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

Acceptance Sampling Webinar Acceptance Sampling I

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 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 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 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 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 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 Distribution of Means The Distribution of Means obeys normal distribution – regardless of distribution of parent population.

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

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 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 The O-C Curve Operating Characteristic Curve PaPa Percent Defective Ideal O-C Curve

Acceptance Sampling Webinar The Typical O-C Curve

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 “A c ” 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 * P a

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 “R e ”  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 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 Sampling Risks (cont)

Acceptance Sampling Webinar Acceptance Sampling II

Acceptance Sampling Webinar 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, A c = 2 N=200, n=20, A c = 2 N=1000, n=100, A c = 2 N=1000, n=6, A c = 2 Let’s do k ( A c, c - # of successes ) = 0 first

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

Acceptance Sampling Webinar Hyper-geometric (cont) Construct the following Table p D=NpP(k=0)P(k=1)P(k=2)P(k ≤ 2) 0 % 1% 2% 3% etc. A Hyper-geometric calculator can be found at 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. = ( D C k Nq C n-k ) / N C n

Acceptance Sampling Webinar Hypergeometric Calculator

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

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

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

Acceptance Sampling Webinar

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

Acceptance Sampling Webinar Poisson Construct the following Table, using the Poisson Cumulative Table p npP (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 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 Poisson Calculator Example: p=0.02, n=10, c=0 Mean = np

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

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

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

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

Acceptance Sampling Webinar OC Curve & AOQ (2)

Acceptance Sampling Webinar OC Curve & AOQ (3)

Acceptance Sampling Webinar Acceptance Sampling III

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

Acceptance Sampling Webinar Hypergeometric Multiple Sampling N =500 n =3060 pD=NpNq=N-NpP(k=0) P(k=1)P(k ≤ 1) K

Acceptance Sampling Webinar Hypergeometric Multiple Sampling

Acceptance Sampling Webinar ANSI/ASQC Z Mil-Std 105  Sampling for Attributes; 95 page Document  P a ’s from 83% to 99%  Information necessary: N, AQL, Inspection Level  How to Use  Code Letters  Single, Double, Multiple Plans  Switching Rules  Obtain: n, A c, R e,  O-C Curves

Acceptance Sampling Webinar ANSI/ASQC Z Exercises  N=475, AQL = 0.1%, Single Plan, Normal  What is Code Letter  What is Sample Size,  What is A c, R e  Repeat for Tightened Inspection  Repeat for Reduced Inspection Note: 0.1% is 1000 ppm

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

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

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

Acceptance Sampling Webinar Z1.4 Switching Rules

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

Acceptance Sampling Webinar 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 ANSI/ASQC Z 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 Q calc > k, there is less of distribution above Q calc than above “k” and lot is accepted. (Compare to “Z” table) Increasing (USL - X-bar) increases P a

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

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

Acceptance Sampling Webinar ANSI/ASQC Z Another Exercise  Same information as before  AQL = 0.1  Find Code Letter, n, k  Accept or Reject Lot?

Acceptance Sampling Webinar Solution – 2 nd 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  Q calc = (USL – X-bar)/s = 1.91  Because Q calc = 1.91 is less than k=2.22, lot is rejected

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

Acceptance Sampling Webinar 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 Inspection Economics Exercise Solution N1000 n c012 PaPa n P a N(1- P a ) ATI AFI AOQ

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

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

Acceptance Sampling Webinar Resources  American Society for Quality  Quality Press   ASQ/NC A&T partnership quality courses  CQIA, CMI, CQT, CQA, CQMgr, CQE, CSSBB  Quality Progress Magazine  And others  Web-Sites  – excellent basic stat site  - greaqt math and stat site