A C C E P T A N C E S A M P L I N G
Statistical ThinkingSession 2 of 410:15 to Noon
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Acceptance SamplingNotebook pages 31 to 40
S A M P L I N G C O N C E P T SB Y
J O H N A . C O N T E , P . E .
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The ASQ CQE BoK
IV Product and Process Control C. Acceptance Sampling
1. Sampling Concepts Define, describe, and apply the concepts of producer and
consumer risk and related terms, including operating characteristic (OC) curves, acceptable quality limit (AQL), lot tolerance percent defective (LTPD), average outgoing quality (AOQ), average outgoing quality limit (AOQL), etc. (Analyze)
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The ASQ CQE BoK
IV Product and Process Control C. Acceptance Sampling
2. Sampling standards and plans Interpret and apply ANSI/ASQ Z1.4 and Z1.9 standards
for attributes and variables sampling. Identify and distinguish between single, double, multiple, sequential, and continuous sampling methods.
Identify characteristics of Dodge-Romig sampling tables and when they should be used. (Analyze)
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Acceptance Sampling
Inspection of “Lots” of either incoming raw materials or outgoing product
Decision to accept or reject based on sample All sampling plans have two components The sample size (n) The maximum number of defectives allowed in the sample (c
or Ac) for acceptance of the lot
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Producer’s Risk
The risk of making an incorrect decision by rejecting a “good” lot
For sampling plans in Z1.4 that risk is set at about 5% for a given AQL
AQL is the Acceptable Quality Limit
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Consumer’s Risk
The probability of making an incorrect decision by accepting a “bad” lot
For Dodge-Romig sampling plans that risk is set at about 10% for a given LTPD
LTPD - Lot Tolerance Percent Defective
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Operating Characteristic Curve
First text given to me as a new engineer at Western Electric Company in 1966
“Sampling Inspection Tables” by Harold Dodge and Harry Romig
115 out of 200 pages devoted to OC curves
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The Operating Characteristic Curve
Allows a sampling plan to be almost completely evaluated at a glace
Gives a pictorial view of the probabilities of accepting lots submitted a varying levels of percent non-conforming.
The OC Curve illustrate the risks of acceptance sampling
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The Operating Characteristic Curve
Curves are based either the Binomial or Poisson distribution
Poisson approximation to the binomial was often used as the binomial computations were quite complex before the age of electronic calculators
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Probability of Acceptance (Pa)
Formula or statistical tables
Most Poisson tables are cumulative and provide the probability for c or fewer given np (λ)
∑=
=
−
=cx
x
npx
a xenpP
0 !)(
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Percent defective
np or λ P(Acceptance)=P(finding 3 or fewer defectives in the sample)
.02 40(.02)=.80 0.991
.04 40(.04)=1.6 0.921
.06 40(.06)=2.4 0.779
.08 40(.08)=3.2 0.603
.10 40(.10)=4.0 0.433
.12 40(.12)=4.8 0.294
.14 40(.14)=5.6 0.191
OC Curve with Poisson Probabilities(λ=0.8, c ≤ 3)
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Percent defective np or λ P(Acceptance)=P(finding 3 or fewer defectives in the sample)
.02 40(.02)=.80 0.991
.04 40(.04)=1.6 0.921
.06 40(.06)=2.4 0.779
.08 40(.08)=3.2 0.603
.10 40(.10)=4.0 0.433
.12 40(.12)=4.8 0.294
.14 40(.14)=5.6 0.191
OC Curve with Poisson Probabilities(λ=2.4, 3.2, 4.0, 4.8, 5.6, c ≤ 3)
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Operating Characteristic (OC) CurveSample Size = 40, Acceptance Number = 3
Draw the OC Curve using probabilities
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Acceptable Quality Level (AQL)
From Z1.4 – Acceptance Quality Limit (AQL) “4.2 Definition: The AQL is the quality level that is
the worst tolerable process average average whena continuing series of lots is submitted for acceptance sample.”
“Note: The use of the abbreviation AQL to mean Acceptable Quality level is no longer recommended.”
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Acceptable Quality Level
From Z1.4 - “Although individual lots with quality as bad as the AQL can be accepted with a fairly high probability, the designation of a n AQL does not suggest that this is necessary a desirable quality level. The AQL is a parameter of the sampling scheme …”
The probability of acceptance of a lot with a process average equal to the AQL is normally set at 0.95
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Minitab SoftwareCreate or Compare Sampling Plan
Stat Quality Tools
Acceptance Sampling by Attributes
Default Screen Create based on input of AQL, LTPD, and N Or Compare based on input of n, c, and N Measurement type percent defective Acceptable Quality Level (AQL) 0.05 Rejectable Quality Level (RQL or LTPD) 0.10
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Operating Characteristic (OC) Curve Average Outgoing Quality (AOQ) Curve
Average Total Inspection (ATI) Curve
Sample Size = 40, Acceptance Number = 3
Minitab Example – CompareInput of n, c, and N
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Mintab Example – Create Plan
Acceptance Sampling by Attributes Measurement type: Go/no go Lot quality in percent defective Lot size: 1000 Use binomial distribution to calculate probability of acceptance Acceptable Quality Level (AQL) 1 Producer's Risk (Alpha) 0.05 Rejectable Quality Level (RQL or LTPD) 5 Consumer's Risk (Beta) 0.1 Generated Plan(s) Sample Size 132 Acceptance Number 3 Accept lot if defective items in 132 sampled <= 3; Otherwise reject. Percent Probability Probability Defective Accepting Rejecting AOQ ATI 1 0.956 0.044 0.830 170.4 5 0.099 0.901 0.431 913.9 Average outgoing quality limit (AOQL) = 1.278 at 2.220 percent defective.
Graphs - Acceptance Sampling by Attributes
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Operating Characteristic (OC) Curve Average Outgoing Quality (AOQ) Curve
Average Total Inspection (ATI) Curve
Sample Size = 132, Acceptance Number = 3
Minitab Create (AQL=1%, LTPD=5%)
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AOQ and AOQL
Average Outgoing Quality (AOQ) AOQ dependent on the Incoming Quality Average Outgoing Quality Limit AOQL highest value of AOQ
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AOQ - Average Outgoing Quality
The expected average quality of all products, including all accepted lots, plus all rejected lots that have been sorted 10% and have had all defective units replaced
Will always be less than the submitted quality AOQ = Pa * p ( 1 –n/N ) AOQL is the maximum value of AAOQ
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