Knowing What to Do Knowing How to Do It Getting Better Every Day Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Acceptance Sampling I Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar What you will learn The purpose of Sampling How to draw a statistically valid Sample How to Develop a Sampling Plan How to construct an O-C curve for your sampling plan How to use (and understand) ANSI/ASQ Z1.4 How to use ANSI/ASQ Z1.9 Assessing Inspection Economics Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar What is Sampling Sampling refers to the practice of evaluating (inspecting) a portion -the sample - of a lot – the population – for the purpose of inferring information about the lot. Statistically speaking, the properties of the sample distribution are used to infer the properties of the population (lot) distribution. An accept/reject decision is normally made based on the results of the sample Sampling is an Audit practice Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Why Sample? Economy Less inspection labor Less time Less handling damage Provides check on process control Fewer errors ??? i.e. inspection accuracy Acceptance Sampling Webinar 20101129

What does Sampling not do? Does not provide detailed information of lot quality Does not provide judgment of fitness for use (of rejected items) Does not guarantee elimination of defectives – any AQL permits defectives Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Sampling Caveats Size of sample is more important than percentage of lot Only random samples are statistically valid Access to samples does not guarantee randomness Acceptance sampling can place focus on wrong place Supplier should provide evidence of quality Focus should be on process control Misuse of sampling plans can be costly and misleading. No such thing as a single representative sample Acceptance Sampling Webinar 20101129

Representative Sample? There is no such thing as a single representative sample Why? Draw repeated samples of 5 from a normally distributed population. Record the X-bar (mean) and s (std.dev) for each sample What is the result? Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Distribution of Means The Distribution of Means obeys normal distribution – regardless of distribution of parent population. Acceptance Sampling Webinar 20101129

Standard Error of the Mean Central Limit Theorem The relationship of the standard deviation of sample means to the standard deviation of the population Note: For a uniform distribution, Underestimates error by 25% with n=2, but only by 5% with n=6 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar The Random Sample At any one time, each of the remaining items in the population has an equal chance of being the next item selected One method is to use a table of Random Numbers (handout from Grant & Leavenworth) Enter the table Randomly ( like pin-the-tail-on-the-donkey) Proceed in a predetermined direction – up, down, across Discard numbers which cannot be applied to the sample Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Random Number Table Source: Statistical Quality Control by Grant & Leavenworth Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Stratified Sampling Random samples are selected from a “homogeneous lot”. Often, the parts may not be homogeneous because they were produced on different machines, by different operators, in different plants, etc. With stratified sampling, random samples are drawn from each “group” of processes that are different from other groups. Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Selecting the Sample Wrong way to select sample Judgement: often leads to Bias Convenience Right ways to select sample Randomly Systematically: e.g. every nth unit; risk of bias occurs when selection routine matches a process pattern Acceptance Sampling Webinar 20101129

The O-C Curve Operating Characteristic Curve Ideal O-C Curve Pa Percent Defective Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar The Typical O-C Curve Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Sampling Terms AQL – Acceptable Quality Level: The worst quality level that can be considered acceptable. Acceptance Number: the largest number of defective units permitted in the sample to accept a lot – usually designated as “Ac” or “c” AOQ – Average Outgoing Quality: The expected quality of outgoing product, after sampling, for a given value of percent defective in the incoming product. AOQ = p * Pa Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Sampling Terms (cont.) AOQL – Average Outgoing Quality Level: For a given O-C curve, the maximum value of AOQ. Rejection Number – smallest number of defective units in the sample which will cause the lot to be rejected – usually designated as “Re” Sample Size – number of items in sample – usually designated by “n” Lot Size – number of items in the lot (population) – usually designated by “N” Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Sampling Risks Producers Risk – α: calling the population bad when it is good; also called Type I error Consumers Risk – β: calling the population good when it is bad; also called Type II error Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Sampling Risks (cont) Acceptance Sampling Webinar 20101129

Acceptance Sampling II Acceptance Sampling Webinar 20101129

Constructing the O-C curve We will do the following O-C curves Use Hyper-geometric and Poisson for each of the following N=60, n=6, Ac = 2 N=200, n=20, Ac = 2 N=1000, n=100, Ac = 2 N=1000, n=6, Ac = 2 Let’s do k (Ac, c - # of successes) = 0 first Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Hyper-geometric The number of distinct combination of “n” items taken “r” at a time is Acceptance Sampling Webinar 20101129

Hyper-geometric (cont) = (DCk NqCn-k) / NCn Construct the following Table p D=Np P(k=0) P(k=1) P(k=2) P(k ≤ 2) 0 % 1% 2% 3% etc. A Hyper-geometric calculator can be found at www.stattrek.com Note: The Hyper-geometric distribution applies when the population, N, is small compared to the sample size, however, it can always be used. Sampling is done without replacement. Acceptance Sampling Webinar 20101129

Hypergeometric Calculator Acceptance Sampling Webinar 20101129

Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10 Acceptance Sampling Webinar 20101129

Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10 Acceptance Sampling Webinar 20101129

Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10 P (k=0) = 0.809091 P (k=1) = 0.181818 P (k=2) = 0.009091 ----------------------- P(k≤2) = 1.0 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar From QCI-CQE Primer 2005, pVI-9 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Poisson Construct the following Table, using the Poisson Cumulative Table p np P (k ≤ 2) 0% 1% 2% 3% 4% etc. Compare. When is Poisson a good approximation Use the Poisson when n/N˂0.1 and np ˂5. Acceptance Sampling Webinar 20101129

Poisson Calculator Example: p=0.02, n=10, c=0 X=k, the number of successes in the sample, i.e. “c” Acceptance Sampling Webinar 20101129

Poisson Calculator Example: p=0.02, n=10, c=0 Mean = np Acceptance Sampling Webinar 20101129

Poisson Calculator Example: p=0.02, n=10, c=0 TRUE for cumulative, i.e. Σk; FALSE for probability mass function, i.e.p(x=k) Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar From QCI-CQE Primer 2005, pVI-8 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar From QCI-CQE Primer 2005, pVI-8 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar From QCI-CQE Primer 2005, pVI-9 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar O-C Curve & AOQ Determine the O-C curve. Prepare the following Table using the Poisson distribution p Pa AOQ = p * Pa 0% 1% 2% 3% etc Graph the results: Pa and AOQ vs p. Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar OC Curve & AOQ (2) Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar OC Curve & AOQ (3) Acceptance Sampling Webinar 20101129

Acceptance Sampling III Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Questions What if this AOQ is not adequate? What if you would like to add a 2nd sample when the first sample fails? Example OC curve after 1st Sample: p=0.02, n=30, N=500, c (Ac)=0, Re=2 OC curve after 2nd Sample (of 30 more): p=0.02, n=60, N=500, c (Ac)= 1, Re=2 Acceptance Sampling Webinar 20101129

Hypergeometric Multiple Sampling   N = 500 n = 30 60 p D=Np Nq=N-Np P(k=0) P(k=1) P(k ≤ 1) K 1 0.00 0.01 5 495 0.73 0.53 0.36 0.89 0.02 10 490 0.54 0.28 0.38 0.66 0.03 15 485 0.39 0.14 0.30 0.44 0.04 20 480 0.07 0.21 0.05 25 475 0.20 0.17 0.06 470 0.15 0.08 0.10 35 465 0.11 Acceptance Sampling Webinar 20101129

Hypergeometric Multiple Sampling Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar ANSI/ASQC Z1.4-1993 Mil-Std 105 Sampling for Attributes; 95 page Document Pa’s from 83% to 99% Information necessary: N, AQL, Inspection Level How to Use Code Letters Single, Double, Multiple Plans Switching Rules Obtain: n, Ac, Re, O-C Curves Acceptance Sampling Webinar 20101129

ANSI/ASQC Z1.4-1993 Exercises N=475, AQL = 0.1%, Single Plan, Normal What is Code Letter What is Sample Size, What is Ac, Re Repeat for Tightened Inspection Repeat for Reduced Inspection Note: 0.1% is 1000 ppm Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Z1.4 Code Letters I-Reduced, II-Normal, III-tightened |||| For N=475, Normal, code letter is “H” Acceptance Sampling Webinar 20101129

Z1.4 Single Plan – Normal Insp. Table II-A n=125, New code Letter “K” Acceptance Sampling Webinar 20101129

Z1.4 O-C Curve for Code Letter “K” Table X-K Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Z1.4 Switching Rules Acceptance Sampling Webinar 20101129

What happens when AQL = . 1% isn’t good enough ANSI/ASQC Z1.4-1993 What happens when AQL = . 1% isn’t good enough AQL = 0.1% => 1000 ppm Is Z1.4 Adequate? How would you decide? If not, what would you do? Construct O-C curve for n=1000, c=0 (Poisson). Use 100ppm < p < 5000 ppm (see slides 38 & 39) Acceptance Sampling Webinar 20101129

ANSI/ASQC Z1.9-1993 Mil-Std 414 Sampling for Variables; 110 page Document Four Sections in the document Section A: General description of Plans Section B: Plans used when variability is unknown (Std. deviation method is used) Section C: Plans used when variability is unknown (range method is used) Section D: Plans used when the variability is known. Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar ANSI/ASQC Z1.9-1993 Mil-Std 414 Information necessary: N, AQL, Inspection Level How to Use Code Letters Single or Double Limit, Std. Dev or Range Method Plans Switching Rules Obtain: Code Letter, n, Accept/Reject criteria, critical statistic (k) O-C Curves Acceptance Sampling Webinar 20101129

ANSI/ASQC Z1.9-1993 Exercise (From QCI, CQE Primer, pVI-37) The specified max. temp for operation of a device is 209F. A lot of 40 is submitted for inspection. Use Normal (Level II) with AQL = 0.75%. The Std. Dev. is unknown. Use Std. Dev. Method, variation unknown Find Code Letter, Sample Size, k Should lot be accepted or rejected Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Z1.9 Code Letters For N=40, AQL=0.75 |||||| Use AQL=1.0 & Code Letter “D” Acceptance Sampling Webinar 20101129

Z1.9 – Finding Decision Criteria Std. Dev method – Table B-1 For Code Letter “D”, n=5 & AQL=1, k=1.52 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar ANSI/ASQC Z1.9-1993 What is “k” “k” is a critical statistic (term used in hypothesis testing). It defines the maximum area of the distribution which can be above the USL. When Qcalc > k, there is less of distribution above Qcalc than above “k” and lot is accepted. (Compare to “Z” table) Increasing (USL - X-bar) increases Pa Acceptance Sampling Webinar 20101129

ANSI/ASQC Z1.9-1993 Exercise Solution The five reading are 197F, 188F, 184F, 205F, 201F. X-bar (mean) = 195F S (Std. Dev) = 8.8F Qcalc = (USL – X-bar)/s = 1.59 Because Qcalc = 1.59 is greater than k=1.52, lot is accepted Acceptance Sampling Webinar 20101129

Z1.9 – OC Curve for “D” Table A-3 (p9) Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar ANSI/ASQC Z1.9-1993 Another Exercise Same information as before AQL = 0.1 Find Code Letter, n, k Accept or Reject Lot? Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Solution – 2nd Exercise New code letter is “E”, n=7, & k=2.22 The seven reading are 197F, 188F, 184F, 205F, 201F, 193F & 197F. X-bar (mean) = 195F S (std. Dev) = 7.3F Qcalc = (USL – X-bar)/s = 1.91 Because Qcalc = 1.91 is less than k=2.22, lot is rejected Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Inspection Economics Average Total Inspection: The average number of devices inspected per lot by the defined sampling plan ATI = n Pa + N(1- Pa) which assumes each rejected lot is 100% inspected. Average Fraction Inspected: AFI = ATI/N Average Outgoing Quality: AOQ = AQL (1 – AFI) Acceptance Sampling Webinar 20101129

Inspection Economics Exercise (from Grant & Leavenworth, p395) AQL = 0.5%, N=1000 Which sampling plan would have least ATI. n = 100, c = 0 n = 170, c = 1 n = 240, c = 2 Acceptance Sampling Webinar 20101129

Inspection Economics Exercise Solution 1000 n 100 170 240 c 1 2 Pa 0.59 0.8 0.92 n Pa 59 136 220.8 N(1- Pa) 410 200 80 ATI 460 336 300.8 AFI 0.460 0.336 0.301 AOQ 0.0027 0.00332 .00349 Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Inspection Economics Comparison of Cost Alternatives No Inspection NpD 100% Inspection NC Sampling nC + (N-n)pDPa + (N-n)(1-Pa)C D = Cost if defective passes; C = Inspection cost/item Acceptance Sampling Webinar 20101129

Sample Size Break-Even Point Inspection Economics Sample Size Break-Even Point nBE = D/C D = Cost if defective passes; C = Inspection cost/item Acceptance Sampling Webinar 20101129

Acceptance Sampling Webinar Resources American Society for Quality Quality Press www.asq.org ASQ/NC A&T partnership quality courses CQIA, CMI, CQT, CQA, CQMgr, CQE, CSSBB Quality Progress Magazine And others Web-Sites www.stattrek.com – excellent basic stat site http://mathworld.wolfram.com/ - greaqt math and stat site Acceptance Sampling Webinar 20101129