TUV德国莱茵技术六西格码培训资料BasicStatisticsforSPC课件

统计学基础BasicStatistics统计学基础1Whatisstatistics?什么是统计Definitionofstatistics:统计的定义*Statisticsarefactsandfigures.统计是事实和数据*Statisticsconsistofasetofmethodsandrulesfororganizingandinterpretingobservationsfrompopulationsandsamples统计通过一系列方法和规则来组织、解释来自总体和样本的观测值PopulationsandSamples总体和样本*Populationistheentiregrouporsetofallpossibleeventsofinterestintheparticularstudy.总体是被关注、研究的对象的全部*Sampleisasubsetofthepopulation样本是从总体中抽出的一部分ENTIREPOPULATION总体SAMPLEWITHIN(subset)样本AStatistic:统计值Anumericalvaluethatdescribesasample用来描述样本的数值Whatisstatistics?什么是统计Defini2TwoTypesofData

统计获得的两种数据AttributeDataCategoriesYes,NoGo,NogoMachine1,Machine2,Machine3Pass/FailVariableDataDiscrete(Count)DataMaintenanceEquipmentFailures,NumberofClogsNumberofcustomerreturnsContinuousDataDecimalsubdivisionsaremeaningfulTime,Pressure,ConveyorSpeed特性数据(定性)等级是非通止个体（如1号机器,2号机器,3号机器)成败变量数据（定量)间断型数据（计数）如设备维修次数、阻塞次数等客户退货次数连续型数据(计量)可有小数点如时间、压力、传送速度等TwoTypesofData

统计获得的两种数据Att3DescriptionofContinuousData-Graphical计量型数据的描述-图形HistogramHeightof90ladies#ofocurrenceHeight(inch)

数据分布图在实际统计中，统计结果是分段表示的，因此作出的分布图为柱形图。在分析数据时，通常将它拟合成连续的曲线。DescriptionofContinuousData4Examplesofdistributions不同的分布NegativeSkew负斜PositiveSkew正斜SymmetricDistribution对称分布Left-tailedRight-tailedTwo-tailedExamplesofdistributions不同的分5Mean:Arithmeticaverageofasetofvalues均值：算数平均值Reflectstheinfluenceofallvalues反映全部数据的影响StronglyInfluencedbyextremevalues受特殊值干扰大Median:Reflectsthe50%rank-thecenternumberafterasetofnumbers

hasbeensortedfromlowtohigh.中位数：反映系列的一半—将一组数据按大小顺序排列，取中间的一个数据Doesnotincludeallvaluesincalculation计算中未包含全部数据Is“robust”toextremeoutlierscores.对特殊值的干扰有抵抗Themeanandmedianwillbeaffectedbythenatureofthedistributionofnumbers.均值和中值都受数据分布的影响Mode:Mostfrequentlyoccurringvalueinadataset.InaParetothisis thelargestbaronthechart.众数：数据中重复次数最多的值，在柏拉图上表现为最高的那条柱DescriptionofQuantitativeData-CentralTendency计量型数据的描述-中心位置Mean:Arithmeticaverageofa6Relationship:meanandmedian

均值和中位数的比较1101009080706050403020100500NormalFrequencyMean,Median807060504030201003002001000Neg

SkewFrequencyMedianMean130120110100908070603002001000Pos

SkewFrequencyMedianMeanRelationship:meanandmedian

7Samplemeanofadistribution样本均值Mean=Average=xi/Ni=1N

=X1+X2+....XN

NExamples:例Partweights零件重量:8.47,8.67,9.34,7.99AVERAGE=8.47+8.67+9.34+7.99=8.62平均值 4

Samplemeanofadistribution8Don’tWorry.Thatropeisoneinchthickontheaverage.不要担心。绳子是平均一英寸粗Don’tWorry.Thatropeisone9Range=maximumvalue-minimumvalue范围=样本内最大值-最小值Variance=meansquareddistancefromthemean方差=数据与均值差距的平方之均值Standarddeviation=isthesquarerootofthevarianceandprovidesameasureofthestandarddistancefromthemean.标准偏差=方差的开方DescriptionofQuantitativeData-Dispersionor‘Spread’计量型数据的描述-离散度Range=maximumvalue-mini10lDeviation（偏差）isthedistancefromthemean.是离开均值的距离lDeviationscore（偏差值）=observation-truemean观测值-均值lVariance方差=meanoraverageofsquareddeviationscores偏差值的平方均值.isthesymbolforvariance方差的符号.lStandardDeviation标准偏差=squarerootofvariance方差的开方.isthesymbolforthestandarddeviation.标准偏差的符号TheStandardDeviationisaMeasureofVariability标准偏差是对变异的描述m=PopulationMean总体的均值iDeviation(distancefrommean)偏差s2sStandardDeviation

标准偏差lDeviation（偏差）isthedistance11^s==SampleStandardDeviation样本标准偏差X=SampleMean样本均值=PopulationMean总体均值StatisticsEstimateParameters=PopulationStandardDeviation总体标准偏差SamplePopulationSAMPLE样本POPULATION总体Statisticsorparameters?样本统计值与总体参数？统计活动的实质：用样本统计值来估计总体参数，从而了解总体^s==SampleStandardX=Sampl12Populationvs.sample

总体和样本计算公式PopulationMean总体均值SampleMean样本均值PopulationStandardDeviation总体标准偏差SampleStandardDeviation样本标准偏差=x=xnii=1n

=s=(X)n-1i2i=1nX^^Populationvs.sample

总体和样本计算公13Example:Calculating“sigma”计算练习Usingtheformabove,calculatethestandarddeviationforthenumbers用上列的表计算以下5个数据的标准偏差：2，1，3，5，4XX-X(X-)2XExample:Calculating“sigma”计算14ExerciseSolution计算结果:X X-X(X-)2XExerciseSolution计算结果:15数据的描述总览分布的位置LocationMean均值Median中值Mode代表值Quantiles分位数 Q1四分之一处 Q2二分之一处 Q3四分之三处P＃％机率位置离散度SpreadRange范围StandardDeviation标准偏差Variance变差StabilityFactor稳定因子Span跨度InterquartileRange内分位宽度SumofSquares平方和 Shape形状Histograms直方图RunCharts运行图TimePlots时序图ScatterPlots散点图BoxPlots盒状图BlockChart块图NormalityPlot正态性图NumercialGraphical数据的描述总览分布的位置Location离散度Sprea16NormalDistribution正态分布Mean均值Bell-shapeSymmetricDistribution‘倒钟’状对称分布fx(x)=12ps2e-(x-m)2/2s2NormalDistribution正态分布17MeasuredbyStandardDeviation

用标准偏差为尺度mean68.27%15.865%15.865%MeasuredbyStandardDeviation18MeasuredbyStandardDeviation

用标准偏差为尺度mean95.45%MeasuredbyStandardDeviation1968.2%的数据落在±1s以内95.4%的数据落在±2s以内99.7%的数据落在±3s以内99.99999975%的数据落在±6s以内MeasuredbyStandardDeviation

用标准偏差为尺度+4+5+6+1+2+3-2-1-4-3-6-5068.27%95.45%99.73%99.9937%99.999943%99.9999998%68.2%的数据落在±1s以内MeasuredbyS20总体任意抽取4组样品，每组3个样品总体的参数样品的统计值总体与样品在统计上的关系总体任意抽取4组样品，每组3个样品总体的参数样品的统计值总体21样品之间的统计分布样品之间的统计分布22中心极限定理（CentralLimitTheorem）*条件:

X1,X2,…,Xn是从总体中随机抽取样品的某特性的测量值，总体关于该特性的均值为

,总体的标准偏差为,结论: 该组样品的均值所属分布（假定有多组这样的样品，多组的均值形成一个分布）的均值和标准偏差为: 另外样品大小n越大，组均值的分布越接近正态分布.nXXXnX+++=...21使用正态分布来讨论大多数问题的基础中心极限定理（CentralLimitTheorem）*23(a)正态型l(b)规则型(c)指数型(d)偶次方型总体分布形状样品大小n=2的均值分布n=5n=30不论其总体的分布如何，其样品的均值的分布趋向正态分布实践经验如果总体是正态分布,样品均值一定为正态分布，无论样品大小如何如果总体的分布不够对称，5–20的样品大小即可最差的情形:30的样品大小可以应付一切形状的总体分布，无论总体的分布离正态分布相差多远(a)正态型l(b)规则型(c)指数型(d)偶次方型总体分布24二项分布与泊松分布计数型数据（合格与不合格）的分布遵从二项分布或泊松分布的规律：假定一批产品的不合格的几率为p，从中随机抽出一个容量为n的样本，那么n件产品中不合格品数X是一个离散型的随机变量,它服从二项分布（BinomialorBernoulli)的规律。P{X=r}=Crnpr(1-p)n-rr=1,2,…..,n当n很大，不合格数nu很小时，不合格数C＝nu的分布趋向于泊松分布(Poisson)：

lc

Pc=C-l

C!当n充分大（n>50），nu不小于5时，二项分布和泊松分布都趋于正态分布。二项分布与泊松分布25Variation&Capability变差与过程能力Variation变差与过程能力26IntroductiontoVariation什么是变差IntroductiontoVariation27160155150145165170175Willtheballalwaysgothesamedistance?

球会总是落在同一点吗？160155150145165170175Willthe28Therewillalwaysbevariabilitypresentinanyprocess变差存在于任何过程Wecantoleratevariabilityif我们在下列条件下可以容忍变差ThetotalvariabilityoftheOutputisrelativelysmallcomparedtotheprocessspecificationsandtheprocessisontarget与过程的工程规范相比较而言过程的变差很小,并且过程对中于目标值Theprocessisstableovertime过程长时间稳定LSLUSLNomUSLCost成本Traditional传统观念Acceptable可接受LSLUSLNomCost成本New新观念CanWeTolerateVariability?我们能容忍变差吗?Therewillalwaysbevariabili29Asthestandarddeviationincreasesprobabilityofdefectincrease标准偏差越大,缺陷几率越大1stdistribution2nddistribution3rddistributionLowerspecUpperspec.Defects.Defects&Distributions

缺陷与分布形状p(d)缺陷几率规范上限规范下限Asthestandarddeviationincr30ProcessCapability过程能力ProcessCapability31过程输入的变差已知有两种类型的过程变差:统计不受控:与特殊原因有关统计受控:与普通原因有关在研究过程表现时:特殊原因造成的变差往往表现为严重地偏离目标值，或不随机的平均值漂移普通原因造成的变差表现为正态的随机分布过程能力分析适用于普通原因造成的变差过程输入的变差已知有两种类型的过程变差:32ProcessCapabilityIndicators过程能力指数ProcessPotential:Ratioofthespecificationwidthto6times工艺潜能 ameasureoftheprocessvariation工程规 范的宽度与六倍的过程标准偏差的比值Cp=USL-LSL6sPPp=USL-LSL6sTCpisusedwhentheprocessisinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.Cp用于统计稳定的过程,过程是否稳定可由统计控制图来确定Cpusesthepooledstandarddeviation.Cp使用分组汇合的标准偏差PpisusedwhentheprocessisNOTinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.PpusesthetotalstandarddeviationPp用于统计不稳定的过程,它使用整体的标准偏差ProcessCapabilityIndicatorsP33Notindicatingifcenteredhere...不论中心在哪Orhere...或这儿Orhere...或这儿Indicatorswithvariablesdata:变量数据的指示Cp PpNotindicatingOrhere...O34TUV德国莱茵技术六西格码培训资料BasicStatisticsforSPC课件35sPsTPooledstandarddeviation分组汇合标准偏差Takenoverarelativelyshorttimeperiod.相对较短时间内的数据Takesintoaccountonlythevariationwithinasubset.只计入小组内的变差Containsonlycommoncausesofvariation.只包含普通原因的变差TotalorOverallstandarddeviation整体标准偏差Takenoversufficientsubsetstoshowthevariationduetoallcommonandspecialcausesofvariation.足够长时间内的数据以展示普通原因的变差和特殊原因的变差Calculatedfrommanysamplesthatrepresenttheshiftanddriftthatoccursinthepopulationduetoallcausesofvariation.从很多样品中计算包含所有变差所带来的总体漂移PooledVsOverallStandardDeviation分组汇合标准偏差与整体标准偏差sPsTPooledstandarddeviation36ActualProcessPerformance:Ratioofthedifferencebetweentheprocessaverageandthenearestspecificationto3timesameasureoftheprocessvariation实际过程的表现:过程均值与最近的规范线的距离跟3倍的过程标准偏差之比值ProcessCapabilityIndicators过程能力指数CPK=Min{CPL,CPU}CPL=X-LSL3sPCPU=USL-X3sPPPL=X-LSL3sTPPU=USL-X3sTPPK=Min{PPL,PPU}Cpkisusedwhentheprocessisinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.Cp用于统计稳定的过程,过程是否稳定可由统计控制图来确定Cpkusesthepooledstandarddeviation.Cpk使用分组汇合的标准偏差PpkisusedwhentheprocessisNOTinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.PpusesthetotalstandarddeviationPpk用于统计不稳定的过程,它使用整体的标准偏差ActualProcessPerformance:R37IndicatesSpread&Distance

分布宽度和距离的指示Width&positionareindicatedhere指示了分布宽度和位置andhere...andhere...Cpk PpkIndicatesSpread&Distance

分布38TUV德国莱茵技术六西格码培训资料BasicStatisticsforSPC课件39计算Cp/Cpk

Cp=LSL140USL260200Cp=LSL140USL260200Cp=LSL140USL260200Cp=LSL140USL260200215sst=20sst=20sst=10sst=10Cpk=Cpk=Cpk=Cpk=计算Cp/CpkCp=LSLUSL200Cp=LSL40Cp

PPCPK

PPKCapability能力OverallStdDevNotinControl长期整体标准偏差统计不受控PooledStdDevInControl短期分组汇合标准偏差统计受控Performance表现CPrepresents“entitlement”!Cp代表改进的目标Relatesstd.deviationtotolerance考察标准偏差与公差宽度Relatesmean&std.deviationtospec.考察均值/标准偏差和工程规范ShortTermLongTerm“Snapshot”Look照相“Video”Look录像Cp PPCapability能力Performance表41Sample3Sample4Sample5

s4

s3

s2

s1s5TotalsT

Inaperfectstateofcontrolstotal=spPooledStandardDeviation分组汇合标准差“Average”StandardDeviation‘平均’的标准偏差AssumingequalsubgroupsizeSample2Sample1

Sample3Sample4Sample5s4s342MondaySubgroup1子组1MondaySubgroup1子组143TuesdayFromMondaySubgroup2子组2TuesdayFromSubgroup2子组244WednesdaySubgroup3子组3FromMonday&TuesdayWednesdaySubgroup3子组3From45ThursdaySubgroup4子组4FromMon,Tues,WedThursdaySubgroup4子组4From46Thisdistributionismadeupofmanysmallertimeperiods.这一分布是由一段时间内许多小分布组成LongTermDistributionSubgroups1-4子组1-4Thisdistributionismadeupo47MondayTuesdayWednesdayThursdayOVERTIME…时间一长…SHIFTHAPPENS漂移发生了MondayTuesdayWednesdayThursday48Whichstandarddeviationshouldweuse?应该选用哪种标准偏差?Itdependsonwhatwearetryingtodo…取决于我们要作什么Ifwewanttoknowthebestourprocessiscapableof,如果我们想知道过程的最佳状态如何theprocessentitlement亦即过程的天赋能力usesst-theshorttermprocesscapability使用短期标准差计算短期过程能力Ifwewanttoknowwhatourcustomerssee,如果想了解客户是怎么看的ouractualperformance,我们的表现useslt-thelongtermprocesscapability使用长期标准差计算长期过程能力IfwewanttoknowouropportunityforimprovementUSEBOTH!如果我们想知道改进的机会,两者都用Whichstandarddeviationshoul490ThedifferencebetweensstandsltrepresentsTheAreaforProcessImprovement短期标准差与长期标准差的差距代表改进的空间TheFocusofMAIC改进的关注点Anotherwayofthinkingaboutit…另一中思考角度00Thedifferencebetweensstan50YB8207批号Y4092BY4092BY4092BY4092BY4092BY4092BY4092BY4092BY4092BY4095BY4095BY4095BY4095BY4095BY4095BY4095BY4095BY4095BY4095BY4095BMFI13.0612.9512.3712.6013.1913.3312.7612.9712.7113.1912.2013.7012.0013.1612.2712.5712.7213.0213.0212.55批号Y4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4228BY4228BY4228BY4228BY4228BMFI12.8412.7413.0012.6012.6212.6413.0312.7413.1012.6012.6012.5612.8612.6014.0013.40批号Y4240BY4240BY4240BY4240BY4240BY4240BY4240BY4240BY4241BY4241BY4241BY4241BY4241BY4241BMFI12.5912.6011.9311.9712.9012.7512.3712.7013.0412.6012.9013.7012.4913.18MeltFlowIndexofPlastics某塑胶材料的流动性数据子组大小不同时YB8207MeltFlowIndexof5115.0014.5014.0013.5013.0012.5012.0011.50MFI15.0014.5014.0013.5013.0012.5012.0011.50MFIAverage:12.8015.0014.5014.0013.5013.0012.5012.0011.50MFI15.00MFI15.00MFIAverage:15.00M52第一张图显示的是所有的原始数据（按批号分开）假设每一组都为正态分布。“第二张图显示的是各批的变差情况，其“平均”的标准偏差为0.400.计算方法如下 Sst=gå(nj-1)sj2=[8(0.301)2+10(0.508)2+10(0.190)2+4(0.612)2+7(0.358)2+5(0.437)2

][50-6]=0.400 其中: g为数据组的个数(批); n为测量值的个数第三张图显示的是将50个数据全部合并一起，计算总的标准偏差为（长期标准偏差): Slt=0.418j=1(n-g)第一张图显示的是所有的原始数据（按批号分开）假设每一组都为正53Discussionquestion:WhatinformationisprovidedbythedifferenceinZstandZlt?实例计算练习假定规范为11.0—14.0Thesamplemeanofthe50observationsisy=__________.Theoverallsamplestandarddeviationisslt=_______.Theweightedaverageofthelotstandarddeviations(variances)is

sst=______.ThetargetspecificationvalueisassumedtobecenteredatT=_____.Thecapabilityindex(potential)isCp=_______________.TheshorttermZst=3Cp=________________.TheuppercapabilityindexisCpu=_____________________.ThelowercapabilityindexisCpl=______________________.ThecapabilityindexisCpk=min{Cpu,Cpl}=___________________.TheupperperformanceindexisPpu=____________________.ThelowerperformanceindexisPpl=____________________.TheperformanceindexPpk=min{Ppu,Ppl}=________.Thelong-termZlt=3Ppk=_____________.

Discussionquestion:Whatinfo54Thesamplemeanofthe50observationsisy=12.7998.Theoverallsamplestandarddeviationisslt=0.418.Theweightedaverageofthelotstandarddeviations(variances)issst=0.401.ThetargetspecificationvalueisassumedtobecenteredatT=12.5.Thecapabilityindex(potential)isCp==1.25.TheshorttermZst=3x(1.25)=3.74.TheuppercapabilityindexisCpu=ThelowercapabilityindexisCpl=ThecapabilityindexisCpk=min{0.998,1.496}=0.998.TheupperperformanceindexisPpu=ThelowerperformanceindexisPpl=TheperformanceindexPpk=min{Ppu,Ppl}=0.96.ThelongtermZlt=3(0.96)=2.88.

(14.0-11.0)6(0.401)(14.0-12.7998)3(0.401)=0.998.(12.7998-11.0)3(0.401)=1.496.(14.0-12.7998)3(0.418)=0.96.(12.7998-11.0)3(0.418)=1.44.Discussionquestion:WhatinformationisprovidedbythedifferenceinZstandZlt?习题答案Thesamplemeanofthe50obse55统计学基础BasicStatistics统计学基础56Whatisstatistics?什么是统计Definitionofstatistics:统计的定义*Statisticsarefactsandfigures.统计是事实和数据*Statisticsconsistofasetofmethodsandrulesfororganizingandinterpretingobservationsfrompopulationsandsamples统计通过一系列方法和规则来组织、解释来自总体和样本的观测值PopulationsandSamples总体和样本*Populationistheentiregrouporsetofallpossibleeventsofinterestintheparticularstudy.总体是被关注、研究的对象的全部*Sampleisasubsetofthepopulation样本是从总体中抽出的一部分ENTIREPOPULATION总体SAMPLEWITHIN(subset)样本AStatistic:统计值Anumericalvaluethatdescribesasample用来描述样本的数值Whatisstatistics?什么是统计Defini57TwoTypesofData

统计获得的两种数据AttributeDataCategoriesYes,NoGo,NogoMachine1,Machine2,Machine3Pass/FailVariableDataDiscrete(Count)DataMaintenanceEquipmentFailures,NumberofClogsNumberofcustomerreturnsContinuousDataDecimalsubdivisionsaremeaningfulTime,Pressure,ConveyorSpeed特性数据(定性)等级是非通止个体（如1号机器,2号机器,3号机器)成败变量数据（定量)间断型数据（计数）如设备维修次数、阻塞次数等客户退货次数连续型数据(计量)可有小数点如时间、压力、传送速度等TwoTypesofData

统计获得的两种数据Att58DescriptionofContinuousData-Graphical计量型数据的描述-图形HistogramHeightof90ladies#ofocurrenceHeight(inch)

数据分布图在实际统计中，统计结果是分段表示的，因此作出的分布图为柱形图。在分析数据时，通常将它拟合成连续的曲线。DescriptionofContinuousData59Examplesofdistributions不同的分布NegativeSkew负斜PositiveSkew正斜SymmetricDistribution对称分布Left-tailedRight-tailedTwo-tailedExamplesofdistributions不同的分60Mean:Arithmeticaverageofasetofvalues均值：算数平均值Reflectstheinfluenceofallvalues反映全部数据的影响StronglyInfluencedbyextremevalues受特殊值干扰大Median:Reflectsthe50%rank-thecenternumberafterasetofnumbers

hasbeensortedfromlowtohigh.中位数：反映系列的一半—将一组数据按大小顺序排列，取中间的一个数据Doesnotincludeallvaluesincalculation计算中未包含全部数据Is“robust”toextremeoutlierscores.对特殊值的干扰有抵抗Themeanandmedianwillbeaffectedbythenatureofthedistributionofnumbers.均值和中值都受数据分布的影响Mode:Mostfrequentlyoccurringvalueinadataset.InaParetothisis thelargestbaronthechart.众数：数据中重复次数最多的值，在柏拉图上表现为最高的那条柱DescriptionofQuantitativeData-CentralTendency计量型数据的描述-中心位置Mean:Arithmeticaverageofa61Relationship:meanandmedian

均值和中位数的比较1101009080706050403020100500NormalFrequencyMean,Median807060504030201003002001000Neg

SkewFrequencyMedianMean130120110100908070603002001000Pos

SkewFrequencyMedianMeanRelationship:meanandmedian

62Samplemeanofadistribution样本均值Mean=Average=xi/Ni=1N

=X1+X2+....XN

NExamples:例Partweights零件重量:8.47,8.67,9.34,7.99AVERAGE=8.47+8.67+9.34+7.99=8.62平均值 4

Samplemeanofadistribution63Don’tWorry.Thatropeisoneinchthickontheaverage.不要担心。绳子是平均一英寸粗Don’tWorry.Thatropeisone64Range=maximumvalue-minimumvalue范围=样本内最大值-最小值Variance=meansquareddistancefromthemean方差=数据与均值差距的平方之均值Standarddeviation=isthesquarerootofthevarianceandprovidesameasureofthestandarddistancefromthemean.标准偏差=方差的开方DescriptionofQuantitativeData-Dispersionor‘Spread’计量型数据的描述-离散度Range=maximumvalue-mini65lDeviation（偏差）isthedistancefromthemean.是离开均值的距离lDeviationscore（偏差值）=observation-truemean观测值-均值lVariance方差=meanoraverageofsquareddeviationscores偏差值的平方均值.isthesymbolforvariance方差的符号.lStandardDeviation标准偏差=squarerootofvariance方差的开方.isthesymbolforthestandarddeviation.标准偏差的符号TheStandardDeviationisaMeasureofVariability标准偏差是对变异的描述m=PopulationMean总体的均值iDeviation(distancefrommean)偏差s2sStandardDeviation

标准偏差lDeviation（偏差）isthedistance66^s==SampleStandardDeviation样本标准偏差X=SampleMean样本均值=PopulationMean总体均值StatisticsEstimateParameters=PopulationStandardDeviation总体标准偏差SamplePopulationSAMPLE样本POPULATION总体Statisticsorparameters?样本统计值与总体参数？统计活动的实质：用样本统计值来估计总体参数，从而了解总体^s==SampleStandardX=Sampl67Populationvs.sample

总体和样本计算公式PopulationMean总体均值SampleMean样本均值PopulationStandardDeviation总体标准偏差SampleStandardDeviation样本标准偏差=x=xnii=1n

=s=(X)n-1i2i=1nX^^Populationvs.sample

总体和样本计算公68Example:Calculating“sigma”计算练习Usingtheformabove,calculatethestandarddeviationforthenumbers用上列的表计算以下5个数据的标准偏差：2，1，3，5，4XX-X(X-)2XExample:Calculating“sigma”计算69ExerciseSolution计算结果:X X-X(X-)2XExerciseSolution计算结果:70数据的描述总览分布的位置LocationMean均值Median中值Mode代表值Quantiles分位数 Q1四分之一处 Q2二分之一处 Q3四分之三处P＃％机率位置离散度SpreadRange范围StandardDeviation标准偏差Variance变差StabilityFactor稳定因子Span跨度InterquartileRange内分位宽度SumofSquares平方和 Shape形状Histograms直方图RunCharts运行图TimePlots时序图ScatterPlots散点图BoxPlots盒状图BlockChart块图NormalityPlot正态性图NumercialGraphical数据的描述总览分布的位置Location离散度Sprea71NormalDistribution正态分布Mean均值Bell-shapeSymmetricDistribution‘倒钟’状对称分布fx(x)=12ps2e-(x-m)2/2s2NormalDistribution正态分布72MeasuredbyStandardDeviation

用标准偏差为尺度mean68.27%15.865%15.865%MeasuredbyStandardDeviation73MeasuredbyStandardDeviation

用标准偏差为尺度mean95.45%MeasuredbyStandardDeviation7468.2%的数据落在±1s以内95.4%的数据落在±2s以内99.7%的数据落在±3s以内99.99999975%的数据落在±6s以内MeasuredbyStandardDeviation

用标准偏差为尺度+4+5+6+1+2+3-2-1-4-3-6-5068.27%95.45%99.73%99.9937%99.999943%99.9999998%68.2%的数据落在±1s以内MeasuredbyS75总体任意抽取4组样品，每组3个样品总体的参数样品的统计值总体与样品在统计上的关系总体任意抽取4组样品，每组3个样品总体的参数样品的统计值总体76样品之间的统计分布样品之间的统计分布77中心极限定理（CentralLimitTheorem）*条件:

X1,X2,…,Xn是从总体中随机抽取样品的某特性的测量值，总体关于该特性的均值为

,总体的标准偏差为,结论: 该组样品的均值所属分布（假定有多组这样的样品，多组的均值形成一个分布）的均值和标准偏差为: 另外样品大小n越大，组均值的分布越接近正态分布.nXXXnX+++=...21使用正态分布来讨论大多数问题的基础中心极限定理（CentralLimitTheorem）*78(a)正态型l(b)规则型(c)指数型(d)偶次方型总体分布形状样品大小n=2的均值分布n=5n=30不论其总体的分布如何，其样品的均值的分布趋向正态分布实践经验如果总体是正态分布,样品均值一定为正态分布，无论样品大小如何如果总体的分布不够对称，5–20的样品大小即可最差的情形:30的样品大小可以应付一切形状的总体分布，无论总体的分布离正态分布相差多远(a)正态型l(b)规则型(c)指数型(d)偶次方型总体分布79二项分布与泊松分布计数型数据（合格与不合格）的分布遵从二项分布或泊松分布的规律：假定一批产品的不合格的几率为p，从中随机抽出一个容量为n的样本，那么n件产品中不合格品数X是一个离散型的随机变量,它服从二项分布（BinomialorBernoulli)的规律。P{X=r}=Crnpr(1-p)n-rr=1,2,…..,n当n很大，不合格数nu很小时，不合格数C＝nu的分布趋向于泊松分布(Poisson)：

lc

Pc=C-l

C!当n充分大（n>50），nu不小于5时，二项分布和泊松分布都趋于正态分布。二项分布与泊松分布80Variation&Capability变差与过程能力Variation变差与过程能力81IntroductiontoVariation什么是变差IntroductiontoVariation82160155150145165170175Willtheballalwaysgothesamedistance?

球会总是落在同一点吗？160155150145165170175Willthe83Therewillalwaysbevariabilitypresentinanyprocess变差存在于任何过程Wecantoleratevariabilityif我们在下列条件下可以容忍变差ThetotalvariabilityoftheOutputisrelativelysmallcomparedtotheprocessspecificationsandtheprocessisontarget与过程的工程规范相比较而言过程的变差很小,并且过程对中于目标值Theprocessisstableovertime过程长时间稳定LSLUSLNomUSLCost成本Traditional传统观念Acceptable可接受LSLUSLNomCost成本New新观念CanWeTolerateVariability?我们能容忍变差吗?Therewillalwaysbevariabili84Asthestandarddeviationincreasesprobabilityofdefectincrease标准偏差越大,缺陷几率越大1stdistribution2nddistribution3rddistributionLowerspecUpperspec.Defects.Defects&Distributions

缺陷与分布形状p(d)缺陷几率规范上限规范下限Asthestandarddeviationincr85ProcessCapability过程能力ProcessCapability86过程输入的变差已知有两种类型的过程变差:统计不受控:与特殊原因有关统计受控:与普通原因有关在研究过程表现时:特殊原因造成的变差往往表现为严重地偏离目标值，或不随机的平均值漂移普通原因造成的变差表现为正态的随机分布过程能力分析适用于普通原因造成的变差过程输入的变差已知有两种类型的过程变差:87ProcessCapabilityIndicators过程能力指数ProcessPotential:Ratioofthespecificationwidthto6times工艺潜能 ameasureoftheprocessvariation工程规 范的宽度与六倍的过程标准偏差的比值Cp=USL-LSL6sPPp=USL-LSL6sTCpisusedwhentheprocessisinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.Cp用于统计稳定的过程,过程是否稳定可由统计控制图来确定Cpusesthepooledstandarddeviation.Cp使用分组汇合的标准偏差PpisusedwhentheprocessisNOTinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.PpusesthetotalstandarddeviationPp用于统计不稳定的过程,它使用整体的标准偏差ProcessCapabilityIndicatorsP88Notindicatingifcenteredhere...不论中心在哪Orhere...或这儿Orhere...或这儿Indicatorswithvariablesdata:变量数据的指示Cp PpNotindicatingOrhere...O89TUV德国莱茵技术六西格码培训资料BasicStatisticsforSPC课件90sPsTPooledstandarddeviation分组汇合标准偏差Takenoverarelativelyshorttimeperiod.相对较短时间内的数据Takesintoaccountonlythevariationwithinasubset.只计入小组内的变差Containsonlycommoncausesofvariation.只包含普通原因的变差TotalorOverallstandarddeviation整体标准偏差Takenoversufficientsubsetstoshowthevariationduetoallcommonandspecialcausesofvariation.足够长时间内的数据以展示普通原因的变差和特殊原因的变差Calculatedfrommanysamplesthatrepresenttheshiftanddriftthatoccursinthepopulationduetoallcausesofvariation.从很多样品中计算包含所有变差所带来的总体漂移PooledVsOverallStandardDeviation分组汇合标准偏差与整体标准偏差sPsTPooledstandarddeviation91ActualProcessPerformance:Ratioofthedifferencebetweentheprocessaverageandthenearestspecificationto3timesameasureoftheprocessvariation实际过程的表现:过程均值与最近的规范线的距离跟3倍的过程标准偏差之比值ProcessCapabilityIndicators过程能力指数CPK=Min{CPL,CPU}CPL=X-LSL3sPCPU=USL-X3sPPPL=X-LSL3sTPPU=USL-X3sTPPK=Min{PPL,PPU}Cpkisusedwhentheprocessisinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.Cp用于统计稳定的过程,过程是否稳定可由统计控制图来确定Cpkusesthepooledstandarddeviation.Cpk使用分组汇合的标准偏差PpkisusedwhentheprocessisNOTinastateofstatisticalcontrolasdefinedbystandardcontrolchartingmethods.PpusesthetotalstandarddeviationPpk用于统计不稳定的过程,它使用整体的标准偏差ActualProcessPerformance:R92IndicatesSpread&Distance

分布宽度和距离的指示Width&positionareindicatedhere指示了分布宽度和位置andhere...andhere...Cpk PpkIndicatesSpread&Distance

分布93TUV德国莱茵技术六西格码培训资料BasicStatisticsforSPC课件94计算Cp/Cpk

Cp=LSL140USL260200Cp=LSL140USL260200Cp=LSL140USL260200Cp=LSL140USL260200215sst=20sst=20sst=10sst=10Cpk=Cpk=Cpk=Cpk=计算