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Modern Acceptance Sampling 
A tutorial on how to develop sampling plans that detect and prevent the use of unacceptable materials and services.
By Stan Hilliard
© 1999 H & H Servicco Corp.
Outline PAGE TUTORIAL CONTENTS 2 Abstract 2 Using Process Knowledge and Market Requirements 3 Applying OC Curves
1. To Evaluate Sampling Plans
2. To Compare Sampling Plans
3. To Design/Develop Sampling Plans
4. Guidelines for Application
5. For Matching various types of sampling plan
a. variables & attributes
b. fixedn, & sequential
c. standard deviation known & unknown13 Relationship of Acceptance Sampling to Control Charts
Relationship of TwoPoint Plans to MilStandard Plans15 Conclusion: Review of Application Considerations 16 Appendix
1. A Nomograph for Determining Variables Plans
2. Sequential Acceptance Sampling Plans
3. References20 Software Programs Links to examples of sampling plans
InPerson Presentations of this Tutorial Date Location Event 03/05/96 Minneapolis, MN 43^{rd} Annual Minnesota Quality Conference (ASQC) 10/30/96 Madison, WI 13^{th} Annual Maintenance & Engineering Exposition (ASQC,IMI,SME)
Modern acceptance sampling involves a system of principles and methods. Their purpose is to develop decision rules to accept or reject product based on sample data. Factors are:
 The quality requirements of the product in the marketplace
 The capability of the process
 The cost and logistics of sample taking
This article uses the TwoPoint method of sampling plan design, whereby the person designing the plan specifies two points on the operating characteristic curve. The following important visualization tools of this method will be explained:
 Operating characteristic curves (OCCurves)
 Decision rules
It will show you how you can benefit by matching different types of sampling plan to the same occurve: attributes to variables, fixedn to sequential, unknown to known .
You can avoid the horrors of the product recall experience by applying the following principles. They involve methods of product acceptance and process control.
 Rational Lots
 Define homogeneous lots of product items. Use your knowledge of the patterns of the process to group the items into "natural lots". Lots based on purchase order alone must be subdivided into such natural lots for accurate acceptance decisions.
 Known Standard Deviation
 Apply known plans for variables where feasible. These plans use knowledge of the producer's process to reduce inspection cost. [ (sigma) = standard deviation].
 Process Performance and Capability
 Set an acceptable quality level (AQL) that "recognizes" the current process capability. This requires that you know the attainable levels of the process.
 Market Requirement
 Set a rejectable quality level (RQL) that enforces the customer requirement.
 At rejection, you will know that:
 The lot is unacceptable, and
 The process has deviated from its known capability
(Therefore start corrective action on the process.)
Figure 1. OCCurve. (Operating Characteristic)
The Operating Characteristic (OC) curve shows the probability of acceptance, Pa, for any level of lot quality. See Figure 1. On the horizontal axis is the quality characteristic.
This occurve enables you to evaluate the probability of acceptance for any true lot quality levelon a whatif basis. This way, you can design sampling plans that perform the way you want.
 Interpret the curve according to this example:
 If the lot quality is 0.093 fraction defective, then the probability of acceptance, Pa, is 0.05.
 If the lot quality is 0.018 fraction defective, then the probability of acceptance, Pa, is 0.95.
Figure 2. Comparing Alternative Plans, A and B
You can use occurves to compare alternative plans. See Figure 2. Choose between the plans by their relative ability to detect rejectable lots. You should expect that the steeper the curve, the larger the sample size.
Complete this picture by comparing the costs of the sampling to the resulting performance.
Use program TP414 or the Variables Nomograph
Decision Rule of Current Known Sampling Plan
Sample Size: n = 10
Decision Limit: K =1.80Your Task
Complete the occurve in the table below.
Use the program TP414 or the Variables Nomograph in the Appendix
Fraction Nonconforming p' Probability of Acceptance Pa 0.95 0.70 0.50 0.30 0.05 Useful definitions: K = (XbarLISL) / sigma
and: K = 3 * C_{pk}Note: If you use the nomograph, use vertical nlines for known sigma plans. (See scale at bottom of nomograph.)
A NOMOGRAPH FOR DETERMINING VARIABLES SAMPLING PLANS
See: "Nomograph for Determining Variables Sampling Plans", Lloyd S. Nelson,Journal of Quality Technology, Vol. 13, No. 4, October 1981. (Consult the references below)
The TwoPoint method for developing acceptance sampling plans requires that you specify two points of the operating characteristic curve (occurve).
Producer's Point
Figure 3. The TwoPoint MethodThe producer's point controls the acceptance of lots that are at an acceptable quality level. (See figure 3) The goal: prevent good lots from being rejected.
Consumer's Point
The consumer's point controls the rejection of lots that are at a rejectable quality level. (See figure 3) The goal: prevent bad lots from being accepted.
Decision Table Defines the Two Points
Figure 4. Decision TableAn OC Curve helps you to address the producer's and consumer's points in a visual, right brain way. On the other hand, the decision table (See figure 4) defines the situation in a logical, left brain way. The Type I and Type II errors correspond logically to the two points on the occurve.
Type I error  Wrongful Rejection
A type I error is associated with the producer's point  to reject when the true value of the quality characteristic is AQL. The risk of rejecting an AQL lot is the producer's risk (alpha risk)
Type II error  Wrongful acceptance
A Type II error is to accept when the true value of the quality characteristic is RQL  at the consumer's point. The risk of accepting a lot, if it is an RQL lot, is the consumer's risk (ß = beta risk).
Summary of The TwoPoint Method:
Choose alpha=0.05, beta=0.05, AQL, and RQL. These determine the sample size (n) and the decision limit(s). Then apply any process knowledge (withinlot standard deviation) and market requirements (ISLs).
WORKSHEET #2:
Develop a Fixedn Sampling Plan(Use of the program TP414 or the Variables Nomograph )
Determine the variables fixedn sampling plan.
Product Requirement: Lower ISL=0
Process Capability: = 1 (known historical withinlot standard deviation)Sampling Requirement
Producers Point: AQL=0.01, alpha=0.05
Consumers Point: RQL=0.10, beta=0.05Your Task: use the program TP414 or the Variables Nomograph to determine:
Sample Size: n =_____
Decision Limit: K =_____Useful definitions: K = (XbarLISL) / sigma
and: K = 3 * C_{pk}Note: If you use the nomograph, use vertical nlines for known sigma plans. (See scale at bottom of nomograph.)
No principle is without exceptions, but the following guidelines are generally useful to get you started in a sampling application.
How to choose the Producer's Point in practice
 You should expect that lots at the producer's point quality level (AQL) will be accepted most of the time. Define AQL accordingly. Take into account historical quality levels.
 Choose the producers risk of rejecting a lot that is of AQL quality. Typical: = 0.05.
How to choose the Consumer's Point in practice
 You should expect that lots at the consumer's point quality level (RQL) will be rejected most of the time. Define RQL accordingly.
 Choose the consumers risk of accepting a lot that is of RQL quality. Typical: = 0.05.
 Classify Quality Characteristics
 You do not have to inspect all quality characteristics to the same sampling requirement. Some characteristics need less inspection than others. Some will not require testing at all, and others might require 100% inspection.
 Resources for TwoPoint Calculation
 The Appendix contains references and nomographs for developing twopoint variables and attribute sampling plans. The bibliography below refers to the TwoPoint software programs TP105, TP414, and TP781 that facilitate these calculations.
 The Advantages of TwoPoint Decision Rules
 The TwoPoint method develops plans based on desired performance of the decision rule. Unlike MilStandard AQL plans, their method of selection does not involve esoteric 'inspection levels' and 'code letters'. The underlying statistics is the same.
 A Philosophical Issues with MilStandard AQL Plans
 The MilStandard AQL plans require you only to choose the producers pointnot the consumer's. Actually, the consumer's point is usually more important. As a producer your main purpose is to prevent acceptance of rejectable or recallable quality lots.
 A Technical Issue with Lot Size
 Many sampling plan techniques do not use lot size to calculate the occurve. This includes MilStandard 105 and 414, and the software programs TP105 and TP414.. This practice has negligible effect when sampling large lots with small samples. A rule of thumb for accuracy of the occurve calculation is: N>10*n. (n=sample size, N=lot size)
An important ability is to match sampling plans by their occurves. Two matched plans have the same operating characteristic curve, but different decision rules. You can safely choose between matched plans for economy, knowing they offer equal protection. The following table shows useful matches.
Plan A Plan B Attribute Variables Fixedn Double, Multiple, Sequential Variables  Known standard deviation Variables  Unknown standard deviation The following series of sampling plan examples shows how various types of plan are matched to the same occurve. The occurve of this example is characterized by the two points: (AQL=0.01, =0.05) and (RQL=0.10, =0.05). For convenience we will refer to the curve as "OCCurve X".
Attribute Fixedn Plan
(Matched to OCCurve X)Decision rule produced by: TP105 for Attributes
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)n = 43 (sample size)
C = 1 (acceptance number)Variables Fixedn Plan, Unknown Sigma
(Matched to OCCurve X)Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05), (RQL=0.10, =0.05)n = 27 (sample size)
K=1.80Variables Fixedn Plan, Known Sigma = 2.7
(Matched to OCCurve X)Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)n = 10
A= 4.94Attribute Sequential Sampling Plan
(Matched to OCCurve X)Sequential Probability Ratio method (SPR)
Decision rule produced by: TP105 for Attributes
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)
Sequential Attributes n Decision From /To (Ac) (Re) 1
1
*
**
2
19
*
2
20
30
*
3
31
51
0
3
52
52
1
3
53
53
2
3
Variables Sequential Sampling Plan (SPR)
(Matched to OCCurve X)Known Sigma=2.7
SPR = Sequential Probability Ratio.Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)
SPR Sequential (n) (Ac) (Re) 1 ### 12.55 2 1.13 8.74 3 2.40 7.48 4 3.04 6.84 5 3.42 6.46 6 3.67 6.21 7 3.85 6.03 8 3.99 5.89 9 4.09 5.78 10 4.18 5.70 11 4.25 5.63 12 4.31 5.57 13 4.36 5.52 Variables Sequential Sampling Plan (TSS)
(Matched to OCCurve X)Known =2.7
TSS = Truncatable Single SampleDecision rule produced by: TP414 for variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)
TSS Sequential (n) (Ac) (Re) 1 ### 11.87 2 0.38 9.50 3 1.52 8.36 4 2.25 7.63 5 2.79 7.09 6 3.24 6.64 7 3.62 6.26 8 3.98 5.89 9 4.35 5.53 10 4.94 4.94 Variables Sequential Sampling Plan, unknown Sigma
(Matched to OCCurve X)TSS = Truncatable Single Sample
Decision rule produced by: TP414 for Variables
(AQL=0.01, =0.05) and (RQL=0.10, =0.05)
Variables
TSS Sequential
Unknown Sigma(n) K (Re) K (Ac)
2 3 3.83 4 1.03 5 0.20 6 0.21 7 0.46 8 0.63 8.94 9 0.76 6.86 10 0.87 5.66 11 0.96 4.88 12 1.03 4.33 13 1.10 3.92 14 1.16 3.61 15 1.21 3.36 16 1.26 3.15 17 1.30 2.98 18 1.35 2.83 19 1.39 2.69 20 1.43 2.58 21 1.47 2.47 22 1.51 2.37 23 1.55 2.27 24 1.59 2.18 25 1.64 2.08 26 1.70 1.96 27 1.80 1.80 Translated Sampling Plans
Translated sampling plans have identical decision rules, but differently defined quality characteristics.
Plan A Plan B Variables: ISL & Percent Nonconforming Variables: Mean Exponential: MTBF Exponential: Reliable Life, & Reliability
Shewhart Control Charts are not Acceptance Plans
Shewhart Xbar and R charts analyze processes that involve a series of lots produced over time. They concern the relationship of the subgroups to each other, and not to any externally imposed specification. Use Xbar and R charts to discover the factors that contribute to process variability.
Shewhart charts cannot ensure against accepting poor or recallable lots. Even an incontrol characteristic can have a substantial fraction of nonconformities. Xbar and R charts do not control the consumers risk (ß) of accepting RQL or recallable lots.
Acceptance Control Charts.
Acceptance control charts are acceptance sampling plans that you convert into chart form for implementation. They control the producers point and the consumers point of the occurve. Acceptance charts provide a valid visible means for making acceptance sampling decisions.
Acceptance Sampling for Acceptance Decisions
The best way to make the accept/reject decision  whether a process is outofcontrol or
incontrolis to use both the producer's risk and the consumer's risk. In other words, honor both the process capability and the product specifications.
You can match various types of sampling plan to MilStandard plans. For example, you can match a variables TSS sequential with known to a MilStandard 105E attribute plan. Match them at, say, the Pa=0.05 and Pa=0.95 points.
In instances like this, the twopoint method provides a means to convert to more efficient and/or less costly sampling plans.
Regulatory, Consumer, and Litigation Issues
From the consumer, regulatory, and litigation standpoints, it may be better to know the two probabilities of a decision rule than simply to be able to say that you meet a published standard. Auditors may require that you know the probabilities of the occurve and that the calculations be valid. They do not specify what those probability levels must be.
The Equivalence of Various Types of Sampling plans
The twopoint system for developing sampling plans differs from the "cook book" approach of the Military Standards. But the difference is in how you arrived at a plan,  the statistical calculations are the same. The table below compares sampling plan development methods.
Equivalent Statistical Calculations for Acceptance Sampling Plans Statistical Calculation Development Method Type of Decision Rule Attribute, Binomial and Poisson MilStd105E
ASQC/ANSI Z1.4
ISO28591
ISO8422
TP105 SoftwareFixedn, double, multiple, switching
Fixedn, switching
Fixedn, double, multiple, switching
Sequential
Fixedn, double, multiple, sequential
Variables, Normal, Known , Unknown MilStd414
ASQC/ANSI Z1.9
ISO3951
ISO8423
TP414 softwareFixedn, switching
Fixedn, switching
Fixedn, switching
Sequential
Fixedn, double, multiple, sequentialMTBF Exponential MilStd781
TP781 softwareFixedn, Sequential
Fixedn, double, multiple, sequential
Know the Performance of the Decision Rule
You should use the twopoint method to focus on the performance of the accept/reject decision.
How does the decision rule effect the customer if the lot quality is poor?
(Know the consumer's risk .)How does the decision rule effect the producer if the lot quality is good?
(Know the producer's risk.)Consider Variables Sampling Plans
Consider variables sampling plans based on ISLs. They require smaller n than attribute plans.
Consider Sequential Decision Rules
Sequential sampling is the most statistically efficient type of acceptance sampling plan.
Use Known Plans
Consider the advantages of known versus unknown plans. If you choose known , use a test for variability to check that has not changed.
Choose Between Matched Plans to Minimize Sample Size and Inspection Cost
The following strategies will reduce sample size without increasing the and risks:
 Convert attribute plans to variables plans.
 Convert fixedn plans to sequential plans.
 Convert unknown to known plans.
 Set the Target for the process mean to a region of lower average sample numbers.
For onesided sequential sampling plans for either a lower or upper limit. For plans with both lower and upper limits. Nomograph
See the document "Modern Acceptance Sampling" in the references for a nomograph for designing variables sampling plans.
Sequential Acceptance Sampling Plans
Sequential acceptance sampling plans are the most statistically efficient type of sampling plan. They are used when you do not want to take any more sample items than is necessary, but want to be confident that you have enough data to make a decision. For each decision, they allow the operator to increase the sample size one item at a time, or to form it into groups to match the logistics of the situation.
MilStd105 for attributes contains double and multiple plans that are related to sequential plans. MilStd 414 for variables does not have a comparable scheme. Software programs TP105, TP414, and TP781 develop sequential and fixedn plans for attributes, variables, and reliability.
Two Kinds of Sequential Plan for Variables: SPR and TSS
 SPR Sequential Method
 The SPR "sequential probability ratio" methodology was developed by the Statistical Research Group at Columbia University under the leadership of Abraham Wald. It was originally a Restricted war secret but was declassified in 1945 after world war II. See the bibliography: Abraham Wald, Thomas McWilliams.
 TSS Sequential Method
 The TSS "truncatable single sample" methodology was developed by Stan Hilliard (this author) during the 1970s and 1980s. TSS plans are a computerized numerical method that is implemented through software program TP414. See the bibliography for TP414.
 The sample size of a TSS plan can not exceed the sample size of the fixedn plan whose operating characteristic it matches.
 The occurves of TSS plans match fixedn occurves more closely than SPR.
Comparison of SPR and TSS Sequential Methods
We generally favor TSS plans over SPR plans, though both offer sound statistical risk protection. Simulations have demonstrated this. Our main reason for favoring the TSS method is its sample size limit of fixedn. This property of TSS plans improves their acceptance among operators and inspectors, especially in manufacturing. The software program TP414 implements both TSS and SPR sequential plans.
References
 Acceptance Sampling in Quality Control
 Edward Schilling
 Marcel Dekker, Inc.
 Quality Control and Industrial Statistics
 Acheson J. Duncan
 Richard D. Irwin, Inc.
 Volume 13, How to Use Sequential Statistical Methods
 Thomas P. McWilliams
 American Society for Quality Control
 Sequential Analysis
 Abraham Wald
 New York: John Wiley & Sons
 Nomograph for Determining Variables Sampling Plans
 Lloyd S. Nelson
 Journal of Quality Technology, Vol13, No. 4, October 1981
 A Nomograph of the Cumulative Binomial Distribution
 Harry R. Larson
 Industrial Quality Control, Dec. 1966
 Are Acceptance Sampling and SPC Complementary or Incompatible?
 Sower, Motwani, and Savoie.
 Quality Progress, September 1993
 Product Acceptance Design Journey
 Stan Hilliard, H&H Servicco Corp.
 PO Box 9340
 North St. Paul, MN 551090340
 (612) 7770152 Voice & Fax
 Modern Acceptance Sampling
 Stan Hilliard, H&H Servicco Corp.
 PO Box 9340
 North St. Paul, MN 551090340
 (612) 7770152 Voice & Fax
 Software Programs TP105, TP414, TP781
 H&H Servicco Corp.
 PO Box 9340
 North St. Paul, MN 551090340
 (612) 7770152 Voice & Fax
Software Programs
H & H Servicco Corp.
PO Box 9340
North St. Paul MN 551090340
Phone: (651) 7770152 (Voice & Fax)
Email: service@samplingplans.com
http://www.samplingplans.com
H & H Servicco Corp. provides PC software that designs product acceptance sampling plans, process control plans, reliability test plans, and audit sample plans. These programs run on IBM PC or compatible computers.
The programs are available directly from H & H Servicco Corp.
TP105  TwoPoint Sampling plans for Attributes
Price: $245 (60 day satisfaction guarantee)
Software program TP105 designs and evaluates attribute sampling plans. Both fixedn and sequential sampling plans meet userspecified consumer and producer risks. You input Alpha, Beta, AQL, and RQL, or alternatively, n and Ac. Users evaluate plan performance with OC, ASN, AOQ and ARL curves; evaluate sample data with confidence limits.
TP414  TwoPoint Sampling plans for Variables
Price: $245 (60 day satisfaction guarantee)
Software program TP414 designs and evaluates variables sampling plans. Users control either the lot/process mean or the percent nonconforming to specification(s). You input Alpha, Beta, AQL, and RQL, or alternatively, n and decision limit(s). Includes various types of plan: sequential, fixedn, sigmaknown, and sigmaunknown. Users evaluate performance with OC, ASN, AOQ, and ARL curves.
TP781  TwoPoint Sampling plans for Reliability
Price: $245 (60 day satisfaction guarantee)
Software program TP781 designs test plans for reliability qualification. Users specify acceptable and rejectable levels of either: (1) mean time between failures(MTBF) or, (2) reliability in reaching reliable life. TP781 calculates time/unit or number of units for fixedn and sequential sampling plans. It uses the exponential distribution for OC, ASN, and ARL curves and confidence limits.
ASP  Audit Sample Planner
Price: $245 (60 day satisfaction guarantee)
Software program ASP helps you to plan audits and surveys for attribute (pass/fail) data:
 Planning: You develop a sample size depending on: 1) the margin of error, 2) the expected fraction of items "in error", and 3) the expected reply rate.
 Selection: ASP produces a random sample printed on a data sheet.
 Analysis: A report shows onesided or twosided confidence limits for the fraction and the total "in error" items in a population.
These links will take you to examples of sampling plans:
Example attribute sampling plans:
defectives using the binomial distribution
defects using the poisson distributionExample variables sampling plans for the mean:
Sampling Bulk Liquids, Powders, Pellets
Numerical Color MeasurementExamples variables sampling plans for fraction nonconforming to specification
Fixedn and sequential variables examples
Relationship of software TP414 to MilStd414 and Z1.9Example reliability sampling plans for MTBF and for reliability at reliable life
Strategy for reliability growth testing
Wearout and sample size for MTBF test plans