ECE-580-DOE : Statistical Process Control and Design of Experiments Steve Brainerd 27 Distributions:

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1 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Steve Brainerd 27 Distributions: 1/29/03 Other Distributions Steve Brainerd 1

2 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Steve Brainerd Other Statistical Distribution: Types: Discrete A statistical distribution whose variables can take on only discrete values. Used for data that is always positive like defect counts! There are 18 discrete type distributions! Each has a slightly different shape, property and application. They include: Bernoulli Distribution, Binomial Distribution, Continuous Distribution, Geometric Distribution, Hypergeometric Distribution, Negative Binomial Distribution, Poisson Distribution, Practical applications: Defect analysis, gambling, sampling plans, and radiation counts EXCEL: BINOMDIST(x, n, p, 0); HYPGEOMDIST(x,n,M,N); POISSION(x,λ,TRUE or FALSE) 1/29/03 Other Distributions Steve Brainerd 2

3 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Steve Brainerd Permutations: n objects taken r at a time. The number of arrangements that are possible for n objects taken r at a time in a specific order. In Excel: PERMUT(n,r) for ABC PERMUT(3.3) = 6 Take the letters ABC: ABC, ACB,BAC, BCA,CAB, CBA = 6 permutations 1/29/03 Other Distributions Steve Brainerd 3

4 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Steve Brainerd Combinations: The number of arrangements that are possible for n objects taken r at a time without concern for order. In Excel: COMBIN(n,r) for ABC COMBIN(3.3) =1 Take the letters ABC: ABC= 1 combination 1/29/03 Other Distributions Steve Brainerd 4

5 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Steve Brainerd Compare: Permutations and Combinations: In Excel: PERMUT(n,r); COMBIN(n,r) What are the number of Permutations and Combinations for the five letters ABCDE, 3 letters ABC, and two letters AB? Total n Taken as r Permutations Combinations /29/03 Other Distributions Steve Brainerd 5

6 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Binomial Distribution EXCEL function: BINOMDIST(x, n, p, 0) BINOMDIST(number_s,trials,probability_s,cumulative) x: Number_s is the number of successes in trials. n: Trials is the number of independent trials. p: Probability_s is the probability of success on each trial. 0 or 1: Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, then BINOMDIST returns the cumulative distribution function, which is the probability that there are at most number_s successes; if FALSE, it returns the probability mass function, which is the probability that there are number_s successes. Example The flip of a coin can only result in heads or tails. The probability of the first flip being heads is 0.5, and the probability of exactly 6 of 10 flips being heads is: BINOMDIST(6,10,0.5,FALSE) equals /29/03 Other Distributions Steve Brainerd 6

7 Basic Statistics Binomial Distribution Binomial Distribution The binomial distribution is used when the lot is very large. For large lots, the nonreplacement of the sampled product does not affect the probabilities. The hypergeometric takes into consideration that each sample taken affects the probability associated with the next sample. This is called sampling without replacement. The binomial assumes that the probabilities associated with all samples are equal. This is sometimes referred to as sampling with replacement although the parts are not physically replaced. The binomial is used extensively in the construction of sampling plans. The sampling plans in the Dodge-Romig Sampling Tables were derived from the binomial distribution. The probability of exactly x defective parts in a sample n: The symbol p represents the value of incoming quality expressed as a decimal. (1% =.01, 2% =.02, etc.) 1/29/03 Other Distributions Steve Brainerd 7

8 Basic Statistics Binomial Distribution Example Binomial Distribution The probability of exactly x defective parts in a sample size n: The symbol p represents the value of incoming quality expressed as a decimal. (1% =.01, 2% =.02, etc.) What is probability of wafer Fab Develop check inspector finding 1 bad wafer out of sample of 2 out of a lot of 25 wafers with a known rework rate of 3.5%? N =2; x=1; p = P(X = 1): BINOMDIST(1, 2, 0.035, 0): =0.067 = 6.7% chance of getting a defective wafer! 1/29/03 Other Distributions Steve Brainerd 8

9 Basic Statistics Binomial Distribution Example Binomial Distribution The probability of exactly x defective parts in a sample size n: We have 10 balls in a bowl, 3 of the balls are red and 7 are blue. What is likelihood of picking 2 red balls from a sample of 4? N =4; x=2; p =0.30 P(X = 2): BINOMDIST(2, 4, 0.30, 0): = % chance of getting 2 red balls from a sample of 4 balls! 1/29/03 Other Distributions Steve Brainerd 9

10 Basic Statistics Binomial Distribution Example Coin Tosses The Binomial Distribution Formula shows some interesting facts. For example, the probability to toss EXACTLY 1 heads in 10 tosses is only 0.98%. It is quite difficult to get only 1 heads and 9 tails in 10 tosses. The probability to toss EXACTLY 5 heads in 10 tosses is 24.6%. It is not that usual to get exactly 5 heads in 10 trials, even if the individual chance of heads is 50%! We might have thought that we would get quite often 5 heads and 5 tails in 10 coin tosses. NOT! The chance is even slimmer to get 500 heads and 500 tails in 1000 tosses: 2.52%. The probability to get 5 heads in 5 tosses represents, actually, the probability of 5 heads in a row (3.125%). 1/29/03 Other Distributions Steve Brainerd 10

11 Basic Statistics Binomial Distribution Example Coin Tosses # Heads exactly Tosses Probability % % % % % % % % % % 1/29/03 Other Distributions Steve Brainerd 11

12 Basic Statistics Binomial Distribution Example # Heads in a coin Toss # HEADS in a coin toss (x) P(X) in 20 tosses P(X) in 10 tosses P(X) in 5 tosses P(X = # Heads): BINOMDIST(#heads, # tosses, 0.5, 0): % 0.098% 3.125% % 0.977% % % 4.395% % % % % % % % % % 3.125% % % % % % 4.395% % 0.977% % 0.098% % % % % % % % % % % 1/29/03 Other Distributions Steve Brainerd 12

13 Basic Statistics Binomial Distribution Example Binomial Distribution for Coin Toss: # Heads Probability % % % % % % % 5.000% 0.000% # Heads P(X) in 20 tosses P(X) in 10 tosses P(X) in 5 tosses 1/29/03 Other Distributions Steve Brainerd 13

14 Basic Statistics Normal Distribution Example Coin Toss Example: Excel NORMDIST(x,µ,σ,0) Looks like binomial. Does this make sense? Normal Distribution for Coin Toss example: # Heads Probability % % % % % 5.000% Normal P(X) for 20 tosses(avg = 10;SD = 6.20) Normal P(X) for 10 tosses(avg = 5;SD = 3.32) Normal P(X) for 5 tosses(avg = 2.5;SD = 1.87) 0.000% # Heads 1/29/03 Other Distributions Steve Brainerd 14

15 Basic Statistics Normal Distribution Example Coin Toss Example: Excel NORMDIST(x,µ,σ,0) # HEADS in a coin toss (x) # HEADS in a coin toss (x) # HEADS in a coin toss (x) Normal P(X) for 20 tosses(avg = 10;SD = 6.20) Normal P(X) for 10 tosses(avg = 5;SD = 3.32) Normal P(X) for 5 tosses(avg = 2.5;SD = 1.87) % 3.861% 8.732% % 5.813% % % 7.990% % % % % % % % % % 8.732% % % % % % 7.990% % 5.813% % 3.861% % % % % % % % % % % Average SD /29/03 Other Distributions Steve Brainerd 15

16 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Hypergeometric Distribution EXCEL function: HYPGEOMDIST(x,n,M,N) HYPGEOMDIST(sample_s,number_sample,population_s,number_population) x: Sample_s is the number of successes in the sample. n:number_sample is the size of the sample. M: Population_s is the number of successes in the population. N: Number_population is the population size. HYPGEOMDIST is used in sampling without replacement from a finite population. Example A sampler of chocolates contains 20 pieces. Eight pieces are caramels, and the remaining 12 are nuts. If a person selects 4 pieces at random, the following function returns the probability that exactly 1 piece is a caramel: HYPGEOMDIST(1,4,8,20) equals /29/03 Other Distributions Steve Brainerd 16

17 Hypergeometric Distribution Hypergeometric Application: Gives probability of picking exactly x good units in a sample of n units from a population of N units when there are k bad units in the population. Used in quality control and related applications. Example: Given a lot with 21 good units and four defective. What is the probability that a sample of five will yield not more than one defective? Comments: May be approximated by binomial distribution when n is small related to N. ( i.e. Large population N) HYPGEOMDIST(1,5,4,21) equals 46.78% 1/29/03 Other Distributions Steve Brainerd 17

18 Basic Statistics Hypergeometric Distribution Hypergeometric Distribution The hypergeometric distribution is used to calculate the probability of acceptance of a sampling plan when the lot is relatively small. It can be defined as the true basic probability distribution of attribute data but the calculations could become quite cumbersome for large lot sizes. The probability of exactly x defective parts in a sample n: 1/29/03 Other Distributions Steve Brainerd 18

19 Basic Statistics Hypergeometric Distribution Example For the hypergeometric distribution the following parameters are used: N = the total number of the population. k = the total number of successes (S) in the population (diamonds). Therefore, in the population there are 'k' successes, (S) and (N - k) failures (F). n = the number of random samples selected from the population. x = the number of successes selected in the sample. X = is the hypergeometric random variable. The probability distribution of the hypergeometric random variable X, the number of successes in a random sample of size n selected from the total population N items of which k are labelled success and (N - k) labelled failure is: 1/29/03 Other Distributions Steve Brainerd 19

20 Basic Statistics Hypergeometric Distribution Example 1 A carton contains 24 light bulbs, three of which are defective. What is the probability that, if a sample of six is chosen at random from the carton of bulbs, x will be defective? Solution: 1/29/03 Other Distributions Steve Brainerd 20

21 Basic Statistics Hypergeometric Distribution Example 2 A deck of 52 cards contains 4 aces. What is the probability that I will pick 4 aces if I randomly choose 5 cards from a randomly shuffled deck?? L = lot size = 52; n = sample size = 5; D = defects = aces = 4; L-D = 52 4 = 48 = non defectives in lot = non- ace cards; x = defects (aces) desired in sample = 4 ; n-x = 5-4 = 1 Solution: Pv (4 aces) = x n x = 4 1 = 1 x 48 = 1.85 x e6 D L D L n So you have 1 chance in 185,000 of picking 5 cards from a deck of 52 cards and having 4 of the cards be aces! NOTE: This is a simple example, which gets more complex if the cards are dealt to more than 1 player! 1/29/03 Other Distributions Steve Brainerd 21

22 Basic Statistics Hypergeometric Distribution Example 3 Pinochle Example 4 players and you are the 1st one dealt HYPGEOMDIST(x,n,M,N) Pinochle deck # in deck % Prob Deal Player 1 Cards left N Prob of A K Q J 10 player 1 Player 4 Cards left N Prob of A K Q J 10 player 4 A % HYPGEOMDIST(1,1,8,52) % K % HYPGEOMDIST(1,1,8,48) % Q % HYPGEOMDIST(1,1,8,44) % J % HYPGEOMDIST(1,1,8,40) % % HYPGEOMDIST(1,1,8,36) % Simple case Total 48 Prob of a run 0.02% Assume 8 alwaysv available Prob of a 0.03% Use Hypergeometric Distribution x # sucesses in sample 1 HYPGEOMDIST(x,n,M,N) n Sample 1 card at a time M # sucesses in Population 8 N Size of Population Changes as cards are dealt out 1/29/03 Other Distributions Steve Brainerd 22

23 Distributions ECE-580-DOE : Statistical Process Control and Design of Experiments Poisson Distribution EXCEL function: POISSION(x,λ,TRUE or FALSE) POISSON(x,mean,cumulative) x:x is the number of events. λ = np= Mean is the expected numeric value. 0 or 1: Cumulative is a logical value that determines the form of the probability distribution returned. If cumulative is TRUE, POISSON returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive; if FALSE, it returns the Poisson probability mass function that the number of events occurring will be exactly x. Examples POISSON(2,5,FALSE) equals << We will use this one! POISSON(2,5,TRUE) equals /29/03 Other Distributions Steve Brainerd 23

24 Poisson Distribution Poisson Application: Gives probability of exactly x independent occurrences during a given period of time if events take place independently and at a constant rate. May also represent number of occurrences over constant areas or volumes. Used frequently in quality control, reliability, queuing theory, and so on. Example: Used to represent distribution of number of defects in a piece of material, customer arrivals, insurance claims, incoming telephone calls, alpha particles emitted, and so on. Comments: Frequently used as approximation to binomial distribution. n = sample size p = % defective x = # defects acceptable 1/29/03 Other Distributions Steve Brainerd 24

25 Basic Statistics Poisson Distribution Poisson Distribution The Poisson distribution is used for sampling plans involving the number of defects or defects per unit rather than the number of defective parts. It is also used to approximate the binomial probabilities involving the number of defective parts when the sample (n) is large and p is very small. When n is large and p is small, the Poisson distribution formula may be used to approximate the binomial. Using the Poisson to calculate probabilities associated with various sampling plans is relatively simple because the Poisson tables can be used. The probability of exactly x defects or defective parts in a sample n: The letter e represents the value of the base of the natural logarithm system. It is a constant value (e = ). 1/29/03 Other Distributions Steve Brainerd 25

26 Basic Statistics Poisson Distribution Example Poisson Distribution In EXCEL Can be used for sampling plans and acceptance! Probability % % % % % % % % Poisson Distribution: n = 20 sample; P = % defectives = 0.02, 0.05,0.10, 0.25, 0.50 Poisson P(X) POISSON (x,.4,0) PoissonP(X) POISSON (x,1,0) Poisson P(X) POISSON (x,2,0) PoissonP(X) POISSON (x,5,0) PoissonP(X) POISSON (x,10,0) % 0.000% X # defects acceptable 1/29/03 Other Distributions Steve Brainerd 26

27 Basic Statistics Weibull Distribution Weibull Application: The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in analysis of systems involving a "weakest link." The Weibull distribution is often used to model "time until failure." In this manner, it is applied in actuarial science and in engineering work. It is also an appropriate distribution for describing data corresponding to resonance behavior, such as the variation with energy of the cross section of a nuclear reaction or the variation with velocity of the absorption of radiation in the Mossbauer effect. Example: Life distribution for some capacitors, ball bearings, relays, and so on. Comments: Rayleigh and exponential distribution are special cases. Good website: 1/29/03 Other Distributions Steve Brainerd 27

28 Basic Statistics Weibull Distribution Example Application: The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in analysis of systems involving a "weakest link." 1/29/03 Other Distributions Steve Brainerd 28

29 Basic Statistics Weibull Distribution Example The Weibull distribution EXCEL: 1/29/03 Other Distributions Steve Brainerd 29

30 Basic Statistics Weibull Distribution Example The Weibull distribution EXCEL: 1/29/03 Other Distributions Steve Brainerd 30

31 Weibull Distribution Example Lot Cycletime Analysis Weibull Plots ln(-ln(1-p)) Vs ln(x) 05/20/01 to 06/08/01 Ln(-Ln(1-%fail)) GST 40 DUV Photo Lot Cycletime Weibull Plot X- Theoretical ( 1X =2.88hr) Comets X std Co mets X non-std 99.9% 93.4% 63.2% 30.8% 12.7% 4.8% 1.8% 0.7% 1/29/03 Other Distributions Steve Brainerd 31