TUV德國(guó)萊茵技術(shù)六西格碼培訓(xùn)資料BasicStatisticsforSPC

1、統(tǒng)計(jì)學(xué)基礎(chǔ)Basic StatisticsWhat is statistics?什么是統(tǒng)統(tǒng)計(jì)Definition of statistics:統(tǒng)計(jì)的定定義* Statisticsare facts andfigures.統(tǒng)計(jì)是事事實(shí)和數(shù)數(shù)據(jù)* Statisticsconsistofa setofmethodsandrulesfororganizingandinterpretingobservationsfrompopulationsandsamples統(tǒng)計(jì)通過(guò)過(guò)一系列列方法和和規(guī)則來(lái)來(lái)組織、解釋來(lái)來(lái)自總體體和樣本本的觀(guān)測(cè)測(cè)值PopulationsandSamples總體和樣樣本*Populat

2、ionistheentiregrouporsetofallpossibleeventsofinterestinthe particularstudy.總體是被被關(guān)注、研究的的對(duì)象的的全部*Sampleisasubsetofthepopulation樣本是從從總體中中抽出的的一部分分ENTIREPOPULATION總體SAMPLEWITHIN(subset)樣本A Statistic:統(tǒng)計(jì)值A(chǔ) numericalvaluethat describesasample用來(lái)描述述樣本的的數(shù)值TwoTypesofData統(tǒng)計(jì)獲得得的兩種種數(shù)據(jù)AttributeDataCategoriesYes, NoG

3、o,NogoMachine 1, Machine2,Machine3Pass/FailVariableDataDiscrete(Count) DataMaintenanceEquipmentFailures, NumberofClogsNumberofcustomerreturnsContinuous DataDecimal subdivisionsaremeaningfulTime,Pressure,ConveyorSpeed特性數(shù)據(jù)據(jù)(定性性)等級(jí)是非 通止 個(gè)體（如如 1號(hào)號(hào)機(jī)器, 2號(hào)號(hào)機(jī)器, 3號(hào)號(hào)機(jī)器)成敗變量數(shù)據(jù)據(jù)（定量量)間斷型數(shù)數(shù)據(jù)（計(jì)計(jì)數(shù)）如設(shè)備維維修次數(shù)數(shù)、阻塞塞次數(shù)等等

4、客戶(hù)退貨貨次數(shù)連續(xù)型數(shù)數(shù)據(jù)(計(jì)量)可有小數(shù)數(shù)點(diǎn)如時(shí)間、壓力、傳送速速度等DescriptionofContinuous Data-Graphical計(jì)量型數(shù)數(shù)據(jù)的描描述-圖圖形HistogramHeightof90ladies# of ocurrenceHeight(inch)數(shù)據(jù)分布布圖在實(shí)際統(tǒng)統(tǒng)計(jì)中， 統(tǒng)計(jì)計(jì)結(jié)果是是分段表表示的，因此作作出的分分布圖為為柱形圖圖。在在分析數(shù)數(shù)據(jù)時(shí)，通常將將它擬合合成連續(xù)續(xù)的曲線(xiàn)線(xiàn)。Examplesofdistributions不同的分分布NegativeSkew負(fù)斜P(pán)ositiveSkew正斜SymmetricDistribution對(duì)稱(chēng)分布布Left-t

5、ailedRight-tailedTwo-tailedMean:Arithmeticaverageofa setofvalues均值：算數(shù)平均值Reflectstheinfluence of allvalues反映全部部數(shù)據(jù)的的影響StronglyInfluenced by extremevalues受特殊值值干擾大大Median:Reflectsthe50% rank-thecenternumberaftera setofnumbershasbeensortedfromlowtohigh.中位數(shù)：反映系列的一一半將將一組數(shù)數(shù)據(jù)按大大小順序序排列，取中間的一個(gè)數(shù)據(jù)據(jù)Does notinclud

6、e allvaluesincalculation計(jì)算中未未包含全全部數(shù)據(jù)據(jù)Is“robust”toextreme outlierscores.對(duì)特殊值值的干擾擾有抵抗抗Themeanandmedian willbeaffectedbythe natureofthedistribution of numbers.均值和中中值都受受數(shù)據(jù)分分布的影影響Mode:Most frequentlyoccurringvalueina dataset.Ina Paretothis isthe largestbar on thechart.眾數(shù)：數(shù)數(shù)據(jù)中重重復(fù)次數(shù)數(shù)最多的的值，在柏拉圖圖上表現(xiàn)現(xiàn)為最高高的那條條

7、柱DescriptionofQuantitativeData-Central Tendency計(jì)量型數(shù)數(shù)據(jù)的描描述-中中心位置置Relationship: meanand median均值和中中位數(shù)的的比較1101009080706050403020100500NormalFrequencyMean,Median807060504030201003002001000Neg SkewFrequencyMedianMean130120110100908070603002001000Pos SkewFrequencyMedianMeanSamplemeanofa distribution樣本均值值

8、Mean = Average= xi /Ni=1N = X1 + X2 +.XN NExamples:例Part weights零件重量: 8.47, 8.67, 9.34, 7.99 AVERAGE = 8.47 +8.67 + 9.34 + 7.99 = 8.62平均值4DontWorry.Thatropeisoneinchthickontheaverage.不要擔(dān)心心。繩繩子是平平均一英英寸粗Range=maximumvalue-minimumvalue范圍=樣樣本內(nèi)最最大值-最小值值Variance= meansquareddistancefrom themean方差=數(shù)數(shù)據(jù)與均均值差

9、距距的平方方之均值值Standarddeviation= is thesquarerootofthevarianceand provides ameasureofthestandarddistancefrom themean.標(biāo)準(zhǔn)偏差差=方差差的開(kāi)方方DescriptionofQuantitativeData-Dispersion or Spread計(jì)量型數(shù)數(shù)據(jù)的描描述-離離散度lDeviation（偏差）isthedistancefromthemean.是離開(kāi)均均值的距距離lDeviationscore（偏差值）= observation -truemean觀(guān)測(cè)值-均值lVariance方差

10、= meanoraverageofsquared deviationscores偏差值的的平方均均值.isthesymbol forvariance方差的符符號(hào).lStandardDeviation標(biāo)準(zhǔn)偏差差= squareroot of variance方差的開(kāi)開(kāi)方.isthesymbol forthestandarddeviation.標(biāo)準(zhǔn)偏差差的符號(hào)號(hào)TheStandardDeviation is aMeasureofVariability標(biāo)準(zhǔn)偏差差是對(duì)變變異的描描述m= PopulationMean 總體的均值iDeviation (distance from mean)偏差s2sSt

11、andardDeviation標(biāo)標(biāo)準(zhǔn)偏偏差s = SampleStandardDeviation樣本標(biāo)準(zhǔn)準(zhǔn)偏差X= Sample Mean樣本均值= Population Mean總體均值StatisticsEstimateParameters= PopulationStandardDeviation總體標(biāo)準(zhǔn)準(zhǔn)偏差SamplePopulationSAMPLE樣本POPULATION總體Statistics or parameters?樣本統(tǒng)計(jì)計(jì)值與總總體參數(shù)數(shù)？統(tǒng)計(jì)活動(dòng)動(dòng)的實(shí)質(zhì)質(zhì)：用樣樣本統(tǒng)計(jì)計(jì)值來(lái)估估計(jì)總體體參數(shù)，從而了了解總體體Population vs.sample總體和樣樣本計(jì)算算公式

12、Population Mean總體均值值SampleMean樣本均值值Population Standard Deviation總體標(biāo)準(zhǔn)準(zhǔn)偏差SampleStandardDeviation樣本標(biāo)準(zhǔn)準(zhǔn)偏差=x=xnii=1n =s=(X )n-1i2i=1nX Example:Calculating“sigma”計(jì)算練習(xí)習(xí)Using the form above, calculate the standard deviation for the numbers用上列的表計(jì)算以下5個(gè)數(shù)據(jù)的標(biāo)準(zhǔn)偏差：2， 1， 3， 5 ，4 X X-X (X-)2XExerciseSolution計(jì)算結(jié)果果: X

13、 X-X (X-)2X數(shù)據(jù)的描描述總覽分布的位位置LocationMean 均值值Median中中值Mode代表值Quantiles分分位數(shù)數(shù)Q1四分之一一處Q2二分之一一處Q3四分之三三處P機(jī)機(jī)率位位置離散度SpreadRange范范圍StandardDeviation標(biāo)標(biāo)準(zhǔn)偏偏差Variance變變差StabilityFactor 穩(wěn)定定因子Span 跨度度Interquartile Range 內(nèi)分分位寬度度SumofSquares平平方和Shape形形狀Histograms 直方方圖RunCharts 運(yùn)行行圖Time Plots 時(shí)序序圖Scatter Plots 散點(diǎn)點(diǎn)圖BoxPl

14、ots盒盒狀圖圖BlockChart塊塊圖NormalityPlot正正態(tài)性圖圖NumercialGraphicalNormalDistribution正態(tài)分布布Mean均值Bell-shape SymmetricDistribution倒鐘狀狀對(duì)稱(chēng)分分布fx(x)=12ps2e-(x-m)2/2s2MeasuredbyStandardDeviation用標(biāo)準(zhǔn)偏偏差為尺尺度mean68.27%15.865%15.865%MeasuredbyStandardDeviation用標(biāo)準(zhǔn)偏偏差為尺尺度mean95.45%68.2%的的數(shù)據(jù)落落在1s以?xún)?nèi)95.4%的的數(shù)據(jù)落落在2s以?xún)?nèi)99.7%的的數(shù)據(jù)

15、落落在3s以?xún)?nèi)99.99999975%的的數(shù)據(jù)據(jù)落在6s以?xún)?nèi)MeasuredbyStandardDeviation用標(biāo)準(zhǔn)偏偏差為尺尺度+4+5+6+1+2+3-2-1-4-3-6-5068.27%95.45%99.73%99.9937%99.999943%99.9999998%總體任意抽取4組樣品，每組3個(gè)樣品總體的參參數(shù)樣品的統(tǒng)統(tǒng)計(jì)值總體與樣樣品在統(tǒng)統(tǒng)計(jì)上的的關(guān)系樣品之間間的統(tǒng)計(jì)計(jì)分布中心極限限定理（CentralLimitTheorem）*條件:X1, X2, , Xn是從總體體中隨機(jī)機(jī)抽取樣樣品的某某特性的的測(cè)量值值，總體體關(guān)于該該特性的的均值為為,總體的標(biāo)標(biāo)準(zhǔn)偏差差為,結(jié)論:該組樣品品

16、的均值值所所屬分分布（假假定有多多組這樣樣的樣品品，多組組的均值值形成一一個(gè)分布布）的均均值和標(biāo)標(biāo)準(zhǔn)偏差差為:另外樣品品大小n越越大，組組均值的的分布越越接近正正態(tài)分布布.nXXXnX+=.21使用正態(tài)態(tài)分布來(lái)來(lái)討論大大多數(shù)問(wèn)問(wèn)題的基基礎(chǔ)(a)正態(tài)型l(b)規(guī)則型(c)指數(shù)型(d)偶次方型型總體分布布形狀樣品大小小n =2的均值分分布n =5n =30不論其總總體的分分布如何何，其樣樣品的均均值的分分布趨向向正態(tài)分分布實(shí)踐經(jīng)驗(yàn)驗(yàn)如果總體體是正態(tài)態(tài)分布,樣樣品均值值一定為為正態(tài)分分布，無(wú)無(wú)論樣品品大小如如何如果總體體的分布布不夠?qū)?duì)稱(chēng)，520的樣樣品大小小即可最差的情情形:30的樣品品大小可可以

17、應(yīng)付付一切形形狀的總總體分布布，無(wú)論論總體的的分布離離正態(tài)分分布相差差多遠(yuǎn)二項(xiàng)分布布與泊松松分布計(jì)數(shù)型數(shù)數(shù)據(jù)（合合格與不不合格）的分布布遵從二二項(xiàng)分布布或泊松松分布的的規(guī)律：假定一批批產(chǎn)品的的不合格格的幾率率為p，從中隨隨機(jī)抽出出一個(gè)容容量為n的樣本本，那么么n件產(chǎn)產(chǎn)品中不不合格品品數(shù)X是一個(gè)離離散型的的隨機(jī)變變量,它它服服從二項(xiàng)項(xiàng)分布（BinomialorBernoulli)的規(guī)規(guī)律。PX=r=Crnpr(1-p)n-rr=1, 2, .,n當(dāng)n很很大，不不合格數(shù)數(shù)nu很很小時(shí)，不合格格數(shù)Cnu的的分布趨趨向于泊泊松分布布(Poisson)：lcPc=C-lC!當(dāng)n充分大（n50），nu不不

18、小于5時(shí)，二二項(xiàng)分布布和泊松松分布都都趨于正正態(tài)分布布。Variation&Capability變差與過(guò)過(guò)程能力力IntroductiontoVariation什么是變變差160155150145165170175Will theball alwaysgothesamedistance?球會(huì)總是是落在同同一點(diǎn)嗎嗎？Therewill alwaysbevariabilitypresent in anyprocess變差存在在于任何何過(guò)程Wecantoleratevariabilityif我們?cè)谙孪铝袟l件件下可以以容忍變變差Thetotalvariabilityofthe Outputisrela

19、tively small compared to theprocess specifications andtheprocessisontarget與過(guò)程的的工程規(guī)規(guī)范相比比較而言言過(guò)程的的變差很很小,并且過(guò)程程對(duì)中于于目標(biāo)值值Theprocessisstableovertime過(guò)程長(zhǎng)時(shí)時(shí)間穩(wěn)定定LSLUSLNomUSLCost成本Traditional傳統(tǒng)觀(guān)念A(yù)cceptable可接受LSLUSLNomCost成本New新觀(guān)念CanWeTolerateVariability?我們能容容忍變差差嗎?Asthestandarddeviation increasesprobabilityofdef

20、ectincrease標(biāo)準(zhǔn)偏差差越大, 缺陷陷幾率越越大1stdistribution2nddistribution3rddistributionLowerspecUpperspec.Defects.Defects&Distributions缺陷陷與分布布形狀p(d)缺陷幾率率規(guī)范上限限規(guī)范下限限Process Capability過(guò)程能力力過(guò)程輸入入的變差差已知有兩兩種類(lèi)型型的過(guò)程程變差:統(tǒng)計(jì)不受受控:與與特殊殊原因有有關(guān)統(tǒng)計(jì)受控控:與與普通原原因有關(guān)關(guān)在研究過(guò)過(guò)程表現(xiàn)現(xiàn)時(shí):特殊原因因造成的的變差往往往表現(xiàn)現(xiàn)為嚴(yán)重重地偏離離目標(biāo)值值，或不不隨機(jī)的的平均值值漂移普通原因因造成的的變差表表現(xiàn)為正

21、正態(tài)的隨隨機(jī)分布布過(guò)程能力力分析適適用于普通原因因造成的變變差Process CapabilityIndicators過(guò)程能力力指數(shù)Process Potential: Ratio of thespecification width to 6times工藝潛能能a measureofthe processvariation工程規(guī)范的寬度度與六倍倍的過(guò)程程標(biāo)準(zhǔn)偏偏差的比比值Cp = USL-LSL 6sPPp = USL-LSL 6sTCpisused whenthe processisinastateofstatisticalcontrolasdefined by standard contr

22、olchartingmethods.Cp用于統(tǒng)計(jì)計(jì)穩(wěn)定的的過(guò)程,過(guò)程是否否穩(wěn)定可可由統(tǒng)計(jì)計(jì)控制圖圖來(lái)確定定Cpuses thepooledstandarddeviation.Cp使用分組組匯合的的標(biāo)準(zhǔn)偏偏差Ppisused whenthe processisNOTina state of statistical controlasdefined by standard controlchartingmethods.Pp usesthe total standard deviationPp用于統(tǒng)計(jì)計(jì)不穩(wěn)定定的過(guò)程程,它使用整整體的標(biāo)標(biāo)準(zhǔn)偏差差Notindicatingifcenteredhere

23、 .不論中心心在哪Orhere .或這兒Orhere .或這兒Indicators withvariables data:變量數(shù)據(jù)據(jù)的指示示CpPpsPsTPooledstandarddeviation分組匯合合標(biāo)準(zhǔn)偏偏差Takenover arelativelyshorttime period.相對(duì)較短短時(shí)間內(nèi)內(nèi)的數(shù)據(jù)據(jù)Takesinto accountonlythevariationwithin asubset.只計(jì)入小小組內(nèi)的的變差Containsonly commoncausesofvariation.只包含普普通原因因的變差差TotalorOverallstandarddeviati

24、on整體標(biāo)準(zhǔn)準(zhǔn)偏差Takenover sufficientsubsetstoshowthevariationdue toallcommonand specialcausesofvariation.足夠長(zhǎng)時(shí)時(shí)間內(nèi)的的數(shù)據(jù)以展示普普通原因因的變差差和特殊殊原因的的變差Calculated frommanysamples thatrepresent theshiftanddriftthat occursinthepopulationduetoall causesofvariation.從很多樣樣品中計(jì)計(jì)算包含含所有變變差所帶帶來(lái)的總總體漂移移PooledVsOverallStandardDeviat

25、ion分組匯合合標(biāo)準(zhǔn)偏偏差與整整體標(biāo)準(zhǔn)準(zhǔn)偏差A(yù)ctualProcessPerformance:Ratioofthedifferencebetween theprocess averageand thenearest specificationto3timesa measureofthe processvariation實(shí)際過(guò)程程的表現(xiàn)現(xiàn):過(guò)程均值值與最近近的規(guī)范范線(xiàn)的距距離跟3倍的過(guò)程程標(biāo)準(zhǔn)偏偏差之比比值Process CapabilityIndicators過(guò)程能力力指數(shù)CPK = Min CPL, CPUCPL = X-LSL 3sPCPU = USL-X 3sPPPL = X-LSL 3

26、sTPPU = USL-X 3sTPPK = Min PPL, PPUCpkisused whenthe processisinastateofstatisticalcontrolasdefined by standard controlchartingmethods.Cp用于統(tǒng)計(jì)計(jì)穩(wěn)定的的過(guò)程,過(guò)程是否否穩(wěn)定可可由統(tǒng)計(jì)計(jì)控制圖圖來(lái)確定定Cpkuses thepooledstandarddeviation.Cpk使用分組組匯合的的標(biāo)準(zhǔn)偏偏差Ppkisused whenthe processisNOTina state of statistical controlasdefined by sta

27、ndard controlchartingmethods.Pp usesthe total standard deviationPpk用于統(tǒng)計(jì)計(jì)不穩(wěn)定定的過(guò)程程,它使用整整體的標(biāo)標(biāo)準(zhǔn)偏差差I(lǐng)ndicatesSpread &Distance分布寬度度和距離離的指示示W(wǎng)idth& positionareindicated here指示了分分布寬度度和位置andhere.andhere.CpkPpk計(jì)算Cp/ CpkCp=LSL140USL260200Cp=LSL140USL260200Cp=LSL140USL260200Cp=LSL140USL260200215sst= 20sst= 20sst=

28、 10sst= 10Cpk=Cpk=Cpk=Cpk=CpPPCPKPPKCapability能力Overall StdDevNotinControl長(zhǎng)期整體標(biāo)準(zhǔn)準(zhǔn)偏差統(tǒng)計(jì)不受受控PooledStd DevInControl短期分組匯合合標(biāo)準(zhǔn)偏偏差統(tǒng)計(jì)受控控Performance表現(xiàn)CPrepresents“entitlement”!Cp代表改進(jìn)進(jìn)的目標(biāo)Relates std.deviationtotolerance考察標(biāo)準(zhǔn)準(zhǔn)偏差與與公差寬寬度Relates mean&std. deviationtospec.考察均值/標(biāo)準(zhǔn)偏差差和工程程規(guī)范ShortTermLong Term“Snapshot

29、” Look照相“Video” Look錄像Sample3Sample4Sample5 s4 s3 s2 s1s5Total sT Ina perfectstateofcontrolstotal= spPooled Standard Deviation分組匯合標(biāo)準(zhǔn)差“Average” Standard Deviation平均的標(biāo)準(zhǔn)偏差A(yù)ssuming equal subgroup sizeSample2Sample1 MondaySubgroup1子組1TuesdayFromMondaySubgroup2子組2WednesdaySubgroup3子組3FromMonday&TuesdayThu

30、rsdaySubgroup4子組4FromMon, Tues,WedThis distributionismade up of manysmallertime periods.這一分布布是由一一段時(shí)間間內(nèi)許多多小分布布組成Long TermDistributionSubgroups1-4子組1-4MondayTuesdayWednesdayThursdayOVER TIME時(shí)間一長(zhǎng)長(zhǎng)SHIFTHAPPENS漂移發(fā)生生了Whichstandarddeviationshould we use?應(yīng)該選用用哪種標(biāo)標(biāo)準(zhǔn)偏差差?Itdepends on whatweare tryingtodo取決于我我們

31、要作作什么Ifwewant to knowthebestourprocessiscapable of,如果我們們想知道道過(guò)程的的最佳狀狀態(tài)如何何theprocessentitlement亦即過(guò)程程的天賦賦能力usesst- theshortterm processcapability使用短期期標(biāo)準(zhǔn)差差計(jì)算短短期過(guò)程程能力Ifwewant to knowwhatourcustomerssee,如果想了了解客戶(hù)戶(hù)是怎么么看的ouractual performance,我們的表表現(xiàn)useslt- thelong termprocesscapability使用長(zhǎng)期期標(biāo)準(zhǔn)差差計(jì)算長(zhǎng)長(zhǎng)期過(guò)程程能力Ifwe

32、want to knowouropportunityforimprovementUSEBOTH!如果我們們想知道道改進(jìn)的的機(jī)會(huì),兩者都用用0Thedifferencebetween sstandsltrepresentsTheAreaforProcessImprovement短期標(biāo)準(zhǔn)準(zhǔn)差與長(zhǎng)長(zhǎng)期標(biāo)準(zhǔn)準(zhǔn)差的差差距代表表改進(jìn)的的空間TheFocusofMAIC改進(jìn)的關(guān)關(guān)注點(diǎn)Another wayofthinkingaboutit另一中思思考角度度0YB8207批號(hào)Y4092BY4092BY4092BY4092BY4092BY4092BY4092BY4092BY4092BY4095BY4095BY4

33、095BY4095BY4095BY4095BY4095BY4095BY4095BY4095BY4095BMFI13.0612.9512.3712.6013.1913.3312.7612.9712.7113.1912.2013.7012.0013.1612.2712.5712.7213.0213.0212.55批號(hào)Y4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4167BY4228BY4228BY4228BY4228BY4228BMFI12.8412.7413.0012.6012.6212.6413.0312.7413.1

34、012.6012.6012.5612.8612.6014.0013.40批號(hào)Y4240BY4240BY4240BY4240BY4240BY4240BY4240BY4240BY4241BY4241BY4241BY4241BY4241BY4241BMFI12.5912.6011.9311.9712.9012.7512.3712.7013.0412.6012.9013.7012.4913.18Melt FlowIndexofPlastics某塑膠材材料的流流動(dòng)性數(shù)數(shù)據(jù)子組大小小不同時(shí)時(shí)15.0014.5014.0013.5013.0012.5012.0011.50MFI15.0014.5014.00

35、13.5013.0012.5012.0011.50MFIAverage:12.8015.0014.5014.0013.5013.0012.5012.0011.50MFI第一張圖圖顯示的的是所有有的原始始數(shù)據(jù)（按批號(hào)號(hào)分開(kāi)）假設(shè)每每一組都都為正態(tài)態(tài)分布?！暗诙垙垐D顯示示的是各各批的變變差情況況，其“平平均”的的標(biāo)準(zhǔn)偏偏差為0.400.計(jì)計(jì)算方法法如下Sst=g(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)250-6=0.400其中:g為為數(shù)據(jù)據(jù)組的個(gè)個(gè)數(shù)(批);n 為測(cè)測(cè)量值的的個(gè)數(shù)第三張圖圖顯示的的是將50個(gè)個(gè)數(shù)據(jù)據(jù)全部合合并一起起，計(jì)算算總的標(biāo)標(biāo)準(zhǔn)偏差差為（長(zhǎng)期標(biāo)標(biāo)準(zhǔn)偏差差):Slt=0.418j =1(n-g)Discussion question:What information is provided by thedifference in ZstandZlt?實(shí)例計(jì)算算練習(xí)假定規(guī)范范為11.014.0Thesample meanofthe 50 observationsisy=_.Theoverallsamplestan