商务与经济统计第20章课件

StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter20

StatisticalMethodsforQualityControlAcceptanceSampling|||||||||||||||||||||||||UCLCLLCLStatisticalProcessControlPhilosophiesandFrameworksChapter20

StatisticalMethodsQualityQualityis“thetotalityoffeaturesandcharacteristicsofaproductorservicethatbearsonitsabilitytosatisfygivenneeds.”Organizationsrecognizethattheymuststriveforhighlevelsofquality.Theyhaveincreasedtheemphasisonmethodsformonitoringandmaintainingquality.QualityQualityis“thetotalitTotalQuality(TQ)isapeople-focusedmanagementsystemthataimsatcontinualincreaseincustomersatisfactionatcontinuallylowerrealcost.TQisatotalsystemapproach(notaseparateworkprogram)andanintegralpartofhigh-levelstrategy.TQworkshorizontallyacrossfunctions,involvesallemployees,toptobottom,andextendsbackwardandforwardtoincludeboththesupplyandcustomerchains.TotalQualityTQstresseslearningandadaptationtocontinualchangeaskeystoorganizationsuccess.TotalQuality(TQ)isapeopleTotalQualityRegardlessofhowitisimplementedindifferentorganizations,TotalQualityisbasedonthreefundamentalprinciples:afocusoncustomersandstakeholdersparticipationandteamworkthroughouttheorganizationafocusoncontinuousimprovementandlearningTotalQualityRegardlessofhowQualityPhilosophiesDr.WalterA.Shewhart

Developedasetofprinciplesthatarethebasisforwhatisknowntodayasprocesscontrol

Constructedadiagramthatwouldnowberecognizedasastatisticalcontrolchart

Broughttogetherthedisciplinesofstatistics,engineering,andeconomicsandchangedthecourseofindustrialhistory

RecognizedasthefatherofstatisticalqualitycontrolFirsthonorarymemberofASQQualityPhilosophiesDr.WalterQualityPhilosophiesDr.W.EdwardsDemingHelpededucatetheJapaneseonqualitymanagementshortlyafterWorldWarIIStressedthatthefocusonqualitymustbeledbymanagersDevelopedalistof14pointshebelievedrepresentthekeyresponsibilitiesofmanagersJapannameditsnationalqualityawardthe

DemingPrizeinhishonorQualityPhilosophiesDr.W.EdwQualityPhilosophiesJosephJuranHelpededucatetheJapaneseonqualitymanagementshortlyafterWorldWarIIProposedasimpledefinitionofquality:

fitnessforuseHisapproachtoqualityfocusedonthreequalityprocesses:qualityplanning,quality

control,andqualityimprovementQualityPhilosophiesJosephJurQualityPhilosophiesOtherSignificantIndividualsPhilipB.CrosbyA.V.FeigenbaumKarouIshikawaGenichiTaguchiQualityPhilosophiesOtherSignQualityFrameworks

1.Leadership

2.StrategicPlanning

3.CustomerandMarketFocusIn2003,the“BaldrigeIndex”(ahypotheticalstockindexcomprisedofBaldrigeAwardwinningcompanies)outperformedtheS&P500by4.4to1.Establishedin1987andgivenbytheU.S.presidenttoorganizationsthatapplyandarejudgedtobeoutstandinginsevenareas:

4.Measurement,AnalysisandKnowledgeMgmt.

5.HumanResourceFocus

6.ProcessManagement

7.BusinessResultsMalcolmBaldrigeNationalQualityAwardQualityFrameworks1.LeadersQualityFrameworksThestandardsdescribetheneedfor:aneffectivequalitysystem,ensuringthatmeasuringandtestingequipmentiscalibratedregularly,andmaintaininganadequaterecord-keepingsystem.Aseriesoffivestandardspublishedin1987bytheInternationalOrganizationforStandardizationinGeneva,Switzerland.ISO9000ISO9000registrationdetermineswhetheracompanycomplieswithitsownqualitysystem.QualityFrameworksThestandarQualityFrameworksSixSigmaThemethodologycreatedtoreachthisqualitygoalisreferredtoasSixSigma.SixSigmaisamajortoolinhelpingorganizationsachieveBaldrigelevelsofbusinessperformanceandprocessquality.

Sixsigmalevelofqualitymeansthatforeverymillionopportunitiesnomorethan3.4defectswilloccur.QualityFrameworksSixSigmaThQualityFrameworksSixSigma(continued)x-6s-2s-4s+2s+4s+6s99.9999998%mLowerQualityLimitUpperQualityLimit-5s-3s-1s+1s+3s+5sRoughly2defectivesin10millionQualityFrameworksSixSigma(c

QCconsistsofmakingaseriesofinspectionsandmeasure-mentstodeterminewhetherqualitystandardsarebeingmet.QAreferstotheentiresystemofpolicies,procedures,andguide-linesestablishedbyanorganizationtoachieveandmaintainquality.QAconsistsoftwofunctions...

Itsobjectiveistoincludequalityinthedesignofproductsandprocessesandtoidentifypotentialqualityproblemspriortoproduction.QualityTerminologyQualityAssuranceQualityEngineeringQualityControl

StatisticalProcessControl(SPC)Outputoftheproductionprocessissampledandinspected.UsingSPCmethods,itcanbedeterminedwhether variationsinoutputareduetocommoncausesor assignablecauses.Thegoalisdecidewhethertheprocesscanbe continuedorshouldbeadjustedtoachieveadesired qualitylevel.StatisticalProcessControl(SCausesofProcessOutputVariationCommonCausesrandomlyoccurringvariationsinmaterials,humidity,temperature,...variationstheproducercannotcontrolprocessisinstatisticalcontrolprocessdoesnotneedtobeadjustedCausesofProcessOutputVariaCausesofProcessOutputVariationAssignableCausesnon-randomvariationsinoutputduetotoolswearingout,operatorerror,incorrectmachinesettings,poorqualityrawmaterial,...variationstheproducercancontrolprocessisoutofcontrolcorrectiveactionshouldbetakenCausesofProcessOutputVaria

Haisformulatedintermsoftheproductionprocessbeingoutofcontrol.NullHypothesisAlternativeHypothesisSPCHypotheses SPCproceduresarebasedonhypothesis-testingmethodology.

H0isformulatedintermsoftheproductionprocessbeingincontrol.HaisformulatedintermsofDecisionsandStateoftheProcessCorrectDecisionTypeIIErrorAllowout-of-controlprocesstocontinueCorrectDecisionTypeIErrorAdjustin-controlprocessReject

H0AdjustProcess

Accept

H0ContinueProcessH0TrueIn-ControlH0FalseOut-of-ControlDecisionStateofProductionProcessTypeIandTypeIIErrors

DecisionsandStateoftheProControlChartsSPCusesgraphicaldisplaysknownascontrolchartstomonitoraproductionprocess.Controlchartsprovideabasisfordecidingwhetherthevariationintheoutputisduetocommoncauses(incontrol)orassignablecauses(outofcontrol).ControlChartsSPCusesgraphicTwoimportantlinesonacontrolchartaretheuppercontrollimit(UCL)

andlowercontrollimit(LCL).ControlChartsTheselinesarechosensothatwhentheprocessisincontrol,therewillbeahighprobabilitythatthesamplefindingwillbebetweenthetwolines.Valuesoutsideofthecontrollimitsprovidestrongevidencethattheprocessisoutofcontrol.TwoimportantlinesonacontrThischartisusedtomonitortherangeofthemeasurementsinthesample.RChartVariablesControlChartsxChartThischartisusedifthequalityoftheoutputismeasuredintermsofavariablesuchaslength,weight,temperature,andsoon.xrepresentsthemeanvaluefoundinasampleoftheoutput.ThischartisusedtomonAttributesControlCharts

Thischartisusedtomonitorthenumberofdefectiveitemsinthesample.npChartpChartThischartisusedtomonitortheproportiondefectiveinthesample.AttributesControlChartsTxChartStructureUCLLCLProcessMeanWheninControlCenterLineTimexUpperControlLimitLowerControlLimitxChartStructureUCLLCLProcessControlLimitsforanxChartProcessMeanmandStandardDeviationsKnown

where:

n=samplesizeControlLimitsforanxChartPExample:GraniteRockCo.ControlLimitsforanxChart:ProcessMeanandStandardDeviationKnown

WhenGraniteRock’spackagingprocessisincontrol,theweightofbagsofcementfilledbytheprocessisnormallydistributedwithameanof50poundsandastandarddeviationof1.5pounds. Whatshouldbethecontrollimitsforsamplesof9bags?Example:GraniteRockCo.Cont=50,=1.5,n=9UCL=50+3(.5)=51.5LCL=50-3(.5)=48.5ControlLimitsforanxChart:ProcessMeanandStandardDeviationKnownCL=50=50,=1.5,n=9=ProcessMeanandStandardDeviationUnknown

ControlLimitsforanxChartwhere:

x=overallsamplemean

R=averagerange

A2=constantthatdependsonn;takenfrom“FactorsforControlCharts”table=_==ProcessMeanandStandardDevFactorsforxControlChart“FactorsforControlCharts”Table(Partial)FactorsforxControlChart“FaControlLimitsforanRChartProcessMeanandStandardDeviationUnknown

where:

R=averagerange

D3,D4=constantsthatdependonn;taken from“FactorsforControlCharts” table_ControlLimitsforanRChartPFactorsforRControlChart“FactorsforControlCharts”Table(Partial)FactorsforRControlChart“FaRChartInpractice,theRchartisusuallyconstructedbeforethexchart.IftheRchartindicatesthattheprocessvariabilityisincontrol,thenthexchartisconstructed.Becausethecontrollimitsforthexchartdependonthevalueoftheaveragerange,theselimitswillnothavemuchmeaningunlesstheprocessvariabilityisincontrol.RChartInpractice,theRcharControlLimitsforanRChart:ProcessMeanandStandardDeviationUnknownExample:GraniteRockCo. Thetwentysamples,collectedwhentheprocesswasincontrol,resultedinanoverallsamplemeanof50.01poundsandanaveragerangeof.322pounds. SupposeGranitedoesnotknowthetruemeanandstandarddeviationforitsbagfillingprocess.ItwantstodevelopxandRchartsbasedontwentysamplesof5bagseach.ControlLimitsforanRChart:UCL=RD4=.322(2.114)=.681_LCL=RD3=.322(0)=0_x=50.01,R=.322,n=5_=ControlLimitsforanRChart:ProcessMeanandStandardDeviationUnknownCL=R=.322_UCL=RD4=.322(2.114)=.6RChartforGraniteRockCo.0.0000.400.500.600.700.8005101520SampleNumberSampleRangeRLCLUCLControlLimitsforanRChart:ProcessMeanandStandardDeviationUnknownRChartforGraniteRockCo.0.x=50.01,R=.322,n=5_=UCL=x+A2R=50.01+.577(.322)=50.196=_LCL=x

-

A2R=50.01-.577(.322)=49.824=_ControlLimitsforanxChart:ProcessMeanandStandardDeviationUnknownCL=x=50.01=x=50.01,R=.322,n=549.749.849.950.005101520SampleNumberSampleMeanUCLLCLxChartforGraniteRockCo.ControlLimitsforanxChart:ProcessMeanandStandardDeviationUnknown49.749.849.950.005ControlLimitsforapChartwhere:assuming:

np

>5

n(1-p)>5Note:IfcomputedLCLisnegative,setLCL=0ControlLimitsforapChartwhControlLimitsforapChartEverycheckcashedordepositedatNorwestBankmustbeencodedwiththeamountofthecheckbeforeitcanbegintheFederalReserveclearingprocess.Theaccuracyofthecheckencodingprocessisofutmostimportance.Ifthereisanydiscrepancybetweentheamountacheckismadeoutforandtheencodedamount,thecheckisdefective.Example:NorwestBankControlLimitsforapChartTwentysamples,eachconsistingof400checks,wereselectedandexaminedwhentheencodingprocesswasknowntobeoperatingcorrectly.Thenumberofdefectivechecksfoundinthe20samplesarelistedbelow.ControlLimitsforapChartExample:NorwestBankTwentysamples,eachconsSupposeNorwestdoesnotknowtheproportionofdefectivechecks,p,fortheencodingprocesswhenitisincontrol.ControlLimitsforapChartWewilltreatthedata(20samples)collectedasonelargesampleandcomputetheaveragenumberofdefectivechecksforallthedata.Thatvaluecanthenbeusedtoestimatep.SupposeNorwestdoesnotknowNotethatthecomputedLCLisnegative.Estimatedp=128/((20)(400))=128/8000=.016ControlLimitsforapChartnp=400(.016)=6.4>5;n(1-p)=400(.984)=393.6>5NotethatthecomputedLCLisControlLimitsforapChart

EncodedChecksProportionDefective0.0000.0050.0100.0150.0200.0250.0300.0350.0400.04505101520SampleNumberSampleProportionpUCLLCLControlLimitsforapChartEControlLimitsforannpChartNote:IfcomputedLCLisnegative,setLCL=0assuming: np

>5

n(1-p)>5ControlLimitsforannpChartThelocationandpatternofpointsinacontrolchartenableustodetermine,withasmallprobabilityoferror,whetheraprocessisinstatisticalcontrol.Aprimaryindicationthataprocessmaybeoutofcontrolisadatapointoutsidethecontrollimits.InterpretationofControlChartsThelocationandpatternofp

Certainpatternsofpointswithinthecontrollimitscanbewarningsignalsofqualityproblems:alargenumberofpointsononesideofcenterlinesixorsevenpointsinarowthatindicateeitheranincreasingordecreasingtrendInterpretationofControlChartsCertainpatternsofpointswiAcceptanceSampling

Acceptancesamplingisastatisticalmethodthatenablesustobasetheaccept-rejectdecisionontheinspectionofasampleofitemsfromthelot.Theitemsofinterestcanbeincomingshipmentsofrawmaterialsorpurchasedpartsaswellasfinishedgoodsfromfinalassembly.AcceptanceSamplingAcceptanceAcceptanceSamplingAcceptancesamplinghasadvantagesover100%inspectionincluding:usuallylessexpensivelessproductdamageduetolesshandlingfewerinspectorsrequiredprovidesonlyapproachpossibleifdestructivetestingmustbeusedAcceptanceSamplingAcceptanceAcceptanceSamplingProcedureLotreceivedSampleselectedSampleditemsinspectedforqualityResultscomparedwithspecifiedqualitycharacteristicsAcceptthelotRejectthelotSendtoproductionorcustomerDecideondispositionofthelotQualityisnotsatisfactoryQualityissatisfactoryAcceptanceSamplingProcedureLAcceptanceSamplingThehypothesesare:

H0:Good-qualitylotAcceptancesamplingisbasedonhypothesis-testingmethodology.

Ha:Poor-qualitylotAcceptanceSamplingThehypothTheOutcomesofAcceptanceSamplingTypeIandTypeIIErrors

CorrectDecisionTypeIIErrorAcceptingaPoor-qualitylotCorrectDecisionTypeIErrorRejectingaGood-qualitylotReject

H0RejecttheLot

Accept

H0AccepttheLotH0TrueGood-QualityLotH0FalsePoor-QualityLotDecisionStateoftheLotTheOutcomesofAcceptanceSamBinomialProbabilityFunctionforAcceptanceSamplingProbabilityofAcceptingaLotwhere:

n=samplesize

p=proportionofdefectiveitemsinlot

x=numberofdefectiveitemsinsample

f(x)=probabilityofxdefectiveitemsinsampleBinomialProbabilityFunctionExample:AcceptanceSampling Aninspectortakesasampleof20itemsfromalot.Hispolicyistoacceptalotifnomorethan2defectiveitemsarefoundinthesample. Assumingthat5percentofalotisdefective,whatistheprobabilitythathewillacceptalot?Rejectalot?Example:AcceptanceSamplingExample:AcceptanceSamplingP(AcceptLot)=.3585+.3774+.1887=.9246P(AcceptLot)=f(0)+f(1)+f(2)n=20,c=2,andp=.05Example:Acceptanc

n=20,c=2,andp=.05Example:AcceptanceSamplingP(RejectLot)=1–P(AcceptLot) =1-.9246 =.0754 n=20,c=SelectinganAcceptanceSamplingPlanInformulatingaplan,managersmustspecifytwovaluesforthefractiondefectiveinthelot.a=theprobabilitythatalotwithp0defectiveswillberejectedb

=theprobabilitythatalotwith

p1defectiveswillbeacceptedSelectinganAcceptanceSampliThen,thevaluesofnandcareselectedthatresultinanacceptancesamplingplanthatcomesclosesttomeetingboththeaandb

requirementsspecified.SelectinganAcceptanceSamplingPlanThen,thevaluesofnandcaProbabilityofAcceptingtheLot0

510152001.00PercentDefectiveintheLotp0p1b(1-a)an=15,c=0p0=.03,p1=.15a=.3667,b=.0874OperatingCharacteristicCurveProbabilityofAcceptingtheLMultipleSamplingPlansAmultiplesamplingplanusestwoormorestagesofsampling.Multiplesamplingplansoftenresultinasmallertotalsamplesizethansingle-sampleplanswiththesameTypeIerrorandTypeIIerrorprobabilities.Ateachstagethedecisionpossibilitiesare:stopsamplingandacceptthelot,stopsamplingandrejectthelot,orcontinuesampling.MultipleSamplingPlansAmultiATwo-StageAcceptanceSamplingPlanInspectn1itemsFindx1defectiveitemsinthissamplex1

<

c1?x1

>

c2?Inspectn2additionalitemsAcceptthelotRejectthelotx1+x2

<

c3?Findx2defectiveitemsinthissampleYesYesNoNoNoYesFirstStageSecondStageATwo-StageAcceptanceSamplinEndofChapter20EndofChapter20StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter20

StatisticalMethodsforQualityControlAcceptanceSampling|||||||||||||||||||||||||UCLCLLCLStatisticalProcessControlPhilosophiesandFrameworksChapter20

StatisticalMethodsQualityQualityis“thetotalityoffeaturesandcharacteristicsofaproductorservicethatbearsonitsabilitytosatisfygivenneeds.”Organizationsrecognizethattheymuststriveforhighlevelsofquality.Theyhaveincreasedtheemphasisonmethodsformonitoringandmaintainingquality.QualityQualityis“thetotalitTotalQuality(TQ)isapeople-focusedmanagementsystemthataimsatcontinualincreaseincustomersatisfactionatcontinuallylowerrealcost.TQisatotalsystemapproach(notaseparateworkprogram)andanintegralpartofhigh-levelstrategy.TQworkshorizontallyacrossfunctions,involvesallemployees,toptobottom,andextendsbackwardandforwardtoincludeboththesupplyandcustomerchains.TotalQualityTQstresseslearningandadaptationtocontinualchangeaskeystoorganizationsuccess.TotalQuality(TQ)isapeopleTotalQualityRegardlessofhowitisimplementedindifferentorganizations,TotalQualityisbasedonthreefundamentalprinciples:afocusoncustomersandstakeholdersparticipationandteamworkthroughouttheorganizationafocusoncontinuousimprovementandlearningTotalQualityRegardlessofhowQualityPhilosophiesDr.WalterA.Shewhart

Developedasetofprinciplesthatarethebasisforwhatisknowntodayasprocesscontrol

Constructedadiagramthatwouldnowberecognizedasastatisticalcontrolchart

Broughttogetherthedisciplinesofstatistics,engineering,andeconomicsandchangedthecourseofindustrialhistory

RecognizedasthefatherofstatisticalqualitycontrolFirsthonorarymemberofASQQualityPhilosophiesDr.WalterQualityPhilosophiesDr.W.EdwardsDemingHelpededucatetheJapaneseonqualitymanagementshortlyafterWorldWarIIStressedthatthefocusonqualitymustbeledbymanagersDevelopedalistof14pointshebelievedrepresentthekeyresponsibilitiesofmanagersJapannameditsnationalqualityawardthe

DemingPrizeinhishonorQualityPhilosophiesDr.W.EdwQualityPhilosophiesJosephJuranHelpededucatetheJapaneseonqualitymanagementshortlyafterWorldWarIIProposedasimpledefinitionofquality:

fitnessforuseHisapproachtoqualityfocusedonthreequalityprocesses:qualityplanning,quality

control,andqualityimprovementQualityPhilosophiesJosephJurQualityPhilosophiesOtherSignificantIndividualsPhilipB.CrosbyA.V.FeigenbaumKarouIshikawaGenichiTaguchiQualityPhilosophiesOtherSignQualityFrameworks

1.Leadership

2.StrategicPlanning

3.CustomerandMarketFocusIn2003,the“BaldrigeIndex”(ahypotheticalstockindexcomprisedofBaldrigeAwardwinningcompanies)outperformedtheS&P500by4.4to1.Establishedin1987andgivenbytheU.S.presidenttoorganizationsthatapplyandarejudgedtobeoutstandinginsevenareas:

4.Measurement,AnalysisandKnowledgeMgmt.

5.HumanResourceFocus

6.ProcessManagement

7.BusinessResultsMalcolmBaldrigeNationalQualityAwardQualityFrameworks1.LeadersQualityFrameworksThestandardsdescribetheneedfor:aneffectivequalitysystem,ensuringthatmeasuringandtestingequipmentiscalibratedregularly,andmaintaininganadequaterecord-keepingsystem.Aseriesoffivestandardspublishedin1987bytheInternationalOrganizationforStandardizationinGeneva,Switzerland.ISO9000ISO9000registrationdetermineswhetheracompanycomplieswithitsownqualitysystem.QualityFrameworksThestandarQualityFrameworksSixSigmaThemethodologycreatedtoreachthisqualitygoalisreferredtoasSixSigma.SixSigmaisamajortoolinhelpingorganizationsachieveBaldrigelevelsofbusinessperformanceandprocessquality.

Sixsigmalevelofqualitymeansthatforeverymillionopportunitiesnomorethan3.4defectswilloccur.QualityFrameworksSixSigmaThQualityFrameworksSixSigma(continued)x-6s-2s-4s+2s+4s+6s99.9999998%mLowerQualityLimitUpperQualityLimit-5s-3s-1s+1s+3s+5sRoughly2defectivesin10millionQualityFrameworksSixSigma(c

QCconsistsofmakingaseriesofinspectionsandmeasure-mentstodeterminewhetherqualitystandardsarebeingmet.QAreferstotheentiresystemofpolicies,procedures,andguide-linesestablishedbyanorganizationtoachieveandmaintainquality.QAconsistsoftwofunctions...

Itsobjectiveistoincludequalityinthedesignofproductsandprocessesandtoidentifypotentialqualityproblemspriortoproduction.QualityTerminologyQualityAssuranceQualityEngineeringQualityControl

StatisticalProcessControl(SPC)Outputoftheproductionprocessissampledandinspected.UsingSPCmethods,itcanbedeterminedwhether variationsinoutputareduetocommoncausesor assignablecauses.Thegoalisdecidewhethertheprocesscanbe continuedorshouldbeadjustedtoachieveadesired qualitylevel.StatisticalProcessControl(SCausesofProcessOutputVariationCommonCausesrandomlyoccurringvariationsinmaterials,humidity,temperature,...variationstheproducercannotcontrolprocessisinstatisticalcontrolprocessdoesnotneedtobeadjustedCausesofProcessOutputVariaCausesofProcessOutputVariationAssignableCausesnon-randomvariationsinoutputduetotoolswearingout,operatorerror,incorrectmachinesettings,poorqualityrawmaterial,...variationstheproducercancontrolprocessisoutofcontrolcorrectiveactionshouldbetakenCausesofProcessOutputVaria

Haisformulatedintermsoftheproductionprocessbeingoutofcontrol.NullHypothesisAlternativeHypothesisSPCHypotheses SPCproceduresarebasedonhypothesis-testingmethodology.

H0isformulatedintermsoftheproductionprocessbeingincontrol.HaisformulatedintermsofDecisionsandStateoftheProcessCorrectDecisionTypeIIErrorAllowout-of-controlprocesstocontinueCorrectDecisionTypeIErrorAdjustin-controlprocessReject

H0AdjustProcess

Accept

H0ContinueProcessH0TrueIn-ControlH0FalseOut-of-ControlDecisionStateofProductionProcessTypeIandTypeIIErrors

DecisionsandStateoftheProControlChartsSPCusesgraphicaldisplaysknownascontrolchartstomonitoraproductionprocess.Controlchartsprovideabasisfordecidingwhetherthevariationintheoutputisduetocommoncauses(incontrol)orassignablecauses(outofcontrol).ControlChartsSPCusesgraphicTwoimportantlinesonacontrolchartaretheuppercontrollimit(UCL)

andlowercontrollimit(LCL).ControlChartsTheselinesarechosensothatwhentheprocessisincontrol,therewillbeahighprobabilitythatthesamplefindingwillbebetweenthetwolines.Valuesoutsideofthecontrollimitsprovidestrongevidencethattheprocessisoutofcontrol.TwoimportantlinesonacontrThischartisusedtomonitortherangeofthemeasurementsinthesample.RChartVariablesControlChartsxChartThischartisusedifthequalityoftheoutputismeasuredintermsofavariablesuchaslength,weight,