六西格玛黑带培训教材课件

04十一月2022六西格玛黑带培训教材01十一月2022六西格玛黑带培训教材1ScopeofModuleProcessVariationProcessCapabilitySpecification,ProcessandControlLimitsProcessPotentialvsProcessPerformanceShort-TermvsLong-TermProcessCapabilityProcessCapabilityforNon-NormalDataCycle-Time (ExponentialDistribution)RejectRate (BinomialDistribution)DefectRate (PoissonDistribution)ScopeofModuleProcessVariati2ProcessVariationProcessVariationistheinevitabledifferencesamongindividualmeasurementsorunitsproducedbyaprocess.SourcesofVariationwithinunit (positionalvariation)betweenunits (unit-unitvariation)betweenlots (lot-lotvariation)betweenlines (line-linevariation)acrosstime (time-timevariation)measurementerror (repeatability&reproducibility)ProcessVariationProcessVaria3TypesofVariationInherentorNaturalVariationDuetothecumulativeeffectofmanysmallunavoidablecausesAprocessoperatingwithonlychancecausesofvariationpresentissaidtobe“instatisticalcontrol”TypesofVariationInherentor4TypesofVariationSpecialorAssignableVariationMaybedueto a)improperlyadjustedmachine b)operatorerror c)defectiverawmaterialAprocessoperatinginthepresenceofassignablecausesofvariationissaidtobe“out-of-control”TypesofVariationSpecialorA5ProcessCapabilityProcessCapabilityistheinherentreproducibilityofaprocess’soutput.Itmeasureshowwelltheprocessiscurrentlybehavingwithrespecttotheoutputspecifications.Itreferstotheuniformityoftheprocess.Capabilityisoftenthoughtofintermsoftheproportionofoutputthatwillbewithinproductspecificationtolerances.Thefrequencyofdefectivesproducedmaybemeasuredina) percentage(%)b) partspermillion(ppm)c) partsperbillion(ppb)ProcessCapabilityProcessCapa6ProcessCapabilityProcessCapabilitystudiescan

indicatetheconsistencyoftheprocessoutputindicatethedegreetowhichtheoutputmeetsspecificationsbeusedforcomparisonwithanotherprocessorcompetitorProcessCapabilityProcessCapa7ProcessCapabilityvsSpecificationLimitsa)b)c)a)Processishighlycapableb)Processismarginallycapablec)ProcessisnotcapableProcessCapabilityvsSpecific8ThreeTypesofLimitsSpecificationLimits(LSLandUSL)createdbydesignengineeringinresponsetocustomerrequirementstospecifythetoleranceforaproduct’scharacteristicProcessLimits(LPLandUPL)measuresthevariationofaprocessthenatural6limitsofthemeasuredcharacteristicControlLimits(LCLandUCL)measuresthevariationofasamplestatistic(mean,variance,proportion,etc)ThreeTypesofLimitsSpecifica9ThreeTypesofLimitsDistributionofIndividualValuesDistributionofSampleAveragesThreeTypesofLimits10ProcessCapabilityIndicesTwomeasuresofprocesscapabilityProcessPotentialCp

ProcessPerformanceCpu

Cpl

Cpk

ProcessCapabilityIndicesTwo11ProcessPotentialTheCpindexassesseswhetherthenaturaltolerance(6)ofaprocessiswithinthespecificationlimits.ProcessPotentialTheCpindex12ProcessPotentialACpof1.0indicatesthataprocessisjudgedtobe“capable”,i.e.iftheprocessiscenteredwithinitsengineeringtolerance,0.27%ofpartsproducedwillbebeyondspecificationlimits. Cp RejectRate 1.00 0.270% 1.33 0.007% 1.50 6.8ppm 2.00 2.0ppbProcessPotentialACpof1.0i13ProcessPotentiala)b)c)a)Processishighlycapable(Cp>2)b)Processiscapable(Cp=1to2)c)Processisnotcapable(Cp<1)ProcessPotentiala)b)c)14ProcessPotentialTheCpindexcomparestheallowablespread(USL-LSL)againsttheprocessspread(6).Itfailstotakeintoaccountiftheprocessisnotcenteredbetweenthespecificationlimits.ProcessiscenteredProcessisnotcenteredProcessPotentialTheCpindex15ProcessPerformanceTheCpkindexrelatesthescaleddistancebetweentheprocessmeanandthenearestspecificationlimit.ProcessPerformanceTheCpkind16ProcessPerformance Cpk RejectRate 1.0 0.13–0.27% 1.1 0.05–0.10% 1.2 0.02–0.03% 1.3 48.1–96.2ppm 1.4 13.4–26.7ppm 1.5 3.4–6.8ppm 1.6 794–1589ppb 1.7 170–340ppb 1.8 33–67ppb 1.9 6–12ppb 2.0 1–2ppbProcessPerformance Cpk Reje17ProcessPerformancea)Processishighlycapable(Cpk>1.5)b)Processiscapable(Cpk=1to1.5)c)Processisnotcapable(Cpk<1)a)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk<1ProcessPerformancea)Cp=2b)C18Example1SpecificationLimits : 4to16gMachine Mean StdDev (a) 10 4 (b) 10 2 (c) 7 2 (d) 13 1DeterminethecorrespondingCpandCpkforeachmachine.Example1SpecificationLimits 19Example1AExample1A20Example1BExample1B21Example1CExample1C22Example1DExample1D23ProcessCapabilityForanormallydistributedcharacteristic,thedefectiverateF(x)maybeestimatedviathefollowing:Forcharacteristicswithonlyonespecificationlimit:a) LSLonlyb) USLonlyLSLUSLProcessCapabilityForanormal24Example2SpecificationLimits : 4to16gMachine Mean StdDev (a) 10 4 (b) 10 2 (c) 7 2 (d) 13 1Determinethedefectiverateforeachmachine.Example2SpecificationLimits 25Example2MeanStdDevZLSLZUSLF(x<LSL)F(x>USL)F(x) 10 4 -1.5 1.5 66,807 66,807 133,614 10 2 -3.0 3.0 1,350 1,350 2,700 7 2 -1.5 4.5 66,8073 66,811 13 1 -9.0 3.0 0 1,350 1,350 LowerSpecLimit =4gUpperSpecLimit =16gExample2MeanStdDevZL26ProcessPotentialvsProcessPerformance(a)PoorProcessPotential (b)PoorProcessPerformanceLSLUSLLSLUSLExperimentalDesigntoreducevariationExperimentalDesigntocentermeantoreducevariationProcessPotentialvsProcessP27ProcessPotentialvsProcessPerformance ProcessPotentialIndex(Cp)Cpk

1.0 1.2 1.4 1.6 1.8 2.0

1.0

2,699.9 1,363.3 1,350.0 1,350.0 1,350.0 1,350.0

1.2

318.3 159.9 159.1 159.1 159.1

1.4

26.7 13.4 13.4 13.4

1.6

1.6 0.8 0.8

1.8

0.1 0.0

2.0

0.0

DefectiveRate(measuredindppm)isdependentontheactualcombinationofCpandCpk..ProcessPotentialvsProcessP28ProcessPotentialvsProcessPerformancea)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk<1Cp–Cpk

MissedOpportunityProcessPotentialvsProcessP29AlternativeProcessPerformanceIndexProcesscapabilitystatisticsmeasureprocessvariationrelativetospecificationlimits.TheCpstatisticcomparestheengineeringtoleranceagainsttheprocess’snaturalvariation.TheCpkstatistictakesintoaccountthelocationoftheprocessrelativetothemidpointbetweenspecifications.Iftheprocesstargetisnotcenteredbetweenspecifications,theCpmstatisticispreferred.AlternativeProcessPerformanc30ProcessStabilityAprocessisstableifthedistributionofmeasurementsmadeonthegivenfeatureisconsistentovertime.TimeStableProcessTimeUnstableProcessucllclucllclProcessStabilityAprocessis31WithinvsOverallCapabilityWithinCapability(previouslycalledshort-termcapability)showstheinherentvariabilityofamachine/processoperatingwithinabriefperiodoftime.OverallCapability(previouslycalledlong-termcapability)showsthevariabilityofamachine/processoperatingoveraperiodoftime.Itincludessourcesofvariationinadditiontotheshort-termvariability.WithinvsOverallCapabilityWi32WithinvsOverallCapability Within OverallSampleSize 30–50units 100unitsNumberofLots singlelot severallotsPeriodofTime hoursordays weeksormonthsNumberofOperators singleoperator differentoperatorsProcessPotential Cp Pp

ProcessPerformance Cpk Ppk

WithinvsOverallCapability 33WithinvsOverallCapability WithinCapability OverallCapability ThekeydifferencebetweenthetwosetsofindicesliesintheestimatesforWithinandOverall.WithinvsOverallCapability W34EstimatingWithinandOverall

ConsiderthefollowingobservationsfromaControlChart:

S/N X1 X2 …Xk Mean Range StdDev 1 x1,1 x2,1 …xk,1 X1 R1 S1 2 x1,2 x2,2 …xk,2 X2 R2 S2 : : : : : : : m x1,m x2,m …xk,m Xm Rm SmTheoverallvariationOverall

isestimatedby–––EstimatingWithinandOveral35EstimatingWithinandOverall

ThewithinvariationWithin

maybeestimatedbyoneofthefollowing:(a) R-barMethod where d2isaShewhartconstant=(k)(b) S-barMethod where c4isaShewhartconstant=(k)(c) PooledStandardDeviationMethodInMiniTab,thePooledStandardDeviationisthedefaultmethod.EstimatingWithinandOveral36EstimatingWithinandOverall

Incaseswherethereisonly1observationpersub-group(i.e.k=1),theMovingRangeMethodisused,where .ThewithinvariationWithin

isthenestimatedusingeithera) theAverageMovingRange:b) theMedianMovingRange:EstimatingWithinandOveral37Example3Thelengthofacamshaftforanautomobileengineisspecifiedat600±2mm.Controlofthelengthofthecamshaftiscriticaltoavoidscrap/rework.Thecamshaftisprovidedbyanexternalsupplier.Assesstheprocesscapabilityforthissupplier.ThedataisavailableinProcessCapabilityAnalysis.MTW.Example3Thelengthofacamsh38Example3StatQualityToolsCapabilityAnalysis(Normal)Example3StatQualityTools39Example3Example340Example3AHistogramofcamshaftlengthsuggestsmixedpopulations.Furtherinvestigationrevealedthattherearetwosuppliersforthecamshaft.Datawascollectedovercamshaftsfrombothsources.Arethetwosupplierssimilarinperformance?Ifnot,whatareyourrecommendations?Example3AHistogramofcamshaf41Example3AStatQualityToolsCapabilitySixpack(Normal)Example3AStatQualityTools42Example3AExample3A43Example3AExample3A44What’sSixSigmaQuality—ThenOriginalDefinitionbyMotorola:ifthespecificationlimitsareatleast±6awayfromtheprocessmean,i.e.Cp2,andtheprocessshiftsbylessthan1.5,i.e.Cpk1.5,thentheprocesswillyieldlessthan3.4dppmrejects.66Shift1.54.5What’sSixSigmaQuality—T45What’sSixSigmaQuality—NowMikelJHarryclaimsthattheprocessmeanbetweenlotswillvary,withanaverageprocessshiftof1.5.k=z+1.5k=z+1.5Shift1.5zNote: SigmaCapability=ƒ(dpmo)ƒ(dppm)What’sSixSigmaQuality—N46ProcessCapabilityforNon-NormalDataNoteverymeasuredcharacteristicisnormallydistributed. Characteristic Distribution CycleTime Exponential RejectRate Binomial DefectRate Poisson ProcessCapabilityforNon-Nor47ProcessCapabilityforCycleTimeTheWeibullDistributionisageneralfamilyofdistributionwithwhere scaleparameter

isthevalueatwhichCDF=68.17%,and shapeparameterdeterminestheshapeofthePDF.ProcessCapabilityforCycleT48ProcessCapabilityforCycleTimeAt=1, theWeibullDistributionisreducedto ForanExponentialDistribution,TheExponentialDistributionisthusaWeibullDistributionwith=1.Weibull(x;=1,)Exponential(x;)ProcessCapabilityforCycleT49Example4Acustomerservicemanagerwantstodeterminetheprocesscapabilityforhisdepartment.Aprimaryperformanceindexisthetimetakentocloseacustomercomplaint.Thegoalforthisindexistocloseacomplaintwithinonecalendarweek.Performanceoverthelast400complaintswasreviewed.Example4Acustomerservicema50Example4StatQualityToolsCapabilityAnalysis(Weibull)Example4StatQualityTools51Example4Example452Example4AStatQualityToolsCapabilitySixpack(Weibull)Example4AStatQualityTools53Example4AExample4A54ProcessCapabilityforRejectRateForaNormalDistribution,theproportionofpartsproducedbeyondaspecificationlimitisRejectRateProcessCapabilityforReject55ProcessCapabilityforRejectRateThus,foreveryrejectratethereisanaccompanyingZ-Score,whereRecallthatHenceProcessCapabilityforReject56ProcessCapabilityforRejectRateEstimationofPpkforRejectRateDeterminethelong-termrejectrate(p)Determinetheinversecumulativeprobabilityforp, usingCalcProbabilityDistributionNormalZ-ScoreisthemagnitudeofthereturnedvaluePpkisone-thirdoftheZ-ScoreProcessCapabilityforReject57Example5Asalesmanagerplanstoassesstheprocesscapabilityofhistelephonesalesdepartment’shandlingofincomingcalls.Thefollowingdatawascollectedoveraperiodof20days:numberofincomingcallsperdaynumberofunansweredcallsperdaysExample5Asalesmanagerplans58Example5StatQualityToolsCapabilityAnalysis(Binomial)Example5StatQualityTools59Example5Ppk=0.25Example5Ppk=0.2560ProcessCapabilityforDefectRateOtherapplications,approximatingaPoissonDistribution:errorratesparticlecountchemicalconcentrationProcessCapabilityforDefect61ProcessCapabilityforDefectRateEstimationofYtpforDefectRateDefinesizeofaninspectionunitDeterminethelong-termdefectsperunit(DPU) DPU =TotalDefectsTotalUnitsDeterminethethroughputyield(Ytp) Ytp =exp{–DPU}ProcessCapabilityforDefect62ProcessCapabilityforDefectRateEstimationofSigma-CapabilityforDefectRateDeterminetheopportunitiesperunitDeterminethelong-termdefectsperopportunity(d)

d =defectsperunitopportunitiesperunitDeterminetheinversecumulativeprobabilityford, usingCalcProbabilityDistributionNormalZ-ScoreisthemagnitudeofthereturnedvalueSigma-Capability=Z-Score+1.5ProcessCapabilityforDefect63Example6Theprocessmanagerforawiremanufacturerisconcernedabouttheeffectivenessofthewireinsulationprocess.Randomlengthsofelectricalwiringaretakenandtestedforweakspotsintheirinsulationbymeansofatestvoltage.Thenumberofweakspotsandthelengthofeachpieceofwirearerecorded.Example6Theprocessmanagerf64Example6StatQualityToolsCapabilityAnalysis(Poisson)Example6StatQualityTools65Example6DefectsperUnit=0.0265194ThroughputYield=exp{–DPU}=exp{–0.0265194}=0.9738c.f.First-TimeYield=2/100=0.02Example6DefectsperUnit66Example6Define 1InspectionUnit =125unitlengthofwirei.e. Units =Length

125Example6Define67Example6AStatQualityToolsCapabilityAnalysis(Poisson)Example6AStatQualityTools68Example6ADefectsperUnit=3.31493ThroughputYield=exp{–DPU}=exp{–3.31493}=0.0363c.f.First-TimeYield=2/100=0.02Example6ADefectsperUnit69Example6BDefectsperUnit=3.31493OpportunitiesperUnit=1DefectsperOpportunity=3.31493Z-Score=???Example6BDefectsperUnit70Example6B1inspectionunit=1unitlengthofwireOpportunitiesperUnit=1DefectsperOpportunity=32912,406=0.0265Z-Score=Abs{–1(0.0265)}=1.935Sigma-Capability=Z-Score+1.5=3.435Example6B1inspectionunit7104十一月2022六西格玛黑带培训教材01十一月2022六西格玛黑带培训教材72ScopeofModuleProcessVariationProcessCapabilitySpecification,ProcessandControlLimitsProcessPotentialvsProcessPerformanceShort-TermvsLong-TermProcessCapabilityProcessCapabilityforNon-NormalDataCycle-Time (ExponentialDistribution)RejectRate (BinomialDistribution)DefectRate (PoissonDistribution)ScopeofModuleProcessVariati73ProcessVariationProcessVariationistheinevitabledifferencesamongindividualmeasurementsorunitsproducedbyaprocess.SourcesofVariationwithinunit (positionalvariation)betweenunits (unit-unitvariation)betweenlots (lot-lotvariation)betweenlines (line-linevariation)acrosstime (time-timevariation)measurementerror (repeatability&reproducibility)ProcessVariationProcessVaria74TypesofVariationInherentorNaturalVariationDuetothecumulativeeffectofmanysmallunavoidablecausesAprocessoperatingwithonlychancecausesofvariationpresentissaidtobe“instatisticalcontrol”TypesofVariationInherentor75TypesofVariationSpecialorAssignableVariationMaybedueto a)improperlyadjustedmachine b)operatorerror c)defectiverawmaterialAprocessoperatinginthepresenceofassignablecausesofvariationissaidtobe“out-of-control”TypesofVariationSpecialorA76ProcessCapabilityProcessCapabilityistheinherentreproducibilityofaprocess’soutput.Itmeasureshowwelltheprocessiscurrentlybehavingwithrespecttotheoutputspecifications.Itreferstotheuniformityoftheprocess.Capabilityisoftenthoughtofintermsoftheproportionofoutputthatwillbewithinproductspecificationtolerances.Thefrequencyofdefectivesproducedmaybemeasuredina) percentage(%)b) partspermillion(ppm)c) partsperbillion(ppb)ProcessCapabilityProcessCapa77ProcessCapabilityProcessCapabilitystudiescan

indicatetheconsistencyoftheprocessoutputindicatethedegreetowhichtheoutputmeetsspecificationsbeusedforcomparisonwithanotherprocessorcompetitorProcessCapabilityProcessCapa78ProcessCapabilityvsSpecificationLimitsa)b)c)a)Processishighlycapableb)Processismarginallycapablec)ProcessisnotcapableProcessCapabilityvsSpecific79ThreeTypesofLimitsSpecificationLimits(LSLandUSL)createdbydesignengineeringinresponsetocustomerrequirementstospecifythetoleranceforaproduct’scharacteristicProcessLimits(LPLandUPL)measuresthevariationofaprocessthenatural6limitsofthemeasuredcharacteristicControlLimits(LCLandUCL)measuresthevariationofasamplestatistic(mean,variance,proportion,etc)ThreeTypesofLimitsSpecifica80ThreeTypesofLimitsDistributionofIndividualValuesDistributionofSampleAveragesThreeTypesofLimits81ProcessCapabilityIndicesTwomeasuresofprocesscapabilityProcessPotentialCp

ProcessPerformanceCpu

Cpl

Cpk

ProcessCapabilityIndicesTwo82ProcessPotentialTheCpindexassesseswhetherthenaturaltolerance(6)ofaprocessiswithinthespecificationlimits.ProcessPotentialTheCpindex83ProcessPotentialACpof1.0indicatesthataprocessisjudgedtobe“capable”,i.e.iftheprocessiscenteredwithinitsengineeringtolerance,0.27%ofpartsproducedwillbebeyondspecificationlimits. Cp RejectRate 1.00 0.270% 1.33 0.007% 1.50 6.8ppm 2.00 2.0ppbProcessPotentialACpof1.0i84ProcessPotentiala)b)c)a)Processishighlycapable(Cp>2)b)Processiscapable(Cp=1to2)c)Processisnotcapable(Cp<1)ProcessPotentiala)b)c)85ProcessPotentialTheCpindexcomparestheallowablespread(USL-LSL)againsttheprocessspread(6).Itfailstotakeintoaccountiftheprocessisnotcenteredbetweenthespecificationlimits.ProcessiscenteredProcessisnotcenteredProcessPotentialTheCpindex86ProcessPerformanceTheCpkindexrelatesthescaleddistancebetweentheprocessmeanandthenearestspecificationlimit.ProcessPerformanceTheCpkind87ProcessPerformance Cpk RejectRate 1.0 0.13–0.27% 1.1 0.05–0.10% 1.2 0.02–0.03% 1.3 48.1–96.2ppm 1.4 13.4–26.7ppm 1.5 3.4–6.8ppm 1.6 794–1589ppb 1.7 170–340ppb 1.8 33–67ppb 1.9 6–12ppb 2.0 1–2ppbProcessPerformance Cpk Reje88ProcessPerformancea)Processishighlycapable(Cpk>1.5)b)Processiscapable(Cpk=1to1.5)c)Processisnotcapable(Cpk<1)a)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk<1ProcessPerformancea)Cp=2b)C89Example1SpecificationLimits : 4to16gMachine Mean StdDev (a) 10 4 (b) 10 2 (c) 7 2 (d) 13 1DeterminethecorrespondingCpandCpkforeachmachine.Example1SpecificationLimits 90Example1AExample1A91Example1BExample1B92Example1CExample1C93Example1DExample1D94ProcessCapabilityForanormallydistributedcharacteristic,thedefectiverateF(x)maybeestimatedviathefollowing:Forcharacteristicswithonlyonespecificationlimit:a) LSLonlyb) USLonlyLSLUSLProcessCapabilityForanormal95Example2SpecificationLimits : 4to16gMachine Mean StdDev (a) 10 4 (b) 10 2 (c) 7 2 (d) 13 1Determinethedefectiverateforeachmachine.Example2SpecificationLimits 96Example2MeanStdDevZLSLZUSLF(x<LSL)F(x>USL)F(x) 10 4 -1.5 1.5 66,807 66,807 133,614 10 2 -3.0 3.0 1,350 1,350 2,700 7 2 -1.5 4.5 66,8073 66,811 13 1 -9.0 3.0 0 1,350 1,350 LowerSpecLimit =4gUpperSpecLimit =16gExample2MeanStdDevZL97ProcessPotentialvsProcessPerformance(a)PoorProcessPotential (b)PoorProcessPerformanceLSLUSLLSLUSLExperimentalDesigntoreducevariationExperimentalDesigntocentermeantoreducevariationProcessPotentialvsProcessP98ProcessPotentialvsProcessPerformance ProcessPotentialIndex(Cp)Cpk

1.0 1.2 1.4 1.6 1.8 2.0

1.0

2,699.9 1,363.3 1,350.0 1,350.0 1,350.0 1,350.0

1.2

318.3 159.9 159.1 159.1 159.1

1.4

26.7 13.4 13.4 13.4

1.6

1.6 0.8 0.8

1.8

0.1 0.0

2.0

0.0

DefectiveRate(measuredindppm)isdependentontheactualcombinationofCpandCpk..ProcessPotentialvsProcessP99ProcessPotentialvsProcessPerformancea)Cp=2Cpk=2b)Cp=2Cpk=1c)Cp=2Cpk<1Cp–Cpk

MissedOpportunityProcessPotentialvsProcessP100AlternativeProcessPerformanceIndexProcesscapabilitystatisticsmeasureprocessvariationrelativetospecificationlimits.TheCpstatisticcomparestheengineeringtoleranceagainsttheprocess’snaturalvariation.TheCpkstatistictakesintoaccountthelocationoftheprocessrelativetothemidpointbetweenspecifications.Iftheprocesstargetisnotcenteredbetweenspecifications,theCpmstatisticispreferred.AlternativeProcessPerformanc101ProcessStabilityAprocessisstableifthedistributionofmeasurementsmadeonthegivenfeatureisconsistentovertime.TimeStableProcessTimeUnstableProcessucllclucllclProcessStabilityAprocessis102WithinvsOverallCapabilityWithinCapability(previouslycalledshort-termcapability)showstheinherentvariabilityofamachine/processoperatingwithinabriefperiodoftime.OverallCapability(previouslycalledlong-termcapability)showsthevariabilityofamachine/processoperatingoveraperiodoftime.Itincludessourcesofvariationinadditiontotheshort-termvariability.WithinvsOverallCapabilityWi103WithinvsOverallCapability Within OverallSampleSize 30–50units 100unitsNumberofLots singlelot severallotsPeriodofTime hoursordays weeksormonthsNumberofOperators singleoperator differentoperatorsProcessPotential Cp Pp

ProcessPerformance Cpk Ppk

WithinvsOverallCapability 104WithinvsOverallCapability WithinCapability OverallCapability ThekeydifferencebetweenthetwosetsofindicesliesintheestimatesforWithinandOverall.WithinvsOverallCapability W105EstimatingWithinandOverall

ConsiderthefollowingobservationsfromaControlChart:

S/N X1 X2 …Xk Mean Range StdDev 1 x1,1 x2,1 …xk,1 X1 R1 S1 2 x1,2 x2,2 …xk,2 X2 R2 S2 : : : : : : : m x1,m x2,m …xk,m Xm Rm SmTheoverallvariationOverall

isestimatedby–––EstimatingWithinandOveral106EstimatingWithinandOverall

ThewithinvariationWithin

maybeestimatedbyoneofthefollowing:(a) R-barMethod where d2isaShewhartconstant=(k)(b) S-barMethod where c4isaShewhartconstant=(k)(c) PooledStandardDeviationMethodInMiniTab,thePooledStandardDeviationisthedefaultmethod.EstimatingWithinandOveral107EstimatingWithinandOverall

Incaseswherethereisonly1observationpersub-group(i.e.k=1),theMovingRangeMethodisused,where .ThewithinvariationWithin

isthenestimatedusingeithera) theAverageMovingRange:b) theMedianMovingRange:EstimatingWithinandOveral108Example3Thelengthofacamshaftforanautomobileengineisspecifiedat600±2mm.Controlofthelengthofthecamshaftiscriticaltoavoidscrap/rework.Thecamshaftisprovidedbyanexternalsupplier.Assesstheprocesscapabilityforthissupplier.ThedataisavailableinProcessCapabilityAnalysis.MTW.Example3Thelengthofacamsh109Example3StatQualityToolsCapabilityAnalysis(Normal)Example3StatQualityTools110Example3Example3111Example3AHistogramofcamshaftlengthsuggestsmixedpopulations.Furtherinvestigationrevealedthattherearetwosuppliersforthecamshaft.Datawascollectedovercamshaftsfrombothsources.Arethetwosupplierssimilarinperformance?Ifnot,whatareyourrecommendations?Example3AHistogramofcamsha