IV. BASIC STATISTICS

Transcription

1 IV. BASI STATISTIS A STATE OF STATISTIAL ONTROL IS NOT A NATURAL STATE FOR A MANUFATURING PROESS. IT IS INSTEAD AN AHIEVEMENT, ARRIVED AT BY ELIMINATING ONE BY ONE, BY DETERMINED EFFORT, THE SPEIAL AUSES OF EXESSIVE VARIATION. W. EDWARDS DEMING QT 2011 IV - 1 QUALITY OUNIL OF INDIANA

2 IV. BASI STATISTIS GENERAL ONEPTS/TERMINOLOGY II.A.1 Intoduction Basic Statistics is pesented in two majo categoy aeas: Geneal oncepts alculations Geneal oncepts is eviewed in the following topic aeas: Teminology Fequency Distibutions Teminology To follow ae a numbe of basic quality and statistical tems. Most of these ae included in the QT. Souces of the definitions ae identified in the efeences at the end of this Pime Section. Paamete The tue population value, often unknown, estimated by a statistic. (Omdahl, 2010) 7 Population All possible obsevations of simila items fom which a sample is dawn. (Omdahl, 2010) 7 Quality The degee to which a set of inheent chaacteistics fulfill equiements. Also, the totality of featues and chaacteistics of a poduct, activity, o system that beas on its ability to satisfy stated o implied needs. (ASQ, 1992) 1, (ISO 8402, 1994) 5 QT 2011 IV - 2 QUALITY OUNIL OF INDIANA

3 IV. BASI STATISTIS GENERAL ONEPTS/TERMINOLOGY II.A.1 Teminology (ontinued) Quality Assuance All those planned and systematic actions necessay to povide adequate confidence that a poduct o sevice will satisfy given quality equiements. (ANSI/ASQ ISO 9000:2005) 1 Quality ontol The opeational techniques and activities that ae used to fulfill equiements fo quality. (ANSI/ASQ ISO 9000:2005) 1 Quality System The oganizational stuctue, esponsibilities, pocedues, pocesses, and esouces fo implementing quality management. (ISO 8402, 1992) 5 Sample A goup of units o obsevations taken fom a lage collection of units o obsevations that seves to povide an infomation basis fo making a decision concening the lage quantity. (Omdahl, 2010) 7 Statistic A numeical data measuement taken fom a sample that may be used to make an infeence about a population. (Omdahl, 2010) 7 Random Vaiable A andom vaiable is any obsevation that can vay. It can epesent eithe discete o continuous data. (Gant, 1988) 4 QT 2011 IV - 3 QUALITY OUNIL OF INDIANA

4 IV. BASI STATISTIS GENERAL ONEPTS/FREQUENY DISTRIBUTIONS II.A.2 ontinuous Fequency Distibutions A continuous distibution contains infinite (vaiable) data points that may be displayed on a continuous measuement scale. Examples: include nomal, exponential and Weibull distibutions. The exponential and Weibull distibutions will not be on the QT exam. They ae widely used in eliability aeas. Only statisticians wok with the nomal distibution fomula. Most pofessionals use Z values to detemine failue ates. A discussion of Z values occus late in this Pime Section. NORMAL (GAUSSIAN) EXPONENTIAL WEIBULL SHAPE =1/2 =1 µ µo =3 FORMULAS P x = μ = Mean 1 e 2 x x - 1 Px = e o P x = e -x - 1 Px = x - e η = Scale paamete -x - σ = Standad deviation μ = θ = Mean β = Shape paamete e = X = X axis eading γ = Location paamete λ = failue ate APPLIATIONS Numeous applications. Useful when it is equally likely that eadings will fall above o below the aveage. Descibes constant failue ate conditions. Applies fo the useful life cycle of many poducts. Often, time(t) is used fo X. Used fo many eliability applications. an test fo the end of the infant motality peiod. an also descibe the nomal and exponential distibutions. Table 4.1 A ompaison of ontinuous Distibutions QT 2011 IV - 4 QUALITY OUNIL OF INDIANA

5 IV. BASI STATISTIS GENERAL ONEPTS/FREQUENY DISTRIBUTIONS II.A.2 Discete Fequency Distibutions A discete distibution esults fom countable (attibute) data that has a finite numbe of possible values. Examples include binomial, Poisson, and hypegeometic distibutions. POISSON BINOMIAL HYPERGEOMETRI SHAPE p=0.1 n=30 p=0.3 p=0.5 p=0.1 p=0.3 n=30 p=0.5 p=0.1 N=60 n=30 p=0.3 p=0.5 FORMULAS P = n = Sample size -np np e! P = n! n - p q! n -! N - d d n - P = N n n = Sample size n = Sample size = Numbe of occuences = Numbe of occuences = Numbe of occuences p = Pobability np = μ = Aveage p = Pobability q = 1 - p d = Occuences in population N = Population size APPLIATIONS The Poisson is used as a distibution fo defect counts and can be used as an appoximation to the binomial. Fo np < 5 the binomial is bette appoximated by the Poisson than the Nomal. The binomial is an appoximation to the hypegeometic. Sampling is with eplacement. The sample size is less than 10 % of N (n < 10 % of N). The nomal distibution appoximates the binomial when np $ 5. Used when the numbe of defects (d) is known. Sampling is without eplacement. The population size (N) is fequently small. Applied when the sample (n) is a elatively lage popotion of the population (n > 10 % of N). Table 4.2 A ompaison of Discete Distibutions QT 2011 IV - 5 QUALITY OUNIL OF INDIANA

6 IV. BASI STATISTIS GENERAL ONEPTS/FREQUENY DISTRIBUTIONS II.A.2 Hypegeometic Fequency Distibution The hypegeometic distibution will not be on the QT exam. It pobably should be eviewed fom a compehension standpoint. ANSI/ASQ Z1.4 (2008) 3 fo small populations is based on the hypegeometic distibution. The hypegeometic distibution applies when the population is small compaed to the sample size. Sampling is done without eplacement. The hypegeometic distibution is a complex combination calculation. The numbe of occuences ()* in the sample follows the hypegeometic function: P = d N - d n - N n Whee: N = Population size n = Sample size d = Numbe of occuences in the population N - d = Numbe of non occuences in the population = Numbe of occuences* in the sample * can also equal the numbe of defectives o successes in a sample. The tem X is used instead of in many texts. Example 4.1: Fom a goup of 20 poducts, containing 5 defectives, 10 ae selected at andom. What is the pobability that these 10 contain the 5 defectives? N = 20, n = 10, d = 5, (N-d) = 15 and = 5 n n! P = note that =! n -! ! 15! 5!0! 5!10! 15! 10!10! P = = 20! 5!10! 20! 10!10! Answe P = = 1.63 % QT 2011 IV - 6 QUALITY OUNIL OF INDIANA

7 IV. BASI STATISTIS GENERAL ONEPTS/FREQUENY DISTRIBUTIONS II.A.2 Binomial Fequency Distibution The binomial distibution applies when the population is lage (N > 50) and the sample size is small compaed to the population. Geneally, n is less than 10 % of N. It is most appopiate to use when the popotion defective is equal to o geate than (0.1). n n - n! P = p 1 - p = p 1 - p! n -! n - Whee: n = Sample size Note: 4! = 1x 2 x 3 x 4 = 24 = Numbe of defectives p = Popotion defective Example 4.2: A andom sample of 10 units is taken fom a steady steam of poduct fom a pess. Past expeience has shown 10 % defective pats. Find the pobability of exactly one bad pat. n = 10 = 1 p = 0.1 n 1 - p n - P = p 10! 1 9 P = !9! P = Answe P = = % Solve fo 2 bad pats (answe = %). Solve fo 0 bad pats (answe = %) Note: Thee is a limited binomial pobability table in the Appendix. This table will save consideable calculation time if the sample size is 10 o less. QT 2011 IV - 7 QUALITY OUNIL OF INDIANA

8 IV. BASI STATISTIS GENERAL ONEPTS/FREQUENY DISTRIBUTIONS II.A.2 Binomial Fequency Distibution (ontinued) The binomial distibution aveage and sigma can be obtained fom the following calculations: The binomial aveage = = np The binomial sigma = = np 1 - p Example 4.3: If one tosses an honest coin 100 times, what is expected to be the aveage numbe of heads? What will be the 3 sigma vaiation? n = 100 p = 0.5 Answe: = np = = 50 heads Sigma = np 1 - p = = 25 = 5 Answe: 3s = heads Poisson Fequency Distibution The Poisson is anothe discete distibution that has numeous applications in industy. The Poisson is also an appoximation to the binomial distibution when p is equal to o less than 0.1, and the sample size n is faily lage. P = e! - Whee: = np = the population mean = numbe of defectives e = the base of natual logaithms QT 2011 IV - 8 QUALITY OUNIL OF INDIANA

9 IV. BASI STATISTIS GENERAL ONEPTS/FREQUENY DISTRIBUTIONS II.A.2 Poisson Fequency Distibution (ontinued) Example 4.4: A continuous pocess is unning a 2 % defective ate. What is the pobability that a 100 piece sample will contain exactly 2 defectives? = np = = 2 = e 2 e 2 P = = = 2! 2! 2!e 4 Answe P = = 0.27 = 27 % Solve fo = 0 Answe (13.5 %) Solve fo = 1 Answe 0.27 (27 %) Solve fo = 3 Answe 0.18 (18 %) The vey good news is that Poisson tables exist which allow an easy detemination of pobabilities. To use the Poisson table, a quick calculation of np is needed. In a pevious poblem, np = = 2 Fom the Poisson table in the Appendix: The pobability of stated o fewe defectives equals: np Thus, fo np = 2.0 the pobability of stated defects only equals: np The Poisson distibution aveage and sigma can be obtained fom the following calculations: The Poisson aveage = = np = c * The Poisson sigma = = = np = c * * Fom the attibute c chat. QT 2011 IV - 9 QUALITY OUNIL OF INDIANA

10 IV. BASI STATISTIS ALULATIONS/ENTRAL TENDENY II.B.1 alculations alculations ae pesented in the following topic aeas: Measues of ental Tendency Measues of Dispesion Statistical Infeence onfidence Limits Pobability Measues of ental Tendency Measues of cental tendency epesent diffeent ways of chaacteizing the cental value of a collection of data. Thee of these measues will be addessed hee: mean, mode and median. The Mean (X-ba, X ) The mean is the sum total of all data values divided by the numbe of data points. X Fomula: X = n X is the mean X epesents each numbe means summation n is the sample size Example 4.5: (9 Numbes) Find X : Answe: 6 The aithmetic mean is the most widely used measue of cental tendency. Advantages of using the mean: It is the cente of gavity of the data It uses all data No soting is needed QT 2011 IV - 10 QUALITY OUNIL OF INDIANA

11 IV. BASI STATISTIS ALULATIONS/ENTRAL TENDENY II.B.1 Measues of ental Tendency (ontinued) Disadvantages of using the mean: The Mean (X-ba, X ) (ontinued) Exteme data values may distot the pictue It can be time consuming The mean may not be the actual value of any data points The Mode The mode is the most fequently occuing numbe in a data set. Example 4.6: (9 Numbes) Find the mode: Answe: 5 Note: It is possible fo goups of data to have moe than one mode. Advantages of using the mode: No calculations o soting is necessay It is not influenced by exteme values It is an actual value It can be detected visually in distibution plots Disadvantage of using the mode: The data may not have a mode QT 2011 IV - 11 QUALITY OUNIL OF INDIANA

12 IV. BASI STATISTIS ALULATIONS/ENTRAL TENDENY II.B.1 Measues of ental Tendency (ontinued) The Median (Midpoint) The median is the middle value when the data is aanged in ascending o descending ode. Fo an even set of data, the median is the aveage of the middle two values. Examples 4.7: (10 Numbes) (9 Numbes) Find the median: Answe: 5 fo both examples Advantages of using the median: Povides an idea whee most data ae located Little calculation equied Insensitivity to exteme values Disadvantages of using the median: The data must be soted and aanged Exteme values may be impotant Two medians cannot be aveaged to obtain a combined median The median will have moe vaiation (between samples) than the aveage (X ) Fo a Nomal Distibution Fo a Right Skewed Distibution MODE MEDIAN MEAN Figue 4.3 A ompaison of ental Tendency in Nomal and Skewed Distibutions QT 2011 IV - 12 QUALITY OUNIL OF INDIANA

13 IV. BASI STATISTIS ALULATIONS/MEASURES OF DISPERSION II.B.2 2 X - X - X Population, = Sample, S = Measues of Dispesion Othe than cental tendency, the othe impotant paamete to descibe a set of data is spead o dispesion. Thee main measues of dispesion will be eviewed: ange, vaiance, and standad deviation. Range (R) The ange of a set of data is the diffeence between the lagest and smallest values. Example 4.8: (9 Numbes) Find R: Answe: 6 Vaiance (σ 2, S 2 ) The vaiance, σ 2 o S 2, equal to the sum of the squaed deviations fom the mean, divided by the sample size. The fomulas fo vaiance ae: N n - 1 The vaiance is equal to the standad deviation squaed. Standad Deviation (σ, s) The standad deviation is the squae oot of the vaiance. 2 X - X - X 2 Population, = Sample, S = N n - 1 Note: N is used fo a population, and n - 1 fo a sample (to emove potential bias in elatively small samples - less than 30) oefficient of Vaiation (OV) The coefficient of vaiation equals the standad deviation divided by the mean and is expessed as a pecentage. S OV = 100 o OV = 100 X QT 2011 IV - 13 QUALITY OUNIL OF INDIANA