西格瑪應(yīng)用PPT課件

1、We are crossing the bridge now .第1頁/共96頁Statistics OverviewDot PlotBox PlotHistogramBar ChartTrend ChartChartsFrequency DistributionTablesCharts & TablesMeanMedianModeLocationRangeStandard DeviationVarianceDispersionSkewnessKurtosisShapeNumerical MeasuresDescriptive StatisticsPoint EstimateInter

2、val EstimateParameter EstimationParametric MethodsNonparametric MethodsHypothesis TestingInferential StatisticsStatistics第2頁/共96頁What will be taught.Tools to be taught Parameter Estimation Hypothesis Testing ANOVAInferential Statisticscomprises those methods concerned with the analysis of a subset o

3、f data leading to predictions or inferences about the entire set of data第3頁/共96頁Learning ObjectiveslWhat is a Probability Distribution?Experiment, Sample Space, EventRandom Variable, Probability Functions (pmf, pdf, cdf)lDiscrete DistributionsBinomial DistributionPoisson DistributionlContinuous Dist

4、ributionsNormal DistributionExponential DistributionlSampling DistributionsZ Distributiont Distributionc2 DistributionF Distribution第4頁/共96頁lAs we progress from description of data towards inference of data, an important concept is the idea of a probability distribution.lTo appreciate the notion of

5、a probability distribution, we need to review various fundamental concepts related to it:Experiment, Sample Space, EventRandom VariableWhat do we mean by inference of data?第5頁/共96頁ExperimentlAn experiment is any activity that generates a set of data, which may be numerical or not numerical.1, 2, .,

6、6(a)Throwing a diceExperiment generates numerical / discrete dataPinsStainsRejectAccept(b)Inspecting for stain marksExperiment generates attribute dataPins(c)Measuring shaft 10.53 mm10.49 mm10.22 mm10.29 mm11.20 mmExperiment generates continuous data第6頁/共96頁Random ExperimentlIf we throw the dice aga

7、in and again, or produce many shafts from the same process, the outcomes will generally be different, and cannot be predicted in advance with total certainty.lAn experiment which can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment

8、.第7頁/共96頁Sample SpacelThe collection of all possible outcomes of an experiment is called its sample space.1, 2, ., 6- Tossing of a dicePass, Fail- Inspecting for stain marksAll possible values- Measuring shaft between 0 and 10mm)EventlAn outcome, or a set of outcomes, from a random experiment is cal

9、led an event, i.e. it is a subset of the sample space.第8頁/共96頁EventlExample 1: Some events from tossing of a dice.Event 1: the outcome is an odd numberEvent 2: the outcome is a number 4 lExample 2: Some events from measuring shaft :Event 1: the outcome is a diameter meanEvent 2: the outcome is a par

10、t failing specs. E2 = x USL E2 = 5, 6 E1 = 1, 3, 5 E1= x m第9頁/共96頁Random VariablelFrom a same experiment, different events can be derived depending on which aspects of the experiment we consider important.lIn many cases, it is useful and convenient to define the aspect of the experiment we are inter

11、ested in by denoting the event of interest with a symbol (usually an uppercase letter), e.g.: Let X be the event “the number of a dice is odd”.Let W be the event “the shaft is within specs.”.第10頁/共96頁Random VariablelWe have defined a function that assigns a real number to an experimental outcome wit

12、hin the sample space of the random experiment.lThis function (X or W in our examples) is called a random variable because: The outcomes of the same event are clearly uncertain and are variable from one outcome to another.Each outcome has an equal chance of being selected.PinsMeasuring shaft X = Part

13、s out of specs.(LSL = 8 mm,USL = 10 mm)0.,7.99998, 7.99999, 8, 8,00001,9.99999, 10, 10.00001, 10.00002, LSLUSL第11頁/共96頁ProbabilitylTo quantify how likely a particular outcome of a random variable can occur, we typically assign a numerical value between 0 and 1 (or 0 to 100%).This numerical value is

14、called the probability of the outcome.lThere are a few ways of interpreting probability. A common way is to interpret probability as a fraction (or proportion) of times the outcome occurs in many repetitions of the same random experiment.This method is the relative frequency approach or frequentist

15、approach to interpreting probability.第12頁/共96頁Probability DistributionlWhen we are able to assign a probability to each possible outcome of a random variable X, the full description of all the probabilities associated with the possible outcomes is called a probability distribution of X.lA probabilit

16、y distribution is typically presented as a curve or plot that has:All the possible outcomes of X on the horizontal axisThe probability of each outcome on the vertical axis第13頁/共96頁Normal DistributionExponential DistributionUniform DistributionBinomial DistributionDiscrete Probability Distributions (

17、Theoretical)Continuous Probability Distributions (Theoretical)第14頁/共96頁Empirical Distributions Created from actual observations. Usually represented as histograms. Empirical distributions, like theoretical distributions, apply to both discrete and continuous distributions.第15頁/共96頁 Three common impo

18、rtant characteristics:Shape- defines nature of distributionCenter- defines central tendency of dataSpread- defines dispersion of data(or Dispersion, or Scale)Properties of DistributionsExponential DistributionUniform Distribution第16頁/共96頁ShapelDescribes how the probabilities of all the possible outc

19、omes are distributed.lCan be described mathematically with an equation called a probability function, e.g:Probability functionLowercase letter represents a specific value of random variable X xexfx22121m f(x) means P(X = x)第17頁/共96頁Probability Functions For a discrete distribution,f(x) called is the

20、 probabilitymass function (pmf), e.g.: For a continuous distribution,f(x) is called the probabilitydensity function (pdf), e.g.: n,0,1,2,xp1pxnxPxnx 0,1tettft00f(t)123 = 4210.5第18頁/共96頁Properties of Distributions The total probability for any distribution sums to 1.In a discrete distribution,probabi

21、lity is representedas height of the bar.In a continuous distribution,probability is representedas area under the curve(pdf), between two points.Binomial DistributionNormal Distribution第19頁/共96頁Properties of DistributionsProbability of An Exact Value Under PDF is Zero! For a continuous random variabl

22、e, the probability of an exact value occurring is theoretically 0 because a line on a pdf has 0 width, implying: In practice, if we obtain a particular value, e.g. 12.57, of a random variable X, how do we interpret the probability of 12.57 happening?It is interpreted as the probability of X assuming

23、 a value within a small interval around 12.57, i.e. 12.565, 12.575.This is obtained by integrating the area under the pdf between 12.565 and 12.575.P(X = x) = 0for a continuousrandom variable第20頁/共96頁Properties of DistributionsExponential DistributionArea of a line is zero!f(9.5) = P(X = 9.5) = 0To

24、get probability of 20.0, integrate area between 19.995 and 20.005, i.e.P(19.995 X 10n) for inspection.lEach part is classified asaccept or reject.Binomial DistributionReject rate = pSample size (n)第28頁/共96頁Binomial ExperimentlAssuming we have a process that is historically known to produce p reject

25、rate.p can be used as the probability of finding a failed unit each time we draw a part from the process for inspection.lLets pull a sample of n partsrandomly from a large population ( 10n) for inspection.lEach part is classified asaccept or reject.Binomial Distribution1. For each trial (drawing a u

26、nit), the probability of success is constant.2. Trials are independent; result of a unit does not influence outcome of next unit3. Each trial results in only two possible outcomes.A binomial experiment!第29頁/共96頁Probability Mass FunctionlIf each binomial experiment (pulling n parts randomly for pass/

27、fail inspection) is repeated several times, do we see the same x defective units all the time?lThe pmf that describes how the x defective units (called successes) are distributed is given as:Binomial Distribution n ,0,1,2,xp1pxnxPxnxProbability of getting x defective units(x successes)Using a sample

28、 size of n units(n trials)Given that the overall defective rate is p(probability of success is p) 第30頁/共96頁ApplicationslThe binomial distribution is extensively used to model results of experiments that generate binary outcomes, e.g. pass/fail, go/nogo, accept/reject, etc.In industrial practice, it

29、is used for data generated from counting of defectives, e.g.:1. Acceptance Sampling2. p-chartBinomial DistributionBinomial Distribution0.000.050.100.150.200.250.30012345678Number of Rejects (X)Probability of Finding X Rejects xnxp1pxnxP第31頁/共96頁Example 1If a process historically gives 10% reject rat

30、e (p = 0.10), what is the chance of finding 0, 1, 2 or 3 defectives within a sample of 20 units (n = 20)?1.Binomial Distribution n ,0,1,2,xp1pxnxPxnx .,0020101100200P1xfor .,0etc101101201P2xfor121第32頁/共96頁Example 1 (contd)2. These probabilities can be obtained from Minitab:Calc Probability Distribut

31、ions BinomialBinomial DistributionP(x)n = 20p = 0.1Specify column containing x defectivesSpecify column to store results第33頁/共96頁Example 1 (contd)3. Create its pmf from Minitab and read off the answers:Binomial DistributionlSpecify column of possible outcomes, x: 0 to 20 defectives.lCompute and stor

32、e results, P(x), as shown previously:Calc Probability Distributions Binomial.lCreate a chart of the pmf:lGraph ChartDisplay data labelsSelect range of x to plotSpecify axis titles第34頁/共96頁Example 1 (contd)Binomial DistributionBinomial Distribution0.1220.2700.2850.1900.0900.0320.0090.0020.0000.000.05

33、0.100.150.200.250.30012345678No. of Defectives (x)Probability of Finding x Defectives n ,0,1,2,xp1pxnxPxnxFrom Excel:From Minitab:What is the probability of getting 2 defectives or less?第35頁/共96頁Example 1 (contd)lFor the 2 previous charts, the x-axis denotes the number of defective units, x.lIf we d

34、ivide each x valueby constant sample size, n,and re-express the x-axisas a proportion defectivep-axis, the probabilitiesdo not change.Binomial Distribution第36頁/共96頁The location, dispersion and shape of a binomial distribution are affected by the sample size, n, and defective rate, p.Parameters of Bi

35、nomial DistributionParameters of the distribution第37頁/共96頁Binomial DistributionNormal Approximation to the BinomialDepending on the values of n and p, the binomial distributions are a family of distributions that can be skewed to the left or right.Under certain conditions (combinations of n and p),

36、the binomial distribution approximately approaches the shape of a normal distribution:For p 0.5,np 5For p far from 0.5 (smaller or larger),np 10第38頁/共96頁Binomial DistributionMean and Variance Although n and p pin down a specific binomial distribution, often the mean and variance of the distribution

37、are used in practical applications such as the p-chart.The mean and variance of a binomial distribution:ornpnpp12ppmpnnpppnnp12第39頁/共96頁40第40頁/共96頁ApplicationslThe Poisson distribution is a useful model for any random phenomenon that occurs on a per unit basis:Per unit areaPer unit volumePer unit ti

38、me, etc.lA typical application is as amodel of number of defectsin a unit of product, e.g.:Number of cracks per 10m rollinspection in production of continuous rolls of sheetmetal.Number of particles per cm2 of partPoisson DistributionDefect rate = Inspection units (n)# defectsper unit第41頁/共96頁Poisso

39、n ProcesslThe Poisson distribution is derived based on a random experiment called a Poisson process.lLets look at the inspectionof 10m roll of sheetmetal again:Each 10m roll represents asubinterval of a continuousroll (interval) of real numbers.In the entire production of thecontinuous roll, defects

40、 occurrandomly and results of aninspected unit does not influenceoutcome of the next unit.Conceptually, the sheetmetal can be partitioned into sub-intervals until each is small enough that:1. There is either 1 defect or none within the subinterval, i.e. probability of more than 1 defect is zero.2. P

41、robability of 1 count in any subinterval is the same.3. Probability of 1 count increases proportionately as subinterval size increases. 4. Outcome in each.subinterval is independent of other subintervals第42頁/共96頁Probability Mass FunctionlIf each Poisson process (pulling an inspection unit randomly t

42、o count defects) is repeated over time, do we see the same x defects per unit all the time?lThe pmf that describes how the x defects (called counts) per unit are distributed is given as: 0,1,2,xx!exPxProbability of getting x defects per inspection unit(x counts)Given that the overall defects per uni

43、t is (defect rate is )第43頁/共96頁Example 2If a process is historically known to give 4.0 defects per unit ( = 4), what is the chance of finding 0, 1, 2 or 3 defects per unit?1. 0,1,2,xx!exPx .0!e0P0,xfor04 .etc1!e1P1,xfor14第44頁/共96頁Example 2 (contd)2. These probabilities can be obtained from Minitab:C

44、alc Probability Distributions Poisson = 4.0P(x)Specify column containing x defectsSpecify column to store results第45頁/共96頁Example 2 (contd)3. Create its pmf from Minitab and read off the answers:Poisson DistributionlSpecify column of possible outcomes, x: 0 to 20 defects.lCompute and store results,

45、P(x), as shown previously:Calc Probability Distributions Binomial.lCreate a chart of the pmf:lGraph ChartDisplay data labelsSelect range of x to plotSpecify axis titles第46頁/共96頁Example 2 (contd)Poisson DistributionFrom Excel:From Minitab:What is the probability of getting 2 defects or less? 0,1,2,xx

46、!exPx第47頁/共96頁The location, dispersion and shape of a Poisson distribution are affected by the mean, .Parameter of the Poisson DistributionParameter of the distribution第48頁/共96頁Poisson DistributionNormal Approximation to the PoissonThe Poisson distributions are generally skewed to the right. For 15,

47、 the Poisson distribution approximately approaches the normal distribution.Poisson Approximation to the BinomialThe binomial distribution can be shown to approach the Poisson distribution in its limiting conditions, i.e.:when p is very small (approaching zero) n is large (approaching infinity)This a

48、llows the Poisson pmf to be used (easier) when a binomial experiment assumes above conditionsnp = = constant第49頁/共96頁Summary of ApproximationsBinomial p 10 p Poisson Normal 第50頁/共96頁Poisson DistributionMean and Variance Although pins down a specific Poisson distribution, often the mean and variance

49、of the distribution are used in practical applications such as the c-chart.The mean and variance of a Poisson distribution:2What happens to the variability as the mean of the Poisson distribution increases?第51頁/共96頁ExercisesA process yields a defective rate of 10%. For a sampling plan of 10 units, d

50、etermine the probability distribution (pmf and cdf).A certain process yields a defect rate of 2.8 dpmo. For a million opportunities inspected, determine the probability distribution (pmf and cdf).第52頁/共96頁53lNormal DistributionlExponential DistributionlWeibull Distribution第53頁/共96頁54Normal Distribut

51、ion第54頁/共96頁The most widely used model for the distribution of continuous random variables.Arises in the study of numerous natural physical phenomena, such as the velocity of molecules, as well as in one of the most important findings, the Central Limit Theorem.第55頁/共96頁Many natural phenomena and ma

52、n-made processes are observed to have normal distributions, or can be closely represented as normally distributed.For example, the length of a machined part is observed to vary about its mean due to:temperature drift, humidity change, vibrations, cutting angle variations, cutting tool wear, bearing

53、wear, rotational speed variations, fixturing variations, raw material changes and contamination level changesIf these sources of variation are small, independent and equally likely to be positive or negative about the mean value, the length will closely approximate a normal distribution.第56頁/共96頁 dx

54、xfxXPxFx)(Cumulative Distribution Function xforexfx22121mNormal DistributionProbability Density FunctionNormal Distributionaa0.5dxexx22121m第57頁/共96頁 First introduced by French mathematician Abraham DeMoivre in 1733. Made famous in 1809 by German mathematician K.F. Gauss when he also developed a norm

55、al distribution independently and used it in his study of astronomy. As a result, it is also known as the Gaussian distribution.Karl Friedrich GausslDuring mid to late nineteenth century, many statisticians believed that it was “normal” for most well-behaved data to follow this curve, hence the “nor

56、mal distribution”.第58頁/共96頁A normal distribution can be completely described by knowing only the:Mean (m)Variance (2)Some Properties of the Normal DistributionDistribution OneDistribution TwoDistribution ThreeWhat is the difference between the 3 normal distributions?X N(m, 2)1Parameters of the distr

57、ibution第59頁/共96頁ANormal(mA,A)BNormal(mB,B)ANormal(mA,A)BNormal(mB,B)ANormal(mA,A)BNormal(mB,B)What is the difference between process A & B for each case?第60頁/共96頁The mean, median and mode all coincide at the same value m. There is perfect symmetry.Some Properties of the Normal Distribution+ - Do

58、es it mean that any data setwhich has mean, median and modeat the same value will automaticallybe a normal distribution?MeanMedianMode2第61頁/共96頁The area under sections of the curve can be used to estimate the probability of a certain “event” occurring:Some Properties of the Normal DistributionPoint

59、of Inflection1+ - 68.27%95.45%99.73%m +/- 3 is often referred to as the width of a normal distribution3第62頁/共96頁Lets compute the cumulative probabilities of the following distributions:Some Properties of the Normal Distribution+ - m = 3.5 = 0.61.8+ - 20.0m = 16.6 = 2.8+ - m = -1.5 = 0.9-2.80.5F(1.8)

60、 = P(X 20.0) = 1 F(20.0)?P(-2.8 X 1) = 1 P(T1) = 1 0.3297(b) P(1T2) = P(T2) P(T1) = 0.5507 0.3297Example 3 (contd)第72頁/共96頁Exponential DistributionExponential and PoissonlThe exponential and Poisson distributions are related:In a Poisson process (recall the 10m roll sheetmetal inspection), we observe the nu