C=0 Sampling Plans Fifth Edition

Zero Acceptance Number Sampling Plans

Fifth Edition

SINGLE-USER LICENSE ONLY, COPYING AND NETWORKING PROHIBITED.

Licensed to Insydex/Angelica Astrain by ASQ (www.asq.org) Order# 1001297427 Downloaded: 10/20/2009

Also available from ASQ Quality Press:

The Handbook of Applied Acceptance Sampling: Plans, Procedures and PrinciplesKenneth S. Stephens

Process Quality Control: Troubleshooting and Interpretation of Data, Fourth EditionEllis R. Ott, Edward G. Schilling, and Dean V. Neubauer

Glossary and Tables for Statistical Quality Control, Fourth EditionASQ Statistics Division

Failure Mode and Effect Analysis: FMEA From Theory to Execution, Second EditionD.H. Stamatis

The Weibull Analysis Handbook, Second EditionBryan Dodson

Statistical Engineering: An Algorithm for Reducing Variation in Manufacturing ProcessesStefan H. Steiner and R. Jock MacKay

Integrating Inspection Management into Your Quality Improvement SystemWilliam D. Mawby

Make Your Destructive, Dynamic, and Attribute Measurement System Work For YouWilliam D. Mawby

Lean Kaizen: A Simplified Approach to Process ImprovementsGeorge Alukal and Anthony Manos

The Certified Quality Inspector HandbookH. Fred Walker, Ahmad Elshennawy, Bhisham C. Gupta, and Mary McShane Vaughn

Root Cause Analysis: Simplified Tools and Techniques, Second EditionBjørn Andersen and Tom Fagerhaug

The Certified Manager of Quality/Organizational Excellence Handbook: Third EditionRussell T. Westcott, editor

To request a complimentary catalog of ASQ Quality Press publications, call 800-248-1946, or visit our Web site at http://www.asq.org/quality-press.



Zero Acceptance Number Sampling Plans

Fifth Edition

Nicholas L. Squeglia

ASQ Quality Press Milwaukee, Wisconsin



American Society for Quality, Quality Press, Milwaukee 53203© 2008 by American Society for QualityAll rights reserved. Published 2008Printed in the United States of America14 13 12 11 10 09 08 5 4 3 2 1

Library of Congress Cataloging-in-Publication Data

Squeglia, Nicholas L. Zero acceptance number sampling plans / Nicholas L. Squeglia.—5th ed. p. cm. ISBN 978-0-87389-739-6 (alk. paper) 1. Acceptance sampling. I. Title. TS156.4.S68 2008 658.5′62—dc22 2008009352

No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

Publisher: William A. TonyAcquisitions Editor: Matt MeinholzProject Editor: Paul O’MaraProduction Administrator: Randall Benson

ASQ Mission: The American Society for Quality advances individual, organizational, and community excellence world-wide through learning, quality improvement, and knowledge exchange.

Attention Bookstores, Wholesalers, Schools, and Corporations: ASQ Quality Press books, videotapes, audiotapes, and software are available at quantity discounts with bulk purchases for business, educational, or instructional use. For information, please contact ASQ Quality Press at 800-248-1946, or write to ASQ Quality Press, P.O. Box 3005, Milwaukee, WI 53201-3005.

To place orders or to request a free copy of the ASQ Quality Press Publications Catalog, including ASQ membership information, call 800-248-1946. Visit our Web site at www.asq.org or http://www.asq.org/quality-press.

Printed in the United States of America

Printed on acid-free paper



This book is dedicated to my wife Joan; my children Vanessa, Nicholas, and Jacqueline; and my grandson Shane.



vii



vii

CONTENTS

List of Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Attribute Sampling Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Nonstatistical Sampling Plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Relationship of c=0 Plans to ANSI Z1.4 Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Estimating Potential Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Why Constant Sample Sizes Are Not Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Use of the c=0 Plans Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Physically Taking the Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Comments of the AOQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Adjustments from MIL-STD-105E/ANSI Z1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Sampling Plan “Switching” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Switching Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Operating Characteristic Curves and Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Small Lot Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33



ix



ix

LiST Of figurES aND TabLES

figure 1 Operating characteristic curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

figure 2 Effects of acceptance numbers on OC curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Table 1 c=0 sampling plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

figure 3 AOQL curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Table 2 AOQLs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

figure 4 Curves for determining AOQL values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Table 3 Inspection results from a large receiving inspection department over a one-month period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Table 4 Receiving inspection figures for one month. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13



OC curves for single sampling plans, acceptance number equal to zero (various sample sizes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Table 7 Small lot size supplement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31



xi



xi

PrEfaCEFor many years, the acceptable quality level (AQL)1 concept was used largely because of the influence of MIL-STD-105 and its revisions. However, in the current business climate of immense worldwide competition and greater demands by customers, more companies are realizing that quality control does not cost—it pays. As a result, the prevailing wisdom has moved toward zero defects, and AQLs are no longer the rule, but the exception; they are simply not compatible with today’s environment. Many companies are striving for zero defects through statistical process control, improved processes, closed loop inspection systems, and other means.

The use of sampling plans with zero acceptance numbers is the norm today. The sampling plans in this book (c=0) actually represent a revision in 1963 of similar plans I developed in 1961. Because of the wide-spread use of MIL-STD-105C in 1961, the only way to depart from this standard was to develop a set of plans that could be compared favorably with the military standard. The c=0 plans were developed and originally implemented in a medium-sized plant that did both military and commercial work. Although the plans were not formally approved, there was no opposition to them.

In 1963, MIL-STD-105D was introduced, and the c=0 plans were updated and revised. This time, the plans were proposed to a large aerospace manufacturer with a staff of resident government quality control representatives. It was necessary to deliver a formal presentation and explain the c=0 plans in great detail. The aerospace manufacturer and the government representatives agreed to accept the plans on a trial basis. While the plans were targeted essentially to the limiting quality (LQ) percentages in the military standard tables, there were departures from these targets in several instances. These special adjustments were neces-sary to maintain the logic of the c=0 plans. These adjustments were highlighted during the presentation to the aerospace manufacturer.

The results of the trial period at this company were excellent. Not only were the savings significant, but there was a significant reduction in assembly problems as well. A check with the company in 1983 (20 years later) revealed the c=0 plans were still being used.

The c=0 sampling plans were presented in a national publication in 1965 (N. L. Squeglia, “Sampling Plans for Zero Defects,” Quality Assurance 4 [August 1965]: 28). The inquiries and interest generated by the article prompted me to write the first edition of this book. Published in 1969, it described the plans in more detail and contained operating characteristic (OC) curves. The continual interest in the c=0 plans and encouragement from Professor N. L. Enrick of Kent State University resulted in the publication of the second edition in 1981.

In 1983, I conducted an informal survey to get some idea of the extent of the savings realized by users of the c=0 plans who had switched from MIL-STD-105D. A few said it was too early to tell, but the majority re-ported a range of savings from 8 percent to 30 percent, with an average of 18 percent. Of course, the extent of savings is based on the lot sizes and index value (associated AQL) used. The larger the lot and index value, the greater the savings. It is not necessary to implement the plans to determine the savings. The savings potential can be evaluated from past data as described later, in the section titled “Estimating Potential Savings.”

While the hypergeometric distribution was used originally to maximize mathematical accuracy, it is my humble opinion that the most important feature of the plans is the philosophy of zero defects.

The c=0 sampling plans are now in wide use throughout the United States and in other countries. In 1983, the c=0 plans became a part of the Department of Defense’s DLAM 8200.2 for use by government Defense Contract Administration Services quality assurance representatives. In 1989, MIL-STD-105E super-seded 105D. This revision placed emphasis on the use of 105E as a guide in developing inspection strategies, and it recognized the limitation of the AQL concept. The sampling plans were not changed. As a result of this

1. ANSI Z1.4 changed “acceptable quality level” to “acceptance quality limit” and provided an explanation. The c=0 plans are not AQL plans. For comparison purposes, they are associated with particular ANSI plans. See Table 1; the numbers are index values.



xii 1

revision, the fourth edition of this book was published in 1994. That edition included important updates and a small-lot sampling table.

In the earliest editions it was necessary to compare the c=0 plans to the MIL-STD plans in order to show the advantages of the c=0 plans. The comparisons provide important information as to their derivation that provided for a smooth transition from the MIL-STD plans to the c=0 plans, as discussed in the section titled “Relationship of c=0 Plans to ANSI Z1.4 Plans.” After years of extensive application by government contrac-tors, commercial manufacturing, and service industries, the c=0 sampling plans presented in this book are now considered stand-alone sampling plans, although the c=0 plans provide equal or greater protection than the MIL-STD plans.

In 2000 the Department of Defense declared MIL-STD-105E obsolete and recommended that the c=0 plans from this book be used in its place. Upon cancellation of MIL-STD-105E, the c=0 plans were authorized for use by the Defense Contract Management Agency/Department of Defense. This correspondence in part reads:

Zero Acceptance Sampling Plans By Nicholas L. Squeglia This book gives a number of zero based sampling plans and their corresponding Operating Characteristic (OC) Curves and values. It is the state of the art in zero based sampling plans. (DCMA Guidebook)

The companies who were using the 105E plans at the time of their obsolescence switched to ANSI Z1.4 (2003), which, ironically, is a virtual copy of the AQL-oriented 105E plans.

In the early 2000s, United Technologies issued the c=0 table to their suppliers as part of their supplier Quality Requirements. The table was reprinted with the permission of ASQ.

In May 2005, I was awarded ASQ’s Shainin Medal at the World Quality Congress in Milwaukee for my c=0 sampling plans. The citation read:

Nicholas L. Squeglia is credited as one of the most significant contributors to the effort for driving Zero Defects by developing a set of zero acceptance number sampling plans (c=0) forcing preventive actions, thus saving millions of dollars in military and commercial applications. There was wide-spread acceptance of the plans because they proved to be practical, simple to use, and economical.

The c=0 plans have been continually gaining in popularity for more than 45 years.

Nicholas L. Squeglia



xii 1

iNTrODuCTiONThe zero acceptance number plans developed by the author were originally designed and used to provide over-all equal or greater consumer protection with less inspection than the corresponding MIL-STD-105 sampling plans. In addition to the economic advantages they offer, these plans are simple to use and administer. Because of these advantages and because greater emphasis is now being placed on zero defects and product liability prevention, these plans have found their place in many commercial industries, although they were originally developed for military products. Comparisons can now be made to ANSI Z1.4 (2003) because it is a virtual copy of the cancelled MIL-STD-105E.

The derivation of these plans is covered in detail. It is important, however, to emphasize that although the derivation involves considerable comparison with MIL-STD-105E/ANSI Z1.4, the c=0 plans are not limited to applications involving industries that are using those plans.

There is no specific sampling plan or procedure that can be considered best suited for all applications. It is impractical to cite all of the applications in which these c=0 plans are used. Some examples are machined, formed, cast, powered metal, plastic, and stamped parts; and electrical, electronic, and mechanical compo-nents. They have found application in receiving inspection, in-process inspection, and final inspection in many industries. Regardless of the product, wherever the potential for lot-by-lot sampling exists, the c=0 plans may be applicable.

Quite often, the basic objective of sampling is overlooked. The primary objective is derived from the question, “Why sample?” Most of us are aware that sampling is employed simply to provide a degree of qual-ity protection against accepting nonconforming material. Further, we know that what we are continually striv-ing for is 100 percent good product. Assuming our inspection capability is 100 percent efficient in detecting nonconformances, the only way to assure 100 percent good product is to inspect everything 100 percent. This, then, is the reason for sampling: We sample because it is impractical in most cases to perform 100 percent inspection. What we are seeking, therefore, are sampling plans that economically provide us with a reason-able amount of protection to ensure 100 percent good quality. There are times when something less than 100 percent good product is considered acceptable; in other words, there are times when we knowingly accept defective product. Such cases, however, should be treated on an exception basis.

This book provides a set of attribute plans for lot-by-lot inspection. The acceptance number in all cases is zero. This means that for some level of protection you select a certain size sample and withhold the lot if the sample contains one or more nonconforming pieces.

The phrase “withhold the lot” is significant in that it does not necessarily mean rejection. Under these plans, the inspector does not automatically reject the lot if one or more nonconformances are found. The inspector only accepts the lot if zero nonconformances are found in the sample. Withholding the lot forces a review and disposition by engineering or management personnel in regard to the extent and seriousness of the nonconformance.

From this point on, we will use the terms “defective” and “defect” to describe nonconformances, re-gardless of whether the defective is fit for use. The term “defective” is commonly used in quality control to describe a part, component, item, or any other unit of product that contains one or more defects. The word “defect” is commonly used to describe a particular nonconforming characteristic on a unit of product. For example, a particular item contains a slot of a certain width and length, and it also contains a hole. Everything about the item conforms to specifications except the diameter of the hole. This nonconforming diameter is a defect, and the item is therefore defective.



2 3

aTTribuTE SamPLiNg PLaNSThe following section provides a brief overview of lot-by-lot attribute plans while relating them to other plans.

Two broad categories of sampling are (1) continuous and (2) lot-by-lot.

ContinuousContinuous sampling is often used when units of product are submitted one at a time. This can apply to assem-bly lines, for example. Product moving along a conveyor can also be thought of as a candidate for continuous sampling. Consider, for example, a functional test. Units are moved along a line, and assembly progresses to completion at the end of the line. Upon completion, we have a functional test. The continuous sampling plan may call for a frequency check of, say, one unit out of five. It basically starts off with 100 percent inspection until a designated consecutive number of accepted items is reached. When that number is reached, the fre-quency check starts and remains in effect until a defective unit is observed. When we find a defective unit on the frequency inspection, we go back to 100 percent inspection until we get the specified number of consecu-tive good items. We then go back to frequency inspection, and so on. MIL-STD-1235C provides continuous sampling plans.

Lot-by-Lot: attributesLot-by-lot sampling involves units of products that are presented in a group, batch, or lot for inspection, as opposed to being presented one at a time.

In these cases, a sample of a specified quantity is drawn and compared with acceptance criteria—for instance, the quantity of defectives allowed in the sample by the sampling plan. ANSI Z1.4 allows defectives in the sample to a great extent. For example, a particular sampling plan may call for a sample size of 125 and an acceptance number of two. If two or fewer defectives are detected in the sample of 125 pieces, the lot is accepted. If more than two defectives are detected in the 125-piece sample, the lot is rejected. ANSI Z1.4, like the c=0 plans, contains attribute sampling plans. Measurements of characteristics are not required.

The characteristics evaluated either conform or do not conform. Go/no go type gages are often used in attribute plans.

Lot-by-Lot: VariablesAnother lot-by-lot sampling procedure involves the analysis of measured characteristics. MIL-STD-414 (ANSI/ASQC Z1.9) is a variables-type sampling plan procedure. Variables sampling compared with attribute sampling essentially involves the inspection of a smaller sample size to obtain the same protection afforded by an attribute plan. The economics of the smaller sample size, however, are quite often offset by the calcu-lating involved and the need for obtaining and recording measurements. Where variables data are required from an inspection operation, however, variables plans should definitely be considered. It should be noted that MIL-STD-414 plans assume normality in all cases, and that if a lot is rejected on the basis of calculation but no defectives are detected in the sample, the inspector is required to use the corresponding MIL-STD/ANSI attribute sampling plans. The c=0 plans can also be used at this point.

NONSTaTiSTiCaL SamPLiNg PLaNSThere are cases where we can virtually assure zero defects although the sample size cannot logically be de-fined in terms of statistical risks. Such sample sizes are generally exceptionally low for the more important characteristics and, therefore, knowledge of the process control features is required.

In order to avoid any confusion in justifying such sample sizes on inspection plans, specific notations should be used to avoid any tie-in with statistical risks. The reason for such a selection should be noted, either directly on the plan or in the quality engineering standards.



2 3

An example might be a stamping operation where just the first and last pieces from a lot are inspected dimensionally. A statistical sample may be taken for visual inspection only.

The higher index values in the c=0 plans are also used where favorable process control has been demon-strated and just an audit is required. Although the statistical risks seem high, the risks from a practical stand-point would be exceptionally low.

rELaTiONShiP Of C=0 PLaNS TO aNSi Z1.4 PLaNSThe ANSI Z1.4 sampling plans are AQL-oriented. Essentially the AQL is a specified percentage that is con-sidered to be good quality. In any sampling plan, an OC curve can be generated to define the risks of accepting lots with varying degrees of percentage defective.

When we use the AQL concept, as in the ANSI Z1.4 plans, we have a high probability of acceptance associated with this AQL percentage. Normally, this is on the order of a 0.90–0.98 probability of acceptance level. The risk of rejecting this AQL percentage is on the order of a 0.10–0.02 probability level. This rejection risk is called the producer’s risk.

The assumption in employing the AQL concept is that some agreement has been reached between the producer and the consumer. And, because sampling is used, the producer must assume the risk of having a lot rejected even though the actual percentage defective in the lot is equal to or less than the specified AQL.

If no prior AQL agreement exists, and sampling is to be performed simply because 100 percent inspection is impractical, then overinspection is usually the result when the ANSI Z1.4 plans are used. Also, when we sample-inspect because 100 percent inspection is impractical, we would expect to inspect a smaller number of pieces on less critical characteristics.

To illustrate this point, consider the following example. Let us assume we are presently using the MIL-STD/ANSI plans. A 1.0 percent AQL is used for major characteristics, and a 4.0 percent AQL is used for mi-nor characteristics, although no consumer/producer AQL agreement exists. Let the lot size equal 1300. From ANSI Z1.4 based on the lot size, we must take a sample size of 125 pieces, with 3 defectives allowed for the 1.0 percent AQL and 10 defectives allowed for the 4.0 percent AQL:

aQL Sample size acceptance number

1.0% 125 3

4.0% 125 10

We see that for a less critical characteristic, the sample size remains the same. The difference is in the acceptance number. That is, for the 1.0 percent AQL we can accept the lot if the sample contains 3 or fewer defectives. For the minor characteristic, we can accept the lot if the sample contains 10 or fewer defectives.

Because the sample size is the same, there is no reduction in pieces inspected on the minor characteristics.

Now, there is another part of the OC curve for a sampling plan that is often overlooked. (A sampling plan consists of a sample size and acceptance criteria.) An OC curve for a sample size of 125 and an acceptance number of 10 is shown in Figure 1. The AQL and producer’s risk, as previously described, are shown. The LQ here is considered poor quality. Several sampling plans can have OC curves passing through the same AQL/producer’s risk point. For each of these plans, however, there will be a different LQ at some constant probability of acceptance level. This probability of acceptance level corresponding to the LQ is usually low, with 0.10 being widely accepted. This probability level is called the consumer’s risk.

The user of the sampling plan, therefore, must select a plan that will provide reasonably good protection against accepting lots with a percent defective not too much greater than the AQL.

With the AQL/producer’s risk point fixed, the closer the LQ gets to the AQL, the larger the sample size and acceptance number become. This is illustrated in Figure 2.



4 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

13.0

12.0

11.0

10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

Prod

ucer

’s r

isk

(0.0

14)

a =

(1

– Pa

)

n =

sam

ple

size

= 1

25c

= a

ccep

tanc

e nu

mbe

r =

10

Probability of acceptance (Pa)

AQL (4.0%)

LQ (12.3%) Con

sum

er’s

ris

kb

= (

0.10

Pa)

Inco

min

g qu

ality

(pe

rcen

t def

ectiv

e) P

´

fig

ure

1 O

pera

ting

char

acte

rist

ic c

urve

.



4 5

0.1 0

0.5

0.9

1.0

105

1

n =

125

c =

10

x

x

n =

18

c =

0

Pa

fig

ure

2 E

ffec

ts o

f ac

cept

ance

num

bers

on

OC

cur

ve.

Not

e: W

here

an

AQ

L is

invo

lved

, the

nar

row

er th

e di

stan

ce b

etw

een

the

AQ

L a

nd th

e L

Q, t

he h

ighe

r th

e sa

mpl

e si

ze a

nd a

ccep

tanc

e nu

mbe

r ar

e. 1

00 p

erce

nt in

spec

tion

will

occ

ur if

the

AQ

L=

LQ

. As

the

targ

et is

the

LQ

in th

e c=

0 pl

ans,

we

sim

ply

incr

ease

the

dist

ance

as

muc

h as

pos

sibl

e. W

ith n

o A

QL

in th

e c=

0 pl

ans,

this

equ

ates

to z

ero

on th

e x-

axis

. 100

per

cent

insp

ectio

n w

ould

take

pla

ce if

the

LQ

=0.



6 7

However, if we are satisfied with the specified LQ of ANSI Z1.4 and are not concerned with the AQL concept, we can get equal or greater protection than the ANSI Z1.4 plan by using corresponding c=0 plans.

Figure 2 shows the effects of acceptance numbers on OC curves. With the acceptance number set to zero, we have greater protection at the LQ level with a sample size of 18, as compared with a sampling plan from ANSI Z1.4 that has a sample size of 125 with an acceptance number of 10.

Now, let us compare a set of c=0 plans from Table 1 with the previous ANSI Z1.4 example used.

aNSi Z1.4

aQL Sample size acceptance number

1.0% 125 3

4.0% 125 10

c=0 plans

associated aQL Sample size acceptance number

1.0% 42 0

4.0% 18 0

The c=0 plans provide essentially equal or greater LQ protection at the 0.10 consumer’s risk level. We also see that less inspection is performed on less important characteristics.

The c=0 plans in Table 1 are “associated” with the AQLs of ANSI Z1.4. In all of these plans, the protec-tion afforded to the consumer is equal to or greater than the ANSI Z1.4 plans. This method of developing the plans provides for simple conversion from the ANSI Z1.4 plans to the c=0 plans. Table 1 labels these associ-ated AQLs as “index values” because, of course, they are not AQLs.

It should be pointed out that the question of rejecting more lots under these plans may arise because of the zero acceptance numbers. Aside from experience, which has shown that in fact considerable savings can be derived from using these plans, consider the following:

If your quality is very bad, acceptance numbers of greater than zero will not be much help.When you allow acceptance numbers of greater than zero in your plan, you are in effect authorizing an inspector to accept parts that may not be usable. The zero acceptance number forces a review of any defectives by qualified personnel so that a proper disposition can take place.If you are striving for zero defects, how can you allow defectives in your sampling plans?

ESTimaTiNg POTENTiaL SaViNgSYou can obtain an estimate of the cost reduction from switching from the ANSI Z1.4 plans to the c=0 plans by examining your past records. Compare these records with the c=0 plans. For example, if ANSI Z1.4 called for a sample size of 125 and an acceptance number of 3, and the records show you inspected 125 pieces and found no defectives, then your comparison is the 125 pieces against the 42 pieces in the c=0 plans. Also, of course, if you found 4 bad pieces in the 125-piece sample, the chances are pretty high that 1 of these 4 was detected in the first 42 inspected. In other words, the lot would have been withheld with fewer pieces inspected.

Some actual inspection results are shown and discussed in the “Background Information” section.

••

•



6 7

Tabl

e 1

c=0

sam

plin

g pl

ans.

inde

x va

lues

(a

ssoc

iate

d a

QL

s)

.010

.015

.025

.040

.065

.10

.15

.25

.40

.65

1.0

1.5

2.5

4.0

6.5

10.0

Lot

siz

eSa

mpl

e si

ze

2–8

**

**

**

**

**

**

53

33

9–15

**

**

**

**

**

13

85

33

3

16–2

5*

**

**

**

**

20

13

85

33

3

26–5

0*

**

**

**

* 3

2 2

0 1

38

77

53

51–9

0*

**

**

* 8

0 5

0 3

2 2

0 1

313

118

54

91–1

50*

**

**

125

80

50

32

20

19

1911

96

5

151–

280

**

**

200

125

80

50

32

29

29

1913

107

6

281–

500

**

*31

520

012

5 8

0 5

0 4

8 4

7 2

921

1611

97

501–

1200

*80

050

031

520

012

5 8

0 7

5 7

3 4

7 3

427

1915

118

1201

–320

012

5080

050

031

520

012

512

011

6 7

3 5

3 4

235

2318

139

3201

–10,

000

1250

800

500

315

200

192

189

116

86

68

50

3829

2215

9

10,0

01–3

5,00

012

5080

050

031

530

029

418

913

510

8 7

7 6

046

3529

159

35,0

01–1

50,0

0012

5080

050

049

047

629

421

817

012

3 9

6 7

456

4029

159

150,

001–

500,

000

1250

800

750

715

476

345

270

200

156

119

90

6440

2915

9

500,

001

and

over

1250

1200

1112

715

556

435

303

244

189

143

102

6440

2915

9

*Ind

icat

es e

ntir

e lo

t mus

t be

insp

ecte

d.

Zer

o A

ccep

tanc

e N

umbe

r Sa

mpl

ing

Pla

ns, F

ifth

Edi

tion

by

Nic

hola

s L

. Squ

eglia

Not

e: T

he a

ccep

tanc

e nu

mbe

r in

all

case

s is

zer

o.

Am

eric

an S

ocie

ty f

or Q

ualit

y



8 9

Why CONSTaNT SamPLE SiZES arE NOT uSEDAs discussed in this book, the c=0 plans were designed essentially to be equal or greater in consumer and aver-age outgoing quality limit (AOQL) protection than the corresponding ANSI Z1.4 plans. The c=0 plans contain other features of the ANSI Z1.4 plans. Within a particular AQL column, the OC curves actually differ for the most part as the lot size increases. The reason for this common feature, in addition to satisfying the statistical relationship, is that it is generally considered more practical to obtain greater protection on larger lot sizes.

The use of constant sample sizes often results in a combination of overinspection and underinspection for broad lot size ranges.

In summary, c=0 plans are used when:

1. Manufactured parts are expected to completely conform to specification requirements2. Less inspection is desired on less critical characteristics3. Sampling is performed because 100 percent inspection in general on all characteristics of all parts is

impractical4. Inspectors, as a general rule, are not allowed to knowingly accept nonconforming products5. Interim inspections are needed until a problem is corrected6. Auditing is required for various reasons, such as:

Audit of stock for assuranceAudit of items for potential transit damage

7. Visual inspection in general is being done

uSE Of ThE C=0 PLaNS TabLEAssume you are presently working with ANSI Z1.4 plans and have established AQL levels.

A particular lot of 700 pieces is to be inspected. Normally you would use, for example, a 1.0 percent AQL in accordance with ANSI Z1.4.

Under the c=0 plans, looking at the left-hand column in Table 1, we find that 700 lies within the lot size range of 501–1200. Reading across, and down from the index value (associated AQL) of 1.0, we find our sample size to be 34.

From this lot of 700 pieces we take a sample size of 34, and if one or more defectives are found in the sample, we withhold the lot.

If you are not presently working with ANSI Z1.4, then someone must designate the type of plans to use in terms of index value. This designation depends on your particular situation and requires product process knowledge. However, once you select the desired index value, the procedure for taking the sample is identical to that previously cited.

PhySiCaLLy TakiNg ThE SamPLEA representative sample is necessary to assure reliable results. One of the most common ways to obtain a representative sample is by random sampling. Randomness is achieved only when each piece and combina-tion of pieces in the lot has an equal chance of being selected for the sample. Such a sample may be drawn in several ways. When material is packaged in an orderly manner, or laid out on a bench in groups or rows, each unit may be numbered from 1 to the total in the lot, and the sample selected by use of a random number table. For bulk-packed material, a bench may be marked in numbered areas on which the units are evenly spread. The units for the sample can be selected by reference to the numbered zone in which they are located using a random number table. Many variations of such approaches have been used with success.

If random sampling is not appropriate, common sense should prevail in obtaining a representative sample. Stratified sampling, for example, should be considered.

——



8 9

COmmENTS ON ThE aOQLThe AOQL is the maximum average outgoing quality from an inspection station for a particular sampling plan. For example, Figure 3 shows an AOQ curve for the sampling plan n=50, c=0. The AOQL is designated. The AOQL values for the c=0 plans are shown in Table 2. All of the AOQLs for the c=0 are virtually the same as or less than those of the corresponding ANSI Z1.4 plans.

aOQL ComparisonsFor informational purposes, Figure 4 shows a comparison of AOQL values for various sample sizes and ac-ceptance numbers. Infinite lot sizes were used in constructing this graph, and, with the limitations of most graphs, the AOQL values read approximate.

baCkgrOuND iNfOrmaTiONThe cancelled MIL-Q-9858A permitted the use of other plans, such as the c=0 sampling plans. Reference was made to the following clause that permitted this authorization.

6.6 Statistical Quality Control and analysisThe contractor may employ sampling inspection in accordance with applicable military standards and sampling plans (e.g., from MIL-STD-105, MIL-STD-414, or Handbooks H 106, 107 and 108). If the contractor uses other sampling plans, they shall be subject to review by the cognizant Gov-ernment Representative. Any sampling plan used shall provide valid confidence and quality levels. (MIL-Q-9858A)

AOQL0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.0

0.73A

vera

ge o

utgo

ing

qual

ity(p

erce

nt)

Incoming quality (percent)

0 1 2 3 4 5 6 7 8

n = 50c = 0Lot size = 10,000

figure 3 AOQL curve.



10 11

Tabl

e 2

AO

QL

s.

inde

x va

lues

(a

ssoc

iate

d a

QL

s)

.010

.015

.025

.040

.065

.10

.15

.25

.40

.65

1.0

1.5

2.5

4.0

6.5

10.0

Lot

siz

ea

OQ

L

(acc

epta

nce

num

ber

in a

ll ca

ses

is z

ero)

2–8

2.80

7.7

7.7

7.7

9–15

0.38

2.10

4.90

9.8

9.8

9.8

16–2

50.

371.

403.

105.

9010

.810

.810

.8

26–5

00.

411.

102.

103.

904.

50 4

.5 6

.611

.5

51–9

00.

050.

330.

741.

402.

402.

402.

90 4

.2 6

.9 8

.8

91–1

500.

050.

210.

490.

901.

601.

701.

703.

10 3

.8 5

.9 7

.1

151–

280

0.05

0.16

0.33

0.60

1.00

1.20

1.20

1.80

2.70

3.5

5.1

6.0

281–

500

0.04

0.11

0.22

0.39

0.66

0.69

0.71

1.20

1.70

2.20

3.3

4.0

5.2

501–

1200

0.02

0.04

0.09

0.15

0.26

0.43

0.46

0.47

0.75

1.10

1.30

1.90

2.4

3.3

4.6

1201

–320

00.

020.

030.

060.

110.

170.

280.

300.

310.

490.

680.

861.

001.

60 2

.0 2

.8 4

.1

3201

–10,

000

0.03

0.04

0.07

0.11

0.18

0.19

0.19

0.31

0.42

0.54

0.73

0.96

1.30

1.7

2.4

4.1

10,0

01–3

5,00

00.

030.

040.

070.

120.

120.

120.

190.

270.

340.

480.

610.

801.

10 1

.3 2

.5 4

.1

35,0

01–1

50,0

000.

030.

050.

070.

070.

080.

120.

170.

220.

300.

380.

500.

660.

92 1

.3 2

.5 4

.1

150,

001–

500,

000

0.03

0.05

0.05

0.05

0.08

0.11

0.14

0.18

0.24

0.31

0.41

0.57

0.92

1.3

2.5

4.1

500,

001

and

over

0.03

0.03

0.03

0.05

0.07

0.08

0.12

0.15

0.19

0.26

0.36

0.57

0.92

1.3

2.5

4.1



10 11

It is interesting to note that there was no restriction or requirement to have “other” sampling plans match any specifics of MIL-STD-105 and its later revisions.

The aerospace standard, AS9100 (presently revision B) has essentially replaced MIL-Q-9858A. This standard goes one step further in that it restricts sampling plans to zero acceptance numbers:

When the organization uses sampling inspection as a means of product acceptance, the plan shall be statistically valid and appropriate for use. The plan will preclude the acceptance of lots with whose samples have known nonconformities. (8.2.4)

This book still provides sufficient information on the validity of these plans by means of the OC curves, AOQL tables, and other information.

Certain anomalies existed in MIL-STD-105 and its revisions that are discussed and compensated for in the c=0 plans. The c=0 plans’ OC curves were calculated on the basis of the hypergeometric distribution. MIL-STD-105E, like the previous editions, utilized the binomial and Poisson distributions to calculate OC curves as an approximation of the hypergeometric distribution.

Some actual Data based on the results of a Large receiving inspection Department using the c=0 PlansIn the 1960s, a lengthy presentation was made to government officials on the c=0 plans. Although previ-ous studies had shown lower appraisal costs and lower assembly failure quality costs, it was requested that

figure 4 Curves for determining AOQL values.

AO

QL

(pe

rcen

t def

ectiv

e)

0

1

2

3

4

5

6

7

8

9

10

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

Sample size

c = 15

c = 14

c = 13

c = 12

c = 11

c = 10

c = 9

c = 8

c = 7

c = 6

c = 5

c = 4

c = 3

c = 2

c = 1

c = 0



12 13

one month’s current actual inspection results be taken from a large receiving inspection and the results compared.

Table 3 shows the results of 1141 lots inspected. Essentially, of the 46 lots rejected, 45 would have been rejected anyway if the MIL-STD-105D plans (whose sampling plans were the same as those in the “E” revi-sion) had been used, because the number of allowable defects was exceeded. (We compared the corresponding sample sizes and acceptance numbers.)

Table 4 shows the number of lots in which the sample size was the same as MIL-STD-105D, greater than 105D, and smaller than 105D. The reason for the “greater than” is explained later in light of the anomalies detected in 105D.

Table 5 continues the analysis in terms of pieces received. Although there were more pieces screened, there were lower total inspection costs.

Table 6 shows the total number of characteristics inspected on the 1,250,762 pieces from the 1141 lots. It is here where the savings become more pronounced.

Since the plans had been in use for some time prior to this study, a significant improvement in purchased material was realized. The c=0 plans, unlike plans with allowable defect numbers, do not allow or imply that producers may submit defective material, which eliminates or minimizes any misinterpretation about AQLs.

aDjuSTmENTS frOm miL-STD-105E/aNSi Z1.4There are general sampling plans in MIL-STD-105E/ANSI Z1.4 that contain apparent anomalies that are compensated for in the c=0 plans. These particular plans do not provide for the smooth progression required for less inspection on less critical characteristics. As such, the c=0 plans contain larger samples in these in-stances. Also, of course, where the acceptance number is zero in the MIL-STD-105E/ANSI Z1.4 plans, the sample size is the same in the c=0 plans.

Take for example code letter “M” in the MIL-STD-105E/ANSI Z1.4 plans. Reading from more critical to less critical AQLs we have the following sampling sizes: M=1250, 800, 500, 315, 200, 500, 315. Actually, both the 200 and following 500 are out of balance, but the 500 can be adjusted mathematically to conform to the logic of the c=0 plans. Thus, for AQLs of 0.010–0.040, the sample sizes are the same in both the c=0 and MIL-STD-105E/ANSI Z1.4 sampling plans. At 0.065 AQL when M=200 occurs, the c=0 sampling plan sample size is 300, which is greater than the MIL-STD-105E/ANSI Z1.4 sample size. In 0.10 and up, the c=0 sample sizes are smaller.

The other anomalies in MIL-STD-105E/ANSI Z1.4 occur at H-0.4 AQL, J-0.25 AQL, K-0.15, L-0.10, N-0.04, P-.025, and Q-.015.

In the previous editions of this book, there were five cases where I made a downward adjustment in sample sizes that allowed for a smooth progression but did not meet the targeted LQs. This information was detailed and provided the user with replacement sample sizes if meeting the targeted LQs was necessary. It was also explained that in order to have a smooth progression, the preceding index sample sizes would have to increase, resulting in them being larger than the corresponding ANSI sample sizes. Considering the relative overall slight changes, I made the changes to meet the LQ targets and increased the preceding index sample sizes in this edition.

SamPLiNg PLaN “SWiTChiNg”Considerable thought was given to tightened and reduced inspection. A table of c=0 plans for tightened in-spection was calculated and prepared, but was never made a part of the c=0 plans.

Actually, different OC curves are involved for tightened, normal, and reduced inspection. There were three reasons for not “switching” in the c=0 plans: Switching would have taken away from the simplicity of the plans; most companies were using the “normal” table, and perhaps correctly so; and switching assumes a continuing series of lots.



12 13

Table 3 Inspection results from a large receiving inspection department over a one-month period.

Number of lots inspected

Number of lots rejected

Number of lots in which number of defectives in c=0 sample exceeded acceptance number in miL-STD-105D

Number of lots in which number of defectives in

c=0 sample did not exceed acceptance number in

miL-STD-105D

1141 46 45 1*

*One defective in c=0, sample size 11. Corresponding MIL-STD-105D c=3, n=50.

Table 4 Receiving inspection figures for one month.

Total number of lots inspected

Number of lots in which c=0 sample size was

The same as miL-STD-105D

greater than miL-STD-105D

Smaller than miL-STD-105D

1141430 30 681

37.7% 2.6% 59.7%


Total number of pieces received Sample plan

Total number of pieces inspected Percent

inspection due to sampling

inspection due to screening Percent

1,250,762c=0 50,032 4.0 30,672 19,360 1.5

MIL-STD-105D

80,436 6.4 62,332 18,104 1.4


Total number of characteristics on 1,250,762

pieces receivedSample

plan

Total number of

characteristics inspected Percent

Number of characteristics inspected due to sampling Percent

Number of characteristics inspected due to screening Percent

24,907,674c=0 304,122 1.2 282,866 1.1 21,256 0.1

MIL-STD-105D

1,013,489 4.1 993,179 4.0 20,310 0.1



14 15

The original intent of going to tightened inspection is questionable in these days of quality programs. Tightened inspection has been viewed as a means to force corrective action by increasing the chance of lot rejections.

On reduced inspection, for all practical purposes, MIL-STD-105E/ANSI Z1.4 plans are getting into non-statistical sampling plans. The main problem is that lots can be accepted that are much worse than the original intended plan.

However, several requests were made for switching guidelines, and as a result, I have included tightened and reduced sampling inspection switching guidelines.

SWiTChiNg guiDELiNESIn general, judgment is often used to determine tighter or reduced inspections when using the c=0 sampling plans. This could be accomplished by a periodic review of inspection results. Some items or characteristics, for example, might be considered so important that although there is no history of rejections, you may want to keep the same sample size. There may be other cases where you may experience just one rejected lot and want to go to 100 percent inspection until the cause of the problem is eliminated and a sufficient number of items, or means used, are taken to verify that the problem was corrected. In other cases, because of verifiable excellent process controls, you may just want to start off with an audit-type sample essentially equivalent to a reduced inspection. Based on your particular situation, including your process/product knowledge and other evidence of control, you can come up with your own guidelines.

If, however, it is desired for any reason to use a tightened and reduced inspection sampling switching scheme, or it is required by contract or a customer representative, then a solution would be to use the next higher or lower index value of the c=0 sampling plans, and use the sampling plans in that particular column.

For example, for switching to tightened, if the present normal index level you are working with is 2.5, then the tightened index level would be 1.5.

For switching from normal to reduced using the same index of 2.5 as an example, the reduced index value would be 4.0.

The switching guidelines compatible with the MIL and ANSI plans are:

1. Normal to tightened when two out of five consecutive lots are not acceptable2. Tightened back to normal when five consecutive lots are accepted3. Normal to reduced when preceding 10 lots are accepted4. Reduced to tightened when a lot is not acceptable



14 15

OPEraTiNg CharaCTEriSTiC CurVES aND VaLuES



17



17

0102030405060708090100

8070

6050

4030

2010

0

63.7

46.9

27.5

12.5

05.

002.

500.

50

46.7

32.5

18.3

8.33

3.33

1.67

0.33

26.0

18.3

10.0

5.00

2.00

1.00

0.20

2 3 5

2

35 Sa

mpl

esi

zeP

roba

bilit

y of

acc

epta

nce

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e2–

8

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



18 19

0102030405060708090100

8070

6050

4030

2010

0

66.0

048

.30

28.3

012

.90

5.00

2.50

0.50

50.0

034

.50

19.2

08.

613.

331.

670.

33

31.8

020

.80

11.3

05.

002.

001.

000.

20

2 3 518

.70

12.1

06.

253.

131.

250.

620.

12 8

8.46

5.77

3.85

1.92

0.76

0.38

0.07

13

23

58

13

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e9–

15

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



18 19

0102030405060708090100

8070

6050

4030

2010

0

67.0

049

.00

28.7

013

.10

5.04

2.50

0.50

51.4

035

.50

19.8

08.

803.

331.

670.

33

33.9

022

.30

11.9

05.

202.

001.

000.

20

2 3 521

.40

13.7

07.

183.

131.

250.

620.

12 8

11.9

07.

543.

851.

920.

760.

380.

0713

6.40

3.75

2.50

1.25

0.50

0.25

0.05

20

23

58

1320

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e16

–25

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



20 21

0102030405060708090100

8070

6050

4030

2010

0

52.5

036

.30

20.2

08.

973.

391.

670.

33

35.4

023

.20

12.4

05.

382.

001.

000.

20

3 523

.20

14.8

07.

723.

311.

250.

620.

12 8

14.2

08.

914.

591.

920.

760.

380.

0713

8.73

5.42

2.82

1.25

0.50

0.25

0.05

20

4.60

2.94

1.56

0.78

0.31

0.15

0.03

32

35

813

20

32

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e26

–50

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



20 21

0102030405060708090100

4836

2412

0

36.1

023

.70

12.7

05.

472.

041.

000.

19 5

24.0

015

.30

7.98

3.39

1.26

0.62

0.12

8

15.1

09.

444.

862.

050.

760.

380.

0713

9.70

5.99

3.06

1.29

0.49

0.25

0.50

20

5.68

3.48

1.80

0.78

0.31

0.15

0.03

32

3.17

1.98

1.00

0.50

0.20

0.10

0.02

50

1.23

0.93

0.62

0.31

0.12

0.06

0.01

80

58

1320

3250

80

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e51

–90

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



22 23

0102030405060708090100

4836

2412

0

36.4

023

.90

12.8

05.

522.

061.

010.

19 5

27.5

017

.60

9.24

3.95

1.47

0.71

0.14

7

15.6

09.

714.

992.

100.

770.

380.

0713

10.2

06.

273.

191.

340.

490.

240.

0520

6.21

3.80

1.92

0.81

0.31

0.15

0.03

32

2.00

1.24

0.62

0.31

0.12

0.06

0.01

80

57

1320

3280

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e91

–150

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



22 23

0102030405060708090100

4836

2412

0

31.6

020

.40

10.8

04.

641.

720.

840.

16

6

20.2

012

.70

6.59

2.79

1.03

0.50

0.09

10

15.9

09.

905.

082.

140.

790.

380.

07 1

3

10.5

06.

473.

291.

380.

510.

240.

05 2

0

6.55

4.00

2.03

0.84

0.31

0.15

0.03

32

4.10

2.49

1.26

0.53

0.19

0.09

0.02

50

1.39

0.85

0.43

0.20

0.08

0.04

0.00

125

6

10

1320

3250

125

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e15

1–28

0

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



24 25

0102030405060708090100

4836

2412

0

27.8

017

.80

9.36

4.00

1.48

0.72

0.14

7

18.7

011

.70

6.04

2.55

0.94

0.46

0.09

11

13.2

08.

174.

171.

760.

640.

310.

06 1

6

7.41

4.54

2.30

0.95

0.35

0.17

0.03

29

4.28

2.60

1.31

0.54

0.20

0.10

0.02

50

1.59

0.96

0.48

0.20

0.08

0.04

0.00

125

711

1629

5012

5

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e28

1–50

0

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



24 25

0102030405060708090100

4030

2010

0

18.8

011

.80

6.07

2.57

0.94

0.46

0.09

11

11.3

06.

983.

551.

490.

540.

260.

05 1

9

8.08

4.95

2.51

1.05

0.38

0.18

0.03

27

4.69

2.85

1.44

0.59

0.21

0.10

0.02

47

2.93

1.77

0.89

0.37

0.13

0.06

0.01

75

1.73

1.05

0.52

0.21

0.07

0.03

0.00

125

1119

2747

7512

5

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e50

1–12

00

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



26 27

0102030405060708090100

4030

2010

0

22.4

014

.20

7.37

3.13

1.16

0.56

0.11

9

16.1

010

.10

5.17

2.18

0.80

0.39

0.07

13

9.46

5.82

2.95

1.24

0.45

0.22

0.04

23

5.29

3.22

1.62

0.67

0.24

0.12

0.02

42

3.07

1.86

0.93

0.38

0.14

0.06

0.01

73

1.11

0.66

0.33

0.13

0.05

0.02

0.00

200

913

2942

73

200

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e12

01–3

200

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



26 27

0102030405060708090100

4030

2010

0

22.1

014

.10

7.32

3.11

1.15

0.56

0.11

9

14.0

08.

744.

481.

880.

690.

330.

06 1

5

7.56

4.64

2.35

0.98

0.36

0.17

0.03

29

4.47

2.72

1.37

0.57

0.20

0.10

0.02

50

2.62

1.59

0.79

0.33

0.12

0.05

0.01

86

1.20

0.72

0.36

0.15

0.05

0.02

0.00

189

915

2950

8618

9

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e32

01–1

0,00

0

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



28 29

0102030405060708090100

4030

2010

0

25.6

015

.40

7.38

3.02

1.14

0.55

0.10

9

15.4

08.

544.

401.

850.

680.

330.

06 1

5

7.61

4.58

2.33

0.97

0.35

0.17

0.03

29

4.80

2.93

1.48

0.61

0.22

0.11

0.02

46

2.91

1.77

0.89

0.37

0.13

0.06

0.01

77

1.20

0.72

0.36

0.15

0.05

0.02

0.00

189

915

2946

7718

9

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e10

,001

–35,

000

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



28 29

0102030405060708090100

4030

2010

0

25.6

015

.40

7.70

3.20

1.14

0.55

0.10

9

15.4

09.

244.

621.

920.

690.

330.

06 1

5

7.94

4.78

2.39

0.98

0.36

0.17

0.03

29

4.11

2.48

1.24

0.51

0.18

0.09

0.01

56

2.40

1.44

0.71

0.29

0.10

0.05

0.01

96

1.35

0.81

0.40

0.16

0.06

0.03

0.00

170

915

2956

9617

0

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e35

,001

–150

,000

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



30 31

0102030405060708090100

4030

2010

0

25.6

015

.40

7.70

3.20

1.08

0.52

0.10

9

15.4

09.

244.

621.

920.

660.

320.

06 1

5

7.94

4.78

2.39

0.99

0.35

0.17

0.03

29

3.60

2.17

1.07

0.44

0.10

0.07

0.01

64

1.48

0.88

0.44

0.18

0.06

0.03

0.00

156

915

2964

156

Sam

ple

size

Pro

babi

lity

of a

ccep

tanc

e

.10

.25

.50

.75

.90

.95

.99

Per

cent

def

ecti

ve (

P´)


Lot

siz

e15

0,00

1–50

0,00

0

OC

cur

ves

for

sing

le s

ampl

ing

plan

sA

ccep

tanc

e nu

mbe

r eq

ual t

o ze

ro(S

ampl

e si

ze a

s in

dica

ted)



30 31

SmaLL LOT SuPPLEmENTMany people have expressed a need for a special sampling plan table for small lots, when the associated AQL values they are using are 1.5 and below. Above 1.5, the main c=0 table works out well for small lot sizes. For the most part, any sampling plans developed for use with associated AQLs of less than 0.25 would not be valid. Therefore, in general, if you have broad lot size ranges and are using associated AQLs in the 0.25–1.5 range, the small lot size sampling table given in Table 7 should be of interest.

DerivationThe LQ for the small lot sizes was targeted to the LQ of the largest lot size range in which a constant sample size appeared. For example, looking at Table 1 we see that for an associated AQL of 1.0, 13 is used at the 51–90 lot size range; for 0.65, 20 is used at the 91–150 lot size range; and so forth.

The sample sizes for the smaller lots shown in the supplement have essentially the same LQs as the tar-geted LQs. As in Table 1, the hypergeometric distribution is used.

The hypergeometric is a discrete distribution and necessitates interpolation to arrive at a beta (consumer’s risk) of essentially 0.10.

The small lot size supplement with lot sizes up to 35 does not give the same protection as MIL-STD-105E/ANSI Z1.4. However, it does give you the same protection as you would get on the targeted or larger lot sizes.

Obviously, if we look at, for example, the 16–25 lot size in Table 1, we find that 100 percent inspection is required for the associated AQL of 0.40, and therefore our protection is 100 percent. Yet, the larger lot sizes involve samples. In Table 7, the sample size for a lot size of 21–25 with an associated AQL of 0.40 is 17. Thus, if we had a lot size of 25, we would inspect only 17 pieces—32 percent fewer—and get the same protection as if we were sampling in the lot size range of 151–280 of Table 1 or the MIL-STD-105E/ANSI Z1.4 plans.

Table 7 Small lot size supplement.

associated aQLS

Lot size .25 .40 .65 1.0 1.5

5–10 * * * 8 5

11–15 * * 11 8 6

16–20 * 16 12 9 6

21–25 22 17 13 10 6

26–30 25 20 16 11 7

31–35 28 23 18 12 8

*Inspect 100 percent.



33



33

iNDEx

Page numbers followed by f or t refer to figures or tables, respectively.

a

acceptable quality level (AQL), xiii

defined, 3acceptance numbers, 1ANSI Z1.4, 2, 8

adjustments from, 12c=0 sampling plans and, 3–6

AOQL. See average outgoing quality limits (AOQLs)

AQL. See acceptable quality level (AQL)

attribute sampling plans, 2average outgoing quality limits

(AOQLs), 9, 9f, 10t

C

c=0 sampling plans, xiii–xiv, 7tANSI Z1.4 plans and, 3–6

c=0 sampling plans table, use of, 8

constant sample sizes, reasons for not using, 8

consumer’s risk, 3, 4f, 6, 31continuous sampling, 2

D

defect, defined, 1defective, defined, 1

i

inspection results, 13t

L

limiting quality (LQ) percentages, xiii

lot-by-lot attributes, 2lot-by-lot variables, 2

m

MIL-Q-9858A, 9MIL-STD-105, xiii, xiv

adjustments from, 12MIL-STD-414, 2MIL-STD-1235C, 2

N

nonstatistical sampling plans, 2–3

O

operating characteristic (OC) curves, 3, 4f, 5f

for single sampling plans, 17–30

lot size 2–8, 17lot size 9–15, 18lot size 16–25, 19lot size 26–50, 20lot size 51–90, 21lot size 91–150, 22lot size 151–280, 23lot size 281–500, 24lot size 501–1200, 25lot size 1201–3200, 26lot size 3201–10,000, 27lot size 10,001–35,000, 28lot size 35,001–150,000, 29lot size 150,001–500,000, 30

P

potential savings, estimating, 6producer’s risk, 3

r

risk, producer’s, 3

S

samples, physically taking, 8sample sizes, reasons for not

using constant, 8sampling

continuous, 2

lot-by-lot, 2objective of, 1

sampling plans. See also single sampling plans, OC curves for

attribute, 2nonstatistical, 2–3for small lots, 31switching, 12

savings, estimating potential, 6single sampling plans, OC curves

for, 17–30. See also sampling plans

lot size 2–8, 17lot size 9–15, 18lot size 16–25, 19lot size 26–50, 20lot size 51–90, 21lot size 91–150, 22lot size 151–280, 23lot size 281–500, 24lot size 501–1200, 25lot size 1201–3200, 26lot size 3201–10,000, 27lot size 10,001–35,000, 28lot size 35,001–150,000, 29lot size 150,001–500,000, 30

small lots, sampling for, 31switching

guidelines, 14sampling plans, 12

V

variables-type sampling plan, 2

W

“withhold the lot” phrase, significance of, 1

Z

zero acceptance number plans, derivation of, 1

zero defects, xiii



Documents

C=0 Sampling Plans Fifth Edition