT ROADMAPUses of Correlation AnalysisBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlDetermine and quantify the relationship between factors (x) and output characteristics (Y).C
99、ommon UsesAlways plot the dataRemember: Correlation does not always imply cause & effectUse correlation as a follow up to the Fishbone DiagramKeep it simple and do not let the tool take on a life of its ownKEYS TO SUCCESSWHAT IS CORRELATION?Input or x variable (independent)Output or y variable (depe
100、ndent)CorrelationY= f(x) As the input variable changes, there is an influence or bias on the output variable. vA measurable relationship between two variable data characteristics.Not necessarily Cause & Effect (Y=f(x)vCorrelation requires paired data sets (ie (Y1,x1), (Y2,x2), etc)vThe input variabl
101、e is called the independent variable (x or KPIV) since it is independent of any other constraintsvThe output variable is called the dependent variable (Y or KPOV) since it is (theoretically) dependent on the value of x.vThe coefficient of linear correlation “r” is the measure of the strength of the
102、relationship.vThe square of “r” is the percent of the response (Y) which is related to the input (x).WHAT IS CORRELATION?TYPES OF CORRELATIONStrongWeakNonePositiveNegativeY=f(x)Y=f(x)Y=f(x)xxxCALCULATING “r” Coefficient of Linear CorrelationCalculate sample covariance ( )Calculate sx and sy for each
103、 data setUse the calculated values to compute rCALC.Add a + for positive correlation and - for a negative correlation.()()sxxyynxyii=-1rs sCALCsxyxy=sxyWhile this is the most precise method to calculate Pearsons r, there is an easier way to come up with a fairly close approximation.APPROXIMATING “r”
104、Coefficient of Linear CorrelationWLY=f(x)xrWL -1r - = -16712 647.+ = positive slope- = negative slopeWL|1234567891011121314156.712.6Plot the data on orthogonal axisDraw an Oval around the dataMeasure the length and width of the OvalCalculate the coefficient of linear correlation (r) based on the for
105、mulas belowHOW DO I KNOW WHEN I HAVE CORRELATION?OrderedPairsrCRIT5.886.817.758.719.6710.6315.5120.4425.4030.3650.2880.22100.20 The answer should strike a familiar cord at this point We have confidence (95%) that we have correlation when |rCALC| rCRIT.Since sample size is a key determinate of rCRIT
106、we need to use a table to determine the correct rCRIT given the number of ordered pairs which comprise the complete data set.So, in the preceding example we had 60 ordered pairs of data and we computed a rCALC of -.47. Using the table at the left we determine that the rCRIT value for 60 is .26. Comp
107、aring |rCALC| rCRIT we get .47 .26. Therefore the calculated value exceeds the minimum critical value required for significance. Conclusion: We are 95% confident that the observed correlation is significant.Learning ObjectivesvUnderstand how correlation can be used to demonstrate a relationship betw
108、een two factors.vKnow how to perform a correlation analysis and calculate the coefficient of linear correlation (r).vUnderstand how a correlation analysis can be used in a blackbelt story.CENTRAL LIMIT THEOREMLearning ObjectivesvUnderstand the concept of the Central Limit Theorem.vUnderstand the app
109、lication of the Central Limit Theorem to increase the accuracy of measurements.Why do we Care?The Central Limit Theorem is: the key theoretical link between the normal distribution and sampling distributions.the means by which almost any sampling distribution, no matter how irregular, can be approxi
110、mated by a normal distribution if the sample size is large enough.IMPROVEMENT ROADMAPUses of the Central Limit TheoremBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlThe Central Limit Theorem underlies all statistic techniques whic
111、h rely on normality as a fundamental assumptionCommon UsesFocus on the practical application of the conceptKEYS TO SUCCESSWHAT IS THE CENTRAL LIMIT THEOREM?Central Limit TheoremFor almost all populations, the sampling distribution of the mean can be approximated closely by a normal distribution, pro
112、vided the sample size is sufficiently large.NormalWhy do we Care?What this means is that no matter what kind of distribution we sample, if the sample size is big enough, the distribution for the mean is approximately normal.This is the key link that allows us to use much of the inferential statistic
113、s we have been working with so far. This is the reason that only a few probability distributions (Z, t and c2) have such broad application. If a random event happens a great many times, the average results are likely to be predictable.Jacob BernoulliHOW DOES THIS WORK?Parent Populationn=2n=5n=10As y
114、ou average a larger and larger number of samples, you can see how the original sampled population is transformed.ANOTHER PRACTICAL ASPECTsnxx=sThis formula is for the standard error of the mean.What that means in laymans terms is that this formula is the prime driver of the error term of the mean. R
115、educing this error term has a direct impact on improving the precision of our estimate of the mean.The practical aspect of all this is that if you want to improve the precision of any test, increase the sample size. So, if you want to reduce measurement error (for example) to determine a better esti
116、mate of a true value, increase the sample size. The resulting error will be reduced by a factor of . The same goes for any significance testing. Increasing the sample size will reduce the error in a similar manner.1nDICE EXERCISEBreak into 3 teamsTeam one will be using 2 diceTeam two will be using 4
117、 diceTeam three will be using 6 diceEach team will conduct 100 throws of their dice and record the average of each throw.Plot a histogram of the resulting data.Each team presents the results in a 10 min report out.Learning ObjectivesvUnderstand the concept of the Central Limit Theorem.vUnderstand th
118、e application of the Central Limit Theorem to increase the accuracy of measurements.PROCESS CAPABILITY ANALYSISLearning ObjectivesvUnderstand the role that process capability analysis plays in the successful completion of an improvement project.vKnow how to perform a process capability analysis.Why
119、do we Care?Process Capability Analysis is necessary to: determine the area of focus which will ensure successful resolution of the project.benchmark a process to enable demonstrated levels of improvement after successful resolution of the project. demonstrate improvement after successful resolution
120、of the project. IMPROVEMENT ROADMAPUses of Process Capability AnalysisBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlBaselining a process primary metric (Y) prior to starting a project.Common UsesCharacterizing the capability of c
121、ausitive factors (x).Characterizing a process primary metric after changes have been implemented to demonstrate the level of improvement.Must have specification limits - Use process targets if no specs availableDont get lost in the mathRelate to Z for comparisons (Cpk x 3 = Z)For Attribute data use
122、PPM conversion to Cpk and ZKEYS TO SUCCESSWHAT IS PROCESS CAPABILITY?Process capability is simply a measure of how good a metric is performing against and established standard(s). Assuming we have a stable process generating the metric, it also allows us to predict the probability of the metric valu
123、e being outside of the established standard(s).SpecOut of SpecIn SpecProbabilitySpec (Lower)Spec (Upper)In SpecOut of SpecOut of SpecProbabilityProbabilityUpper and Lower Standards (Specifications)Single Standard (Specification)WHAT IS PROCESS CAPABILITY?Process capability (Cpk) is a function of how
124、 the population is centered (|m-spec|) and the population spread (s).Process Center (|m-spec|) Spec (Lower)Spec (Upper)In SpecOut of SpecOut of SpecHigh Cpk Poor Cpk Spec (Lower)Spec (Upper)In SpecOut of SpecOut of SpecProcess Spread (s) Spec (Lower)Spec (Upper)In SpecOut of SpecOut of SpecSpec (Low
125、er)Spec (Upper)In SpecOut of SpecOut of SpecHOW IS PROCESS CAPABILITY CALCULATEDSpec (LSL)Spec (USL)mNote: LSL = Lower Spec LimitUSL = Upper Spec LimitDistance between the population mean and the nearest spec limit (|m-USL |). This distance divided by 3s is Cpk.Expressed mathematically, this looks l
126、ike:PROCESS CAPABILITY EXAMPLEvCalculation Values:vUpper Spec value = $200,000 maximumvNo Lower Specvm = historical average = $250,000vs = $20,000v Calculation:Answer: Cpk= -.83We want to calculate the process capability for our inventory. The historical average monthly inventory is $250,000 with a
127、standard deviation of $20,000. Our inventory target is $200,000 maximum.()()CMINLSL USLPK=-=-=mms,$200,$250,*$20,-.8330000003000ATTRIBUTE PROCESS CAPABILITY TRANSFORMZPPMSTCpkPPMLT (+1.5 )0.0500,0000.0933,1930.1460,1720.0919,2430.2420,7400.1903,1990.3382,0890.1884,9300.4344,5780.1864,3340.5308,5380.
128、2841,3450.6274,2530.2815,9400.7241,9640.2788,1450.8211,8550.3758,0360.9184,0600.3725,7471.0158,6550.3691,4621.1135,6660.4655,4221.2115,0700.4617,9111.396,8010.4579,2601.480,7570.5539,8281.566,8070.5500,0001.654,7990.5460,1721.744,5650.6420,7401.835,9300.6382,0891.928,7160.6344,5782.022,7500.7308,538
129、2.117,8640.7274,2532.213,9030.7241,9642.310,7240.8211,8552.48,1980.8184,0602.56,2100.8158,6552.64,6610.9135,6662.73,4670.9115,0702.82,5550.996,8012.91,8661.080,7573.01,3501.066,8073.19681.054,7993.26871.144,5653.34831.135,9303.43371.128,7163.52331.222,7503.61591.217,8643.71081.213,9033.872.41.310,72
130、43.948.11.38,1984.031.71.36,210()CMINLSL USLPK=-mms,3If we take the Cpk formula below We find that it bears a striking resemblance to the equation for Z which is: ZCALC=-mms0with the value m-m0 substituted for MIN(m-LSL,USL-m). Making this substitution, we get : CMINLSL USLZpkMINLSL USL=-=-133*(,)(,
131、)mmsmmWe can now use a table similar to the one on the left to transform either Z or the associated PPM to an equivalent Cpk value. So, if we have a process which has a short term PPM=136,666 we find that the equivalent Z=1.1 and Cpk=0.4 from the table. Learning ObjectivesvUnderstand the role that p
132、rocess capability analysis plays in the successful completion of an improvement project.vKnow how to perform a process capability analysis.MULTI-VARI ANALYSISLearning ObjectivesvUnderstand how to use multi-vari charts in completing an improvment project.vKnow how to properly gather data to construct
133、 multi-vari charts.vKnow how to construct a multi-vari chart.vKnow how to interpret a multi-vari chart.Why do we Care?Multi-Vari charts are a: Simple, yet powerful way to significantly reduce the number of potential factors which could be impacting your primary metric.Quick and efficient method to s
134、ignificantly reduce the time and resources required to determine the primary components of variation.IMPROVEMENT ROADMAPUses of Multi-Vari ChartsBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlEliminate a large number of factors fr
135、om the universe of potential factors.Common UsesKEYS TO SUCCESSGather data by systematically sampling the existing processPerform “ad hoc” training on the tool for the team prior to useEnsure your sampling plan is complete prior to gathering dataHave team members (or yourself) do the sampling to avo
136、id biasCareful planning before you startThe Goal of the logical searchis to narrow down to 5-6 key variables !techniquetoolingoperator errorhumiditytemperaturesupplierhardnesslubricationold machinerywrong spec.tool wearhandling damagepressureheat treatfatigueA World of Possible Causes (KPIVs).REDUCI
137、NG THE POSSIBILITIES“.The Dictionary Game? ”WebstersDictionaryIs it a Noun? WebstersDictionaryIm thinking of a wordin this book? Can youfigure out what it is?USE A LOGICAL APPROACH TO SEE THE MAJOR SOURCES OF VARIATIONREDUCING THE POSSIBILITIESHow many guesses do you think it will take to find a sin
138、gle word in the text book?Lets try and see. REDUCING THE POSSIBILITIESHow many guesses do you think it will take to find a single word in the text book?Statistically it should take no more than 17 guesses217=2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2= 131,072 Most Unabridged dictionaries have 127,000 words.R
139、eduction of possibilities can be an extremely powerful technique.PLANNING A MULTI-VARI ANALYSISDetermine the possible families of variation.Determine how you will take the samples.Take a stratified sample (in order of creation). DO NOT take random samples.Take a minimum of 3 samples per group and 3
140、groups.The samples must represent the full range of the process.Does one sample or do just a few samples stand out?There could be a main effect or an interaction at the cause.MULTI-VARI ANALYSIS, VARIATION FAMILIESVariation is present upon repeat measurements within the same sample. Variation is pre
141、sent upon measurements of different samples collected within a short time frame.Variation is present upon measurements collected with a significant amount of time between samples.Sources of VariationWithin Individual SamplePiece to PieceTime to TimeThe key is reducing the number of possibilities to
142、a manageable few.MULTI-VARI ANALYSIS, VARIATION SOURCESWithin Individual SampleMeasurement AccuracyOut of RoundIrregularities in PartPiece to PieceMachine fixturingMold cavity differences Time to TimeMaterial ChangesSetup DifferencesTool WearCalibration DriftOperator InfluenceManufacturing(Machining
143、)Transactional(Order Rate)Piece to PieceCustomer DifferencesOrder EditorSales OfficeSales RepWithin Individual SampleMeasurement AccuracyLine Item ComplexityTime to TimeSeasonal VariationManagement ChangesEconomic ShiftsInterest RateHOW TO DRAW THE CHARTRange within a single sampleSample 1Average wi
144、thin a single sample Plot the first sample range with a point for the maximum reading obtained, and a point for the minimum reading. Connect the points and plot a third point at the average of the within sample readingsStep 1Range between two sample averagesSample 1Sample 2Sample 3 Plot the sample r
145、anges for the remaining “piece to piece” data. Connect the averages of the within sample readings. Step 2 Plot the “time to time” groups in the same manner.Time 1Time 2Time 3Step 3READING THE TEA LEAVES.Common Patterns of VariationWithin PieceCharacterized by large variation in readings taken of the
146、 same single sample, often from different positions within the sample.Piece to PieceCharacterized by large variation in readings taken between samples taken within a short time frame.Time to TimeCharacterized by large variation in readings taken between samples taken in groups with a significant amo
147、unt of time elapsed between groups.MULTI-VARI EXERCISEWe have a part dimension which is considered to be impossible to manufacture. A capability study seems to confirm that the process is operating with a Cpk=0 (500,000 ppm). You and your team decide to use a Multi-Vari chart to localize the potenti
148、al sources of variation. You have gathered the following data:SampleDay/TimeBeginningof PartMiddle ofPartEnd ofPart11/0900.015.017.01821/0905.010.012.01531/0910.013.015.01642/1250.014.015.01852/1255.009.012.01762/1300.012.014.01673/1600.013.014.01783/1605.010.013.01593/1610.011.014.0179Construct a m
149、ulti-vari chart of the data and interpret the results.Learning ObjectivesvUnderstand how to use multi-vari charts in completing an improvment project.vKnow how to properly gather data to construct multi-vari charts.vKnow how to construct a multi-vari chart.vKnow how to interpret a multi-vari chart.S
150、AMPLE SIZE CONSIDERATIONSLearning ObjectivesvUnderstand the critical role having the right sample size has on an analysis or study.vKnow how to determine the correct sample size for a specific study.vUnderstand the limitations of different data types on sample size.Why do we Care?The correct sample
151、size is necessary to: ensure any tests you design have a high probability of success.properly utilize the type of data you have chosen or are limited to working with.IMPROVEMENT ROADMAPUses of Sample Size ConsiderationsBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizati
152、onPhase 3:ImprovementPhase 4:ControlSample Size considerations are used in any situation where a sample is being used to infer a population characteristic.Common UsesUse variable data wherever possibleGenerally, more samples are better in any studyWhen there is any doubt, calculate the needed sample
153、 size Use the provided excel spreadsheet to ease sample size calculationsKEYS TO SUCCESSCONFIDENCE INTERVALSThe possibility of error exists in almost every system. This goes for point values as well. While we report a specific value, that value only represents our best estimate from the data at hand
154、. The best way to think about this is to use the form:true value = point estimate +/- errorThe error around the point value follows one of several common probability distributions. As you have seen so far, we can increase our confidence is to go further and further out on the tails of this distribut
155、ion. Point Value+/- 1s = 67% Confidence Band+/- 2s = 95% Confidence BandThis “error band” which exists around the point estimate is called the confidence interval.BUT WHAT IF I MAKE THE WRONG DECISION?Not different (Ho)RealityTest DecisionDifferent (H1)Not different (Ho)Different (H1)Correct Conclus
156、ionCorrect Conclusiona a Risk Type I Error Producer Risk Type II Error b b risk Consumer RiskTestReality = DifferentDecision Pointb b Riska a RiskWHY DO WE CARE IF WE HAVE THE TRUE VALUE?How confident do you want to be that you have made the right decision?Ho: Patient is not sickH1: Patient is sickE
157、rror ImpactType I Error = Treating a patient who is not sick Type II Error = Not treating a sick patientA person does not feel well and checks into a hospital for tests. Not different (Ho)RealityTest DecisionDifferent (H1)Not different (Ho)Different (H1)Correct ConclusionCorrect Conclusiona a Risk T
158、ype I Error Producer Risk Type II Error b b risk Consumer RiskHOW ABOUT ANOTHER EXAMPLE?Ho: Order rate unchangedH1: Order rate is differentError ImpactType I Error = Unnecessary costs Type II Error = Long term loss of salesA change is made to the sales force to save costs. Did it adversely impact th
159、e order receipt rate?Not different (Ho)RealityTest DecisionDifferent (H1)Not different (Ho)Different (H1)Correct ConclusionCorrect Conclusiona a Risk Type I Error Producer Risk Type II Error b b risk Consumer RiskCONFIDENCE INTERVAL FORMULASThese individual formulas are not critical at this point, b
160、ut notice that the only opportunity for decreasing the error band (confidence interval) without decreasing the confidence factor, is to increase the sample size. Xttnanan- - + +- - -/ ,/ ,2121s sm ms snXMeansnsnaa- - - - -1112222c cs sc c/Standard DeviationCpnCpCpnananc cc c121221211- - - - - - -/ ,
161、/ ,Process Capability( () )( () )$/pZppnppZppnaa- - - + +- -2211Percent DefectiveSAMPLE SIZE EQUATIONSnZXa=-/22smnsa=+sc2221/Allowable error = m - (also known as d)XAllowable error = s/s ()nppZE=-$/122aAllowable error = E Standard DeviationMeanPercent DefectiveSAMPLE SIZE EXAMPLEvCalculation Values:
162、vAverage tells you to use the mean formulavSignificance: a = 5% (95% confident)vZa/2 = Z.025 = 1.96vs=10 poundsvm-x = error allowed = 2 poundsv Calculation:vAnswer: n=97 SamplesWe want to estimate the true average weight for a part within 2 pounds. Historically, the part weight has had a standard de
163、viation of 10 pounds. We would like to be 95% confident in the results. nZX=- = =asm/.*222196 10297SAMPLE SIZE EXAMPLEvCalculation Values:vPercent defective tells you to use the percent defect formulavSignificance: a = 5% (95% confident)vZa/2 = Z.025 = 1.96vp = 10% = .1vE = 1% = .01v Calculation:vAn
164、swer: n=3458 SamplesWe want to estimate the true percent defective for a part within 1%. Historically, the part percent defective has been 10%. We would like to be 95% confident in the results. ()()nppZE=-=-=$./11 1 1196013458222aLearning ObjectivesvUnderstand the critical role having the right samp
165、le size has on an analysis or study.vKnow how to determine the correct sample size for a specific study.vUnderstand the limitations of different data types on sample size.CONFIDENCE INTERVALSLearning ObjectivesvUnderstand the concept of the confidence interval and how it impacts an analysis or study
166、.vKnow how to determine the confidence interval for a specific point value.vKnow how to use the confidence interval to test future point values for significant change.Why do we Care?Understanding the confidence interval is key to: understanding the limitations of quotes in point estimate data.being
167、able to quickly and efficiently screen a series of point estimate data for significance.IMPROVEMENT ROADMAPUses of Confidence IntervalsBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlUsed in any situation where data is being evalua
168、ted for significance.Common UsesUse variable data wherever possibleGenerally, more samples are better (limited only by cost)Recalculate confidence intervals frequently Use an excel spreadsheet to ease calculationsKEYS TO SUCCESSWHAT ARE CONFIDENCE INTERVALS?The possibility of error exists in almost
169、every system. This goes for point values as well. While we report a specific value, that value only represents our best estimate from the data at hand. The best way to think about this is to use the form:true value = point estimate +/- errorThe error around the point value follows one of several com
170、mon probability distributions. As you have seen so far, we can increase our confidence is to go further and further out on the tails of this distribution. Point Value+/- 1s = 67% Confidence Band+/- 2s = 95% Confidence BandThis “error band” which exists around the point estimate is called the confide
171、nce interval.So, what does this do for me?The confidence interval establishes a way to test whether or not a significant change has occurred in the sampled population. This concept is called significance or hypothesis testing.Being able to tell when a significant change has occurred helps in prevent
172、ing us from interpreting a significant change from a random event and responding accordingly.REMEMBER OUR OLD FRIEND SHIFT & DRIFT?All of our work was focused in a narrow time frame. Over time, other long term influences come and go which move the population and change some of its characteristics. T
173、imeConfidence Intervals give us the tool to allow us to be able to sort the significant changes from the insignificant.Original StudyUSING CONFIDENCE INTERVALS TO SCREEN DATA234567TIME95% Confidence IntervalSignificant Change?WHAT KIND OF PROBLEM DO YOU HAVE?Analysis for a significant change asks th
174、e question “What happened to make this significantly different from the rest?”Analysis for a series of random events focuses on the process and asks the question “What is designed into this process which causes it to have this characteristic?”. CONFIDENCE INTERVAL FORMULASThese individual formulas e
175、nable us to calculate the confidence interval for many of the most common metrics. Meansnsnaa- - - - -1112222c cs sc c/Standard DeviationCpnCpCpnananc cc c121221211- - - - - - -/ ,/ ,Process Capability( () )( () )$/pZppnppZppnaa- - - + +- -2211Percent DefectiveXttnanan- - + +- - -/ ,/ ,2121m mnXssCO
176、NFIDENCE INTERVAL EXAMPLEvCalculation Values:vAverage defect rate of 14,000 ppm = 14,000/1,000,000 = .014vSignificance: a = 5% (95% confident)vZa/2 = Z.025 = 1.96vn=10,000vComparison defect rate of 23,000 ppm = .023v Calculation:vAnswer: Yes, .023 is significantly outside of the expected 95% confide
177、nce interval of .012 to .016.Over the past 6 months, we have received 10,000 parts from a vendor with an average defect rate of 14,000 dpm. The most recent batch of parts proved to have 23,000 dpm. Should we be concerned? We would like to be 95% confident in the results. ( () )( () )$/pZppnppZppnaa-
178、 - - + +- -2211( () )( () ).,.,0141 96014 101410 0000141 96014 101410 000- - - + +- -p.01400230140023- - + +p.012016 pCONFIDENCE INTERVAL EXERCISEWe are tracking the gas mileage of our late model ford and find that historically, we have averaged 28 MPG. After a tune up at Billy Bobs auto repair we f
179、ind that we only got 24 MPG average with a standard deviation of 3 MPG in the next 16 fillups. Should we be concerned? We would like to be 95% confident in the results. What do you think?Learning ObjectivesvUnderstand the concept of the confidence interval and how it impacts an analysis or study.vKn
180、ow how to determine the confidence interval for a specific point value.vKnow how to use the confidence interval to test future point values for significant change.CONTROL CHARTSLearning ObjectivesvUnderstand how to select the correct control chart for an application.vKnow how to fill out and maintai
181、n a control chart.vKnow how to interpret a control chart to determine the occurrence of “special causes” of variation.Why do we Care?Control charts are useful to: determine the occurrence of “special cause” situations.Utilize the opportunities presented by “special cause” situations” to identify and
182、 correct the occurrence of the “special causes” .IMPROVEMENT ROADMAPUses of Control ChartsBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlControl charts can be effectively used to determine “special cause” situations in the Measure
183、ment and Analysis phasesCommon UsesUse control charts on only a few critical output characteristicsEnsure that you have the means to investigate any “special cause”KEYS TO SUCCESSWhat is a “Special Cause”?Remember our earlier work with confidence intervals? Any occurrence which falls outside the con
184、fidence interval has a low probability of occurring by random chance and therefore is “significantly different”. If we can identify and correct the cause, we have an opportunity to significantly improve the stability of the process. Due to the amount of data involved, control charts have historicall
185、y used 99% confidence for determining the occurrence of these “special causes”Point ValueSpecial cause occurrence.X+/- 3s = 99% Confidence BandWhat is a Control Chart ?A control chart is simply a run chart with confidence intervals calculated and drawn in. These “Statistical control limits” form the
186、 trip wires which enable us to determine when a process characteristic is operating under the influence of a “Special cause”.+/- 3s = 99% Confidence IntervalSo how do I construct a control chart?First things first: Select the metric to be evaluatedSelect the right control chart for the metricGather
187、enough data to calculate the control limitsPlot the data on the chartDraw the control limits (UCL & LCL) onto the chart.Continue the run, investigating and correcting the cause of any “out of control” occurrence.How do I select the correct chart ?What type of data do I have?VariableAttributeCounting
188、 defects or defectives?X-s ChartIMR ChartX-R Chartn 101 n 10n = 1DefectivesDefectsWhat subgroup size is available?Constant Sample Size?Constant Opportunity?yesyesnononp Chartu Chartp Chartc ChartNote: A defective unit can have more than one defect.How do I calculate the control limits?X= average of
189、the subgroup averagesR= average of the subgroup range values2A= a constant function of subgroup size (n)XR Chart -For the averages chart:For the range chart:nD4D3A223.2701.8832.5701.0242.2800.7352.1100.5862.0000.4871.920.080.4281.860.140.3791.820.180.34UCL = upper control limitLCL = lower control li
190、mitHow do I calculate the control limits?p and np ChartsP = number of rejects in the subgroup/number inspected in subgroupP= total number of rejects/total number inspectedn = number inspected in subgroupFor varied sample size:For constant sample size:Note: P charts have an individually calculated co
191、ntrol limit for each point plottedHow do I calculate the control limits?c and u ChartsU= total number of nonconformities/total units evaluatedn = number evaluated in subgroupFor varied opportunity (u):For constant opportunity (c):C= total number of nonconformities/total number of subgroupsNote: U ch
192、arts have an individually calculated control limit for each point plottedHow do I interpret the charts?vThe process is said to be “out of control” if:One or more points fall outside of the control limitsWhen you divide the chart into zones as shown and:v2 out of 3 points on the same side of the cent
193、erline in Zone Av4 out of 5 points on the same side of the centerline in Zone A or Bv9 successive points on one side of the centerlinev6 successive points successively increasing or decreasing v14 points successively alternating up and downv15 points in a row within Zone C (above and/or below center
194、line)Zone AZone BZone CZone CZone BZone AUpper Control Limit (UCL)Lower Control Limit (LCL)Centerline/AverageWhat do I do when its “out of control”?Time to Find and Fix the cause Look for patterns in the dataAnalyze the “out of control” occurrenceFishbone diagrams and Hypothesis tests are valuable “
195、discovery” tools.Learning ObjectivesvUnderstand how to select the correct control chart for an application.vKnow how to fill out and maintain a control chart.vKnow how to interpret a control chart to determine out of control situations.HYPOTHESIS TESTINGLearning ObjectivesvUnderstand the role that h
196、ypothesis testing plays in an improvement project.vKnow how to perform a two sample hypothesis test.vKnow how to perform a hypothesis test to compare a sample statistic to a target value.vKnow how to interpret a hypothesis test.Why do we Care?Hypothesis testing is necessary to: determine when there
197、is a significant difference between two sample populations.determine whether there is a significant difference between a sample population and a target value.IMPROVEMENT ROADMAPUses of Hypothesis TestingBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:Improv
198、ementPhase 4:ControlConfirm sources of variation to determine causative factors (x).Demonstrate a statistically significant difference between baseline data and data taken after improvements were implemented.Common UsesUse hypothesis testing to “explore” the dataUse existing data wherever possibleUs
199、e the teams experience to direct the testingTrust but verify.hypothesis testing is the verifyIf theres any doubt, find a way to hypothesis test itKEYS TO SUCCESSSO WHAT IS HYPOTHESIS TESTING?The theory of probability is nothing more than good sense confirmed by calculation.LaplaceWe think we see som
200、ethingwell, we thinkerr maybe it is could be.But, how do we know for sure?Hypothesis testing is the key by giving us a measure of how confident we can be in our decision.SO HOW DOES THIS HYPOTHESIS STUFF WORK?Statistic ValueCritical PointToto, I dont think were in Kansas anymore.Ho = no difference(N
201、ull hypothesis)H1 = significant difference(Alternate Hypothesis)StatisticCALC StatisticCRITWe determine a critical value from a probability table for the statistic. This value is compared with the calculated value we get from our data. If the calculated value exceeds the critical value, the probabil
202、ity of this occurrence happening due to random variation is less than our test a a.SO WHAT IS THIS NULL” HYPOTHESIS?NullHoFail to Reject the Null Hypothesis Data does not support conclusion that there is a significant differenceAlternativeH1 Reject the Null HypothesisData supports conclusion that th
203、ere is a significant differenceHypothesisSymbolHow you say itWhat it meansMathematicians are like Frenchmen, whatever you say to them they translate into their own language and forth with it is something entirely different. GoetheHYPOTHESIS TESTING ROADMAP.Test UsedF Stat (n30)F Stat (n5)Test UsedZ
204、Stat (n30)Z Stat (p)t Stat (n30) t Stat (n30)Z Stat (p)t Stat (n30) t Stat (n30) c2 Stat (n5)Population AveragePopulation VarianceCompare 2 Population AveragesCompare a Population Average Against a Target ValueCompare 2 Population VariancesCompare a Population Variance Against a Target ValuevDetermi
205、ne the hypothesis to be tested (Ho:=, ).vDetermine whether this is a 1 tail (a a) or 2 tail (a a/2) test.vDetermine the a a risk for the test (typically .05).vDetermine the appropriate test statistic.vDetermine the critical value from the appropriate test statistic table.vGather the data.vUse the da
206、ta to calculate the actual test statistic.vCompare the calculated value with the critical value. vIf the calculated value is larger than the critical value, reject the null hypothesis with confidence of 1-a a (ie there is little probability (pm2Ho: s1nn1230+2 Sample Tau2 Sample Z2 Sample t(DF: n1+n2
207、-2)Use these formulas to calculate the actual statistic for comparison with the critical (table) statistic. Note that the only major determinate here is the sample sizes. It should make sense to utilize the simpler tests (2 Sample Tau) wherever possible unless you have a statistical software package
208、 available or enjoy the challenge. XX-12vHo: m1 = m2vStatistic Summary:vn1 = n2 = 5vSignificance: a/2 = .025 (2 tail)vtaucrit = .613 (From the table for a = .025 & n=5) vCalculation:vR1=337, R2= 577vX1=2868, X2=2896 vtauCALC=2(2868-2896)/(337+577)=|.06|vTest: vHo: tauCALC tauCRITvHo: .06.613 = true?
209、 (yes, therefore we will fail to reject the null hypothesis).vConclusion: Fail to reject the null hypothesis (ie. The data does not support the conclusion that there is a significant difference)Receipts 1 Receipts 23067320027302777284026232913304427892834Hypothesis Testing Example (2 Sample Tau)Seve
210、ral changes were made to the sales organization. The weekly number of orders were tracked both before and after the changes. Determine if the samples have equal means with 95% confidence.tdCALCRR=+12XX-122vHo: m1 = m2vStatistic Summary:vn1 = n2 = 5vDF=n1 + n2 - 2 = 8vSignificance: a/2 = .025 (2 tail
211、)vtcrit = 2.306 (From the table for a=.025 and 8 DF) vCalculation:vs1=130, s2= 227vX1=2868, X2=2896 vtCALC=(2868-2896)/.63*185=|.24|vTest: vHo: tCALC tCRITvHo: .24 n30X -m0Sample Variance vs Sample Variance (s2) Coming up with the calculated statistic. Use these formulas to calculate the actual stat
212、istic for comparison with the critical (table) statistic. Note that the only major determinate again here is the sample size. Here again, it should make sense to utilize the simpler test (Range Test) wherever possible unless you have a statistical software package available. Fsscalc=22=FRRCALCMAX nM
213、IN n,12Range TestF Test(DF1: n1-1, DF2: n2-1)n1 10, n2 30, n2 30MAXMINHypothesis Testing Example (2 Sample Variance)Several changes were made to the sales organization. The number of receipts was gathered both before and after the changes. Determine if the samples have equal variance with 95% confid
214、ence.vHo: s12 = s22vStatistic Summary:vn1 = n2 = 5vSignificance: a/2 = .025 (2 tail)vFcrit = 3.25 (From the table for n1, n2=5) vCalculation:vR1=337, R2= 577vFCALC=577/337=1.7vTest: vHo: FCALC FCRITvHo: 1.7 30, n2 30How about a manufacturing example?We have a process which we have determined has a c
215、ritical characteristic which has a target value of 2.53. Any deviation from this value will sub-optimize the resulting product. We want to sample the process to see how close we are to this value with 95% confidence. We gather 20 data points (shown below). Perform a 1 sample t test on the data to se
216、e how well we are doing.Learning ObjectivesvUnderstand the role that hypothesis testing plays in an improvement project.vKnow how to perform a two sample hypothesis test.vKnow how to perform a hypothesis test to compare a sample statistic to a target value.vKnow how to interpret a hypothesis test.AN
217、alysis Of VArianceANOVALearning ObjectivesvUnderstand the role that ANOVA plays in problem solving tools & methodology.vUnderstand the fundamental assumptions of ANOVA.vKnow how to perform an ANOVA to identify sources of variation & assess their significance. vKnow how to interpret an ANOVA.Why do w
218、e Care?Anova is a powerful method for analyzing process variation:Used when comparing two or more process means. Estimate the relative effect of the input variables on the output variable.IMPROVEMENT ROADMAPUses of Analysis of Variance Methodology-ANOVABreakthroughStrategyCharacterizationPhase 1:Mea
219、surementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlCommon Uses Hypothesis Testing Design of Experiments (DOE)Dont be afraid of the mathMake sure the assumptions of the method are metUse validated data & your process experts to identify key variables.Carefully plan data collection
220、and experimentsUse the teams experience to direct the testingKEYS TO SUCCESSAnalysis Of Variance -ANOVALinear Model1231 52 02 5L ite rs P er H r L i t e r s P e r H r B y F o r m u la t i o nG r o u p M e a n s a r e In d i c a te d b y L i n e sF o r m u la ti o nOv e r a l l M e a n mti n3 Groups
221、= 3 TreatmentsLets say we run a study where we have three groups which we are evaluating for a significant difference in the means. Each one of these groups is called a “treatment” and represents one unique set of experimental conditions. Within each treatments, we have seven values which are called
222、 repetitions. Analysis Of Variance -ANOVALinear Model1231 52 02 5L ite rs P er H r L i t e r s P e r H r B y F o r m u la t i o nG r o u p M e a n s a r e In d i c a te d b y L i n e sF o r m u la ti o nOv e r a l l M e a n mti n3 Groups = 3 TreatmentsThe basic concept of ANOVA is to compare the var
223、iation between the treatments with the variation within each of the treatments. If the variation between the treatments is statistically significant when compared with the “noise” of the variation within the treatments, we can reject the null hypothesis.One Way AnovaSum of Squares (SS):(Variation Be
224、tween Treatments) (Variation of Noise)The first step in being able to perform this analysis is to compute the “Sum of the Squares” to determine the variation between treatments and within treatments.Note:a = # of treatments (i)n = # of repetitions within each treatment (j)One Way AnovaSince we will
225、be comparing two sources of variation, we will use the F test to determine whether there is a significant difference. To use the F test, we need to convert the “Sum of the Squares” to “Mean Squares” by dividing by the Degrees of Freedom (DF). The Degrees of Freedom is different for each of our sourc
226、es of variation.DFbetween = # of treatments - 1DFwithin = (# of treatments)(# of repetitions of each treatment - 1) DFtotal = (# of treatments)(# of repetitions of each treatment) - 1Note that DFtotal = DFbetween + DFwithin DETERMINING SIGNIFICANCEStatistic ValueF Critical ValueHo : All t t s = 0 Al
227、l m m s equal (Null hypothesis)Ha : At least one t t not = 0 At least one m m is different (Alternate Hypothesis)StatisticCALC StatisticCRITWe determine a critical value from a probability table. This value is compared with the calculated value. If the calculated value exceeds the critical value, we
228、 will reject the null hypothesis (concluding that a significant difference exists between 2 or more of the treatment means.Class Example - Evaluate 3 Fuel FormulationsIs there a difference?Here we have an example of three different fuel formulations that are expected to show a significant difference
229、 in average fuel consumption. Is there a significant difference? Lets use ANOVA to test our hypothesis. Class Example - Evaluate 3 Fuel FormulationsIs there a difference?Step 1: Calculating the Mean Squares BetweenThe first step is to come up with the Mean Squares Between. To accomplish this we:find
230、 the average of each treatment and the overall averagefind the difference between the two values for each treatment and square itsum the squares and multiply by the number of replicates in each treatmentdivide this resulting value by the degrees of freedomClass Example - Evaluate 3 Fuel Formulations
231、Is there a difference?Step 2: Calculating the Mean Squares WithinThe next step is to come up with the Mean Squares Within. To accomplish this we:square the difference between each individual within a treatment and the average of that treatmentsum the squares for each treatmentdivide this resulting v
232、alue by the degrees of freedomClass Example - Evaluate 3 Fuel FormulationsIs there a difference?Remaining StepsThe remaining steps to complete the analysis are:find the calculated F value by dividing the Mean Squares Between by the Mean Squares WithinDetermine the critical value from the F tableComp
233、are the calculated F value with the critical value determined from the table and draw our conclusions.Learning ObjectivesvUnderstand the role that ANOVA plays in problem solving tools & methodology.vUnderstand the fundamental assumptions of ANOVA.vKnow how to perform an ANOVA to identify sources of
234、variation & assess their significance. vKnow how to interpret an ANOVA.CONTINGENCY TABLES(CHI SQUARE)Learning ObjectivesvUnderstand how to use a contingency table to support an improvement project.vUnderstand the enabling conditions that determine when to use a contingency table.vUnderstand how to c
235、onstruct a contingency table.vUnderstand how to interpret a contingency table.Why do we Care?Contingency tables are helpful to: Perform statistical significance testing on count or attribute data.Allow comparison of more than one subset of data to help localize KPIV factors.IMPROVEMENT ROADMAPUses o
236、f Contingency TablesBreakthroughStrategyCharacterizationPhase 1:MeasurementPhase 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlConfirm sources of variation to determine causative factors (x).Demonstrate a statistically significant difference between baseline data and data taken after impro
237、vements were implemented.Common UsesConduct “ad hoc” training for your team prior to using the toolGather data, use the tool to test and then move on.Use historical data where ever possibleKeep it simpleEnsure that there are a minimum of 5 units in each cellKEYS TO SUCCESSSO WHAT IS A CONTINGENCY TA
238、BLE?A contingency table is just another way of hypothesis testing. Just like the hypothesis testing we have learned so far, we will obtain a “critical value” from a table (c2 in this case) and use it as a tripwire for significance. We then use the sample data to calculate a c2CALC value. Comparing t
239、his calculated value to our critical value, tells us whether the data groups exhibit no significant difference (null hypothesis or Ho) or whether one or more significant differences exist (alternate hypothesis or H1).c2CRITHoH1c2CALCOne or more significant differencesNo significant differencesHo: P1
240、=P2=P3H1: One or more population (P) is significantly differentSo how do you build a Contingency Table?Vendor AVendor BVendor COrders On TimeOrders LateDefine the hypothesis you want to test. In this example we have 3 vendors from which we have historically purchased parts. We want to eliminate one
241、and would like to see if there is a significant difference in delivery performance. This is stated mathematically as Ho: Pa=Pb=Pc. We want to be 95% confident in this result. We then construct a table which looks like this example. In this table we have order performance in rows and vendors in colum
242、ns.We ensure that we include both good and bad situations so that the sum of each column is the total opportunities which have been grouped into good and bad. We then gather enough data to ensure that each cell will have a count of at least 5. Fill in the data.Add up the totals for each column and r
243、ow.Vendor AVendor BVendor COrders On Time255812Orders Late795Vendor AVendor BVendor CTotalOrders On Time25581295Orders Late79521Total326717116Wait, what if I dont have at least 5 in each cell?Collapse the tableIf we were placing side bets in a number of bars and wondered if there were any nonrandom
244、factors at play. We gather data and construct the following table:Since bar #3 does not meet the “5 or more” criteria, we do not have enough data to evaluate that particular cell for Bar #3. This means that we must combine the data with that of another bar to ensure that we have significance. This i
245、s referred to as “collapsing” the table. The resulting collapsed table looks like the following:We can now proceed to evaluate our hypothesis. Note that the data between Bar #2 and Bar #3 will be aliased and therefore can not be evaluated separately.Bar #1Bar #2Bar #3Won Money572Lost Money794Bar #1B
246、ar #2&3Won Money59Lost Money713Bar #3 Collapsed into Bar #2So how do you build a Contingency Table?Calculate the percentage of the total contained in each row by dividing the row total by the total for all the categories. For example, the Orders on Time row has a total of 95. The overall total for a
247、ll the categories is 116. The percentage for this row will be row/total or 95/116 = .82.The row percentage times the column total gives us the expected occurrence for each cell based on the percentages. For example, for the Orders on time row, .82 x32=26 for the first cell and .82 x 67= 55 for the s
248、econd cell.Vendor AVendor BVendor CTotalPortionOrders On Time255812950.82Orders Late795210.18Total3267171161.00Actual OccurrencesVendor AVendor BVendor CTotalPortionOrders On Time255812950.82Orders Late795210.18Total3267171161.00Expected OccurrencesVendor AVendor BVendor COrders On Time2655Orders La
249、teSo how do you build a Contingency Table?Complete the values for the expected occurrences.Now we need to calculate the c2 value for the data. This is done using the formula (a-e)2/e (where a is the actual count and e is the expected count) for each cell. So, the c2 value for the first column would
250、be (25-26)2/26=.04. Filling in the remaining c2 values we get: Actual OccurrencesExpected OccurrencesVendor AVendor BVendor CTotalPortionVendor AVendor BVendor COrders On Time255812950.82Orders On Time265514Orders Late795210.18Orders Late6123Column Total3267171161.00Calculations (Expected)Vendor AVe
251、ndor BVendor COrders On Time.82x32.82x67.82x17Orders Late.18x32.18x67.18x17Actual OccurrencesExpected OccurrencesVendor AVendor BVendor CTotalPortionVendor AVendor BVendor COrders On Time255812950.82Orders On Time265514Orders Late795210.18Orders Late6123Column Total3267171161.00Calculations ( c2=(a-
252、e)2/e)Calculated c2 ValuesVendor AVendor BVendor CVendor AVendor BVendor COrders On Time(25-26)2/26(58-55)2/55(12-14)2/14Orders On Time0.040.180.27Orders Late(7-6)2/6(9-12)2/12(5-3)2/3Orders Late0.250.811.20Now what do I do?Performing the AnalysisDetermine the critical value from the c2 table. To ge
253、t the value you need 3 pieces of data. The degrees of freedom are obtained by the following equation; DF=(r-1)x(c-1). In our case, we have 3 columns (c) and 2 rows (r) so our DF = (2-1)x(3-1)=1x2=2.The second piece of data is the risk. Since we are looking for .95 (95%) confidence (and a risk = 1 -
254、confidence) we know the a risk will be .05.In the c2 table , we find that the critical value for a = .05 and 2 DF to be 5.99. Therefore, our c2CRIT = 5.99Our calculated c2 value is the sum of the individual cell c2 values. For our example this is .04+.18+.27+.25+.81+1.20=2.75. Therefore, our c2CALC
255、= 2.75.We now have all the pieces to perform our test. Our Ho: is c2CALC c2CRIT . Is this true? Our data shows 2.755.99, therefore we fail to reject the null hypothesis that there is no significant difference between the vendor performance in this area.Contingency Table ExerciseWe have a part which
256、is experiencing high scrap. Your team thinks that since it is manufactured over 3 shifts and on 3 different machines, that the scrap could be caused (Y=f(x) by an off shift workmanship issue or machine capability. Verify with 95% confidence whether either of these hypothesis is supported by the data
257、.Construct a contingency table of the data and interpret the results for each data set.Good PartsScrapActual OccurrencesMachine 1Machine 2Machine 3100350900151823Actual OccurrencesShift 1Shift 2Shift 3Good Parts500400450Scrap201917Learning ObjectivesvUnderstand how to use a contingency table to supp
258、ort an improvement project.vUnderstand the enabling conditions that determine when to use a contingency table.vUnderstand how to construct a contingency table.vUnderstand how to interpret a contingency table.DESIGN OF EXPERIMENTS (DOE) FUNDAMENTALSLearning ObjectivesvHave a broad understanding of th
259、e role that design of experiments (DOE) plays in the successful completion of an improvement project.vUnderstand how to construct a design of experiments.vUnderstand how to analyze a design of experiments.vUnderstand how to interpret the results of a design of experiments.Why do we Care?Design of Ex
260、periments is particularly useful to: evaluate interactions between 2 or more KPIVs and their impact on one or more KPOVs.optimize values for KPIVs to determine the optimum output from a process.IMPROVEMENT ROADMAPUses of Design of ExperimentsBreakthroughStrategyCharacterizationPhase 1:MeasurementPha
261、se 2:AnalysisOptimizationPhase 3:ImprovementPhase 4:ControlVerify the relationship between KPIVs and KPOVs by manipulating the KPIV and observing the corresponding KPOV change.Determine the best KPIV settings to optimize the KPOV output. Keep it simple until you become comfortable with the toolStati
262、stical software helps tremendously with the calculationsMeasurement system analysis should be completed on KPIV/KPOV(s)Even the most clever analysis will not rescue a poorly planned experimentDont be afraid to ask for help until you are comfortable with the toolEnsure a detailed test plan is written
263、 and followedKEYS TO SUCCESSSo What Is a Design of Experiment?where a mathematical reasoning can be had, its as great a folly to make use of any other, as to grope for a thing in the dark, when you have a candle standing by you. Arbuthnot A design of experiment introduces purposeful changes in KPIVs
264、, so that we can methodically observe the corresponding response in the associated KPOVs.Design of Experiments, Full FactorialKey Process Output VariablesProcessA combination of inputs which generate corresponding outputs.Key Process Input VariablesNoise VariablesxY=f(x)f(x)VariablesInput, Controlla
265、ble (KPIV)Input, Uncontrollable (Noise)Output, Controllable (KPOV)How do you know how much a suspected KPIV actually influences a KPOV? You test it!Design of Experiments, Terminology24IV2 levels for each KPIV4 factors evaluatedResolution IVMathematical objects are sometimes as peculiar as the most e
266、xotic beast or bird, and the time spent in examining them may be well employed.H. SteinhausMain Effects - Factors (KPIV) which directly impact outputInteractions - Multiple factors which together have more impact on process output than any factor individually.Factors - Individual Key Process Input V
267、ariables (KPIV)Levels - Multiple conditions which a factor is set at for experimental purposesAliasing - Degree to which an output cannot be clearly associated with an input condition due to test design.Resolution - Degree of aliasing in an experimental designDOE Choices, A confusing array.Full Fact
268、orialTaguchi L16Half Fraction2 level designs3 level designsscreening designsResponse surface designsetc.For the purposes of this training we will teach only full factorial (2k) designs. This will enable you to get a basic understanding of application and use the tool. In addition, the vast majority
269、of problems commonly encountered in improvement projects can be addressed with this design. If you have any question on whether the design is adequate, consult a statistical expert.Mumble, Mumble, blackbelt, Mumble, statistics stuff.The Yates AlgorithmDetermining the number of TreatmentsOne aspect w
270、hich is critical to the design is that they be “balanced”. A balanced design has an equal number of levels represented for each KPIV. We can confirm this in the design on the right by adding up the number of + and - marks in each column. We see that in each case, they equal 4 + and 4- values, theref
271、ore the design is balanced.Yates algorithm is a quick and easy way (honest, trust me) to ensure that we get a balanced design whenever we are building a full factorial DOE. Notice that the number of treatments (unique test mixes of KPIVs) is equal to 23 or 8. Notice that in the “A factor” column, we
272、 have 4 + in a row and then 4 - in a row. This is equal to a group of 22 or 4. Also notice that the grouping in the next column is 21 or 2 + values and 2 - values repeated until all 8 treatments are accounted for.Repeat this pattern for the remaining factors.Treatment A B C1+2+-3+-+4+-5-+6-+-7-+8- 2
273、3Factorial23=822=421=220=1The Yates AlgorithmSetting up the Algorithm for InteractionsNow we can add the columns that reflect the interactions. Remember that the interactions are the main reason we use a DOE over a simple hypothesis test. The DOE is the best tool to study “mix” types of problems. Tr
274、eatmentABCABACBCABC1+2+-+-3+-+-+-4+-+5-+-+-6-+-+-+7-+-+8-+-23FactorialYou can see from the example above we have added additional columns for each of the ways that we can “mix” the 3 factors which are under study. These are our interactions. The sign that goes into the various treatment boxes for th
275、ese interactions is the algebraic product of the main effects treatments. For example, treatment 7 for interaction AB is (- x - = +), so we put a plus in the box. So, in these calculations, the following apply:minus (-) times minus (-) = plus (+) plus (+) times plus (+) = plus (+) minus (-) times pl
276、us (+) = minus (-)plus (+) times minus (-) = minus (-)Yates Algorithm ExerciseWe work for a major “Donut & Coffee” chain. We have been tasked to determine what are the most significant factors in making “the most delicious coffee in the world”. In our work we have identified three factors we conside
277、r to be significant. These factors are coffee brand (maxwell house vs chock full o nuts), water (spring vs tap) and coffee amount (# of scoops).Use the Yates algorithm to design the experiment.vSelect the factors (KPIVs) to be investigated and define the output to be measured (KPOV).vDetermine the 2
278、 levels for each factor. Ensure that the levels are as widely spread apart as the process and circumstance allow. vDraw up the design using the Yates algorithm. So, How do I Conduct a DOE?TreatmentABCABACBCABC1+2+-+-3+-+-+-4+-+5-+-+-6-+-+-+7-+-+8-+-vDetermine how many replications or repetitions you
279、 want to do. A replication is a complete new run of a treatment and a repetition is more than one sample run as part of a single treatment run.vRandomize the order of the treatments and run each. Place the data for each treatment in a column to the right of your matrix.So, How do I Conduct a DOE?Tre
280、atmentABCABACBCABCAVGRUN1RUN2RUN31+18182+-+-12123+-+-+-664+-+995-+-+-336-+-+-+337-+-+448-+-88vCalculate the average output for each treatment.vPlace the average for each treatment after the sign (+ or -) in each cell.Analysis of a DOETreatmentABCABACBCABCAVGRUN1RUN2RUN31+18+18182+12+-+-12123+6-+-+-6
281、64+9-+995-3+-+-336-3+-+-+337-4-+-+448-8-+-88vAdd up the values in each column and put the result under the appropriate column. This is the total estimated effect of the factor or combination of factors.vDivide the total estimated effect of each column by 1/2 the total number of treatments. This is t
282、he average estimated effect.Analysis of a DOETreatmentABCABACBCABCAVG1+18 +18 +18 +18 +18 +18 +18182+12 +12 -12 +12 -12-12-12123+6-6+6-6+6-6-664+9-9-9-9-9+9+995-3+3+3-3-3+3-336-3+3-3-3+3-3+337-4-4+4+4-4-4+448-8-8-8+8+8+8-88SUM279-121713563AVG6.752.25-0.255.251.753.251.25vThese averages represent the
283、 average difference between the factor levels represented by the column. So, in the case of factor “A”, the average difference in the result output between the + level and the - level is 6.75. vWe can now determine the factors (or combination of factors) which have the greatest impact on the output
284、by looking for the magnitude of the respective averages (i.e., ignore the sign).Analysis of a DOETreatmentABCABACBCABCAVG1+18 +18 +18 +18 +18 +18 +18182+12 +12-12+12-12-12-12123+6-6+6-6+6-6-664+9-9-9-9-9+9+995-3+3+3-3-3+3-336-3+3-3-3+3-3+337-4-4+4+4-4-4+448-8-8-8+8+8+8-88SUM279-121713563AVG6.752.25-
285、0.255.251.753.251.25This means that the impact is in the following order:A (6.75)AB (5.25)BC (3.25)B (2.25)AC (1.75)ABC (1.25)C (-0.25)Analysis of a DOERanked Degree of ImpactA (6.75)AB (5.25)BC (3.25)B (2.25)AC (1.75)ABC (1.25)C (-0.25)We can see the impact, but how do we know if these results are
286、significant or just random variation?What tool do you think would be good to use in this situation?Confidence Interval for DOE resultsRanked Degree of ImpactA (6.75)AB (5.25)BC (3.25)B (2.25)AC (1.75)ABC (1.25)C (-0.25)Confidence Interval= Effect +/- ErrorSome of these factors do not seem to have mu
287、ch impact. We can use them to estimate our error.We can be relatively safe using the ABC and the C factors since they offer the greatest chance of being insignificant.Confidence Interval for DOE resultsRanked Degree of ImpactA (6.75)AB (5.25)BC (3.25)B (2.25)AC (1.75)ABC (1.25)C (-0.25)()Confidencet
288、ABCCDFDF= +a/ ,222DF=# of groups usedIn this case we are using 2 groups (ABC and C) so our DF=2For a = .05 and DF =2 we find ta/2,df = t.025,2 = 4.303()Confidence = + -4 30312525222.( .)Confidence = ( .)(.)4 3039235Confidence = 397.ConfidenceSince only 2 groups meet or exceed our 95% confidence inte
289、rval of +/- 3.97. We conclude that they are the only significant treatments.6 What Do I need to do to improve my Game?GUTTER!MEASURE - Average = 140.9IMPROVEMENT PHASEVital Few VariablesEstablish Operating TolerancesHow about another way of looking at a DOE?It looks like the lanes are in good condit
290、ion today, Mark. Tim has brought three differentbowling balls with him but I dont think he will need them all today. You know he seems to have improved his game ever since he started bowling with that wristband. How do I know what works for me.Lane conditions?Ball type?Wristband?How do I set up the
291、Experiment ? Factor A Factor BFactor C1. Wristband (+) hard ball (+)oily lane (+) 2 Wristband (+) hard ball (+)dry lane (-)3. Wristband (+) soft ball (-)oily lane (+)4. Wristband (+) softball (-)dry lane (-)5. No Wristband(-) hard ball (+)oily lane (+)6. No Wristband(-) hard ball (+)dry lane (-)7. N
292、o Wristband(-) soft ball (-)oily lane (+)8. No Wristband(-) softball (-)dry lane (-)What are all possible Combinations?(Remember Yates Algorithm?)Lets Look at it a different way?A 3 factor, 2 level full factorial DOE would have 23=8experimental treatments!Lets look at the data! dry lane oily lanehar
293、d ballsoft ballhard ballsoft ballwristbandno wristbandoily lanehard bowling ballwearing a wristbanddry lanehard bowing ballnot wearing wrist band1 23 45 67 8This is a Full factorial What about the Wristband? Did it help me? dry lane oily lanehard ballsoft ballhard ballsoft ballwristbandwithout wrist
294、band188174183191158154141159Higher Scores ! The Wristband appears Better.This is called a Main Effect!Average of “withwristband” scores =184Average of “withoutwristband” scores =153 dry lane oily lanehard ballsoft ballhard ballsoft ballwristbandno wristband174183154141188191158159Your best Scores ar
295、e when:Dry Lane Hard Ball OR Oily Lane Soft BallThe Ball Type depends on the Lane Condition.This is called an Interaction!What about Ball Type?Where do we go from here? With Wristband andWhen lane is: use: Dry Hard Ball Oily Soft Ball Youre on your way to the PBA !Where do we go from here?Now, evaluate the results using Yates Algorithm.What do you think?.
