ansi asq z1.9-2008

ANSI/ASQ Z1.9-2008

AMERICAN SOCIETY FOR QUALITY600 NORTH PLANKINTON AVENUEMILWAUKEE, WISCONSIN 53201

AMERICAN NATIONAL STANDARD

SAMPLING PROCEDURES AND TABLES FOR INSPECTION BYVARIABLES FOR PERCENT NONCONFORMING

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AMERICAN NATIONAL STANDARD

SAMPLING PROCEDURES AND TABLES FOR INSPECTIONBY VARIABLES FOR PERCENT NONCONFORMING

PREPARED BY

ACCREDITED STANDARDS COMMITTEEZ1 SUBCOMMITTEE ON STATISTICS

SPONSOR AND SECRETARIAT

AMERICAN SOCIETY FOR QUALITY

ANSI/ASQ Z1.9-2008

ABSTRACT

Sampling Procedures and Tables for Inspection by Variables for Percent Nonconforming is an acceptance sampling systemto be used on a continuing stream of lots for AQL specified. It provides tightened, normal, and reduced plans to be used onmeasurements which are normally distributed. Variation may be measured by sample standard deviation, sample range, orknown standard deviation. It is applicable only when the normality of the measurements is assured.

AMERICAN NATIONAL STANDARD: An American National Standard implies a consensus of those substantially con-cerned with its scope and provisions. An American National Standard is intended as a guide to aid the manufacturer, theconsumer, and the general public. The existence of an American National Standard does not in any respect preclude any-one, whether he has approved the standard or not, from manufacturing, marketing, purchasing, or using products, process-es, or procedures not conforming to the standard. American National Standards are subject to periodic review and users arecautioned to obtain the latest edition.

CAUTION NOTICE: This American National Standard may be revised or withdrawn at any time. The procedures of theAmerican National Standards Institute require that action be taken to reaffirm, revise, or withdraw this standard no laterthan five years from the date of publication. Purchasers of American National Standards may receive current informationon all standards by calling or writing the American National Standards Institute.

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Published by:

©2008 by ASQ

Copyright protection notice for the ANSI/ASQ Z1.9-2008 Standard.This Standard is subject to copyright claims of ANSI, and ASQ. Not for resale. No part of this publication may be reproduced in any form, including an electronic retrieval system, without the prior written permissionof ASQ. All requests pertaining to the ANSI/ASQ Z1.9-2008 Standard should be submitted to ASQ.

ASQ will consider requests for change and information on the submittal of such requests regarding this Standard. All requests should be in writing to the attention of the Standards Administrator at theaddress below.

ASQ Mission: The American Society for Quality advances individual and organizational performance excellence worldwide by providing opportunities for learning, quality improvement, and knowledge exchange.

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ANSI/ASQ Z1.9-2008

FOREWORD

(This foreword is not a part of Draft American National Standard—Sampling Procedures and Tables for Inspection byVariables for Percent Nonconforming, ANSI/ASQ Z1.9-2008)

The present revision ANSI/ASQ Z1.9-2008 is identical to its 2003 version and continues to allow complete interchange-ability of the tabulated plans with ISO 3951-1, which provided a graphical means for implementation of the plans.

ANSI/ASQ Z1.9-2008 is also roughly matched to ANSI/ASQ Z1.4-2008, which corresponds directly to the old militarystandard MIL-STD-105E. The matching is sufficient to allow inspection under either standard for stated AQLs and

inspection levels with reasonably equivalent pro¬tection. Tables are given in Section E—Appendix that show differencesin protection between ANSI/ASQ Z1.9-2008 and ANSI/ASQ Z1.4-2008. These are for use in critical applications to

determine whether moving from one standard to the other is appropriate.

iii

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CONTENTS

Page

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiSECTION A GENERAL DESCRIPTION OF SAMPLING PLANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tables:Table A-1 AQL Conversion Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Table A-2 Sample Size Code Letters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Table A-3 Operating Characteristic Curves for Sampling Plans of Sections

B, C, and D (Graphs for Sample Size Code Letters B through P). . . . . . . 6

SECTION B VARIABILITY UNKNOWN—STANDARD DEVIATION METHOD . . . . . . . . . . . . . . . 32

Part I SINGLE SPECIFICATION LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Examples:

Example B-1 Example of Calculations: Single Specification Limit—Form 1 . . . . . . . 34Example B-2 Example of Calculations: Single Specification Limit—Form 2 . . . . . . . 35

Tables:Table B-1 Master Table for Normal and Tightened Inspection

(Single Specification Limit—Form 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table B-2 Master Table for Reduced Inspection (Single Specification

Limit—Form 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Part II DOUBLE SPECIFICATION LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Examples:

Example B-3 Example of Calculations: Double Specification Limit—One AQL Value for Both Upper and Lower Specification Limit Combined . . . . . . 40

Example B-4 Example of Calculations: Double Specification Limit—Different AQL Values for Upper and Lower Specification Limits . . . . . . 41

Tables:Table B-3 Master Table for Normal and Tightened Inspection (Double

Specification Limit and Form 2—Single Specification Limit) . . . . . . . . 42Table B-4 Master Table for Reduced Inspection (Double Specification

Limit and Form 2—Single Specification Limit) . . . . . . . . . . . . . . . . . . . 43Table B-5 Table for Estimating the Lot Percent Nonconforming . . . . . . . . . . . . . . . 44Table B-6 Values of F for Maximum Standard Deviation (MSD) . . . . . . . . . . . . . . 53

Appendix B Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

SECTION C VARIABILITY UNKNOWN—RANGE METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


Example C-1 Example of Calculations: Single Specification Limit—Form I . . . . . . . . 58Example C-2 Example of Calculations: Single Specification Limit—Form 2 . . . . . . . 59

Tables:Table C-1 Master Table for Normal and Tightened Inspection

(Single Specification Limit—Form I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Table C-2 Master Table for Reduced Inspection (Single Specification

Limit—Form I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Part II DOUBLE SPECIFICATION LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

ANSI/ASQ Z1.9-2008

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PageExamples:

Example C-3 Example of Calculations: Double Specification Limit—One AQL Value for Both Upper and Lower Specification Limit Combined . . . . . . 65

Example C-4 Example of Calculations: Double Specification Limit—Different AQL Value for Upper and Lower Specification Limits . . . . . . . . . . . . . . 66

Tables:Table C-3 Master Table for Normal and Tightened Inspection (Double

Specification Limit and Form 2—Single Specification Limit) . . . . . . . . 67Table C-4 Master Table for Reduced Inspection (Double Specification

Limit and Form 2—Single Specification Limit) . . . . . . . . . . . . . . . . . . . 68Table C-5 Table for Estimating the Lot Percent Nonconforming . . . . . . . . . . . . . . . 69Table C-6 Values of f for Maximum Average Range (MAR) . . . . . . . . . . . . . . . . . . 78

Appendix C Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

SECTION D VARIABILITY KNOWN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


Example D-1 Example of Calculations: Single Specification Limit—Form 1 . . . . . . . 82Example D-2 Example of Calculations: Single Specification Limit—Form 2 . . . . . . . 83

Tables:Table D- 1 Master Table for Normal and Tightened Inspection

(Single Specification Limit—Form 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Table D-2 Master Table for Reduced Inspection (Single Specification

Limit—Form 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Part II DOUBLE SPECIFICATION LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Examples:

Example D-3A Example of Calculations: Double Specification Limit—One AQL Value for Both Upper and Lower Specification Limit Combined . . . . . . 90

Example D-3B Example of Calculations: Double Specification Limit—One AQL Value for Both Upper and Lower Specification Limit Combined . . . . . . 91

Example D-4 Example of Calculations: Double Specification Limit—Different AQL Values for Upper and Lower Specification Limits . . . . . . . . . . . . . 92

Tables:Table D-3 Master Table for Normal and Tightened Inspection (Double

Specification Limit and Form 2—Single Specification Limit) . . . . . . . . 93Table D-4 Master Table for Reduced Inspection (Double Specification Limit

and Form 2—Single Specification Limit) . . . . . . . . . . . . . . . . . . . . . . . . 95Table D-5 Table for Estimating the Lot Percentage Nonconforming . . . . . . . . . . . . 97Table D-6 Value of F σ for Maximum Process Standard Deviation . . . . . . . . . . . . . 98

Appendix D Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

SECTION E APPENDIX—MATCH WITH ANSI/ASQ Z1.4-2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Tables:

1 Matching Code Letters and ANSI/ASQC Z1.9-2008 Sample Size . . . . . . . . . . . . . . . 1012 ANSI/ASQC Z1.9-2008 Percentage Points in Terms of Percent Nonconforming . . . . 1023 ANSI/ASQ Z1.4-2008 (MIL-STD-105) Percentage Points in Terms of

Percent Nonconforming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 Difference in Percentage Points—ANSI/ASQC Z1.9-2008 Minus

ANSI/ASQ Z1.4-2008 (MIL-STD-105) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

ANSI/ASQ Z1.9-2008

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INTRODUCTION

This standard was prepared to meet a need for the use of standard sampling plans for inspection by variables in procure-ment, supply and storage, and maintenance inspection operations. The variables sampling plans apply to a single qualitycharacteristic which can be measured on a continuous scale, and for which quality is expressed in terms of percent non-conforming. The theory underlying the development of the variables sampling plans, including the operating characteris-tic curves, assumes that measurements of the quality characteristics are independent, identically normally distributed ran-dom variables.

It is important to note that variables sampling plans are not to be used indiscriminately, simply because it is possible toobtain variables measurement data. The users are advised to consult their technical agency to determine the feasibility ofapplication in considering applications where the normality or independence assumptions may be questioned.

In comparison with attributes sampling plans, variables sampling plans have the advantage of usually resulting in consid-erable savings in sample size for comparable assurance as to the correctness of decisions in judging a single quality char-acteristic, or for the same sample size, greater assurance is obtained using variables plans. Attributes sampling plans havethe advantage of greater simplicity, of being applicable to either single or multiple quality characteristics, and of requiringno knowledge about the distribution of the continuous measurements of any of the quality characteristics.

This standard is divided into four sections. Section A describes general procedures of the sampling plans. Sections B andC describe specific procedures and applications of the sampling plans when variability is unknown. In Section B the stan-dard deviation is used as the basis for an estimate of the unknown variability, and in Section C the average range of the sam-ple is used. Section D describes the plans when variability is known.

Each of Sections B, C, and D is divided into two parts: (I) Sampling Plans for the Single Specification Limit Case, and (II)Sampling Plans for the Double Specification Limit Case. For the single specification limit case, the acceptability criterionis given in two forms: Form 1 and Form 2. Either of the forms may be used, since they are identical as to sample size anddecision for lot acceptability or rejectability. In deciding whether to use Form 1 or Form 2, the following points should beborne in mind. Form 1 provides the lot acceptability criterion without estimating lot percent nonconforming. The Form 2lot acceptability criterion requires estimates of lot percent nonconforming.

Operating characteristic curves in Table A-3 are for the Normal Inspection sampling plans and show the relationship betweenquality and percent of lots expected to be accepted for the quality characteristic inspected. As stated, these operating char-acteristic curves are based on the assumption that measurements are selected at random from a normal distribution.

The corresponding sampling plans in Sections B, C, and D were matched as closely as possible under a system of fixedsample size with respect to their operating characteristic curves. Operating characteristic curves in Table A-3 have beencomputed for the sampling plans based on the standard deviation estimate of unknown variability. They are equally appli-cable for sampling plans based on the average range of the sample and those based on known variability.

Certain characteristics concerning the sampling plans in Section B and C and those in Section D should be noted. Plansbased on the estimate of unknown variability require fewer sample units for comparable assurance when the standard devi-ation is used than when the average range of the sample is used. On the other hand, plans using the average range of thesample require simpler computations and may be better understood by operating personnel. Plans using known variabilityrequire considerably fewer sample units for comparable assurance than either of the plans of unknown variability; however,the requirement of known variability is a stringent one. The user is well advised to consult a statistician before applyingsampling plans using known variability.

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Table B-6 provides values of the factor F to compute the maximum standard deviation: MSD. The MSD serves as a guidefor the maximum allowable magnitude of the estimate of lot standard deviation when using plans for the double specifica-tion limit case, based on the standard deviation. Similarly Table C-6 provides values of the factor f to compute the maxi-mum average range: MAR. The MAR serves as a guide for the maximum allowable magnitude of the average range of thesample when using plans for the double specification limit case, based on the average range. The estimate of lot standarddeviation or average range of the sample, if it is less than the MSD or MAR, respectively, helps to ensure, but does not guar-antee, lot acceptability.

All symbols and their definitions are given in the appendix of the applicable section. An illustration of the computations andprocedures used in the sampling plans is given in the examples in Parts I and II of the applicable section. The computationsinvolve simple arithmetic operations such as addition, subtraction, multiplication, and division of numbers, or at most, thetaking of a square root of a number. The user should become familiar with the general procedures of Section A, and refer tothe applicable section for detailed instructions regarding specific procedures, computations, and tables for the sampling plans.

Section E—Appendix provides information about the match between this variables standard, ANSI/ASQ Z1.9-2008, andthe corresponding attributes standard, ANSI/ASQ Z1.4-2008.

The flowchart in Figure 1 shows the basic procedure for using the standard.

ANSI/ASQ Z1.9-2008

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ANSI/ASQ Z1.9-2008

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STARTDetermine code letter from tables A-1 and A-2

Processσ known

?

Obtain information1. Lot size (N)2. Level (Z1.9[II])3. AQL

Spec. limit Spec. limit

Rangemethod?

Spec. limit

Form?s

>MSD?

M and n fromB-3 and B-4

Compute

k and n fromB-1 or B-2Compute

M and n fromB-3 and B-4

Compute

M, c, and n fromC-3 and C-4

Compute

M, v, and n fromD-3 and D-4

Compute

k and n fromC-1 or C-2Compute

M, c, and n fromC-3 and C-4

Compute

k and n fromD-1 or D-2Compute

M, v, and n fromD-3 and D-4

Compute

Form?R–

>MAR?

Form?σ

>MPSD?

IsPL or PU

≤ M?

IsP ≤ M?

IsPL or PU

≤ M?

IsP ≤ M?

IsP ≤ M?

IsPL or PU

≤ M?

Is

or

≥ k?

Is

or

≥ k?

(X L) –

s

(U–X)s

Is

or

≥ k?

(X L) –σ

(U–X)σ

No

No

Single

Two Two TwoYesR R R

Yes Yes

One One OneNo No No

Std. dev. method

Std. dev. method Range method Sigma method

Std. dev. method Range method Range method Sigma method Sigma method

Double Single Double Single Double

MSD = F(U – L)F from B-6

MAR = f(U – L)f from C-6

MPSD = Fσ(U – L)F from D-6

Yes

Yes

R

AYes

No

R

AYes

No

R

AYes

No

R

AYes

No

R

AYes

No

R

AYes

No

R

AYes

No

R

AYes

No

R

AYes

No

1) X2) s

3)X L

s or

U –X

s

–

1) X2) R

3)X L

or U –X–

1) X

2)X L

or U –X–

1) X2) s

4) Obtain PU and PL from B-55) P = PU + PL

3) QL = , andX L

sU –X

s

–

QU =

1) X

4) Obtain PU and PL from D-55) P = PU + PL

2) QL = , and(X L)v

(U–X)v

–

QU =R R

4) Obtain PU and PL from C-55) P = PU + PL

3) QL = , and(X L)c

(U–X)c

–

QU =

1) X2) R

R

R

σσ σ

σ

1) X2) s

4) Obtain PU and PL from B-5

3) QL = , andX L

sU –X

s

–

R

R

1) X

3) Obtain PU and PL from D-5

2) QL = , and(X L)v

(U X)v

–

–

σ

σ

(X L) –

R

(U–X)

R

QU =

1) X2) R

4) Obtain PU and PL from C-5

3) QL = , and(X L)c

(U–X)c

–

QU =QU =

Figure 1 Flowchart for use of ANSI/ASQ Z1.9-2008.

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A1. SCOPE

A1.1 Purpose. This standard establishes sampling plansand procedures for inspection by variables for use in pro-curement, supply and storage, and maintenance inspectionoperations. When applicable this standard shall be refer-enced in the specification, contract, or inspection instruc-tions, and the provisions set forth herein shall govern.

These acceptance sampling plans for variables are given interms of the percent or proportion of product in a lot orbatch that depart from some requirement. The general ter-minology used within the document will be given in termsof percent of nonconforming units or number of noncon-formities, since these terms are likely to constitute the mostwidely used criteria for acceptance sampling.

A2. DEFINITIONS AND TERMINOLOGY

For the purpose of this American National Standard, thedefinitions given in ANSI/ISO/ASQ A3534-2:2006 andthe following definitions apply. The definitions and termi-nology employed in this standard are in accord withANSI/ ISO/ASQ A3534-2:2006 Statistics—Vocabularyand symbols—Part 2: Applied Statistics. Definitions quotedfrom ANSI/ISO/ASQ A3534-2:2006 reference the clausenumber of that standard.

A2.1 Acceptance Quality Limit (AQL): The AQL is thequality level that is the worst tolerable product averagewhen a continuing series of lots is submitted for accept-ance sampling.

NOTEThe use of the abbreviation AQL to mean Acceptable QualityLevel is no longer recommended.

A2.2 Defect: Nonfulfillment of a requirement related toan intended or specified use [ISO 9000]. (See ANSI/ISO/ASQ A3534-2:2006 3.1.12)

NOTE 1The distinction between the concepts defect and noncon-formity is important as it has legal connotations, particu-larly those associated with product liability issues.Consequently the term defect should be used with extremecaution.

NOTE 2The intended use by the customer can be affected by thenature of the information, such as operating or mainte-nance instructions, provided by the customer.

A2.3 Inspection by Variables: Inspection by measuringthe magnitude(s) of the characteristic(s) of an item. (SeeANSI/ISO/ASQ A3534-2:2006 4.1.4)

A2.4 Lot Size: The number of items in a lot.

A2.5 Mixed Variables-Attributes Inspection: Mixedvariables-attributes inspection is inspection of a sample byattributes, in addition to inspection by variables alreadymade of a previous sample from the lot, before a decisionas to acceptability or rejectability of a lot can be made.

A2.6 Nonconformity: Nonfulfillment of a requirement[ISO 9000]. (See ANSI/ISO/ASQ A3534-2:2006 3.1.11)

NOTESee notes to defect (A2.2)

A2.7 Sampling Plan: A specific plan which states thesample size(s) to be used and the associated criteria foraccepting the lot.

NOTEThe sampling plan does not contain the rules on how totake the sample.

A2.8 Sampling Scheme: A combination of samplingplans with rules for changing from one plan to another.

ANSI/ASQ Z1.9-2008

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SECTION A

GENERAL DESCRIPTION OF SAMPLING PLANS

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NOTESome schemes have switching rules for automatic changeto tightened inspection plans or reduced inspection plansor change to 100 percent inspection.

A2.9 Switching Rule: Instruction within an acceptancesampling scheme for changing from one acceptance sam-pling plan to another of greater or lesser severity of samplingbased on demonstrated quality level (See ANSI/ISO/ASQA3534-2:2006 4.3.4)

EXAMPLENormal, tightened, reduced inspection or discontinuationof inspection are examples of severity of sampling.

A2.10 Specification Limit: Limiting value stated for acharacteristic. (See ANSI/ISO/ASQ A3534-2:2006 3.1.3)

A3. PERCENT NONCONFORMING

A3.1 Expression of Nonconformance. The extent of non-conformance of product shall be expressed in terms of per-cent nonconforming.

A4. ACCEPTANCE QUALITY LIMIT

A4.1 Acceptance Quality Limit. The acceptance qualitylimit (AQL) represents a nominal value expressed in termsof percent nonconforming specified for a single qualitycharacteristic. Certain numerical values of AQL rangingfrom .10 to 10.00 percent are shown in Table A-1. When arange of AQL values is specified, it shall be treated as if itwere equal to the value of AQL for which sampling plansare furnished and which is included within the AQL range.When the specified AQL is a particular value other thanthose for which sampling plans are furnished, the AQL,which is to be used in applying the provisions of thisstandard, shall be as shown in Table A-1.

A4.2 Note on the Meaning of AQL. The concept of AQLonly applies when an acceptance sampling scheme withrules for switching between normal, tightened, and reducedinspection and discontinuance of sampling inspection isused. These rules are designed to encourage suppliers to

have process averages consistently better than the AQL. Ifsuppliers fail to do so, there is a high probability of beingswitched from normal inspection to tightened inspectionwhere lot acceptance becomes more difficult. Once ontightened inspection, unless corrective action is taken toimprove product quality, it is very likely that the rule requir-ing discontinuance of sampling inspection will be invoked.

Although individual lots with quality as bad as the AQLcan be accepted with fairly high probability, the designa-tion of an AQL does not suggest that this is necessarily adesirable quality level. The AQL is a parameter of thesampling scheme and should not be confused with aprocess average, which describes the operating level of amanufacturing process. It is expected that the productquality level will be less than the AQL to avoid excessivenon-accepted lots.

The sampling plans in this standard are so arranged thatthe probability of lot acceptance at the designated AQLdepends upon sample size, being generally higher for largesamples than for small samples for a given AQL. To deter-mine the specific protection to the consumer at a givenAQL, it is necessary to refer to the operating characteristiccurves (which are provided in this standard) of the corre-sponding scheme and its constituent plans.

The AQL alone does not describe the protection to the con-sumer for individual lots or batches, but more directlyrelates to what is expected from a series of lots or batchesprovided the provisions of this standard are satisfied.

A4.3 Specifying AQLs. The particular AQL value to beused for a single quality characteristic of a given productmust be specified. In the case of a double specificationlimit, either an AQL value is specified for the total percentnonconforming outside of both upper and lower specifica-tion limits, or two AQL values are specified, one for theupper limit and another for the lower limit.

A5. SUBMITTAL OF PRODUCT

A5.1 Lot. The term “lot” shall mean “inspection lot.”

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A5.1.1 Formation of Lots. Each lot shall, as far as ispracticable, consist of units of product of a single type,grade, class, size, or composition manufactured underessentially the same conditions.

A5.1.2 Lot Size. The lot size may differ from the quanti-ty designated in the contract or order as a lot for produc-tion, shipment, or other purposes.

A6. LOT ACCEPTABILITY

A6.1 Acceptability Criterion. The acceptability of a lotof material submitted for inspection shall be determined byuse of one of the sampling plans associated with a speci-fied value of the AQL(s). This standard provides samplingplans based on known and unknown variability. In the lat-ter case two alternative methods are provided, one basedon the estimate of lot standard deviation and the other onthe average range of the sample. These are referred to asthe standard deviation method and the range method. Forthe case of a single specification limit, the acceptabilitycriterion is given in two forms. These are identified asForm 1 and Form 2.

A6.2 Choice of Sampling Plans. Sampling plans andprocedures are provided in Section B if variability isunknown and the standard deviation method is used, inSection C if variability is unknown and the range methodis used, and in Section D if variability is known. Unlessotherwise specified, unknown variability, standard devia-tion method sampling plans, and the acceptability criteri-on of Form 2 (for the single specification limit case)shall be used.

A7. SAMPLE SELECTION

A7.1 Determination of Sample Size. The sample sizecode letter depends on the inspection level and the lot size.There are five inspection levels: Special Levels S3, S4, andGeneral Levels I, II, and III. Unless otherwise specified,Inspection Level II shall be used. However, InspectionLevel I may be specified when less discrimination is need-ed, or Level III may be specified for greater discrimina-tion. Levels S3 and S4 may be used when relatively small

sample sizes are necessary and large sampling risks can ormust be tolerated. The sample size code letter applicable tothe specified inspection level and for lots of given sizeshall be obtained from Table A-2.

A7.2 Drawings of Samples. Units of the sample shall beselected at random without regard to their quality.

A8. NORMALITY ASSUMPTION

This standard assumes the underlying distribution of indi-vidual measurements to be normal in shape. Failure of thisassumption to be valid will affect the OC curves and prob-abilities based on these curves. In particular it will affectthe estimate of percent nonconforming calculated from themean and standard deviation of the distribution. Theassumption should be verified prior to use of this standard.

A variety of statistical tests and graphical techniques areavailable for this purpose. A person knowledgeable in sta-tistics should be consulted who can advise whether the dis-tribution appears suitable for sampling by variables.

A9. SPECIAL PROCEDURE FOR APPLICA-TION OF MIXED VARIABLES-ATTRIBUTES SAMPLING PLANS

A9.1 Applicability. A mixed variables and attributessampling plan may be used under either of the two follow-ing conditions: (Note: No operating characteristic curvesare provided for the mixed variables-attributes samplingplans herein and those in Table A-3 are not applicable.)

Condition A. Ample evidence exists that the productsubmitted for inspection is selected by the supplier to meetthe specification limit(s) by a screening process from alarger quantity of product which is not being producedwithin the specification limit(s).

Condition B. Other conditions exist that warrant the useof a variables-attributes sampling plan.

A9.2 Selection of Sampling Plans. The mixed variables-attributes sampling plan shall be selected in accordancewith the following:

ANSI/ASQ Z1.9-2008

3

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 3

A9.2.1 Select the variables sampling plan in accordancewith Section B, C, or D.

A9.2.2 Select the attributes sampling plan from ANSIZ1.4 using a single sampling plan and tightened inspec-tion. The same AQL value(s) shall be used for the attrib-utes sampling plan as used for the variables plan of para-graph A9.2.1 (Additional sample items may be drawn, asnecessary, to satisfy the requirements for sample size ofthe attributes sampling plan. Count as a nonconformingunit each sample item falling outside of specificationlimit(s).)

A9.3 Determination of Acceptability. A lot meets theacceptability criterion if one of the following conditions issatisfied:

Condition A. The lot complies with the appropriate vari-ables acceptability criterion of Section B, C, or D.

Condition B. The lot complies with the acceptability cri-terion of ANSI/ASQ Z1.4-2008.

A9.3.1 If Condition A is not satisfied, proceed in accor-dance with the attributes sampling plan to meet Condition B.

A9.3.2 If Condition B is not satisfied, the lot does notmeet the acceptability criterion.

A10. NORMAL, TIGHTENED, AND REDUCEDINSPECTION

A10.1 Initiation of Inspection. Normal inspection will beused at the start of inspection unless otherwise directed bythe responsible authority.

A10.2 Continuation of Inspection. Normal, tightened, orreduced inspection shall continue unchanged on succes-sive lots or batches except where the following switchingprocedures require change.

A10.3 Switching Procedures.

A10.3.1 Normal to Tightened. When normal inspectionis in effect, tightened inspection shall be instituted whentwo out of five consecutive lots or batches have beenrejected on original inspection (that is, ignoring resubmittedlots or batches for this procedure).

A10.3.2 Tightened to Normal. When tightened inspec-tion is in effect, normal inspection shall be instituted whenfive consecutive lots or batches have been consideredacceptable on original inspection.

A10.3.3 Normal to Reduced. When normal inspection isin effect, reduced inspection shall be instituted providingthat all of the following conditions are satisfied:

a. The preceding 10 lots or batches have been on normal inspection and none has been rejected; and

b. Production is at a steady rate; and c. Reduced inspection is considered desirable by

the responsible authority and is permitted by thecontract or specification.

A10.3.4 Reduced to Normal. When reduced inspection isin effect, normal inspection shall be instituted if any of thefollowing occur on original inspection:

a. A lot or batch is rejected; or b. Production becomes irregular or delayed; or c. Other conditions warrant that normal inspection

shall be instituted.

A10.4 Discontinuation of Inspection. If the cumulativenumber of lots not accepted in a sequence of consecutivelots on tightened inspection reaches 5, the acceptance pro-cedures of this standard shall be discontinued. Inspectionunder the provisions of this standard shall not be resumeduntil corrective action has been taken. Tightened inspec-tion shall then be used as if A10.3.1 had been invoked.

ANSI/ASQ Z1.9-2008

4

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 4

ANSI/ASQ Z1.9-2008

5

–

0.110

0.165

0.280

0.440

0.700

1.10

1.65

2.80

4.40

7.00

to

to

to

to

to

to

to

to

to

to

to

0.109

0.164

0.279

0.439

0.669

1.09

1.64

2.79

4.39

6.99

10.9

0.10

0.15

0.25

0.40

0.65

1.0

1.5

2.5

4.0

6.5

10.0

For specified AOL valuesfalling within these ranges

Use this AQLvalue

Table A-1AQL Conversion Table

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 5

1The theory governing inspection by variables depends onthe properties of the normal distribution and, therefore,this method of inspection is only applicable when there isreason to believe that the frequency distribution is normal.

2Sample size code letters given in body of table are appli-cable when the indicated inspection levels are to be used.

ANSI/ASQ Z1.9-2008

6

2

9

16

26

51

91

151

281

401

501

1,201

3,201

10,001

35,001

150,001

500,001

to

to

to

to

to

to

to

to

to

to

to

to

to

to

to

and

8

15

25

50

90

150

280

400

500

1,200

3,200

10,000

35,000

150,000

500,000

over

B

B

B

B

B

B

B

C

C

D

E

F

G

H

H

H

B

B

B

B

B

C

D

E

E

F

G

H

I

J

K

K

B

B

B

C

D

E

F

G

G

H

I

J

K

L

M

N

B

B

C

D

E

F

G

H

I

J

K

L

M

N

P

P

C

D

E

F

G

H

I

J

J

K

L

M

N

P

P

P

S3 S4 I II III

Lot SizeInspection Levels

Special General

Table A-21

Sample Size Code Letters2

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 6

ANSI/ASQ Z1.9-2008

Table A-3Operating Characteristic Curves for Sampling Plans

of Sections B, C, and D

7

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 7

ANSI/ASQ Z1.9-2008

8

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

B(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

8076

7268

6460

5652

4844

4036

3228

2420

1612

84

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)

PERCENT OF LOTS EXPECTED TO BE ACCEPTED

2.5

6.5

4.0

10.0

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 8

ANSI/ASQ Z1.9-2008

9

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

C(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

8076

7268

6460

5652

4844

4036

3228

2420

1612

84

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

2.5

4.0

6.5

1.0

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 9

ANSI/ASQ Z1.9-2008

10

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

D(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

8076

7268

6460

5652

4844

4036

3228

2420

1612

84

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

2.5

4.0

6.5

1.0

.65

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 10

ANSI/ASQ Z1.9-2008

11

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

E(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

8076

7268

6460

5652

4844

4036

3228

2420

1612

84

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

2.5

4.0

6.5

1.0

.65

.25

.40

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 11

ANSI/ASQ Z1.9-2008

12

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

F(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

2019

1817

1615

1413

1211

109

87

65

43

21

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.15

.25

.40

.65

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 12

ANSI/ASQ Z1.9-2008

13

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

F (C

ontin

ued)

(Cur

ves

for

sam

plin

g pl

ans

base

d on

ran

ge m

etho

d an

d kn

own

vari

abili

ty a

re e

ssen

tially

equ

ival

ent)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 13

ANSI/ASQ Z1.9-2008

14

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

G(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

2019

1817

1615

1413

1211

109

87

65

43

21

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.15

.10

.25

.40

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 14

ANSI/ASQ Z1.9-2008

15

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

G (

Con

tinue

d)(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.65

1.0

1.5

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 15

ANSI/ASQ Z1.9-2008

16

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

H(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

2019

1817

1615

1413

1211

109

87

65

43

21

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.15

.10

.25

.40

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 16

ANSI/ASQ Z1.9-2008

17

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

H (

Con

tinue

d)(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.65

1.0

1.5

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 17

ANSI/ASQ Z1.9-2008

18

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

I(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

10.0

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.15

.10

.25

.40

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 18

ANSI/ASQ Z1.9-2008

19

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

I (C

ontin

ued)

(Cur

ves

for

sam

plin

g pl

ans

base

d on

ran

ge m

etho

d an

d kn

own

vari

abili

ty a

re e

ssen

tially

equ

ival

ent)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

.65

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 19

ANSI/ASQ Z1.9-2008

20

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

J(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

10.0

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.15.10

.25

.40

.65

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 20

ANSI/ASQ Z1.9-2008

21

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

J (C

ontin

ued)

(Cur

ves

for

sam

plin

g pl

ans

base

d on

ran

ge m

etho

d an

d kn

own

vari

abili

ty a

re e

ssen

tially

equ

ival

ent)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 21

ANSI/ASQ Z1.9-2008

22

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

K(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.40

.10

.15

.25

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 22

ANSI/ASQ Z1.9-2008

23

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

K (

Con

tinue

d)(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

.65

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 23

ANSI/ASQ Z1.9-2008

24

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

L(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.40

.65

.10

.15

.25

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 24

ANSI/ASQ Z1.9-2008

25

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

L (

Con

tinue

d)(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 25

ANSI/ASQ Z1.9-2008

26

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

M(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.40

.65

.10

.15

.25

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 26

ANSI/ASQ Z1.9-2008

27

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

M (

Con

tinue

d)(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 27

ANSI/ASQ Z1.9-2008

28

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

N(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.40

.65

.10

.15

.25

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 28

ANSI/ASQ Z1.9-2008

29

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

N (

Con

tinue

d)(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 29

ANSI/ASQ Z1.9-2008

30

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

den

otes

pla

n us

ed e

xclu

sive

ly o

n tig

hten

ed in

spec

tion

and

prov

ides

sym

bol f

or id

entifi

catio

nof

app

ropr

iate

OC

cur

ve. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

re s

elec

ted

at r

ando

m f

rom

a n

orm

al d

istr

ibut

ion.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

P(C

urve

s fo

r sa

mpl

ing

plan

s ba

sed

on r

ange

met

hod

and

know

n va

riab

ility

are

ess

entia

lly e

quiv

alen

t)

100 90 80 70 60 50 40 30 20 10 0

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


.40

.65

.10

.15

.25

T

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 30

ANSI/ASQ Z1.9-2008

31

NO

TE

:Fig

ures

on

curv

es a

re a

ccep

tanc

e qu

ality

lim

its (

AQ

Ls)

for

nor

mal

insp

ectio

n. T

he v

alue

s of

the

perc

ent o

f lo

ts e

xpec

ted

to b

e ac

cept

ed a

re v

alid

onl

y w

hen

mea

sure

men

ts a

rese

lect

ed a

t ran

dom

fro

m a

nor

mal

dis

trib

utio

n.

Tabl

e A

-3O

pera

ting

Cha

ract

eris

tic C

urve

s fo

r Sa

mpl

ing

Plan

s B

ased

on

Stan

dard

Dev

iatio

n M

etho

dSa

mpl

e Si

ze C

ode

Let

ter

P (C

ontin

ued)

(Cur

ves

for

sam

plin

g pl

ans

base

d on

ran

ge m

etho

d an

d kn

own

vari

abili

ty a

re e

ssen

tially

equ

ival

ent)

100 90 80 70 60 50 40 30 20 10 0

4038

3634

3230

2826

2422

2018

1614

1210

86

42

0

QU

AL

ITY

OF

SUB

MIT

TE

D L

OT

S (I

n pe

rcen

t non

conf

orm

ing)


1.5

1.0

2.5

4.0

6.5

10.0

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 31

ANSI/ASQ Z1.9-2008

32

B1. SAMPLING PLAN FOR SINGLESPECIFICATION LIMIT

This part of the standard describes the procedures for usewith plans for a single specification limit when variabilityof the lot with respect to the quality characteristic isunknown and the standard deviation method is used. Theacceptability criterion is given in two equivalent forms.These are identified as Form 1 and Form 2.

B1.1 Use of Sampling Plans. To determine whether thelot meets the acceptability criterion with respect to a par-ticular quality characteristic and AQL value, the applicablesampling plan shall be used in accordance with the provi-sions of Section A, General Description of SamplingPlans, and those in this part of the standard.

B1.2 Drawing of Samples. All samples shall be drawn inaccordance with paragraph A7.2.

B1.3 Determination of Sample Size Code Letter. Thesample size code letter shall be selected from Table A-2 inaccordance with paragraph A7.1.

B2. SELECTING THE SAMPLING PLANWHEN FORM 1 IS USED

B2.1 Master Sampling Tables. The master samplingtables for plans based on variability unknown for a singlespecification limit when using the standard deviationmethod are Tables B-1 and B-2. Table B-1 is used fornormal and tightened inspection and Table B-2 forreduced inspection.

B2.2 Obtaining the Sampling Plan. The sampling planconsists of a sample size and an associated acceptability

constant.1 The sampling plan is obtained from MasterTable B-1 or B-2.

B2.2.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letter.

B2.2.2 Acceptability Constant. The acceptability con-stant k, corresponding to the sample size mentioned inparagraph B2.2.1, is indicated in the column of the mastertable corresponding to the applicable AQL value. Table B-1is entered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans forreduced inspection are provided in Table B-2.

B3. LOT-BY-LOT ACCEPTABILITYPROCEDURES WHEN FORM 1 IS USED2

B3.1 Acceptability Criterion. The degree of confor-mance of a quality characteristic with respect to a singlespecification limit shall be judged by the quantity (U – X

–)/s

or (X–

– L)/s.

B3.2 Computation. The following quantity shall be com-puted: (U – X

–)/s or (X

–– L)/s, depending on whether the

specification limit is an upper or lower limit, where

U is the upper specification limit,L is the lower specification limit,X–

is the sample mean, ands is the estimate of lot standard deviation.

B3.3 Acceptability Criteria. Compare the quantity (U – X–

)/sor (X

–– L)/s with the acceptability constant k. If (U – X

–)/s or

(X–

– L)/s is equal to or greater than k, the lot meets the accept-ability criterion; if (U – X

–)/s or (X

–– L)/s is less than k or neg-

ative, then the lot does not meet the acceptability criterion.

SECTION B

VARIABILITY UNKNOWN—STANDARD DEVIATION METHOD

Part 1SINGLE SPECIFICATION LIMIT

1See Appendix B for definitions of all symbols used in the sampling plans based on variability unknown—standard deviation method.2See Example B-1 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 32

B4. SUMMARY FOR OPERATION OFSAMPLING PLAN WHEN FORM 1 IS USED

The following steps summarize the procedures to befollowed:

(1) Determine the sample size code letter from Table A-2by using the lot size and inspection level.

(2) Obtain plan from Master Table B-1 or B-2 by select-ing the sample size n and the acceptability constant k.

(3) Select at random the sample of n units from the lot;inspect and record the measurement of the quality charac-teristic for each unit of the sample.

(4) Compute the sample mean (X–

) and estimate of lot stan-dard deviation s, and also compute the quantity (U – X

–)/s for

an upper specification limit U or the quantity (X–

– L)/s for alower specification limit L.

(5) If the quantity (U – X–

)/s or (X–

– L)/s is equal to orgreater than k, the lot meets the acceptability criterion; if(U – X

–)/s or (X

–– L)/s is less than k or negative, then the

lot does not meet the acceptability criterion.

B5. SELECTING THE SAMPLING PLANWHEN FORM 2 IS USED

B5.1 Master Sampling Tables. The master sampling tablesfor plans based on variability unknown for a single specifica-tion limit when using the standard deviation method are TableB-3 and B-4 of Part II. Table B-3 is used for normal and tight-ened inspection and Table B-4 for reduced inspection.

B5.2 Obtaining the Sampling Plan. The sampling planconsists of a sample size and an associated maximumallowable percent nonconforming. The sampling plan isobtained from Master Table B-3 or B-4.

B5.2.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letter.

B5.2.2 Maximum Allowable Percent Nonconforming. Themaximum allowable percent nonconforming M for sampleestimates corresponding to the sample size mentioned inparagraph B5.2.1 is indicated in the column of the mastertable corresponding to the applicable AQL value. Table B-3is entered from the top for normal inspection and from the

bottom for tightened inspection. Sampling plans forreduced inspection are provided in Table B-4.

B6. LOT-BY-LOT ACCEPTABILITYPROCEDURES WHEN FORM 2 IS USED3

B6.1 Acceptability Criterion. The degree of confor-mance of a quality characteristic with respect to a singlespecification limit shall be judged by the percent of non-conforming product outside the upper or lower specifica-tion limit. The percentage of nonconforming product isestimated by entering Table B-5 with the quality indexand the sample size.

B6.2 Computation of Quality Index. The quality indexQU = (U – X

–/s shall be computed if the specification limit is

an upper limit U, or QL = (X–

– L)/s if it is a lower limit L. Thequantities, (X

–) and s, are the sample mean and estimate of

lot standard deviation, respectively.

B6.3 Estimate of Percent Nonconforming in Lot. Thequality of a lot shall be expressed by pU, the estimated per-cent nonconforming in the lot above the upper specifica-tion limit, or by pL, the estimated percent nonconformingbelow the lower specification limit. The estimated percentnonconforming pU or pL is obtained by entering Table B-5with QU or QL and the appropriate sample size.

B6.4 Acceptability Criterion. Compare the estimated lotpercent nonconforming pU or pL with the maximum allow-able percent nonconforming M. If pU or pL is equal to orless than M, the lot meets the acceptability criterion; if pU

or pL is greater than M or if QU or QL is negative, then thelot does not meet the acceptability criterion.

B7. SUMMARY FOR OPERATION OFSAMPLING PLAN WHEN FORM 2 IS USED


(1) Determine the sample size code letter from Table A-2by using the lot size and the inspection level.

(2) Obtain plan from Master Table B-3 or B-4 by select-ing the sampling size n and the maximum allowable per-cent nonconforming M.

ANSI/ASQ Z1.9-2008

33

3See Example B-2 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 33

EXAMPLE B-1

Example of Calculations

Single Specification Limit—Form 1

Variability Unknown—Standard Deviation Method

Example The maximum temperature of operation for a certain device is specified as 209°F. A lot of 40 items is submit-ted for inspection. Inspection Level II, normal inspection, with AQL = 1% is to be used. From Tables A-2 andB-1 it is seen that a sample of size 5 is required. Suppose the measurements obtained are as follows: 197°, 188°,184°, 205°, and 201°; and compliance with the acceptability criterion is to be determined.

Line Information Needed Value Obtained Explanation

1 Sample Size: n 5

2 Sum of Measurements: ∑X 975

3 Sum of Squared Measurements: ∑X2 190,435

4 Correction Factor (CF): (∑X)2/n 190,125 (975)2/5

5 Corrected Sum of Squares (SS): ∑X2 – CF 310 190,435 – 190,125

6 Variance (V): SS/(n – 1) 77.5 310/4

7 Estimate of Lot Standard Deviation s: �V� 8.80 �7�7�.5�8 Sample Mean X

–: ∑X/n 195 975/5

9 Specification Limit (Upper): U 209

10 The quantity: (U – X–

)/s 1.59 (209 – 195)/8.80

11 Acceptability Constant: k 1.52 See Table B-1

12 Acceptability Criterion: Compare (U – X–

)/s with k 1.59 > 1.52 See Para. B3.3

The lot meets the acceptability criterion, since (U – X–

)/s is greater than k.

NOTE: If a single lower specification limit L is given, then compute the quantity (X–

– L)/s in line 10 and compare it withk; the lot meets the acceptability criterion if (X

–– L)/s is equal to or greater than k.

(3) Select at random the sample of n units from the lot;inspect and record the measurement of the quality charac-teristic on each unit of the sample.

(4) Compute the sample mean X–

and the estimate of lotstandard deviation s.

(5) Compute the quality index QU = (U – X–

)/s if an upperspecification limit U is specified, or QL = (X

–– L)/s if a

lower specification limit L is specified.

(6) Determine the estimated lot percent nonconformingPU or PL from Table B-5.

(7) If the estimated lot percent nonconforming PU or PL isequal to or less than the maximum allowable percent non-conforming M, the lot meets the acceptability criterion; ifPU or PL is greater than M or if QU or QL is negative, thenthe lot does not meet the acceptability criterion.

ANSI/ASQ Z1.9-2008

34

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 34

EXAMPLE B-2




Example The maximum temperature of operation for a certain device is specified as 209°F. A lot of 40 items is submit-ted for inspection. Inspection Level II, normal inspection, with AQL = 1% is to be used. From Tables A-2 andB-1 it is seen that a sample of size 5 is required. Suppose the measurements obtained are as follows: 197°, 188°,184°, 205°, and 201°; and compliance with the acceptability criterion is to be determined.


1 Sample Size: n 5





6 Variance (V): SS/(n – 1) 77.5 310/4


–: ∑X/n 195 975/5

9 Specification Limit (Upper): U 209

10 Quantity Index: QU = (U – X–

)/s 1.59 (209 – 195)/8.80

11 Est. of Lot Percent Ncf.: pU 2.19% See Table B-5

12 Max. Allowable Percent Ncf.: M 3.33% See Table B-3

13 Acceptability Criterion: Compare pU with M 2.19% < 3.33% See Para. B6.4

The lot meets the acceptability criterion, since pU is less than M.

NOTE: If a single lower specification limit L is given, then compute the quality index QL = (X–

– L)/s in line 10 andobtain the estimate of lot percent nonconforming pL. Compare pL with M; the lot meets the acceptability criterion if pL is equal to or less than M.

ANSI/ASQ Z1.9-2008

35

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 35

ANSI/ASQ Z1.9-2008

36

Tabl

e B

-1St

anda

rd D

evia

tion

Met

hod

Mas

ter

Tabl

e fo

r N

orm

al a

nd T

ight

ened

Ins

pect

ion

for

Plan

s B

ased

on

Var

iabi

lity

Unk

now

n(S

ingl

e Sp

ecifi

catio

n L

imit—

For

m 1

)

Sam

ple

Size

Cod

eL

ette

r

Sam

ple

Size

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)

T k

.10 k

.15 k

.25 k

.40 k

.65 k

1.00 k

1.50 k

2.50 k

4.00 k

6.50 k

10.0

0

k

B C D E F G H I K L M N PJ

3 4 5 7 10 15 20 25 50 75 100

150

20035

.56

6

.61

7

.67

5

.75

5

.82

8

.88

5

.91

6

.93

5

1.00

1.03

1.05

1.07

1.08

.96

8

.76

5

.81

5

.87

4

.95

5

1.03

1.09

1.12

1.14

1.21

1.25

1.26

1.29

1.29

10.0

0

1.18

.95

8

1.01

1.07

1.15

1.23

1.30

1.33

1.35

1.42

1.46

1.48

1.51

1.51

6.50

1.39

1.12

1.17

1.24

1.33

1.41

1.48

1.51

1.53

1.61

1.65

1.67

1.70

1.70

4.00

1.57

1.34

1.40

1.50

1.59

1.65

1.69

1.72

1.80

1.84

1.86

1.89

1.89

2.50

1.76

1.46

1.52

1.62

1.71

1.79

1.83

1.85

1.94

1.98

2.00

2.03

2.04

1.50

1.89

1.65

1.75

1.84

1.92

1.96

1.98

2.07

2.12

2.14

2.18

2.18

1.00

2.03

1.77

1.88

1.98

2.06

2.10

2.13

2.22

2.27

2.29

2.33

2.33 .65

2.18

2.00

2.11

2.19

2.23

2.26

2.36

2.41

2.43

2.47

2.47 .40

2.31

2.13

2.24

2.32

2.37

2.40

2.50

2.55

2.58

2.62

2.62 .25

2.45

2.42

2.47

2.50

2.61

2.66

2.69

2.73

2.73 .15

2.55

2.53

2.58

2.61

2.22

2.34

2.44

2.72

2.77

2.80

2.84

2.85 .10

2.66

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

T d

enot

es p

lan

used

exc

lusi

vely

on

tight

ened

insp

ectio

n an

dpr

ovid

es s

ymbo

l for

iden

tific

atio

n of

app

ropr

iate

OC

cur

ve.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 36

ANSI/ASQ Z1.9-2008

37

Tabl

e B

-2St

anda

rd D

evia

tion

Met

hod

Mas

ter

Tabl

e fo

r R

educ

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n V

aria

bilit

y U

nkno

wn

(Sin

gle

Spec

ifica

tion

Lim

it—F

orm

1)

Sam

ple

Size

Cod

eL

ette

r

Sam

ple

Size

Acc

epta

nce

Qua

lity

Lim

its

.10 k

.15 k

.25 k

.40 k

.65 k

1.00 k

1.50 k

2.50 k

4.00 k

6.50 k

10.0

0

k


3 3 3 3 4 5 7 10 20 25 30 30 7515

.341

.341

.341

.341

.396

.456

.536

.609

.694

.712

.722

.774

.806

.663

.56

6

.56

6

.56

6

.56

6

.61

7

.67

5

.75

5

.82

8

.91

6

.93

5

.94

6

1.00

1.03

.88

5

.76

5

.76

5

.76

5

.76

5

.81

5

.87

4

.95

5

1.03

1.12

1.14

1.15

1.21

1.25

1.09

.95

8

.95

8

.95

8

.95

8

1.01

1.07

1.15

1.23

1.33

1.35

1.36

1.42

1.46

1.30

1.12

1.12

1.12

1.12

1.17

1.24

1.33

1.41

1.51

1.53

1.55

1.61

1.65

1.48

1.34

1.40

1.50

1.59

1.69

1.72

1.73

1.80

1.84

1.65

1.46

1.52

1.62

1.71

1.83

1.85

1.86

1.94

1.98

1.79

1.65

1.75

1.84

1.96

1.98

2.00

2.08

2.12

1.92

1.77

1.88

1.98

2.10

2.13

2.14

2.22

2.27

2.06

2.00

2.11

2.23

2.26

2.28

2.36

2.41

2.19

2.13

2.24

2.37

2.40

2.41

2.50

2.55

2.32

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 37

ANSI/ASQ Z1.9-2008

38

B8. SAMPLING PLAN FOR DOUBLESPECIFICATION LIMIT

This part of the standard describes the procedures for usewith plans for a double specification limit when variabili-ty of the lot with respect to the quality characteristic isunknown and the standard deviation method is used.

B8.1 Use of Sampling Plans. To determine whether thelot meets the acceptability criterion with respect to a par-ticular quality characteristic and AQL value(s) the applica-ble sampling plan shall be used in accordance with the pro-visions of Section A, General Description of SamplingPlans, and those in this part of the standard.

B9. SELECTING THE SAMPLING PLAN

A sampling plan for each AQL value shall be selected fromTable B-3 or B-4 as follows:

B9.1 Determination of Sample Size Code Letter. Thesample size code letter shall be selected from Table A-2 inaccordance with paragraph A7.1.

B9.2 Master Sampling Tables. The master sampling tablesfor plans based on variability unknown for a double specifi-cation limit when using the standard deviation method areTables B-3 and B-4. Table B-3 is used for normal and tight-ened inspection and Table B-4 for reduced inspection.

B9.3 Obtaining Sampling Plan. A sampling plan consistsof a sample size and the associated maximum allowablepercent nonconforming. The sampling plan to be applied ininspection shall be obtained from Master Table B-3 or B-4.

B9.3.1 Sample Size. The sample size n is shown in themaster tables corresponding to each sample size code letter.

B9.3.2 Maximum Allowable Percent Nonconforming.The maximum allowable percent nonconforming for sam-ple estimates of percent nonconforming for the lower,upper, or both specification limits combined, correspon-ding to the sample size mentioned in paragraph B9.3.1, isshown in the column of the master table corresponding to

the applicable AQL value(s). If different AQLs areassigned to each specification limit, designate the maxi-mum allowable percent nonconforming by ML for thelower limit, and by MU for the upper limit. If one AQL isassigned to both limits combined, designate the maximumallowable percent nonconforming by M. Table B-3 isentered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans forreduced inspection are provided in Table B-4.

B10. DRAWING OF SAMPLES

Samples shall be selected in accordance with paragraphA7.2.

B11. LOT-BY-LOT ACCEPTABILITYPROCEDURES

B11.1 Acceptability Criterion. The degree of confor-mance of a quality characteristic with respect to a doublespecification limit shall be judged by the percent of non-conforming product. The percentage of nonconformingproduct is estimated by entering Table B-5 with the quali-ty index and the sample size.

B11.2 Computation of Quality Indices. The quality indicesQU = (U – X

–)/s and QL = (X

–– L)/s shall be computed, where


is the sample mean, ands is the estimate of lot standard deviation.

B11.3 Percent Nonconforming in the Lot. The quality ofa lot shall be expressed in terms of the lot percent noncon-forming. Its estimate will be designated by pL, pU, or p. Theestimate pU indicates conformance with respect to theupper specification limit, pL with respect to the lower spec-ification limit, and p for both specification limits com-bined. The estimates pL and pU shall be determined byentering Table B-5, respectively with QL and QU and thesample size. The estimate p shall be determined by addingthe corresponding estimated percents nonconforming pL

and pU found in the table.

Part IIDOUBLE SPECIFICATION LIMIT

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 38

B12. ACCEPTABILITY CRITERION ANDSUMMARY FOR OPERATION OFSAMPLING PLANS

B12.1 One AQL value for both Upper and LowerSpecification Limit Combined.

B12.1.1 Acceptability Criterion.4 Compare the estimatedlot percent nonconforming p = pL + pU with the maximumallowable percent nonconforming M. If p is equal to or lessthan M, the lot meets the acceptability criterion; if p isgreater than M or if either QU or QL or both are negative,then the lot does not meet the acceptability criterion.

B12.1.2 Summary for Operation of Sampling Plan. Incases where a single AQL value is established for theupper and lower specification limit combined for a singlequality characteristic, the following steps summarize theprocedures to be used:


(2) Select plan from Master Table B-3 or B-4. Obtain thesample size n and the maximum allowable percent non-conforming M.



and estimate a lot ofstandard deviation s.

(5) Compute the quality indices QU = (U – X–

)/s and QL = (X

–– L)/s.

(6) Determine the estimated lot percent nonconforming p= pU + pL from Table B-5.

(7) If the estimated lot percent nonconforming p is equalto or less than the maximum allowable percent noncon-forming M, the lot meets the acceptability criterion; if p isgreater than M or if either QU or QL or both are negative,then the lot does not meet the acceptability criterion.

B12.2 Different AQL Values for Upper and LowerSpecification Limit.

B12.2.1 Acceptability Criteria.5 Compare the estimated lotpercents nonconforming pL and pU with the correspondingmaximum allowable percents nonconforming ML and MU;also compare p = pL + pU with the larger of ML and MU. IfpL is equal to or less than ML, pU is equal to or less than MU,and p is equal to or less than the larger of ML and MU, thelot meets the acceptability criteria; otherwise, the lot doesnot meet the acceptability criteria. If either QL or QU orboth are negative, then the lot does not meet the accept-ability criteria.

B12.2.2 Summary for Operation of Sampling Plan. Incases where a different AQL value is established for theupper and lower specification limit for a single qualitycharacteristic, the following steps summarize the proce-dures to be used:


(2) Select the sampling plan from Master Table B-3 or B-4. Obtain the sample size n and the maximum allowablepercents nonconforming MU or ML, corresponding to theAQL values for the upper and lower specification limits,respectively.

(3) Select at random the sample of n units from the lot;inspect and record the measurement of the quality charac-teristic on each unit in the sample.


and the estimate of lotstandard deviation s.


)/s and QL = (X

–– L)/s.

(6) Determine the estimated lot percents nonconformingpL and pU corresponding to the percents nonconformingabove the upper and below the lower specification limits.Also determine the combined percent nonconforming p = pL + pU.

ANSI/ASQ Z1.9-2008

39

4See Example B-3 for a complete example of this procedure.5See Example B-4 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 39

(7) If all three of the following conditions:(a) pU is equal to or less than MU,(b) pL is equal to or less than ML,(c) p is equal to or less than the larger of ML and MU,

are satisfied, the lot meets the acceptability criteria; other-wise the lot does not meet the acceptability criteria. Ifeither QL or QU or both are negative, then the lot does notmeet the acceptability criteria.

ANSI/ASQ Z1.9-2008

40

EXAMPLE B-3


Double Specification Limit


One AQL Value for Both Upper and Lower Specification Limit Combined

Example The minimum temperature of operation for a certain device is specified as 180°F. The maximum temperature is209°F. A lot of 40 items is submitted for inspection. Inspection Level II, normal inspection, with AQL = 1% isto be used. From Tables A-2 and B-3 it is seen that a sample of size 5 is required. Suppose the measurementsobtained are as follows: 197°, 188°, 184°, 205°, and 201°; and compliance with the acceptability criterion is tobe determined.


1 Sample Size: n 5





6 Variance (V): SS/(n – 1) 77.5 310/4


–: ∑X/n 195 975/5

9 Upper Specification Limit: U 209

10 Lower Specification Limit: L 180

11 Quality Index: QU = (U – X–

)/s 1.59 (209 – 195)/8.80

12 Quality Index: QL = (X–

– L)/s 1.70 (195 – 180)/8.80

13 Est. of Lot Percent Ncf. above U: pU 2.19% See Table B-5

14 Est. of Lot Percent Ncf. below L: pL 0.66% See Table B-5

15 Total Est. Percent Ncf. in Lot: p = pU + pL 2.85% 2.19% + 0.66%

16 Max. Allowable Percent Ncf.: M 3.33% See Table B-3

17 Acceptability Criterion: Compare p = pU + pL with M 2.85% < 3.33% See Para. B12.1.2(7)

The lot meets the acceptability criterion, since p = pU + pL is less than M.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 40

EXAMPLE B-4




Different AQL Values for Upper and Lower Specification Limits

Example The minimum temperature of operation for a certain device is specified as 180°F. The maximum temperature is209°F. A lot of 40 items is submitted for inspection. Inspection Level II, normal inspection, with AQL = 1% forthe upper and AQL = 2.5% for the lower specification limit is to be used. From Tables A-2 and B-3 it is seenthat a sample of size 5 is required. Suppose the measurements obtained are as follows: 197°, 188°, 184°, 205°,and 201°; and compliance with the acceptability criterion is to be determined.


1 Sample Size: n 5





6 Variance (V): SS/(n – 1) 77.5 310/4


–: ∑X/n 195 975/5




)/s 1.59 (209 – 195)/8.80


– L)/s 1.70 (195 – 180)/8.80

13 Est. of Lot Percent Ncf. above U: pU 2.19% See Table B-5

14 Est. of Lot Percent Ncf. below L: pL 0.66% See Table B-5


16 Max. Allowable Percent Ncf. Above U: MU 3.33% See Table B-3

17 Max. Allowable Percent Ncf. below L: ML 9.80% See Table B-3

18 Acceptability Criteria: (a) Compare pU with MU 2.19% < 3.33% See Para. B12.2.2(7)(a)

(b) Compare pL with ML 0.66% < 9.80% See Para. B12.2.2(7)(b)

(c) Compare p with ML 2.85% < 9.80% See Para. B12.2.2(7)(c)

The lot meets the acceptability criteria, since 18(a), (b), and (c) are satisfied; i.e. pU < MU, pL < ML, and p < ML.

ANSI/ASQ Z1.9-2008

41

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 41

ANSI/ASQ Z1.9-2008

42

Tabl

e B

-3St

anda

rd D

evia

tion

Met

hod

Mas

ter

Tabl

e fo

r N

orm

al a

nd T

ight

ened

Ins

pect

ion

for

Plan

s B

ased

on

Var

iabi

lity

Unk

now

n(D

oubl

e Sp

ecifi

catio

n L

imit

and

For

m 2

—Si

ngle

Spe

cific

atio

n L

imit)

Sam

ple

Size

Cod

eL

ette

r

Sam

ple

Size

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)

T M

.10

M

.15

M

.25

M

.40

M

.65

M

1.00 M

1.50 M

2.50 M

4.00 M

6.50 M

10.0

0

M


3 4 5 7 10 15 20 25 50 75 100

150

20035

33.6

9

29.4

3

26.5

5

23.3

0

20.7

3

18.9

7

18.0

7

17.5

5

15.8

7

15.0

7

14.7

1

14.1

8

14.1

1

16.6

7

26.9

4

22.8

4

20.1

9

17.3

4

15.1

7

13.7

4

13.0

1

12.6

0

11.2

3

10.5

8

10.2

9

9.8

6

9.8

0

10.0

0

11.8

9

18.8

6

16.4

1

14.3

7

12.1

9

10.5

3

9.4

8

8.9

5

8.6

5

7.6

1

7.1

0

6.8

8

6.5

6

6.5

2

6.50

8.1

1

7.5

9

10.8

8

9.8

0

8.4

0

7.2

6

6.5

5

6.1

8

5.9

8

5.2

1

4.8

3

4.6

7

4.4

2

4.3

9

4.00

5.5

8

5.46

5.82

5.34

4.72

4.32

4.10

3.97

3.44

3.17

3.06

2.88

2.86

2.50

3.70

1.49

3.33

3.54

3.27

3.06

2.93

2.86

2.47

2.27

2.18

2.05

2.04

1.50

2.66

1.34

2.13

2.14

2.09

2.03

2.00

1.73

1.59

1.52

1.42

1.42

1.00

1.87

0.04

1

1.05

1.27

1.33

1.33

1.32

1.16

1.06

1.02

0.94

6

0.94

5

.65

1.24

0.42

1

0.71

4

0.83

9

0.86

4

0.87

4

0.77

8

0.71

1

0.68

4

0.63

6

0.63

7

.40

0.83

3

0.08

7

0.34

9

0.49

1

0.53

1

0.55

1

0.50

3

0.46

1

0.44

4

0.41

2

0.41

4

.25

0.53

4

0.31

1

0.35

6

0.37

8

0.35

5

0.32

6

0.31

5

0.29

2

0.29

4

.15

0.37

3

0.18

6

0.22

8

0.25

0

0.00

5

0.17

90.

077

0.24

3

0.22

5

0.21

8

0.20

2

0.20

4

.10

0.25

3

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

T d

enot

es p

lan

used

exc

lusi

vely

on

tight

ened

insp

ectio

n an

dpr

ovid

es s

ymbo

l for

iden

tific

atio

n of

app

ropr

iate

OC

cur

ve.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 42

ANSI/ASQ Z1.9-2008

43

Tabl

e B

-4St

anda

rd D

evia

tion

Met

hod

Mas

ter

Tabl

e fo

r R

educ

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n V

aria

bilit

y U

nkno

wn

(Dou

ble

Spec

ifica

tion

Lim

it an

d F

orm

2—

Sing

le S

peci

ficat

ion

Lim

it)

Sam

ple

Size

Cod

eL

ette

r

Sam

ple

Size

Acc

epta

nce

Qua

lity

Lim

its

.10

M

.15

M

.25

M

.40

M

.65

M

1.00 M

1.50 M

2.50 M

4.00 M

6.50 M

10.0

0

M


3 3 3 3 4 5 7 10 20 25 30 50 7515

40.4

7

40.4

7

40.4

7

40.4

7

36.7

9

33.9

4

30.5

0

27.6

5

24.5

8

23.9

7

23.6

1

21.9

9

21.0

5

25.6

3

33.6

9

33.6

9

33.6

9

33.6

9

29.4

3

26.5

5

23.3

0

20.7

3

18.0

7

17.5

5

17.2

5

15.8

7

15.0

7

18.9

7

26.9

4

26.9

4

26.9

4

26.9

4

22.8

4

20.1

9

17.3

4

15.1

7

13.0

1

12.6

0

12.3

7

11.2

3

10.5

8

13.7

4

18.8

6

18.8

6

18.8

6

18.8

6

16.4

1

14.3

7

12.1

9

10.5

3

8.9

5

8.6

5

8.4

8

7.6

1

7.1

0

9.4

8

7.5

9

7.5

9

7.5

9

7.5

9

10.8

8

9.8

0

8.4

0

7.2

6

6.1

8

5.9

8

5.8

7

5.2

1

4.8

3

6.5

5

5.46

5.82

5.34

4.72

4.10

3.97

3.91

3.44

3.17

4.32

1.49

3.33

3.54

3.27

2.93

2.86

2.82

2.47

2.27

3.06

1.34

2.13

2.14

2.03

2.00

1.98

1.73

1.59

2.09

0.04

1

1.06

1.27

1.33

1.32

1.32

1.16

1.06

1.33

0.42

1

0.71

4

0.86

4

0.87

4

0.88

5

0.77

8

0.71

1

0.83

9

0.08

7

0.34

9

0.53

1

0.55

1

0.56

7

0.50

3

0.46

1

0.49

1

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s M

val

ue. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 43

ANSI/ASQ Z1.9-2008

44

Table B-5Table for Estimating the Lot Percent Nonconforming Using Standard Deviation Method1

1Values tabulated are read in percent.

Sample SizeQUorQL

0 .1 .2 .3 .31.32.33.34.35.36.37.38.39

50.0047.2444.4641.6341.3541.0640.7740.4940.2039.9139.6239.3339.03

50.0046.6743.3340.0039.6739.3339.0038.6738.3338.0037.6737.3337.00

50.0046.4442.9039.3739.0238.6738.3237.9737.6237.2836.9336.5836.23

50.0046.2642.5438.8738.5038.1437.7837.4237.0636.6936.3335.9835.62

50.0046.1642.3538.6038.2337.8637.4937.1236.7536.3836.0235.6535.29

50.0046.1042.2438.4438.0637.6937.3136.9436.5736.2035.8335.4635.10

50.0046.0842.1938.3737.9937.6237.2436.8736.4936.1235.7535.3835.02

50.0046.0642.1638.3337.9537.5837.2036.8336.4536.0835.7135.3434.97

50.0046.0542.1538.3137.9337.5537.1836.8036.4336.0535.6835.3134.94

50.0046.0542.1338.2937.9137.5437.1636.7836.4136.0435.6635.2934.93

50.0046.0442.1138.2737.8937.5137.1336.7536.3836.0135.6335.2634.89

50.0046.0342.1038.2537.8737.4937.1136.7336.3635.9835.6135.2434.87

50.0046.0342.0938.2437.8637.4837.1036.7236.3535.9735.6035.2334.86

50.0046.0242.0938.2337.8537.4737.0936.7136.3435.9635.5935.2234.85

50.0046.0242.0838.2237.8437.4637.0836.7136.3335.9635.5835.2134.84

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

38.7438.4538.1537.8537.5637.2636.9636.6636.3536.05

36.6736.3336.0035.6735.3335.0034.6734.3334.0033.67

35.8835.5435.1934.8534.5034.1633.8133.4733.1232.78

35.2634.9034.5534.1933.8433.4933.1332.7832.4332.08

34.9334.5734.2133.8533.4933.1332.7832.4232.0731.72

34.7334.3734.0033.6433.2832.9232.5732.2131.8531.50

34.6534.2833.9233.5633.2032.8432.4832.1231.7731.41

34.6034.2433.8733.5133.1532.7932.4332.0731.7231.36

34.5834.2133.8533.4833.1232.7632.4032.0431.6931.33

34.5634.1933.8333.4633.1032.7432.3832.0231.6731.31

34.5334.1633.7933.4333.0732.7132.3531.9931.6331.28

34.5034.1333.7733.4033.0432.6832.3231.9631.6131.25

34.4934.1233.7633.3933.0332.6732.3131.9531.6031.24

34.4834.1133.7533.3833.0232.6632.3031.9431.5831.23

34.4734.1133.7433.3833.0132.6532.2931.9331.5831.22

.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

35.7535.4435.1334.8234.5134.2033.8833.5733.2532.93

33.3333.0032.6732.3332.0031.6731.3331.0030.6730.33

32.4432.1031.7631.4231.0830.7430.4030.0629.7329.39

31.7431.3931.0430.7030.3630.0129.6729.3328.9928.66

31.3731.0230.6730.3229.9829.6429.2928.9528.6128.28

31.1530.8030.4530.1029.7629.4129.0728.7328.3928.05

31.0630.7130.3630.0129.6729.3228.9828.6428.3027.96

31.0130.6630.3129.9629.6229.2728.9328.5928.2527.92

30.9830.6330.2829.9329.5929.2428.9028.5628.2227.89

30.9630.6130.2629.9129.5729.2228.8828.5428.2027.87

30.9330.5730.2329.8829.5329.1928.8528.5128.1727.83

30.9030.5530.2029.8529.5129.1628.8228.4828.1427.81

30.8930.5430.1929.8429.4929.1528.8128.4728.1327.79

30.8830.5330.1829.8329.4829.1428.8028.4628.1227.78

30.8730.5230.1729.8229.4829.1328.7928.4528.1127.78

.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

32.6132.2831.9631.6331.3030.9730.6330.3029.9629.61

30.0029.6729.3329.0028.6728.3328.0027.6727.3327.00

29.0528.7228.3928.0527.7227.3927.0626.7326.4026.07

28.3227.9827.6527.3226.9926.6626.3326.0025.6825.35

27.9427.6027.2726.9426.6126.2825.9625.6325.3124.99

27.7227.3927.0526.7226.3926.0725.7425.4225.1024.78

27.6327.3026.9626.6326.3125.9825.6625.3325.0124.70

27.5827.2526.9226.5926.2625.9325.6125.2924.9724.65

27.5527.2226.8926.5626.2325.9025.5825.2624.9424.62

27.5327.2026.8726.5426.2125.8825.5625.2424.9224.60

27.5027.1626.8326.5026.1825.8525.5325.2124.8924.57

27.4727.1426.8126.4826.1525.8325.5125.1924.8724.55

27.4627.1326.8026.4726.1425.8225.4925.1724.8624.54

27.4527.1126.7826.4626.1325.8125.4825.1624.8524.53

27.4427.1126.7826.4526.1225.8025.4825.1624.8424.52

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 44

ANSI/ASQ Z1.9-2008

45

Table B-5–ContinuedTable for Estimating the Lot Percent Nonconforming Using Standard Deviation Method1


Sample SizeQUorQL

.70

.71

.72

.73

.74

.75

.76

.77

.78

.79

29.2728.9228.5728.2227.8627.5027.1326.7626.3926.02

26.6726.3326.0025.6725.3325.0024.6724.3324.0023.67

25.7425.4125.0924.7624.4424.1123.7923.4723.1522.83

25.0324.7124.3924.0723.7523.4423.1222.8122.5022.19

24.6724.3524.0323.7223.4123.1022.7922.4822.1821.87

24.4624.1523.8323.5223.2122.9022.6022.3021.9921.70

24.3824.0623.7523.4423.1322.8322.5222.2221.9221.63

24.3324.0223.7123.4023.0922.7922.4822.1821.8921.59

24.3123.9923.6823.3723.0722.7622.4622.1621.8621.57

24.2923.9823.6723.3623.0522.7522.4422.1421.8521.55

24.2623.9523.6423.3323.0222.7222.4222.1221.8221.53

24.2423.9223.6123.3123.0022.7022.4022.1021.8021.51

24.2323.9123.6023.3022.9922.6922.3922.0921.7821.50

24.2223.9023.5923.2922.9822.6822.3822.0821.7921.49

24.2123.9023.5923.2822.9822.6822.3822.0821.7821.49

.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

25.6425.2524.8624.4724.0723.6723.2622.8422.4221.99

23.3323.0022.6722.3322.0021.6721.3321.0020.6720.33

22.5122.1921.8721.5621.2420.9320.6220.3120.0019.69

21.8821.5821.2729.9720.6720.3720.0719.7819.4819.19

21.5721.2720.9829.6820.3920.1019.8119.5219.2318.95

21.4021.1020.8120.5220.2319.9419.6619.3819.1018.82

21.3321.0420.7520.4620.1719.8919.6019.3219.0518.77

21.2921.0020.7120.4220.1419.8619.5719.3019.0218.74

21.2720.9820.6920.4020.1219.8419.5619.2819.0018.73

21.2620.9720.6820.3920.1119.8219.5419.2718.9918.72

21.2320.9420.6520.3720.0919.8019.5319.2518.9818.70

21.2220.9320.6420.3520.0719.7919.5119.2418.9618.69

21.2120.9220.6320.3520.0619.7819.5119.2318.9618.69

21.2020.9120.6220.3420.0619.7819.5019.2318.9518.68

21.2020.9120.6220.3420.0519.7719.5019.2218.9518.68

.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

21.5521.1120.6620.1919.7319.2518.7518.2517.7417.21

20.0019.6719.3319.0018.6718.3318.0017.6717.3317.00

19.3819.0718.7718.4618.1617.8617.5517.2516.9616.66

18.9018.6118.3318.0417.7617.4817.2016.9216.6516.37

18.6718.3918.1117.8417.5617.2917.0316.7616.4916.23

18.5418.2718.0017.7317.4617.2016.9416.6816.4216.16

18.5018.2317.9617.6917.4317.1716.9016.6516.3916.14

18.4718.2017.9417.6717.4117.1616.8916.6316.3816.13

18.4618.1917.9217.6617.4017.1416.8816.6216.3716.12

18.4518.1817.9217.6517.3917.1316.8816.6216.3716.12

18.4318.1717.9017.6417.3817.1216.8716.6116.3616.12

18.4218.1617.8917.6317.3717.1216.8616.6116.3616.11

18.4218.1517.8917.6317.3717.1116.8616.6116.3616.11

18.4118.1517.8917.6217.3717.1116.8616.6116.3616.11

18.4118.1517.8817.6217.3617.1116.8616.6016.3616.11

1.001.011.021.031.041.051.061.071.081.09

16.6716.1115.5314.9314.3113.6612.9812.2711.5110.71

16.6716.3316.0015.6715.3315.0014.6714.3314.0013.67

16.3616.0715.7815.4815.1914.9114.6214.3314.0513.76

16.1015.8315.5615.3015.0314.7714.5114.2614.0013.75

15.9715.7215.4615.2114.9614.7114.4614.2213.9713.73

15.9115.6615.4115.1714.9214.6814.4414.2013.9713.74

15.8915.6415.4015.1514.9114.6714.4414.2013.9713.74

15.8815.6315.3915.1514.9114.6714.4414.2113.9813.75

15.8815.6315.3915.1514.9114.6714.4414.2113.9813.75

15.8715.6315.3815.1514.9114.6714.4414.2113.9813.76

15.8715.6315.3815.1514.9114.6814.4514.2213.9913.77

15.8715.6215.3815.1514.9114.6814.4514.2213.9913.77

15.8715.6215.3815.1514.9114.6814.4514.2214.0013.77

15.8715.6215.3915.1514.9114.6814.4514.2314.0013.78

15.8715.6215.3915.1514.9114.6814.4514.2214.0013.78

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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46

Table B-5—ContinuedTable for Estimating the Lot Percent Nonconforming Using Standard Deviation Method1


Sample SizeQUorQL

1.101.111.121.131.141.151.161.171.181.19

9.848.897.826.605.082.870.000.000.000.00

13.3313.0012.6712.3312.0011.6711.3311.0010.6710.33

13.4813.2012.9312.6512.3712.1011.8311.5611.2911.02

13.4913.2513.0012.7512.5112.2712.0311.7911.5611.33

13.5013.2613.0312.8012.5712.3412.1211.9011.6811.46

13.5113.2813.0512.8312.6112.3912.1811.9611.7511.54

13.5213.2913.0712.8512.6312.4212.2112.0011.7911.58

13.5213.3013.0812.8612.6512.4412.2212.0211.8111.61

13.5313.3113.0912.8712.6612.4512.2412.0311.8211.62

13.5413.3113.1012.8812.6712.4612.2512.0411.8411.63

13.5413.3213.1112.8912.6812.4712.2612.0611.8511.65

13.5513.3313.1212.9012.6912.4812.2812.0711.8711.67

13.5513.3413.1212.9112.7012.4912.2812.0811.8811.68

13.5613.3413.1312.9112.7012.4912.2912.0911.8811.69

13.5613.3413.1312.9212.7112.5012.2912.0911.8911.69

1.201.211.221.231.241.251.261.271.281.29

0.000.000.000.000.000.000.000.000.000.00

10.00 9.67 9.33 9.008.678.338.007.677.337.00

10.7610.5010.239.979.729.469.218.968.718.46

11.1010.8710.6510.4210.209.989.779.559.349.13

11.2411.0310.8210.6110.4110.2110.009.819.619.42

11.3411.1310.9310.7310.5310.3410.15

9.969.779.58

11.3811.1810.9810.7810.5910.4010.2110.029.849.66

11.4111.2111.0110.8110.6210.4310.2510.069.889.70

11.4211.2211.0310.8410.6410.4610.2710.099.909.72

11.4311.2411.0410.8510.6610.4710.2910.109.929.74

11.4611.2611.0710.8810.6910.5010.3210.139.959.78

11.4711.2811.0910.9010.7110.5210.3410.169.989.80

11.4811.2911.0910.9110.7210.5310.3510.179.999.82

11.4911.3011.1010.9210.7310.5410.3610.1810.009.83

11.4911.3011.1110.9210.7310.5510.3710.1910.019.83

1.301.311.321.331.341.351.361.371.381.39

0.000.000.000.000.000.000.000.000.000.00

6.676.336.005.675.335.004.674.334.003.67

8.217.977.737.497.257.026.796.566.336.10

8.938.728.528.328.127.927.737.547.357.17

9.229.038.858.668.488.308.127.957.777.60

9.409.229.048.868.698.528.358.188.017.85

9.489.309.128.958.788.618.448.288.127.96

9.529.349.179.008.838.668.508.338.178.01

9.559.379.209.038.868.698.538.378.218.05

9.579.399.229.058.888.728.558.398.248.08

9.609.439.269.098.928.768.608.448.288.12

9.639.469.299.128.958.798.638.478.318.16

9.649.479.309.138.978.818.658.498.338.18

9.659.489.319.158.988.828.668.508.358.19

9.669.499.329.158.998.838.678.518.368.20

1.401.411.421.431.441.451.461.471.481.49

0.000.000.000.000.000.000.000.000.000.00

3.333.002.672.332.001.671.331.00.67.33

5.885.665.445.235.024.814.604,394.193.99

6.986.806.626.456.276.105.935.775.605.44

7.447.277.106.946.786.636.476.326.176.02

7.697.537.377.227.076.926.776.636.486.34

7.807.647.497.347.197.046.906.756.616.48

7.867.707.557.407.267.116.976.836.696.55

7.907.747.597.447.307.157.016.876.736.60

7.927.777.627.477.337.187.046.906.776.63

7.977.827.677.527.387.247.106.966.826.69

8.017.867.717.567.427.287.147.006.866.73

8.027.877.737.587.447.307.167.026.886.75

8.047.897.747.607.467.327.187.046.906.77

8.057.907.757.617.477.327.197.056.916.78

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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47



Sample SizeQUorQL

1.501.511.521.531.541.551.561.571.581.59

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

3.803.613.423.233.052.872.692.522.352.19

5.285.134.974.824.674.524.384.244.103.96

5.875.735.595.455.315.185.054.924.794.66

6.206.065.935.805.675.545.415.295.165.04

6.346.206.075.945.815.695.565.445.325.20

6.416.286.156.025.895.775.655.535.415.29

6.466.336.206.075.955.825.705.585.465.34

6.506.366.236.115.985.865.745.625.505.38

6.556.426.296.176.045.925.805.685.565.45

6.606.476.346.216.095.975.855.735.615.50

6.626.496.366.246.115.995.875.755.645.52

6.646.516.386.266.136.015.895.785.665.55

6.656.526.396.276.156.025.905.795.675.56

1.601.611.621.631.641.651.661.671.681.69

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

2.031.871.721.571.421.281.151.020.890.77

3.833.693.573.443.313.193.072.952.842.73

4.544.414.304.184.063.953.843.733.623.52

4.924.814.694.584.474.364.254.154.053.94

5.084.974.864.754.644.534.434.324.224.12

5.175.064.954.844.734.624.524.424.324.22

5.235.125.014.904.794.684.584.484.384.28

5.275.165.044.944.834.724.624.524.424.32

5.335.225.115.014.904.794.694.594.494.39

5.385.275.165.064.954.854.744.644.554.45

5.415.305.195.084.984.874.774.674.574.47

5.435.325.215.115.004.904.804.704.604.50

5.445.335.235.125.014.914.814.714.614.51

1.701.711.721.731.741.751.761.771.781.79

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

0.660.550.450.360.270.190.120.060.020.00

2.622.512.412.302.202.112.011.921.831.74

3.413.313.213.113.022.932.832.742.662.57

3.843.753.653.563.463.373.283.203.113.03

4.023.933.833.743.653.563.473.383.303.21

4.124.023.933.843.753.663.573.483.403.32

4.184.093.993.903.813.723.633.553.473.38

4.224.134.043.943.853.773.683.593.513.43

4.304.204.114.023.933.843.763.673.593.51

4.354.264.174.083.993.903.813.733.643.56

4.384.294.194.104.013.933.843.763.673.59

4.414.314.224.134.043.953.873.783.703.62

4.424.324.234.144.053.973.883.803.713.63

1.801.811.821.831.841.851.861.871.881.89

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

1.651.571.491.411.341.261.191.121.060.99

2.492.402.322.252.172.092.021.951.881.81

2.942.862.792.712.632.562.482.412.342.28

3.133.052.982.902.822.752.682.612.542.47

3.2431.63.083.002.932.852.782.712.642.57

3.303.223.153.072.992.922.852.782.712.64

3.353.273.193.113.042.972.892.822.752.69

3.433.353.273.193.123.052.972.902.832.77

3.483.403.333.253.183.103.032.962.892.83

3.513.433.363.283.213.133.062.992.922.85

3.543.463.383.313.233.163.093.022.952.88

3.553.473.403.323.253.173.103.032.962.90

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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48



Sample SizeQUorQL

1.901.911.921.931.941.951.961.971.981.99

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.93 0.87 0.81 0.76 0.70 0.65 0.60 0.56 0.51 0.47

1.75 1.68 1.62 1.56 1.50 1.44 1.38 1.33 1.27 1.22

2.21 2.14 2.08 2.02 1.96 1.90 1.84 1.78 1.73 1.67

2.40 2.34 2.27 2.21 2.15 2.09 2.03 1.97 1.92 1.86

2.51 2.44 2.38 2.32 2.25 2.19 2.14 2.08 2.02 1.97

2.57 2.51 2.45 2.38 2.32 2.26 2.20 2.14 2.09 2.03

2.62 2.56 2.49 2.43 2.37 2.31 2.25 2.19 2.13 2.08

2.70 2.63 2.57 2.51 2.45 2.39 2.33 2.27 2.21 2.16

2.76 2.69 2.63 2.57 2.51 2.45 2.39 2.33 2.27 2.22

2.79 2.72 2.66 2.60 2.54 2.48 2.42 2.36 2.30 2.25

2.82 2.75 2.69 2.63 2.56 2.50 2.44 2.39 2.33 2.27

2.83 2.77 2.70 2.64 2.58 2.52 2.46 2.40 2.34 2.29

2.002.012.022.032.042.052.062.072.082.09

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.43 0.39 0.36 0.32 0.29 0.26 0.23 0.21 0.18 0.16

1.17 1.12 1.07 1.03 0.98 0.94 0.90 0.86 0.82 0.78

1.62 1.57 1.52 1.47 1.42 1.37 1.33 1.28 1.24 1.20

1.81 1.76 1.71 1.66 1.61 1.56 1.51 1.47 1.42 1.38

1.91 1.86 1.81 1.76 1.71 1.66 1.61 1.57 1.52 1.48

1.98 1.93 1.87 1.82 1.77 1.73 1.68 1.63 1.59 1.54

2.03 1.97 1.92 1.87 1.82 1.77 1.72 1.68 1.63 1.59

2.10 2.05 2.00 1.95 1.90 1.85 1.80 1.76 1.71 1.66

2.16 2.11 2.06 2.01 1.96 1.91 1.86 1.81 1.77 1.72

2.19 2.14 2.09 2.04 1.99 1.94 1.89 1.84 1.79 1.75

2.22 2.17 2.11 2.06 2.01 1.96 1.92 1.87 1.82 1.78

2.23 2.18 2.13 2.08 2.03 1.98 1.93 1.88 1.84 1.79

2.102.112.122.132.142.152.162.172.182.19

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.14 0.12 0.10 0.08 0.07 0.06 0.05 0.04 0.03 0.02

0.74 0.71 0.67 0.64 0.61 0.58 0.55 0.52 0.49 0.46

1.16 1.12 1.08 1.04 1.00 0.97 0.93 0.90 0.87 0.83

1.34 1.30 1.26 1.22 1.18 1.14 1.10 1.07 1.03 1.00

1.44 1.39 1.35 1.31 1.28 1.24 1.20 1.16 1.13 1.09

1.50 1.46 1.42 1.38 1.34 1.30 1.26 1.22 1.19 1.15

1.54 1.50 1.46 1.42 1.38 1.34 1.30 1.27 1.23 1.20

1.62 1.58 1.54 1.50 1.46 1.42 1.38 1.34 1.30 1.27

1.68 1.63 1.59 1.55 1.51 1.47 1.43 1.40 1.36 1.32

1.71 1.66 1.62 1.58 1.54 1.50 1.46 1.42 1.39 1.35

1.73 1.69 1.65 1.61 1.57 1.53 1.49 1.45 1.41 1.38

1.75 1.70 1.66 1.62 1.58 1.54 1.50 1.46 1.42 1.39

2.202.212.222.232.242.252.262.272.282.29

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0150.0100.0060.0030.0020.0010.0000.0000.0000.000

0.4370.4130.3890.3660.3450.3240.3040.2850.2670.250

0.8030.7720.7340.7150.6870.6600.6340.6090.5850.561

0.9680.9360.9050.8740.8450.8160.7890.7620.7350.710

1.1601.0280.9960.9650.9350.9050.8760.8480.8210.794

1.1201.0871.0541.0230.9920.9620.9330.9040.8760.849

1.1601.1281.0951.0631.0321.0020.9720.9430.9150.887

1.2331.1991.1661.1341.1021.0711.0411.0110.9820.954

1.2871.2531.2191.1861.1541.1231.0921.0621.0331.004

1.3141.2791.2451.2121.1801.1481.1171.0871.0581.029

1.3401.3051.2711.2381.2051.1731.1421.1121.0821.053

1.3521.3181.2841.2501.2181.1861.1551.1241.0951.065

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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49



Sample SizeQUorQL

2.302.312.322.332.342.352.362.372.382.39

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.2330.2180.2030.1890.1750.1630.1510.1390.1280.118

0.5380.5160.4950.4740.4540.4350.4160.3980.3810.364

0.6850.6610.6370.6140.5920.5710.5500.5300.5100.491

0.7690.7430.7190.6950.6720.6500.6280.6060.5860.566

0.8230.7970.7720.7480.7240.7010.6780.6560.6350.614

0.8610.8340.8090.7840.7600.7360.7140.6910.6700.648

0.9270.9000.8740.8480.8240.7990.7760.7530.7300.709

0.9770.9490.9230.8970.8720.8470.8230.7990.7770.754

1.0010.9740.9470.9210.8950.8700.8460.8220.7990.777

1.0250.9980.9710.9440.9190.8930.8690.8450.8220.799

1.0371.0090.9820.9560.9300.9050.8800.8560.8330.810

2.402.412.422.432.442.452.462.472.482.49

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.1090.1000.0910.0830.0760.0690.0630.0570.0510.046

0.3480.3320.3170.3020.2880.2750.2620.2490.2370.226

0.4730.4550.4370.4210.4040.3890.3730.3590.3450.331

0.5460.5270.5090.4910.4740.4570.4400.4250.4090.394

0.5940.5750.5550.5370.5190.5010.4840.4680.4520.436

0.6280.6080.5880.5690.5510.5330.5160.4990.4820.466

0.6870.6670.6460.6270.6080.5890.5710.5530.5360.519

0.7320.7110.6910.6700.6510.6320.6130.5950.5770.560

0.7550.7330.7120.6920.6720.6530.6340.6150.5970.580

0.7770.7550.7340.7130.6930.6730.6540.6360.6170.600

0.7870.7660.7440.7240.7030.6840.6640.6460.6270.609

2.502.512.522.532.542.552.562.572.582.59

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0410.0370.0330.0290.0260.0230.0200.0170.0150.013

0.2140.2040.1930.1840.1740.1650.1560.1480.1400.133

0.3170.3050.2920.2800.2680.2570.2460.2360.2260.216

0.3800.3660.3520.3390.3260.3140.3020.2910.2790.269

0.4210.4070.3920.3790.3650.3520.3400.3270.3160.304

0.4510.4360.4210.4070.3930.3790.3660.3540.3410.330

0.5030.4870.4720.4570.4420.4280.4140.4010.3880.375

0.5430.5270.5110.4950.4800.4650.4510.4370.4240.410

0.5630.5460.5300.5140.4990.4840.4690.4550.4410.428

0.5820.5650.5490.5330.5170.5020.4870.4730.4590.445

0.5920.5750.5590.5420.5270.5110.4960.4820.4680.454

2.602.612.622.632.642.652.662.672.682.69

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0110.0090.0080.0070.0060.0050.0040.0030.0020.002

0.1250.1180.1120.1050.0990.0940.0880.0830.0780.073

0.2070.1980.1890.1810.1720.1650.1570.1500.1430.136

0.2580.2480.2380.2290.2200.2110.2020.1940.1860.179

0.2930.2820.2720.2620.2520.2420.2330.2240.2160.208

0.3180.3070.2960.2850.2750.2650.2560.2460.2370.229

0.3630.3510.3390.3280.3170.3070.2960.2860.2770.267

0.3980.3850.3730.3610.3500.3390.3280.3170.3070.297

0.4150.4020.3900.3780.3660.3550.3440.3330.3220.312

0.4320.4190.4060.3940.3820.3710.3590.3480.3380.327

0.4410.4280.4150.4020.3900.3790.3670.3560.3450.335

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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50



Sample SizeQUorQL

2.702.712.722.732.742.752.762.772.782.79

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0000.0000.0000.0000.0000.000

0.0690.0640.0600.0570.0530.0490.0460.0430.0400.037

0.1300.1240.1180.1120.1070.1020.0970.0920.0870.083

0.1710.1640.1570.1510.1440.1380.1320.1260.1210.115

0.2000.1920.1840.1770.1700.1630.1570.1510.1450.139

0.2200.2120.2040.1970.1890.1820.1750.1680.1620.156

0.2580.2490.2410.2320.2240.2160.2090.2010.1940.187

0.2880.2780.2690.2600.2520.2430.2350.2270.2200.212

0.3020.2930.2830.2740.2660.2570.2490.2410.2230.220

0.3170.3070.2980.2880.2790.2710.2620.2540.2460.238

0.3250.3150.3050.2960.2860.2770.2690.2600.2520.244

2.802.812.822.832.842.852.862.872.882.89

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0350.0320.0300.0280.0260.0240.0220.0200.0190.017

0.0790.0750.0710.0670.0640.0600.0570.0540.0510.048

0.1100.1050.1010.0960.0920.0880.0840.0800.0760.073

0.1330.1280.1220.1170.1120.1080.1030.0990.0940.090

0.1500.1440.1380.1330.1280.1220.1180.1130.1080.104

0.1810.1740.1680.1620.1560.1500.1450.1390.1340.129

0.2050.1980.1920.1850.1790.1730.1670.1610.1550.150

0.2180.2110.2040.1970.1900.1840.1780.1720.1660.160

0.2300.2230.2160.2090.2020.1950.1890.1830.1770.171

0.2370.2290.2220.2150.2080.2010.1950.1880.1820.176

2.902.912.922.932.942.952.962.972.982.99

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0160.0150.0130.0120.0110.0100.0090.0090.0080.007

0.0460.0430.0410.0380.0360.0340.0320.0300.0280.027

0.0690.0660.0630.0600.0570.0540.0510.0490.0460.044

0.0870.0830.0790.0760.0720.0690.0660.0630.0600.057

0.1000.0960.0920.0880.0840.0810.0770.0740.0710.068

0.1250.1200.1150.1110.1070.1030.0990.0950.0910.088

0.1450.1400.1350.1300.1250.1210.1170.1120.1080.104

0.1550.1500.1450.1400.1350.1300.1260.1210.1170.113

0.1650.1600.1550.1490.1440.1400.1350.1300.1260.122

0.1710.1650.1600.1540.1490.1440.1400.1350.1300.126

3.003.013.023.033.043.053.063.073.083.09

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0060.0060.0050.0050.0040.0040.0030.0030.0030.002

0.0250.0240.0220.0210.0190.0180.0170.0160.0150.014

0.0420.0400.0380.0360.0340.0320.0300.0290.0270.026

0.0550.0520.0500.0480.0450.0430.0410.0390.0370.036

0.0650.0620.0590.0570.0540.0520.0500.0470.0450.043

0.0840.0810.0780.0750.0720.0690.0660.0640.0610.059

0.1010.0970.0930.0900.0870.0830.0800.0770.0740.072

0.1090.1050.1010.0980.0940.0910.0880.0850.0810.079

0.1180.1130.1100.1060.1020.0990.0950.0920.0890.086

0.1220.1180.1140.1100.1060.1030.0990.0960.0920.089

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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51



Sample SizeQUorQL

3.103.113.123.133.143.153.163.173.183.19

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0020.0020.0020.0020.0010.0010.0010.0010.0010.001

0.0130.0120.0110.0110.0100.0090.0090.0080.0070.007

0.0240.0230.0220.0210.0190.0180.0170.0160.0150.015

0.0340.0320.0310.0290.0280.0260.0250.0240.0220.021

0.0410.0390.0380.0360.0340.0330.0310.0300.0280.027

0.0560.0540.0520.0500.0480.0460.0440.0420.0400.038

0.0690.0660.0640.0610.0590.0570.0550.0530.0500.049

0.0760.0730.0700.0680.0650.0630.0600.0580.0560.054

0.0830.0800.0770.0740.0710.0690.0660.0640.0620.059

0.0860.0830.0800.0770.0750.0720.0690.0670.0650.062

3.203.213.223.233.243.253.263.273.283.29

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0060.0060.0050.0050.0050.0040.0040.0040.0030.003

0.0140.0130.0120.0110.0110.0100.0090.0090.0080.008

0.0200.0190.0180.0170.0160.0150.0150.0140.0130.012

0.0260.0240.0230.0220.0210.0200.0190.0180.0170.016

0.0370.0350.0340.0320.0310.0300.0280.0270.0260.025

0.0470.0450.0430.0410.0400.0380.0370.0350.0340.032

0.0520.0500.0480.0460.0440.0430.0420.0400.0380.037

0.0570.0550.0530.0510.0490.0480.0460.0440.0420.041

0.0600.0580.0560.0540.0520.0500.0480.0460.0450.043

3.303.313.323.333.343.353.363.373.383.39

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0030.0030.0020.0020.0020.0020.0020.0020.0010.001

0.0070.0070.0060.0060.0060.0050.0050.0050.0040.004

0.0120.0110.0100.0100.0090.0090.0080.0080.0070.007

0.0150.0150.0140.0130.0130.0120.0110.0110.0100.010

0.0240.0230.0220.0210.0200.0190.0180.0170.0160.016

0.0310.0300.0290.0270.0260.0250.0240.0230.0220.021

0.0350.0340.0320.0310.0300.0290.0280.0260.0250.024

0.0390.0380.0360.0350.0340.0320.0310.0300.0290.028

0.0420.0400.0380.0370.0360.0340.0330.0320.0310.029

3.403.413.423.433.443.453.463.473.483.49

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0010.0010.0010.000

0.0040.0030.0030.0030.0030.0030.0020.0020.0020.002

0.0070.0060.0060.0050.0050.0050.0050.0040.0040.004

0.0090.0090.0080.0080.0070.0070.0070.0060.0060.005

0.0150.0140.0140.0130.0120.0120.0110.0110.0100.010

0.0200.0200.0190.0180.0170.0160.0160.0150.0140.014

0.0230.0220.0220.0210.0200.0190.0180.0180.0170.016

0.0270.0260.0250.0240.0230.0220.0210.0200.0190.019

0.0280.0270.0260.0250.0240.0230.0220.0220.0210.020

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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Sample SizeQUorQL

3.503.513.523.533.543.553.563.573.583.59

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0020.0020.0020.0010.0010.0010.0010.0010.0010.001

0.0030.0030.0030.0030.0030.0030.0020.0020.0020.002

0.0050.0050.0050.0040.0040.0040.0040.0030.0030.003

0.0090.0090.0080.0080.0080.0070.0070.0060.0060.006

0.0130.0130.0120.0110.0110.0110.0100.0100.0090.009

0.0150.0150.0140.0140.0130.0120.0120.0110.0110.010

0.0180.0170.0160.0160.0150.0150.0140.0130.0130.012

0.0190.0180.0180.0170.0160.0160.0150.0140.0140.013

3.603.613.623.633.643.653.663.673.683.69

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0000.0000.0000.000

0.0020.0020.0020.0010.0010.0010.0010.0010.0010.001

0.0030.0030.0030.0020.0020.0020.0020.0020.0020.002

0.0060.0050.0050.0050.0040.0040.0040.0040.0040.003

0.0080.0080.0080.0070.0070.0070.0060.0060.0060.005

0.0100.0100.0090.0090.0080.0080.0080.0070.0070.007

0.0120.0110.0110.0100.0100.0100.0090.0090.0080.008

0.0130.0120.0120.0110.0110.0100.0100.0100.0090.009

3.703.713.723.733.743.753.763.773.783.79

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0010.0010.0000.000

0.0020.0010.0010.0010.0010.0010.0010.0010.0010.001

0.0030.0030.0030.0030.0030.0020.0020.0020.0020.002

0.0050.0050.0050.0050.0040.0040.0040.0040.0040.003

0.0060.0060.0060.0060.0050.0050.0050.0050.0040.004

0.0080.0070.0070.0070.0060.0060.0060.0060.0050.005

0.0080.0080.0080.0070.0070.0070.0070.0060.0060.006

3.803.813.823.833.843.853.863.873.883.893.90

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0000.0000.0000.0000.000

0.0020.0020.0020.0020.0010.0010.0010.0010.0010.0010.001

0.0030.0030.0030.0030.0030.0020.0020.0020.0020.0020.002

0.0040.0040.0040.0040.0030.0030.0030.0030.0030.0030.003

0.0050.0050.0050.0040.0040.0040.0040.0040.0030.0030.003

0.0060.0050.0050.0050.0050.0040.0040.0040.0040.0040.004

3 4 5 7 10 15 20 25 30 35 50 75 100 150 200

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53

Table B-6Values of F for Maximum Standard Deviation (MSD)

The MSD may be obtained by multiplying the factor F by the difference between the upper specification limit U and lowerspecification limit L. The formula is MSD = F (U – L). The MSD serves as a guide for the magnitude of the estimate of lotstandard deviation when using plans for the double specification limit case, based on the estimate of lot standard deviationof unknown variability. The estimate of lot standard deviation, if it is less than the MSD, helps to insure, but does not guar-antee, lot acceptability.

NOTE: There is a corresponding acceptability constant in Table B-1 for each value of F. For reduced inspection, find theacceptability constant of Table B-2 in Table B-1 and use the corresponding value of F.

All AQL values are in percent nonconforming. T denotes plan used exclusively on tightened inspection and provides sym-bol for identification of appropriate OC curve.

SampleSize

Acceptance Quality Limits (normal inspection)

.10 .15 .25 .40 .65 1.00 1.50 2.50 4.00 6.50 10.00

3

4

5

7

10

15

20

25

50

75

100

150

200

.502

.472

.452

.425

.403

.386

.377

.372

.357

.349

.346

.341

.340

.475

.432

.408

.381

.359

.344

.336

.331

.317

.311

.308

.304

.303

.453

.399

.372

.345

.324

.309

.302

.298

.285

.279

.276

.273

.272

.436

.374

.346

.318

.298

.284

.277

.273

.261

.256

.253

.250

.249

.353

.323

.295

.275

.262

.256

.252

.241

.236

.233

.230

.230

.338

.308

.280

.261

.248

.242

.238

.227

.223

.220

.217

.217

.294

.266

.247

.235

.229

.225

.215

.211

.209

.206

.205

.281

.253

.235

.222

.217

.213

.204

.199

.198

.195

.194

.242

.224

.212

.206

.203

.194

.190

.188

.185

.185

.231

.214

.202

.197

.194

.185

.181

.179

.176

.176

.195

.190

.187

.178

.174

.173

.170

.168

.224

.206

T

.188

.183

.180

30

35

.369

.364

.329

.324

.295

.291

.271

.267

.249

.246

.236

.232

.223

.220

.211

.208

.201

.198

.192

.189

.185

.182

.179

.176

.200

.172

.168

.167

.164

.164

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54

APPENDIX B

DefinitionsSymbol Read Definition

n Sample size for a single lot.

X–

X bar Sample mean. Arithmetic mean of sample measurements from a single lot.

X� = �∑nX�

s Estimate of lot standard deviation. Standard deviation of sample measure-ments from a single lot. (See Examples in Section B.)

s = �∑�X�2�–� �(�∑�X

n�)2

��___________n – 1

U Upper specification limit.

L Lower specification limit.

k The acceptability constant given in Tables B-1 and B-2.

QU Q sub U Quality index for use with Table B-5.

QL Q sub L Quality index for use with Table B-5.

pU p sub U Sample estimate of the lot percent nonconforming above U from Table B-5.

pL p sub L Sample estimate of the lot percent nonconforming below L from Table B-5.

p Total sample estimate of the lot percent nonconforming p = pU + pL.

M Maximum allowable percent nonconforming for sample estimates givenin Tables B-3 and B-4.

MU M sub U Maximum allowable percent nonconforming above U given in Tables B-3and B-4. (For use when different AQL values for U and L are specified.)

ML M sub L Maximum allowable percent nonconforming below L given in Tables B-3and B-4. (For use when different AQL values for U and L are specified.)

p– p bar Sample estimate of the process percent nonconforming, i.e., the estimatedprocess average.

p– U p bar sub U The estimated process average for an upper specification limit.

p–L p bar sub L The estimated process average for a lower specification limit.

F A factor used in determining the maximum standard deviation (MSD).The F values are given in Table B-6.

> Greater than Greater than.

< Less than Less than.

∑ Sum of Sum of.

T AQL symbol denoting plan used exclusively on tightened inspection (pro-vides identification of appropriate OC curve).

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 54

C1. SAMPLING PLAN FOR SINGLESPECIFICATION LIMIT

This part of the standard describes the procedures for usewith plans for a single specification limit when variabilityof the lot with respect to the quality characteristic isunknown and the range method is used. The acceptabilitycriterion is given in two equivalent forms. These are iden-tified as Form 1 and Form 2.

C1.1 Use of Sampling Plans. To determine whether thelot meets the acceptability criterion with respect to a par-ticular quality characteristic and AQL value, the applicablesampling plan shall be used in accordance with the provi-sions of Section A, General Description of the SamplingPlans, and those in this part of the Standard.

C1.2 Drawing of Samples. All samples shall be drawn inaccordance with paragraph A7.2.

C1.3 Determination of Sample Size Code Letter. Thesample size code letter shall be selected from Table A-2 inaccordance with paragraph A7.1.

C2. SELECTING THE SAMPLE PLAN WHENFORM 1 IS USED

C2.1 Master Sampling Tables. The master samplingtables for plans based on variability unknown for a singlespecification limit when using the range method are TablesC-1 and C-2. Table C-1 is used for normal and tightenedinspection and Table C-2 for reduced inspection.

C2.2 Obtaining the Sample Plan. The sampling planconsists of a sample size and an associated acceptabilityconstant.1 The sampling plan is obtained from MasterTable C-1 or C-2.

C2.2.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letter.

C2.2.2 Acceptability Constant. The acceptability constantk, corresponding to the sample size mentioned in paragraphC.2.2.1, is indicated in the column of the master table cor-responding to the applicable AQL value. Table C-1 isentered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans forreduced inspection are provided in Table C-2.

C3. LOT-BY-LOT ACCEPTABILITYPROCEDURES WHEN FORM 1 IS USED2

C3.1 Acceptability Criterion. The degree of conformanceof a quality characteristic with respect to a single specifi-cation limit shall be judged by the quantity (U – X

–)/R–

or(X–

– L)/R–

.

C3.2 Computation. The following quantity shall be com-puted: (U – X

–)/R–

or (X–

– L)/R–

, depending on whether thespecification limit is an upper or a lower limit, where


is the sample mean, andR–

is the average range of the sample.

In this standard, R–

is the average range of subgroup ranges.Each of the subgroups consists of 5 measurements, exceptfor those plans with sample size 3, 4, or 7 in which case thesubgroup size is the same as the sample size and the sam-ple range is used as R

–. In computing R

–, the order of the

sample measurements as made must be retained.Subgroups of consecutive measurements must be formedand the range of each subgroup obtained. R

–is the average

of the individual subgroup ranges.

ANSI/ASQ Z1.9-2008

55

SECTION C

VARIABILITY UNKNOWN—RANGE METHOD

Part ISINGLE SPECIFICATION LIMIT

1See Appendix C for definitions of all symbols used in the sampling plans based on variability unknown—range method.2See Example C-1 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 55

C3.3 Acceptability Criteria. Compare the quantity (U – X

–)/R

–or (X

–– L)/R

–with the acceptability constant k.

If (U – X–

)/R–

or (X–

– L)/R–

is equal to or greater than k, the lotmeets the acceptability criterion; if (U – X

–)/R–

or (X–

– L)/R–

is less than k or negative, then the lot does not meet theacceptability criterion.

C4. SUMMARY FOR OPERATION OFSAMPLING PLAN WHEN FORM 1 IS USED



(2) Obtain plan from Master Table C-1 or C-2 by select-ing the sample size n and the acceptability constant k.



and the average range ofthe sample R

–, and also compute the quantity (U – X

–)/R–

foran upper specification limit U or the quantity (X

–– L)/R

–

for a lower specification limit L.


)/R–

or (X–

– L)/R–

is equal to orgreater than k, the lot meets the acceptability criterion; if(U – X

–)/R–

or (X–

– L)/R–

is less than k or negative, then thelot does not meet the acceptability criterion.

C5. SELECTING THE SAMPLING PLANWHEN FORM 2 IS USED

C5.1 Master Sampling Tables. The master samplingtables for plans based on variability unknown for a singlespecification limit when using the range method areTables C-3 and C-4 of Part II. Table C-3 is used for nor-mal and tightened inspection and Table C-4 for reducedinspection.

C5.2 Obtaining the Sampling Plan. The sampling planconsists of a sample size and an associated maximumallowable percent nonconforming. The sampling plan isobtained from Master Table C-3 or C-4.

C5.2.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letter.

C5.2.2 Maximum Allowable Percent Nonconforming. Themaximum allowable percent nonconforming M for sampleestimates corresponding to the sample size mentioned inparagraphs C5.2.1 is indicated in the column of the mastertable corresponding to the applicable AQL value. Table C-3is entered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans for reducedinspection are provided in Table C-4.

C6. LOT-BY-LOT ACCEPTABILITYPROCEDURES WHEN FORM 2 IS USED3

C6.1 Acceptability Criterion. The degree of conformanceof a quality characteristic with respect to a single specifica-tion limit shall be judged by the percent of nonconformingproduct outside the upper or lower specification limit. Thepercentage of nonconforming product is estimated by enter-ing Table C-5 with the quality index and the sample size.

C6.2 Computation of Quality Index. The quality indexQU = (U – X

–)c/R

–shall be computed if the specification

limit is an upper limit U, or QL = (X–

– L)c/R–

if it is alower limit L. The quantities, X

–and R

–, are the sample

mean and average range of the sample, respectively. Thecomputation of R

–is explained in paragraph C3.2. The

factor c is provided in Master Tables C-3 and C-4 corre-sponding to the sample size code letter.

C6.3 Estimate of Percent Nonconforming in Lot. Thequality of a lot shall be expressed by pU, the estimated per-cent nonconforming in the lot above the upper specifica-tion limit, or by pL, the estimated percent nonconformingbelow the lower specification limit. The estimated percentnonconforming pU or pL is obtained by entering Table C-5with QU or QL and the appropriate sample size.

C6.4 Acceptability Criterion. Compare the estimated lotpercent nonconforming pU or pL with the maximum allow-able percent nonconforming M. If pU or pL is equal to orless than M, the lot meets the acceptability criterion; if pU


ANSI/ASQ Z1.9-2008

56

3See Example C-2 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 56

C7. SUMMARY OF OPERATION OFSAMPLING PLAN WHEN FORM 2 IS USED



(2) Obtain plan from Master Table C-3 or C-4 by select-ing the sample size n, the factor c, and the maximumallowable percent nonconforming M.



and the average range ofthe sample R

–.

(5) Compute the quality index QU = (U –X–

)c/R–

if theupper specification limit U is specified, or QL = (X

–– L)c/R

–

if the lower specification limit L is specified.

(6) Determine the estimated lot percent nonconformingpL or pU from Table C-5.

(7) If the estimated lot percent nonconforming pL or pU isequal to or less than the maximum allowable percent non-conforming M, the lot meets the acceptability criterion; ifpL or pU is greater than M or if QU or QL is negative, thenthe lot does not meet the acceptability criterion.

ANSI/ASQ Z1.9-2008

57

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EXAMPLE C-1



Variability Unknown—Range Method

Example: The lower specification limit for electrical resistance of a certain electrical component is 620 ohms. A lot of 100items is submitted for inspection. Inspection Level II, normal inspection, with AQL = .4% is to be used. FromTables A-2 and C-1 it is seen that a sample of size 10 is required. Suppose that values of the sample resistanc-es in the order reading from left to right are as follows:

643, 651, 619, 627, 658, (R1 = 658 – 619 = 39)670, 673, 641, 638, 650, (R2 = 673 – 638 = 35)

and compliance with the acceptability criterion is to be determined.


1 Sample Size n 10


3 Sample Mean (X–

): ∑X/n 647 6470/10

4 Average Range (R–

): ∑R/no. of subgroups 37 (39 + 35)/2

5 Specification Limit (Lower): L 620

6 The quantity: (X–

– L)/R–

.730 (647 – 620)/37

7 Acceptability Constant: k .811 See Table C-1

8 Acceptability Criterion: Compare (X–

– L)/R–

with k .730 < .811 See Para. C3.3

The lot does not meet the acceptability criterion, since (X–

– L)/R–

is less than k.

NOTE: If a single upper specification limit U is given, then compute the quantity (U – X–

)/R–

in line 6 and compare it with k; the lot meets the acceptability criterion if (U – X

–)/R–

is equal to or greater than k.

ANSI/ASQ Z1.9-2008

58

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EXAMPLE C-2



Variability Unknown—Range Method

Example: The lower specification limit for electrical resistance of a certain electrical component is 620 ohms. A lot of 100items is submitted for inspection. Inspection Level II, normal inspection, with AQL = .4% is to be used. FromTables A-2 and C-1 it is seen that a sample of size 10 is required. Suppose the values of the sample resistancesin the order reading from left to right are as follows:

643, 651, 619, 627, 658, (R1 = 658 – 619 = 39)670, 673, 641, 638, 650, (R2 = 673 – 638 = 35)



1 Sample Size n 10


3 Sample Mean X–

: ∑X/n 647 6470/10

4 Average Range R–

: ∑R/no. of subgroups 37 (39 + 35)/2

5 Factor c 2.405 See Table C-3

6 Specification Limit (Lower): L 620


– L)c/R–

1.76 (647 – 620)2.405/37

8 Est. of Lot Percent Ncf.: pL 2.54% See Table C-5

9 Max. Allowable Percent Ncf.: M 1.14% See Table C-3

10 Acceptability Criterion: Compare pL with M 2.54% > 1.14% See Para. C6.4

The lot does not meet the acceptability criterion, since pL is greater than M.

NOTE: If a single upper specification limit U is given, then compute the quality index QU = (U – X–

)c/R–

in line 7 andobtain the estimate of lot percent nonconforming pU. Compare pU with M; the lot meets the acceptability criterion, if pU is equal to or less than M.

ANSI/ASQ Z1.9-2008

59

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ANSI/ASQ Z1.9-2008

60

Tabl

e C

-1R

ange

Met

hod

Mas

ter

Tabl

e fo

r N

orm

al a

nd T

ight

ened

Ins

pect

ion

for

Plan

s B

ased

on

Var

iabi

lity

Unk

now

n(S

ingl

e Sp

ecifi

catio

n L

imit—

For

m 1

)

Sam

ple

Size

Cod

eL

ette

r

Sam

ple

Size

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)

T k

.10 k

.15 k

.25 k

.40 k

.65 k

1.00 k

1.50 k

2.50 k

4.00 k

6.50 k

10.0

0

k


3 4 5 7 10 15 25 30 60 85 115

175

23040

.296

.276

.272

.266

.341

.368

.398

.403

.432

.441

.449

.460

.462

.415

.401

.364

.352

.336

.424

.452

.484

.490

.521

.530

.539

.552

.553

10.0

0

.503

.502

.450

.431

.405

.507

.536

.571

.577

.610

.621

.631

.644

.646

6.50

.591

.587

.525

.498

.465

.579

.610

.647

.654

.689

.701

.711

.726

.728

4.00

.668

.598

.565

.525

.650

.684

.723

.730

.768

.780

.791

.807

.809

2.50

.746

.651

.614

.569

.703

.738

.779

.787

.826

.839

.851

.868

.870

1.50

.803

.663

.613

.755

.792

.835

.843

.885

.899

.911

.929

.931

1.00

.860

.659

.811

.850

.896

.904

.948

.962

.975

.994

.996 .65

.921

.70

2

.86

3

.90

3

.95

1

.95

9

1.00

1.02

1.03

1.05

1.06

.40

.97

8

.91

6

.95

8

1.01

1.02

1.06

1.08

1.09

1.11

1.12

.25

1.04

.99

9

1.05

1.06

1.11

1.13

1.14

1.16

1.16

.15

1.08

1.04

1.10

1.10

1.16

1.17

1.19

1.21

1.21 .10

1.13

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

T d

enot

es p

lan

used

exc

lusi

vely

on

tight

ened

insp

ectio

n an

dpr

ovid

es s

ymbo

l for

iden

tific

atio

n of

app

ropr

iate

OC

cur

ve.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 60

ANSI/ASQ Z1.9-2008

61

Tabl

e C

-2R

ange

Met

hod

Mas

ter

Tabl

e fo

r R

educ

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n V

aria

bilit

y U

nkno

wn

(Sin

gle

Spec

ifica

tion

Lim

it—F

orm

1)

Sam

ple

Size

Cod

eL

ette

r

Sam

ple

Size

Acc

epta

nce

Qua

lity

Lim

its

.10 k

.15 k

.25 k

.40 k

.65 k

1.00 k

1.50 k

2.50 k

4.00 k

6.50 k

10.0

0

k


3 3 3 3 4 5 7 10 25 30 35 60 8515

.178

.178

.178

.178

.176

.184

.189

.252

.305

.310

.313

.336

.345

.276

.296

.296

.296

.296

.276

.272

.266

.341

.398

.403

.406

.432

.441

.368

.401

.401

.401

.401

.364

.352

.336

.424

.484

.490

.494

.521

.530

.452

.502

.502

.502

.502

.450

.431

.405

.507

.571

.577

.581

.610

.621

.536

.587

.587

.587

.587

.525

.498

.465

.579

.647

.654

.658

.689

.701

.610

.598

.565

.525

.650

.723

.730

.734

.768

.780

.684

.651

.614

.569

.703

.779

.787

.791

.826

.839

.738

.663

.613

.755

.835

.843

.848

.885

.899

.792

.659

.811

.896

.904

.908

.948

.962

.850

.70

2

.86

3

.95

1

.95

9

.96

4

1.00

1.02

.90

3

.91

6

1.01

1.02

1.02

1.06

1.08

.95

8

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 61

ANSI/ASQ Z1.9-2008

62

C8. SAMPLING PLAN FOR DOUBLESPECIFICATION LIMIT

This part of the standard describes the procedures for usewith plans for a double specification limit when variabili-ty of the lot with respect to the quality characteristic isunknown and the range method is used.

C8.1 Use of Sampling Plans. To determine whether thelot meets the acceptability criterion with respect to a par-ticular quality characteristic and AQL value(s), the appli-cable sampling plan shall be used in accordance with theprovisions of Section A, General Description of SamplingPlans, and those in this part of the standard.

C9. SELECTING THE SAMPLE PLAN

A sampling plan for each AQL value shall be selected fromTable C-3 or C-4 as follows:

C9.1 Determination of Sample Size Code Letter. Thesample size code letter shall be selected from Table A-2 inaccordance with paragraph A7.1.

C9.2 Master Sampling Tables. The master samplingtables for plans based on variability unknown for a doublespecification limit when using the range method are TablesC-3 and C-4. Table C-3 is used for normal and tightenedinspection and Table C-4 for reduced inspection.

C9.3 Obtaining Sampling Plan. A sampling plan consistsof a sample size and the associated maximum allowablepercent nonconforming. The sampling plan to be applied ininspection shall be obtained from Master Table C-3 or C-4.

C9.3.1 Sample Size. The sample size n is shown in themaster tables corresponding to each sample size code letter.

C9.3.2 Maximum Allowable Percent Nonconforming.The maximum allowable percent nonconforming for sam-ple estimates of percent nonconforming for the lower,upper, or both specification limits combined, correspon-ding to the sample size mentioned in paragraph C9.3.1, isshown in the column of the master table corresponding to

the applicable AQL value(s). If different AQLs areassigned to each specification limit, designate the maxi-mum allowable percent nonconforming by ML for thelower limit, and by MU for the upper limit. If one AQL isassigned to both limits combined, designate the maximumallowable percent nonconforming by M. Table C-3 isentered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans forreduced inspection are provided in Table C-4.

C10. DRAWING OF SAMPLES


C11. LOT-BY-LOT ACCEPTABILITYPROCEDURES

C11.1 Acceptability Criterion. The degree of confor-mance of a quality characteristic with respect to a doublespecification limit shall be judged by the percent of non-conforming product. The percentage of nonconformingproduct is estimated by entering Table C-5 with the quali-ty index and the sample size.

C11.2 Computation of Quality Indices. The qualityindices QU = (U – X

–)c/R

–and QL = (X

–– L)c/R

–shall be com-

puted, where

U is the upper specification limit,L is the lower specification limit,c is a factor provided in Tables C-3 and C-4,X–

is the sample mean, andR–

is the average range of the sample.

In this standard, R–

is the average range of the subgroupranges. Each of the subgroups consists of 5 measurements,except for those plans with sample size 3, 4, or 7 in whichcase the subgroup size is the same as the sample size andthe sample range is used as R

–. In computing R

–, the order of

the sample measurements as made must be retained.Subgroups of consecutive measurements must be formedand the range of each subgroup obtained. R

–is the average

of the individual subgroup ranges.

Part IIDOUBLE SPECIFICATION LIMIT

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 62

C11.3 Percent Nonconforming in the Lot. The quality ofa lot shall be expressed in terms of the lot percent noncon-forming. Its estimate will be designated by pL, pU, or p. Theestimate pU indicates conformance with respect to theupper specification limit, pL with respect to the lower spec-ification limit, and p for both specification limits com-bined. The estimate pL and pU shall be determined by enter-ing Table C-5, respectively with QL and QU and the samplesize. The estimate p shall be determined by adding the cor-responding estimated percents nonconforming pL and pU

found in the table.

C12. ACCEPTABILITY CRITERION ANDSUMMARY FOR OPERATION OFSAMPLING PLANS

C12.1 One AQL Value for Both Upper and LowerSpecification Limit Combined.

C12.1.1 Acceptability Criterion.4 Compare the estimatedlot percent nonconforming p = pU + pL with the maximumallowable percent nonconforming M. If p is equal to or lessthan M, the lot meets the acceptability criterion; if p isgreater than M or if either QU or QL or both are negative,then the lot does not meet the acceptability criterion.

C12.1.2 Summary for Operation of Sampling Plan. Incases where a single AQL value is established for theupper and lower specification limit combined for a singlequality characteristic, the following steps summarize theprocedures to be used:


(2) Select plan from Master Table C-3 or C-4. Obtain thesample size n, the factor c, and the maximum allowablepercent nonconforming M.



and average range of thesample R

–.


)c/R–

andQL = (X

–– L)c/R

–.

(6) Determine the estimated lot percent nonconformingp = pU + pL from Table C-5.

(7) If the estimated lot percent nonconforming p isequal to or less than the maximum allowable percentnonconforming M, the lot meets the acceptability crite-rion; if p is greater than M or if either QU or QL or bothare negative, then the lot does not meet the acceptabilitycriterion.

C12.2 Different AQL Values for Upper and LowerSpecification Limit.

C12.2.1 Acceptability Criteria.5 Compare the estimatedlot percent nonconforming pL and pU with the correspondingmaximum allowable percents nonconforming ML and MU;also compare p = pL + pU with the larger of ML and MU. IfpL is equal to or less than ML, pU is equal to or less thanMU, and p is equal to or less than the larger of ML and MU,the lot meets the acceptability criteria; otherwise, the lot doesnot meet the acceptability criteria. If either QL or QU orboth are negative, then the lot does not meet the accept-ability criteria.

C12.2.2 Summary for Operation of Sampling Plan. Incases where a different AQL value is established for theupper and lower specification limit for a single qualitycharacteristic, the following steps summarize the proce-dures to be used:


(2) Select the sampling plan from Master Table C-3 or C-4.Obtain the sample size n, the factor c, and the maximumallowable percent nonconforming MU and ML, correspondingto AQL values for the upper and lower specification limits,respectively.


ANSI/ASQ Z1.9-2008

63

4See Example C-3 for a complete example of this procedure.5See Example C-4 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 63


and average range of thesample R

–.


)c/R–

andQL = (X

–– L)c/R

–.

(6) Determine the estimated lot percents nonconforming pU

and pL, corresponding to the percents nonconforming abovethe upper and below the lower specification limits. Alsodetermine the combined percent nonconforming p = pU + pL.

(7) If all three of the following conditions:(a) pU is equal to or less than MU,(b) pL is equal to or less than ML,(c) p is equal to or less than the larger of

ML and MU,


ANSI/ASQ Z1.9-2008

64

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EXAMPLE C-3



Variability Unknown—Average Range Method


Example: The specifications for electrical resistance of a certain electrical component are 650 ± 30 ohms. A lot of 100items is submitted for inspection. Inspection Level II, normal inspection, with AQL = .4% is to be used. FromTables A-2 and C-3 it is seen that a sample of size 10 is required. Suppose the values of the sample resistancein the order reading from left to right are as follows:

643, 651, 619, 627, 658, (R1 = 658 – 619 = 39)670, 673, 641, 638, 650, (R2 = 673 – 638 = 35)



1 Sample Size: n 10


3 Sample Mean X–

: ∑X/n 647 6470/10


: ∑R/no. of subgroups 37 (39 + 35)/2





)c/R–

2.15 (680 – 647)2.405/37


– L)c/R–

1.76 (647 – 620)2.405/37

10 Est. of Lot Percent Ncf. above U: pU 0.35% See Table C-5

11 Est. of Lot Percent Ncf. below L: pL 2.54% See Table C-5


13 Max. Allowable Percent Ncf.: M 1.14% See Table C-3

14 Acceptability Criterion: Compare p = pU + pL with M 2.89% > 1.14% See Para. C12.1.2(7)

The lot does not meet the acceptability criterion, since p = pU + pL is greater than M.

ANSI/ASQ Z1.9-2008

65

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EXAMPLE C-4



Variability Unknown—Average Range Method

Different AQL Value for Upper and Lower Specification Limits

Example: The specifications for electrical resistance of a certain electrical component are 650 ± 30 ohms. A lot of 100 itemsis submitted for inspection. Inspection Level II, normal inspection, with AQL = 2.5% for the upper and AQL = 1%for the lower specification limit is to be used. From Tables A-2 and C-3 it is seen that a sample of size 10 isrequired. Suppose the values of the sample resistances in the order reading from left to right are as follows:

643, 651, 619, 627, 658, (R1 = 658 – 619 = 39)670, 673, 641, 638, 650, (R2 = 673 – 638 = 35)

and compliance with the acceptability critera is to be determined.


1 Sample Size: n 10


3 Sample Mean X–

: ∑X/n 647 6470/10


: ∑R/no. of subgroups 37 (39 + 35)/2





)c/R–

2.15 (680 –647)2.405/37


– L)c/R–

1.76 (647 – 620)2.405/37

10 Est. of Lot Percent Ncf. above U: pU 0.35% See Table C-5

11 Est. of Lot Percent Ncf. below L: pL 2.54% See Table C-5


13 Max. Allowable Percent Ncf. above U: MU 7.42% See Table C-3

14 Max. Allowable Percent Ncf. below L: ML 3.23% See Table C-3

15 Acceptability Criteria: (a) Compare pU with MU 0.35% < 7.42% See Para. C12.2.2(7)(a)

(b) Compare pL with ML 2.54% < 3.23% See Para. C12.2.2(7)(b)

(c) Compare p with MU 2.89% < 7.42% See Para. C12.2.2(7)(c)

The lot meets the acceptability criteria, since 15(a), (b), and (c) are satisfied; i.e., pU < MU, pL < ML, and p < MU.

ANSI/ASQ Z1.9-2008

66

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ANSI/ASQ Z1.9-2008

67

Tabl

e C

-3R

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Use

fir

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; tha

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Sam

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Size

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Sam

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Size

Acc

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norm

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tight

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T M

.10

M

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M

.25

M

.40

M

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M

1.00 M

1.50 M

2.50 M

4.00 M

6.50 M

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0

M


3 4 5 7 10 15 25 30 60 85 115

175

23040

cfa

ctor

0.91

0

2.23

4

2.47

4

2.83

0

2.40

5

2.37

9

2.35

8

2.35

3

2.33

9

2.33

5

2.33

3

2.33

1

2.33

0

2.34

6

33.6

9

29.4

5

26.5

9

23.5

0

21.0

6

19.3

0

17.4

8

17.1

9

15.6

4

15.1

7

14.7

4

14.1

5

14.1

0

16.5

5

26.9

4

22.8

6

20.2

7

17.5

4

15.4

9

14.0

9

12.5

9

12.3

6

11.1

0

10.7

3

10.3

7

9.8

9

9.8

4

10.0

0

11.8

4

18.8

6

16.4

5

14.4

7

12.3

5

10.7

9

9.7

6

8.6

5

8.5

0

7.5

4

7.2

7

6.9

9

6.6

0

6.5

7

6.5

0

8.1

1

7.5

9

10.9

2

9.9

0

8.4

7

7.4

2

6.7

6

5.9

8

5.8

8

5.1

7

4.9

7

4.7

6

4.4

7

4.4

6

4.0

0

5.6

1

5.50

5.93

5.32

4.77

4.44

3.96

3.92

3.44

3.30

3.14

2.93

2.92

2.50

3.73

1.53

3.44

3.46

3.23

3.11

2.82

2.81

2.47

2.37

2.25

2.08

2.08

1.50

2.69

1.42

1.99

2.05

2.10

1.95

1.96

1.74

1.67

1.58

1.46

1.47

1.00

1.88

.89

1.14

1.30

1.27

1.29

1.16

1.12

1.06

.97

2

.97

6

.65

1.25

.28

.58

.786

.827

.856

.781

.755

.718

.655

.661

.40

.842

.23

.430

.506

.537

.504

.493

.468

.427

.432

.25

.539

.253

.336

.366

.356

.350

.333

.303

.308

.15

.375

.136

.214

.240

.244

.242

.230

.210

.215 .10

.252

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 67

ANSI/ASQ Z1.9-2008

68

Tabl

e C

-4R

ange

Met

hod

Mas

ter

Tabl

e fo

r R

educ

ed I

nspe

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n fo

r Pl

ans

Bas

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n V

aria

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y U

nkno

wn

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ble

Spec

ifica

tion

Lim

it an

d F

orm

2—

Sing

le S

peci

ficat

ion

Lim

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All

AQ

L v

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s ar

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cent

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ing.

Use

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st s

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M

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M

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2.50 M

4.00 M

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10.0

0

M


3 3 3 3 4 5 7 10 25 30 35 60 8515

cfa

ctor

1.91

0

1.91

0

1.91

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0

2.23

4

2.47

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0

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5

2.35

8

2.35

3

2.34

9

2.33

9

2.33

5

2.37

9

40.4

7

40.4

7

40.4

7

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7

36.9

0

33.9

5

30.6

6

27.9

0

23.7

9

23.4

2

23.2

1

21.6

3

21.0

5

25.9

2

33.6

9

33.6

9

33.6

9

33.6

9

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5

26.5

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8

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4

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7

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0

26.9

4

26.9

4

26.9

4

26.9

4

22.8

6

20.2

7

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4

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9

12.5

9

12.3

6

12.2

4

11.1

0

10.7

3

14.0

9

18.8

6

18.8

6

18.8

6

18.8

6

16.4

5

14.4

7

12.3

5

10.7

9

8.6

5

8.5

0

8.4

2

7.5

4

7.2

7

9.7

6

7.5

9

7.5

9

7.5

9

7.5

9

10.9

2

9.9

0

8.4

7

7.4

2

5.9

8

5.8

8

5.8

5

5.1

7

4.9

7

6.7

6

5.50

5.93

5.32

4.77

3.96

3.92

3.90

3.44

3.30

4.44

1.53

3.44

3.46

3.23

2.82

2.81

2.82

2.47

2.37

3.11

1.42

1.99

2.05

1.95

1.96

1.98

1.74

1.67

2.10

.89

1.14

1.27

1.29

1.33

1.16

1.12

1.30

.28

.58

.827

.856

.883

.781

.755

.786

.23

.506

.537

.564

.504

.493

.430

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 68

ANSI/ASQ Z1.9-2008

69

Table C-5Table for Estimating the Lot Percent Nonconforming Using Range Method1


Sample SizeQUorQL

0 .1 .2 .3 .31.32.33.34.35.36.37.38.39

50.0047.2444.4641.6341.3541.0640.7740.4940.2039.9139.6239.3339.03

50.0046.6743.3340.0039.6739.3339.0038.6738.3338.0037.6737.3337.00

50.0046.4442.9039.3739.0238.6738.3237.9737.6237.2836.9336.5835.62

50.0046.2942.6038.9538.5938.2337.8737.5137.1536.7936.4336.0735.72

50.0046.2042.4238.7038.3337.9637.6037.2336.8736.5036.1435.7835.41

50.0046.1342.2938.5138.1437.7737.3937.0236.6536.2935.9235.5535.19

50.0046.0842.1938.3838.0037.6337.2536.8836.5036.1335.7635.3935.02

50.0046.0742.1738.3437.9637.5937.2136.8436.4636.0935.7235.3534.98

50.0046.0642.1638.3237.9437.5737.1936.8236.4436.0735.7035.3334.96

50.0046.0542.1538.3137.9337.5537.1836.8036.4336.0535.6835.3134.94

50.0046.0442.1238.2737.8937.5137.1436.7636.3936.0135.6435.2734.90

50.0046.0342.1038.2637.8837.5037.1236.7436.3735.9935.6235.2534.88

50.0046.0342.1038.2437.8637.4837.1136.7336.3635.9735.6135.2434.87

50.0046.0242.0838.2337.8537.4737.0936.7136.3435.9635.5935.2234.85

50.0046.0242.0838.2237.8437.4637.0936.7136.3335.9635.5935.2234.85

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

38.7438.4538.1537.8537.5637.2636.9636.6636.3536.05

36.6736.3336.0035.6735.3335.0034.6734.3334.0033.67

35.8835.5435.1934.8534.5034.1633.8133.4733.1232.78

35.3635.0134.6534.3033.9533.6033.2432.8932.5532.20

35.0534.6934.3333.9833.6233.2732.9132.5632.2131.86

34.8234.4634.1033.7433.3833.0232.6632.3131.9631.60

34.6634.2933.9333.5733.2132.8532.4932.1331.7831.42

34.6234.2533.8933.5333.1732.8132.4532.0931.7431.38

34.5934.2333.8633.5033.1432.7832.4232.0631.7131.35

34.5834.2133.8533.4833.1232.7632.4032.0431.6931.33

34.5334.1733.8033.4433.0832.7232.3632.0031.6431.29

34.5134.1433.7833.4133.0532.6932.3331.9731.6231.26

34.4934.1233.7733.3933.0332.6732.3131.9531.6131.24

34.4834.1133.7533.3833.0232.6632.3031.9431.5931.23

34.4834.1133.7433.3833.0232.6632.3031.9431.5831.23

.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

35.7535.4435.1334.8234.5134.2033.8833.5733.2532.93

33.3333.0032.6732.3332.0031.6731.3331.0030.6730.33

32.4432.1031.7631.4231.0830.7430.4030.0629.7329.39

31.8531.5131.1630.8230.4730.1329.7929.4529.1128.77

31.5131.1630.8130.4630.1229.7829.4429.0928.7628.42

31.2530.9030.5530.2129.8629.5229.1828.8328.5028.16

31.0730.7230.3730.0229.6829.3328.9928.6528.3127.97

31.0330.6830.3329.9829.6429.2928.9528.6128.2727.93

31.0030.6530.3029.9529.6129.2628.9228.5828.2427.91

30.9830.6330.2829.9329.5929.2428.9028.5628.2227.89

30.9430.5930.2429.8929.5429.2028.8628.5228.1827.84

30.9130.5530.2129.8629.5229.1728.8328.4928.1527.82

30.8930.5530.1929.8429.5029.1528.8128.4728.1327.80

30.8830.5330.1829.8329.4829.1428.8028.4628.1227.78

30.8730.5230.1729.8329.4829.1428.7928.4528.1227.78

.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

32.6132.2831.9631.6331.3030.9730.6330.3029.9629.61

30.0029.6729.3329.0028.6728.3328.0027.6727.3327.00

29.0528.7228.3928.0527.7227.3927.0626.7326.4026.07

28.4428.1027.7727.4427.1126.7826.4526.1225.7925.47

28.0827.7527.4127.0826.7526.4226.1025.7725.4525.12

27.8227.4927.1626.8226.5026.1725.8425.5225.2024.88

27.6427.3126.9726.6426.3225.9925.6725.3425.0224.71

27.6027.2726.9326.6026.2825.9525.6325.3024.9824.67

27.5727.2426.9126.5826.2525.9225.6025.2824.9624.64

27.5527.2226.8926.5626.2325.9025.5825.2624.9424.62

27.5127.1726.8426.5126.1925.8625.5425.2224.9024.58

27.4827.1526.8226.4926.1625.8425.5225.2024.8824.56

27.4627.1426.8126.4826.1425.8325.5025.1824.8724.55

27.4527.1226.7926.4626.1325.8125.4825.1624.8524.53

27.4527.1126.7826.4526.1325.8025.4825.1624.8424.53

3 4 5 7 10 15 25 30 35 40 60 85 115 175 230

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 69

ANSI/ASQ Z1.9-2008

70

Table C-5—ContinuedTable for Estimating the Lot Percent Nonconforming Using Range Method1


Sample SizeQUorQL

.70

.71

.72

.73

.74

.75

.76

.77

.78

.79

29.2728.9228.5728.2227.8627.5027.1326.7626.3926.02

26.6726.3326.0025.6725.3325.0024.6724.3324.0023.67

25.7425.4125.0924.7624.4424.1123.7923.4723.1522.83

25.1424.8224.5024.1823.8623.5523.2322.9222.6022.29

24.8024.4824.1723.8523.5423.2222.9122.6022.3021.99

24.5624.2423.9323.6123.3022.9922.6922.3822.0821.78

24.3924.0723.7623.4523.1422.8422.5322.2321.9321.64

24.3524.0323.7223.4123.1022.8022.4922.1921.9021.60

24.3224.0123.7023.3923.0822.7822.4722.1721.8821.58

24.3123.9923.6823.3723.0722.7622.4622.1621.8621.57

24.2723.9523.6423.3323.0322.7222.4222.1221.8321.53

24.2523.9323.6223.3223.0122.7122.4122.1121.8121.52

24.2423.9123.6023.3023.0022.6922.3922.0921.8021.50

24.2223.9023.5923.2922.9822.6822.3822.0821.7821.49

24.2123.9023.5923.2922.9822.6822.3822.0821.7821.49

.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

25.6425.2524.8624.4724.0723.6723.2622.8422.4221.99

23.3323.0022.6722.3322.0021.6721.3321.0020.6720.33

22.5122.1921.8721.5621.2420.9320.6220.3120.0019.69

21.9821.6821.3721.0620.7620.4620.1619.8619.5719.27

21.6921.3921.0920.7920.4920.2019.9019.6119.3319.04

21.4821.1820.8920.5920.3020.0119.7319.4419.1618.88

21.3421.0420.7520.4620.1719.8919.6019.3219.0418.77

21.3021.0120.7220.4320.1519.8719.5819.3119.0318.75

21.2820.9920.7020.4120.1319.8519.5719.2919.0118.74

21.2720.9820.6920.4020.1219.8419.5619.2819.0018.73

21.2420.9520.6620.3720.0919.8119.5419.2518.9818.70

21.2220.9320.6420.3620.0819.7919.5219.2418.9718.69

21.2220.9320.6420.3520.0619.7919.5119.2418.9618.69

21.2020.9120.6220.3420.0619.7819.5019.2218.9518.68

21.2020.9120.6220.3420.0619.7819.5019.2218.9518.68

.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

21.5521.1120.6620.2019.7419.2518.7618.2517.7417.21

20.0019.6719.3319.0018.6718.3318.0017.6717.3317.00

19.3819.0718.7718.4618.1617.8617.5617.2516.9616.66

18.9818.6918.4018.1117.8217.5417.2616.9716.7016.42

18.7518.4718.1917.9117.6417.3617.0916.8216.5516.28

18.6018.3218.0517.7817.5117.2416.9816.7116.4516.19

18.5018.2217.9617.6917.4317.1716.9116.6516.3916.14

18.4818.2117.9517.6817.4217.1616.9016.6416.3816.13

18.4718.2017.9317.6717.4117.1516.8916.6316.3816.13

18.4618.1917.9217.6617.4017.1416.8816.6316.3716.12

18.4318.1717.9017.6517.3917.1316.8716.6216.3716.12

18.4218.1717.8917.6317.3717.1216.8616.6116.3616.11

18.4218.1617.8917.6317.3717.1216.8616.6116.3616.11

18.4118.1517.8817.6217.3617.1116.8616.6016.3616.11

18.4118.1517.8817.6217.3617.1116.8616.6016.3616.11

1.001.011.021.031.041.051.061.071.081.09

16.6716.1115.5314.9314.3113.6612.9812.2711.5110.71

16.6716.3316.0015.6715.3315.0014.6714.3314.0013.67

16.3616.0715.7815.4815.1914.9114.6214.3314.0513.76

16.1415.8715.6015.3315.0614.7914.5314.2714.0113.75

16.0215.7615.5015.2414.9814.7314.4814.2313.9813.74

15.9415.6815.4315.1814.9414.6914.4514.2113.9713.73

15.8915.6415.4015.1514.9114.6714.4414.2013.9713.74

15.8815.6315.3915.1514.9114.6714.4414.2113.9813.75

15.8815.6315.3915.1514.9114.6714.4414.2113.9813.75

15.8815.6315.3915.1514.9114.6714.4414.2113.9813.75

15.8715.6315.3915.1514.9114.6814.4414.2113.9913.76

15.8715.6215.3815.1514.9114.6814.4514.2213.9913.77

15.8715.6215.3815.1514.9114.6814.4514.2213.9913.77

15.8715.6215.3815.1514.9114.6814.4514.2214.0013.78

15.8715.6215.3815.1514.9114.6814.4514.2214.0013.78

3 4 5 7 10 15 25 30 35 40 60 85 115 175 230

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 70

ANSI/ASQ Z1.9-2008

71



Sample SizeQUorQL

1.101.111.121.131.141.151.161.171.181.19

9.848.897.826.605.080.290.000.000.000.00

13.3313.0012.6712.3312.0011.6711.3311.0010.6710.33

13.4813.2012.9312.6512.3712.1011.8311.5611.2911.02

13.5013.2412.9912.7412.4912.2512.0011.7611.5211.29

13.4913.2513.0212.7812.5512.3112.0811.8611.6311.41

13.5013.2713.0412.8112.5912.3712.1511.9311.7111.50

13.5213.2913.0712.8512.6312.4212.2112.0011.7911.58

13.5213.3013.0812.8612.6412.4312.2212.0111.8011.60

13.5213.3013.0812.8612.6512.4412.2312.0211.8111.61

13.5313.3113.0912.8712.6612.4512.2412.0311.8211.62

13.5413.3213.1012.8912.6712.4612.2512.0611.8411.65

13.5513.3213.1212.8912.6912.4812.2712.0711.8611.66

13.5513.3313.1212.9012.6912.4812.2812.0711.8811.68

13.5613.3413.1212.9112.7012.4912.2912.0811.8811.69

13.5613.3413.1212.9112.7012.4912.2912.0811.8811.69

1.201.211.221.231.241.251.261.271.281.29

0.000.000.000.000.000.000.000.000.000.00

10.00 9.67 9.33 9.008.678.338.007.677.337.00

10.7610.5010.239.979.729.469.218.968.718.46

11.0510.8210.5910.3610.139.919.699.479.259.04

11.1910.9710.7610.5410.3310.12 9.929.719.519.31

11.2911.0810.8810.6710.4710.2710.08

9.889.699.50

11.3811.1810.9810.7810.5810.3910.2010.019.839.64

11.4011.2011.0010.8010.6110.4210.2410.059.879.68

11.4111.2111.0210.8210.6310.4410.2610.079.899.71

11.4211.2211.0310.8410.6410.4610.2710.099.909.72

11.4511.2611.0610.8710.6810.4910.3110.139.959.77

11.4711.2711.0810.8910.7010.5110.3310.159.979.79

11.4711.2911.0910.9010.7110.5210.3410.179.999.81

11.4911.3011.1010.9110.7310.5410.3610.1810.009.83

11.4911.3011.1010.9110.7310.5410.3610.1810.009.83

1.301.311.321.331.341.351.361.371.381.39

0.000.000.000.000.000.000.000.000.000.00

6.676.336.005.675.335.004.674.334.003.67

8.217.977.737.497.257.026.796.566.336.10

8.838.628.418.208.007.807.607.407.217.02

9.118.928.738.548.358.167.987.807.627.45

9.329.138.958.778.598.418.248.077.907.73

9.479.299.118.948.778.608.438.278.117.95

9.519.339.158.988.818.648.488.318.157.99

9.539.359.189.018.848.678.518.348.188.02

9.559.379.209.038.868.698.538.378.218.05

9.599.429.259.088.918.758.598.438.268.11

9.629.459.289.118.948.788.628.468.308.14

9.649.479.309.138.968.808.648.488.328.17

9.659.489.319.148.988.828.668.508.348.19

9.659.489.319.158.988.828.668.508.358.19

1.401.411.421.431.441.451.461.471.481.49

0.000.000.000.000.000.000.000.000.000.00

3.333.002.672.332.001.671.331.000.670.33

5.885.665.445.235.014.814.604.394.193.99

6.836.656.466.286.105.935.755.585.415.24

7.277.106.936.766.606.446.286.125.965.81

7.577.417.257.096.936.786.636.486.346.19

7.797.637.487.337.187.036.896.746.606.47

7.847.687.537.387.247.096.956.806.666.53

7.887.717.567.417.287.136.996.856.716.57

7.907.747.597.447.307.157.016.876.736.60

7.967.817.667.517.377.237.096.956.816.67

8.007.857.707.547.417.277.136.996.856.72

8.027.877.727.577.437.297.157.016.876.74

8.037.887.747.597.457.307.177.036.896.76

8.047.897.747.607.467.327.187.046.906.77

3 4 5 7 10 15 25 30 35 40 60 85 115 175 230

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Sample SizeQUorQL

1.501.511.521.531.541.551.561.571.581.59

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

3.803.613.423.233.052.872.692.522.352.19

5.084.924.764.604.454.304.154.013.863.72

5.665.515.375.225.084.944.814.674.544.41

6.055.915.775.645.505.375.245.114.994.86

6.336.196.065.935.805.685.555.435.315.19

6.396.256.125.995.865.745.625.505.385.26

6.436.306.176.045.915.795.675.555.435.31

6.466.336.206.075.955.825.705.585.465.34

6.546.416.286.156.035.905.785.665.555.43

6.586.456.326.206.075.955.835.715.595.48

6.616.486.356.226.105.985.865.745.625.51

6.636.506.376.256.126.005.885.775.655.53

6.646.516.386.266.146.015.895.795.665.55

1.601.611.621.631.641.651.661.671.681.69

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

2.031.871.721.571.421.281.151.020.890.77

3.583.453.313.183.062.932.812.692.572.46

4.284.164.033.913.793.683.563.453.343.23

4.744.624.514.394.284.174.063.953.853.74

5.084.964.854.744.634.524.414.314.214.10

5.145.034.924.814.704.594.494.394.294.19

5.195.084.974.864.754.644.544.444.344.24

5.235.125.014.904.794.684.584.484.384.28

5.325.205.094.994.884.774.674.574.474.37

5.365.255.145.044.934.834.724.624.534.43

5.395.285.175.074.964.864.754.654.564.46

5.425.315.205.104.994.894.794.694.594.49

5.435.325.225.125.004.914.814.714.614.51

1.701.711.721.731.741.751.761.771.781.79

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

0.660.550.450.360.270.190.120.060.020.00

2.352.242.132.031.931.831.731.641.551.46

3.133.022.922.822.732.632.542.452.362.27

3.643.543.453.353.263.163.072.992.902.81

4.003.923.823.733.633.543.453.373.283.20

4.093.993.903.813.723.633.543.453.373.28

4.144.053.953.863.773.683.593.513.433.34

4.184.093.993.903.813.723.633.553.473.38

4.284.184.094.003.913.823.743.653.573.49

4.334.244.154.063.973.883.793.713.623.54

4.364.274.184.094.003.913.823.743.653.57

4.404.304.214.124.033.943.863.773.693.61

4.424.314.234.144.053.963.883.793.713.63

1.801.811.821.831.841.851.861.871.881.89

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

0.000.000.000.000.000.000.000.000.000.00

1.381.291.211.141.060.990.920.860.790.73

2.192.102.021.941.871.791.721.651.581.51

2.732.652.572.492.422.342.272.202.132.06

3.113.032.962.882.802.732.662.592.522.45

3.203.123.052.972.892.822.752.682.612.54

3.263.183.113.032.952.882.812.742.672.60

3.303.223.153.072.992.922.852.782.712.64

3.413.333.253.173.103.032.952.882.812.75

3.463.383.313.233.163.083.012.942.872.81

3.493.413.343.263.193.113.042.972.902.84

3.533.453.373.303.223.153.083.012.942.87

3.553.473.393.323.243.173.103.032.962.89

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Sample SizeQUorQL

1.901.911.921.931.941.951.961.971.981.99

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.67 0.62 0.56 0.51 0.46 0.42 0.37 0.33 0.30 0.26

1.45 1.38 1.32 1.26 1.20 1.15 1.09 1.04 0.99 0.94

1.99 1.93 1.86 1.80 1.74 1.68 1.62 1.57 1.51 1.46

2.38 2.32 2.25 2.19 2.13 2.07 2.01 1.95 1.90 1.84

2.47 2.41 2.34 2.28 2.22 2.16 2.10 2.04 1.99 1.93

2.53 2.47 2.41 2.34 2.28 2.22 2.16 2.10 2.05 1.99

2.57 2.51 2.45 2.38 2.32 2.26 2.20 2.14 2.09 2.03

2.68 2.61 2.55 2.49 2.43 2.37 2.31 2.25 2.19 2.14

2.74 2.67 2.61 2.55 2.49 2.43 2.37 2.31 2.25 2.20

2.77 2.70 2.64 2.58 2.52 2.46 2.40 2.34 2.28 2.23

2.81 2.74 2.68 2.61 2.55 2.49 2.43 2.38 2.32 2.26

2.83 2.76 2.70 2.63 2.57 2.51 2.45 2.40 2.34 2.28

2.002.012.022.032.042.052.062.072.082.09

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.23 0.20 0.17 0.14 0.12 0.10 0.08 0.06 0.05 0.03

0.89 0.84 0.80 0.75 0.71 0.67 0.63 0.60 0.56 0.53

1.41 1.36 1.31 1.26 1.21 1.17 1.12 1.08 1.04 1.00

1.79 1.74 1.69 1.64 1.59 1.54 1.49 1.45 1.40 1.36

1.88 1.83 1.78 1.73 1.68 1.63 1.58 1.54 1.49 1.45

1.94 1.89 1.83 1.78 1.73 1.69 1.64 1.59 1.55 1.50

1.98 1.93 1.87 1.82 1.77 1.73 1.68 1.63 1.59 1.54

2.08 2.03 1.98 1.93 1.88 1.83 1.78 1.74 1.69 1.64

2.14 2.09 2.04 1.99 1.94 1.89 1.84 1.79 1.75 1.70

2.17 2.12 2.07 2.02 1.97 1.92 1.87 1.82 1.78 1.73

2.21 2.16 2.10 2.05 2.00 1.95 1.91 1.86 1.81 1.77

2.23 2.18 2.12 2.07 2.02 1.97 1.93 1.88 1.83 1.79

2.102.112.122.132.142.152.162.172.182.19

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.49 0.46 0.43 0.40 0.38 0.35 0.32 0.30 0.28 0.26

0.96 0.92 0.88 0.85 0.81 0.78 0.75 0.71 0.68 0.65

1.32 1.28 1.24 1.20 1.16 1.13 1.09 1.06 1.02 0.99

1.41 1.36 1.32 1.28 1.25 1.21 1.17 1.13 1.10 1.06

1.46 1.42 1.38 1.34 1.30 1.26 1.22 1.18 1.15 1.11

1.50 1.46 1.42 1.38 1.34 1.30 1.26 1.22 1.19 1.15

1.60 1.56 1.52 1.48 1.44 1.40 1.36 1.32 1.28 1.25

1.66 1.61 1.57 1.53 1.49 1.45 1.41 1.38 1.34 1.30

1.69 1.64 1.60 1.56 1.52 1.48 1.44 1.41 1.37 1.33

1.72 1.68 1.64 1.60 1.56 1.52 1.48 1.44 1.40 1.37

1.74 1.70 1.66 1.62 1.58 1.54 1.50 1.46 1.41 1.39

2.202.212.222.232.242.252.262.272.282.29

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.2360.2170.1990.1820.1660.1500.1360.1230.1110.099

0.6250.5970.5700.5440.5190.4950.4710.4490.4270.406

0.9540.9220.8910.8610.8310.8020.7750.7480.7220.697

1.0300.9970.9660.9350.9050.8750.8470.8190.7920.766

1.0831.0581.0180.9860.9560.9260.8970.8690.8410.814

1.1221.0891.0561.0250.9940.9640.9350.9060.8780.851

1.2141.1801.1471.1151.0831.0521.0220.9930.9640.936

1.2671.2331.1991.1671.1351.1041.0731.0431.0140.986

1.2991.2651.2311.1971.1651.1341.1031.0731.0441.015

1.3301.2951.2611.2281.1951.1631.1321.1031.0731.044

1.3461.3111.2771.2441.2111.1791.1481.1181.0881.059

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Sample SizeQUorQL

2.302.312.322.332.342.352.362.372.382.39

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0890.0790.0700.0610.0540.0470.0400.0350.0290.025

0.3860.3670.3480.3300.3130.2960.2800.2650.2500.236

0.6720.6480.6240.6010.5790.5580.5380.5180.4980.479

0.7410.7160.6910.6680.6450.6230.6020.5800.5600.541

0.7890.7630.7390.7150.6910.6690.6460.6240.6040.584

0.8250.7990.7740.7500.7200.7030.6800.6580.6370.616

0.9090.8820.8560.8310.8070.7820.7590.7360.7140.693

0.9590.9310.9050.8790.8540.8290.8060.7820.7590.737

0.9880.9600.9340.9080.8820.8570.8330.8090.7870.764

1.0160.9880.9620.9350.9090.8840.8600.8360.8130.791

1.0311.0030.9760.9500.9240.8890.8740.8500.8270.804

2.402.412.422.432.442.452.462.472.482.49

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0210.0170.0140.0110.0090.0070.0050.0040.0030.002

0.2230.2100.1980.1860.1750.1650.1540.1450.1360.127

0.4610.4430.4260.4100.3930.3780.3620.3480.3330.321

0.5210.5030.4850.4670.4500.4340.4170.4030.3870.372

0.5640.5450.5260.5080.4910.4730.4560.4410.4250.409

0.5960.5770.5570.5390.5210.5030.4860.4700.4540.438

0.6710.6510.6310.6110.5930.5730.5560.5380.5220.504

0.7150.6950.6740.6540.6350.6160.5970.5790.5620.545

0.7420.7210.7010.6790.6600.6410.6220.6040.5860.569

0.7690.7470.7260.7050.6850.6650.6460.6270.6090.593

0.7820.7600.7390.7180.6980.6780.6590.6400.6220.605

2.502.512.522.532.542.552.562.572.582.59

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0000.0000.0000.0000.0000.0000.0000.000

0.1180.1110.1030.0960.0890.0830.0770.0710.0660.061

0.3070.2940.2820.2700.2580.2470.2370.2270.2170.207

0.3580.3450.3310.3190.3060.2940.2830.2720.2610.251

0.3950.3810.3670.3540.3400.3280.3160.3040.2920.282

0.4230.4090.3940.3810.3670.3540.3410.3280.3170.305

0.4890.4730.4580.4440.4280.4150.4010.3880.3760.363

0.5280.5120.4970.4810.4660.4510.4370.4240.4110.397

0.5520.5360.5190.5030.4880.4730.4590.4450.4320.418

0.5750.5580.5420.5260.5100.4950.4800.4660.4520.439

0.5870.5700.5530.5370.5220.5060.4910.4770.4630.449

2.602.612.622.632.642.652.662.672.682.69

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0560.0520.0480.0440.0400.0370.0340.0310.0280.025

0.1980.1890.1810.1730.1640.1570.1490.1430.1360.129

0.2400.2310.2210.2120.2030.1950.1860.1790.1710.164

0.2710.2600.2500.2410.2320.2230.2130.2050.1970.190

0.2940.2830.2730.2630.2530.2440.2340.2250.2170.209

0.3510.3390.3270.3160.3060.2950.2850.2750.2660.257

0.3850.3720.3600.3490.3380.3270.3160.3050.2960.286

0.4060.3930.3810.3680.3570.3460.3350.3240.3140.304

0.4260.4130.4000.3880.3760.3650.3530.3420.3320.321

0.4360.4230.4100.3980.3860.3750.3630.3520.3420.331

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Sample SizeQUorQL

2.702.712.272.732.742.752.762.772.782.79

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0230.0210.0190.0170.0150.0140.0120.0110.0100.008

0.1230.1170.1110.1060.1010.0960.0910.0860.0810.077

0.1560.1500.1430.1370.1310.1250.1200.1140.1090.103

0.1820.1740.1670.1600.1530.1470.1410.1350.1300.124

0.2010.1930.1850.1780.1710.1640.1580.1520.1460.140

0.2480.2390.2310.2220.2150.2070.2000.1920.1850.179

0.2770.2670.2590.2500.2410.2330.2250.2170.2100.202

0.2950.2850.2750.2660.2580.2480.2410.2320.2260.218

0.3110.3020.2920.2830.2740.2660.2570.2490.2410.233

0.3210.3110.3010.2920.2820.2740.2650.2570.2490.241

2.802.812.822.832.842.852.862.872.882.89

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0070.0070.0060.0050.0040.0040.0030.0030.0020.002

0.0740.0700.0660.0620.0590.0550.0530.0500.0470.044

0.0990.0940.0900.0850.0820.0780.0740.0700.0670.064

0.1180.1130.1090.1030.0990.0950.0910.0870.0820.079

0.1340.1290.1230.1180.1130.1090.1040.1000.0950.091

0.1720.1650.1590.1540.1480.1430.1370.1320.1270.122

0.1960.1890.1830.1760.1700.1640.1590.1520.1470.142

0.2100.2040.1940.1900.1840.1780.1720.1660.1600.155

0.2250.2180.2110.2040.1970.1910.1850.1790.1730.167

0.2330.2260.2190.2120.2050.1980.1920.1850.1790.173

2.902.912.922.932.942.952.962.972.982.99

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0020.0010.0010.0010.0010.0010.0010.0000.0000.000

0.0420.0390.0370.0350.0330.0310.0290.0270.0250.024

0.0610.0570.0550.0520.0490.0470.0440.0420.0390.038

0.0750.0720.0690.0660.0620.0590.0560.0540.0510.049

0.0880.0840.0800.0770.0730.0700.0670.0640.0610.058

0.1170.1120.1070.1040.1000.0960.0920.0880.0850.082

0.1380.1320.1270.1230.1180.1140.1100.1050.1010.098

0.1490.1450.1400.1340.1290.1250.1210.1160.1120.108

0.1610.1560.1510.1460.1410.1360.1320.1270.1230.119

0.1680.1620.1570.1510.1460.1420.1370.1320.1280.124

3.003.013.023.033.043.053.063.073.083.09

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0220.0220.0200.0190.0170.0160.0150.0140.0130.012

0.0360.0340.0320.0300.0280.0270.0250.0240.0220.021

0.0470.0440.0420.0400.0380.0360.0340.0320.0300.029

0.0560.0530.0500.0480.0450.0430.0410.0390.0370.036

0.0780.0750.0720.0690.0660.0640.0610.0590.0560.054

0.0940.0910.0870.0840.0810.0780.0750.0720.0690.067

0.1050.1010.0970.0940.0900.0860.0830.0800.0770.075

0.1150.1110.1070.1030.0990.0960.0920.0890.0860.083

0.1200.1160.1120.1080.1040.1010.0970.0940.0910.088

3 4 5 7 10 15 25 30 35 40 60 85 115 175 230

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76



Sample SizeQUorQL

3.103.113.123.133.143.153.163.173.183.19

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0110.0110.0100.0100.0090.0080.0080.0070.0060.006

0.0200.0190.0180.0170.0150.0140.0140.0130.0120.012

0.0270.0260.0250.0240.0220.0210.0200.0190.0170.017

0.0340.0320.0310.0290.0280.0260.0250.0240.0220.021

0.0510.0500.0480.0460.0440.0420.0400.0380.0360.034

0.0640.0610.0600.0570.0550.0530.0510.0490.0460.044

0.0720.0690.0670.0640.0620.0600.0570.0560.0530.052

0.0800.0770.0740.0720.0690.0670.0640.0620.0600.057

0.0850.0820.0790.0750.0730.0700.0670.0650.0630.060

3.203.213.223.233.243.253.263.273.283.29

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0050.0050.0040.0040.0040.0030.0030.0030.0030.003

0.0110.0100.0090.0090.0090.0080.0070.0070.0060.006

0.0160.0150.0140.0130.0130.0120.0110.0110.0100.009

0.0200.0190.0180.0170.0160.0150.0150.0140.0130.012

0.0330.0320.0310.0290.0280.0270.0250.0240.0230.023

0.0430.0410.0400.0370.0370.0350.0330.0320.0310.029

0.0490.0470.0450.0430.0420.0400.0390.0370.0360.034

0.0550.0530.0510.0490.0470.0460.0440.0420.0400.039

0.0580.0560.0540.0520.0500.0490.0470.0450.0430.042

3.303.313.323.333.343.353.363.373.383.39

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0030.0030.0020.0020.0020.0020.0020.0020.0010.001

0.0050.0050.0040.0040.0040.0040.0040.0040.0030.003

0.0090.0080.0070.0070.0070.0060.0060.0060.0050.005

0.0120.0110.0100.0100.0090.0090.0080.0080.0070.007

0.0210.0210.0200.0190.0180.0170.0160.0150.0140.014

0.0280.0270.0260.0250.0240.0230.0220.0210.0190.019

0.0330.0320.0300.0290.0280.0270.0260.0240.0240.022

0.0370.0360.0340.0330.0320.0310.0300.0280.0270.027

0.0400.0390.0370.0360.0350.0330.0320.0310.0300.029

3.403.413.423.433.443.453.463.473.483.49

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0010.0010.0010.000

0.0030.0020.0020.0020.0020.0020.0020.0020.0020.001

0.0050.0040.0040.0040.0040.0040.0030.0030.0030.003

0.0070.0060.0060.0050.0050.0050.0050.0040.0040.004

0.0130.0120.0120.0110.0110.0110.0100.0100.0090.009

0.0180.0180.0170.0160.0150.0140.0140.0140.0130.012

0.0210.0210.0200.0190.0180.0170.0170.0160.0150.015

0.0260.0250.0240.0230.0220.0210.0200.0190.0180.018

0.0280.0270.0260.0250.0240.0230.0220.0210.0200.020

3 4 5 7 10 15 25 30 35 40 60 85 115 175 230

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77



Sample SizeQUorQL

3.503.513.523.533.543.553.563.573.583.59

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0010.0010.0010.001

0.0020.0020.0020.0020.0020.0020.0010.0010.0010.001

0.0030.0030.0030.0030.0030.0030.0020.0020.0020.002

0.0080.0080.0070.0070.0070.0060.0060.0050.0050.005

0.0120.0110.0100.0100.0100.0090.0090.0080.0080.008

0.0140.0140.0130.0130.0120.0120.0110.0110.0100.010

0.0170.0160.0160.0150.0140.0140.0130.0120.0120.011

0.0190.0180.0170.0160.0150.0150.0140.0130.0130.012

3.603.613.623.633.643.653.663.673.683.69

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0000.0000.0000.000

0.0020.0020.0020.0010.0010.0010.0010.0010.0010.001

0.0050.0040.0040.0040.0030.0030.0030.0030.0030.002

0.0070.0070.0070.0060.0060.0060.0050.0050.0050.004

0.0090.0090.0090.0080.0880.0080.0070.0070.0060.006

0.0110.0110.0100.0100.0090.0090.0090.0080.0080.008

0.0120.0110.0110.0100.0100.0100.0090.0090.0080.008

3.703.713.713.733.743.753.763.773.783.79

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0010.0010.0010.0010.0010.0010.0010.0010.0000.000

0.0020.0020.0020.0020.0020.0020.0020.0020.0020.002

0.0040.0040.0040.0040.0040.0030.0030.0030.0030.003

0.0060.0060.0060.0060.0050.0050.0050.0050.0040.003

0.0070.0070.0070.0070.0060.0060.0060.0060.0050.005

0.0080.0070.0070.0070.0070.0060.0060.0060.0050.005

3.803.813.823.833.843.853.863.873.883.893.90

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

0.0020.0020.0020.0020.0010.0010.0010.0010.0010.0010.001

0.0030.0030.0030.0030.0020.0010.0010.0010.0010.0010.001

0.0030.0030.0030.0030.0030.0020.0020.0020.0020.0020.002

0.0050.0050.0050.0040.0040.0040.0040.0040.0040.0030.003

0.0050.0050.0050.0040.0040.0040.0040.0040.0040.0030.003

3 4 5 7 10 15 25 30 35 40 60 85 115 175 230

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78

Table C-6Values of f for Maximum Average Range (MAR)

The MAR may be obtained by multiplying the factor f by the difference between the upper specification limit U and lowerspecification limit L. The formula is MAR = f (U – L). The MAR serves as a guide for the magnitude of the average rangeof the sample when using plans for the double specification limit case, based on the average range of the sample ofunknown variability. The average range of the sample, if it is less than the MAR, helps to insure, but does not guarantee,lot acceptability.

NOTE: There is a corresponding acceptability constant in Table C-1 for each value of f. For reduced inspection, find theacceptability constant of Table C-2 in Table C-1 and use the corresponding value of f.

All AQL values are in percent nonconforming. T denotes plan used exclusively on tightened inspection and provides symbolfor identification of appropriate OC curve.

SampleSize

Acceptance Quality Limits (in percent nonconforming)

.10 .15 .25 .40 .65 1.00 1.50 2.50 4.00 6.50 10.00

3

4

5

7

10

15

25

30

60

85

115

175

230

.958

1.056

1.118

1.209

.977

.928

.879

.870

.830

.818

.808

.794

,792

.907

.965

1.011

1.086

.876

.830

.785

.776

.740

.729

.720

.708

.706

.865

.891

.923

.985

.793

.748

.707

.699

.666

.656

.648

.637

.633

.833

.836

.857

.910

.730

.688

.649

.642

.608

.602

.594

.584

.582

.788

.801

.846

.677

.637

.600

.593

.564

.555

.548

.538

.536

.756

.764

.804

.642

.602

.567

.560

.533

.525

.517

.508

.507

.730

.765

.610

.572

.537

.531

.505

.497

.490

.481

.480

.727

.579

.542

.509

.503

.478

.470

.464

.455

.454

.695

.553

.517

.486

.480

.455

.448

.442

.434

.432

.529

.493

.463

.457

.434

.427

.421

.413

.412

.477

.447

.442

.419

.412

.406

.399

.397

T

.460

.432

.426

35

40

50

.864

.852

.843

.771

.761

.752

.694

.684

.676

.637

.628

.621

.588

.580

.573

.556

.548

.542

.527

.519

.503

.499

.492

.486

.476

.469

.463

.454

.447

.441

.438

.432

.426

.423

.417

.411

.405

.398

.392

.384

.384

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APPENDIX C

Definitions

Symbol Read Definition


X–


R Range. The difference between the largest and smallest measurements in asubgroup. In this standard, the subgroup size is 5 except for those plans inwhich n = 3, 4, or 7, in which case the subgroup is the same as the sample size.

R1 Range of the first subgroup.

R2 Range of the second subgroup.

R–

R bar Average range. The arithmetic mean of the range values of the subgroups ofthe sample measurements from a single lot.



k The acceptability constant given in Tables C-1 and C-2.

c A factor used in determining the quality index when using the range method.The c values are given in Tables C-3 and C-4.

QU Q sub U Quality Index for use with Table C-5.

QL Q sub L Quality Index for use with Table C-5.

pU p sub U Sample estimate of the lot percent nonconforming above U from Table C-5.

pL p sub L Sample estimate of the lot percent nonconforming below L from Table C-5.

p Total sample estimate of the lot percent nonconforming p = pU + pL.

M Maximum allowable percent nonconforming for sample estimates given inTables C-3 and C-4.

MU M sub U Maximum allowable percent nonconforming above U given in Tables C-3and C-4. (For use when different AQL values for U and L are specified.)

ML M sub L Maximum allowable percent nonconforming below L given in Tables C-3and C-4. (For use when different AQL values for U and L are specified.)


p–U p bar sub U The estimated process average for an upper specification limit.


f A factor used in determining the Maximum Average Range (MAR). The f values are given in Table C-6.



∑ Sum of Sum of.

T AQL symbol denoting plan used exclusively on tightened inspection (provides identification of appropriate OC curve).

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80

D1. SAMPLING PLAN FOR SINGLESPECIFICATION LIMIT

This part of the standard describes the procedures for usewith plans for a single specification limit when variabilityof the lot with respect to the quality characteristic isknown. The acceptability criterion is given in two equiva-lent forms. These are identified as Form 1 and Form 2.

D1.1 Use of Sampling Plans. To determine whether thelot meets the acceptability criterion with respect to a par-ticular quality characteristic and AQL value, the applicablesampling plan shall be used in accordance with the provi-sions of Section A, General Description of the SamplingPlans, and those in this part of the standard.

D1.2 Drawing of Samples. All samples shall be drawn inaccordance with paragraph A7.2.

D1.3 Determination of Sample Size Code Letter. Thesample size code letter shall be selected from Table A-2 inaccordance with paragraph A7.1.

D2. SELECTING THE SAMPLING PLANWHEN FORM 1 IS USED

D2.1 Master Sampling Tables. The master samplingtables for plans based on variability known for a singlespecification limit are Tables D-1 and D-2. Table D-1 isused for normal and tightened inspection and Table D-2 forreduced inspection.

D2.2 Obtaining Sampling Plan. The sampling plan con-sists of a sample size and an associated acceptabilityconstant.1 The sampling plan is obtained from MasterTable D-1 and D-2.

D2.2.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letterand AQL.

D2.2.2 Acceptability Constant. The acceptability constantk, corresponding to the sample size mentioned in para-graph D2.2.1, is indicated in the column of the master tablecorresponding to the applicable AQL value. Table D-1 isentered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans forreduced inspection are provided in Table D-2.

D3. LOT-BY-LOT ACCEPTABILITYPROCEDURES WHEN FORM 1 IS USED2

D3.1 Acceptability Criterion. The degree of conformanceof a quality characteristic with respect to a single specifi-cation limit shall be judged by the quantity (U – X

–)/σ or

(X–

– L)/σ.

D3.2 Computation. The following quantity shall becomputed: (U – X

–)/σ or (X

–– L)/σ, depending on

whether the specification limit is an upper or a lowerlimit, where


is the sample mean, andσ is the known variability.

D3.3 Acceptability Criteria. Compare the quantity (U –X–

)/σ or (X–

– L)/σ with the acceptability constant k. If (U – X

–)/σ or (X

–– L)/σ is equal to or greater than k, the lot

meets the acceptability criterion; if (U – X–

)/σ or (X–

– L)/σis less than k or negative, then the lot does not meet theacceptability criterion.

SECTION D

VARIABILITY KNOWN

Part ISINGLE SPECIFICATION LIMIT

1See Appendix D for definitions of all symbols used in the sampling plans based on variability known.2See Example D-1 for a complete example of this procedure.

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D4. SUMMARY FOR OPERATION OFSAMPLING PLAN WHEN FORM 1 IS USED



(2) Obtain plan from Master Table D-1 or D-2 by select-ing the sample size n and the acceptability constant k.



, and also compute thequantity (U – X

–)/σ for an upper specification limit U or the

quantity (X–

– L)/σ for a lower specification limit L.


)/σ or (X–

– L)/σ is equal to orgreater than k, the lot meets the acceptability criterion; if(U – X

–)/σ or (X

–– L)/σ is less than k or negative, then the

lot does not meet the acceptability criterion.

D5. SELECTING THE SAMPLING PLANWHEN FORM 2 IS USED

D5.1 Master Sampling Tables. The master samplingtables for plans based on variability known for a singlespecification limit are Tables D-3 and D-4 of Part II. TableD-3 is used for normal and tightened inspection and TableD-4 for reduced inspection.

D5.2 Obtaining the Sampling Plan. The sampling planconsists of a sample size and an associated maximumallowable percent nonconforming. The sampling plan isobtained from Master Table D-3 or D-4.

D5.2.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letter.

D5.2.2 Maximum Allowable Percent Nonconforming.The maximum allowable percent nonconforming M forsample estimates corresponding to the sample size men-tioned in paragraphs D5.2.1 is indicated in the column of the

master table corresponding to the applicable AQL value.Table D-3 is entered from the top for normal inspection andfrom the bottom for tightened inspection. Sampling plansfor reduced inspection are provided in Table D-4.

D6. LOT-BY-LOT ACCEPTABILITYPROCEDURES WHEN FORM 2 IS USED3

D6.1 Acceptability Criterion. The degree of confor-mance of a quality characteristic with respect to a singlespecification limit shall be judged by the percent of non-conforming product outside the upper or lower specifica-tion limit. The percentage of nonconforming product isestimated by entering Table D-5 with the quality index.

D6.2 Computation of Quality Index. The quality indexQU = (U – X

–)v/σ shall be computed if the specification

limit is an upper limit U, or QL = (X–

– L)v/σ if it is alower limit L. The quantities, X

–and σ, are the sample

mean and known variability, respectively. The factor v isprovided in Tables D-3 and D-4 corresponding to thesample size.

D6.3 Estimate of Percent Nonconforming in Lot. Thequality of a lot shall be expressed by pU, the estimated per-cent nonconforming in the lot above the upper specifica-tion limit, or by pL, the estimated percent nonconformingbelow the lower specification limit. The estimated percentnonconforming pU or pL is obtained by entering Table D-5with QU or QL.

D6.4 Acceptability Criterion. Compare the estimated lotpercent nonconforming pU or pL with the maximum allow-able percent nonconforming M. If pU or pL is equal to orless than M, the lot meets the acceptability criterion; if pU


D7. SUMMARY OF OPERATION OFSAMPLING PLAN WHEN FORM 2 IS USED


ANSI/ASQ Z1.9-2008

81

3See Example D-2 for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 81


(2) Obtain plan from Master Table D-3 or D-4 by select-ing the sample size n, the factor v, and the maximumallowable percent nonconforming M.



.

(5) Compute the quality index QU = (U –X–

)v/σ if anupper specification limit U is specified, or QL = (X

–– L)v/σ

if the lower specification limit L is specified.

(6) Determine the estimated lot percent nonconformingpU or pL from Table D-5.

(7) If the estimated lot percent nonconforming pU or pL isequal to or less than the maximum allowable percent non-conforming M, the lot meets the acceptability criterion; ifpU or pL is greater than M or if QU or QL is negative, thenthe lot does not meet the acceptability criterion.

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82

EXAMPLE D-1



Variability Known

Example: The specified minimum yield point for certain steel castings is 58,000 psi. A lot of 500 items is submitted forinspection. Inspection Level II, normal inspection, with AQL = 1.5% is to be used. The variability σ is knownto be 3000 psi. From Tables A-2 and D-1 it is seen that a sample of size 10 is required. Suppose the yield pointsof the sample specimens are:

62,500; 60,500; 68,000; 59,000; 65,50062,000; 61,000; 69,000; 58,000; 64,500;



1 Sample Size: n 10

2 Known Variability: σ 3,000

3 Sum of Measurements: ∑X 630,000

4 Sample Mean X–

: ∑X/n 63,000 630,000/10

5 Specification Limit (Lower): L 58,000

6 The Quantity: (X–

– L)/σ 1.67 (63,000 – 58,000)/3,000

7 Acceptability Constant: k 1.70 See Table D-1

8 Acceptability Criterion: Compare (X–

– L)/σ with k 1.67 < 1.70 See Para. D3.3

The lot does not meet the acceptability criterion, since (X–

– L)/σ is less than k.

NOTE: If a single upper specification limit U is given, then compute the quantity (U – X–

)/σ in line 6 and compare it with k; the lot meets the acceptability criterion if (U – X

–)/σ is equal to or greater than k.

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EXAMPLE D-2



Variability Known

Example: The specified minimum yield point for certain steel castings is 58,000 psi. A lot of 500 items is submitted forinspection. Inspection Level II, normal inspection, with AQL = 1.5% is to be used. The variability σ is knownto be 3000 psi. From Tables A-2 and D-3 it is seen that a sample of size 10 is required. Suppose the yield pointsof the sample specimens are:

62,500; 60,500; 68,000; 59,000; 65,500;62,000; 61,000; 69,000; 58,000; 64,500;



1 Sample Size: n 10



4 Sample Mean X–

: ∑X/n 63,000 630,000/10

5 Factor: v 1.054 See Table D-3

6 Specification Limit (Lower): L 58,000


– L)v/σ 1.76 (63,000 – 58,000)1.054/3,000

8 Est. of Lot Percent Ncf.: pL 3.92% See Table D-5

9 Max. Allowable Percent Ncf.: M 3.63% See Table D-3

10 Acceptability Criterion: Compare pL with M 3.92% > 3.63% See Para. D6.4

The lot does not meet the acceptability criterion, since pL is greater than M.

NOTE: If a single upper specification limit U is given, then compute the quality index QU = (U – X–

)v/σ in line 7 andobtain the estimate of the percent nonconforming pU. Compare pU with M; the lot meets the acceptability criterion if pU is equal to or less than M.

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83

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 83

ANSI/ASQ Z1.9-2008

84

Tabl

e D

-1M

aste

r Ta

ble

for

Nor

mal

and

Tig

hten

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n V

aria

bilit

y K

now

n(S

ingl

e Sp

ecifi

catio

n L

imit—

For

m 1

)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)

T

nk

.10

nk

.15

nk

.25

nk

.40

nk

.65

nk


1.58

1.69

1.80

1.88

1.95

1.96

2.07

2.12

2.14

2.17

2.18

2.01

2 3 4 5 7 8 16 23 30 44 5911

1.81

1.91

2.05

2.08

2.13

2.21

2.27

2.29

2.33

2.34

2.16

2 3 5 6 8 14 21 27 40 5410

1.94

2.07

2.14

2.23

2.25

2.35

2.41

2.43

2.47

2.48

2.29

2 3 4 6 7 13 19 25 37 49 9

2.19

2.30

2.34

2.37

2.49

2.54

2.57

2.62

2.62

1.00

2.45

3 4 5 6 12 17 23 34 45

.65

9

2.39

2.46

2.49

2.59

2.65

2.69

2.72

2.73

.40

2.54

4 5 6 11 16 22 31 42

.25

8

2.49

2.55

2.59

2.72

2.77

2.80

2.84

2.85

.15

2.63

3 4 6 11 15 20 30 40

.10

7

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

T d

enot

es p

lan

used

exc

lusi

vely

on

tight

ened

insp

ectio

n an

dpr

ovid

es s

ymbo

l for

iden

tific

atio

n of

app

ropr

iate

OC

cur

ve.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 84

ANSI/ASQ Z1.9-2008

85

Tabl

e D

-1—

Con

tinu

edM

aste

r Ta

ble

for

Nor

mal

and

Tig

hten

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n V

aria

bilit

y K

now

n(S

ingl

e Sp

ecifi

catio

n L

imit—

For

m 1

)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)

1.00

nk

1.50

nk

2.50

nk

4.00

nk

6.50

nk

10.0

0

nk


.57

3

.64

1

.72

8

.79

7

.87

7

.90

6

.92

4

.99

5

1.03

1.05

1.07

1.07

.96

4

3

.75

5

3 .

936

21.

09 2

1.25

21.

36 2

4

5

7

11

14

17

33

49

64

95

127

24

.82

5

.91

9

.99

1

1.07

1.11

1.13

1.21

1.24

1.26

1.29

1.29

1.17

3

5

6

9

12

15

29

42

55

82

109

20

1.01

1.11

1.20

1.28

1.31

1.34

1.42

1.46

1.48

1.51

1.51

1.38

3 4 5 8 10 13 25 36 48 70 9318

1.17

1.28

1.39

1.45

1.49

1.51

1.61

1.65

1.67

1.69

1.70

1.56

3 4 5

1.42

1.56

1.69

2 3 4

7 9 11 22 32 42 61 81

10.0

0

15

1.33

1.44

1.53

1.62

1.68

1.70

1.79

1.84

1.86

1.89

1.89

6.50

1.75

2 3 4 6 8 10 19 28 36 54 71

4.00

14

1.78

1.80

1.83

1.93

1.97

2.00

2.03

2.04

2.50

1.88

6 7 9 17 25 33 49 65

1.50

12

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 85

ANSI/ASQ Z1.9-2008

86

Tabl

e D

-2M

aste

r Ta

ble

for

Red

uced

Ins

pect

ion

for

Plan

s B

ased

on

Var

iabi

lity

Kno

wn

(Sin

gle

Spec

ifica

tion

Lim

it—F

orm

1)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its

.10

nk

.15

nk

.25

nk

.40

nk

.65

nk


1.36

1.42

1.56

1.69

1.80

1.83

1.86

1.93

1.97

1.78

2 2 3 4 7 9 11 17 25 6

1.58

1.69

1.80

1.95

1.96

1.99

2.07

2.12

1.88

2 3 4 7 8 10 16 23 5

1.81

1.91

2.08

2.13

2.13

2.21

2.27

2.05

2 3 6 8 9 14 21 5

1.94

2.07

2.23

2.25

2.26

2.35

2.41

2.14

2 3 6 7 8 13 19 4

2.19

2.34

2.37

2.38

2.49

2.54

2.30

3 5 6 7 12 17 4

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 86

ANSI/ASQ Z1.9-2008

87

Tabl

e D

-2—

Con

tinu

edM

aste

r Ta

ble

for

Red

uced

Ins

pect

ion

for

Plan

s B

ased

on

Var

iabi

lity

Kno

wn

(Sin

gle

Spec

ifica

tion

Lim

it—F

orm

1)

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s k

valu

e. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its

1.00

nk

1.50

nk

2.50

nk

4.0

nk

6.5

nk


.57

3

.64

1

.72

8

.79

7

.90

6

.92

4

.94

2

.99

5

1.03

.87

7

3 4 5 7 14 17 21 33 4911

10.0

0

nk

.344

.429

.515

.584

.685

.706

.719

.770

.803

.649

4 4 6 8 16 20 21 38 5612

.75

5

.82

5

.91

9

.99

1

1.11

1.13

1.15

1.21

1.24

1.07

3 3 5 6 12 15 18 29 42 9

.93

6

1.01

1.11

1.20

1.31

1.34

1.35

1.42

1.46

1.28

2 3 4 5 10 13 15 25 36 8

1.09

1.17

1.28

1.39

1.49

1.51

1.53

1.61

1.65

1.45

2 3 4 5 9 11 13 22 32 7

1.25

1.33

1.44

1.53

1.68

1.70

1.72

1.79

1.84

1.62

2 2 3 4 8 10 12 19 28 6

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 87

ANSI/ASQ Z1.9-2008

88

D8. SAMPLING PLAN FOR DOUBLESPECIFICATION LIMIT

This part of the standard describes the procedures for usewith plans for a double specification limit when variabilityof the lot with respect to the quality characteristic is known.

D8.1 Use of Sampling Plans. To determine whether thelot meets the acceptability criterion with respect to a par-ticular quality characteristic and AQL value(s), the appli-cable sampling plan shall be used in accordance with theprovisions of Section A, General Description of SamplingPlans, and those in this part of the standard.

D8.2 Initial Review for Combined Double SpecificationLimits. Before implementing a sampling plan for com-bined double specification limits when the process vari-ability is known, it is mandatory to carry out a check thatthe process standard deviation, σ, does not exceed theMaximum Process Standard Deviation (MPSD) derivedfrom Table D-6 that corresponds to the specified AQL. If σis greater than the MPSD, the process average is known tobe excessive, and lots shall be judged not acceptable with-out samples being drawn. If σ does not exceed the MPSD,it is an indication that the process average may be accept-able, and sampling inspection should be carried out inorder to determine lot acceptability. To make the compari-son, the following procedure is recommended:

(1) Determine the standardized MPSD from Table D-6for the given AQL.

(2) Multiply the standardized MPSD by the specificationinterval U – L to obtain the MPSD.

(3) Compare the process standard deviation, σ, with thiscalculated MPSD.

D9. SELECTING THE SAMPLING PLAN

A sampling plan for each AQL value shall be selected fromTable D-3 or D-4 as follows:

D9.1 Determination of Sample Size Code Letter. Thesample size code letter shall be selected from Table A-2 inaccordance with paragraph A7.1.

D9.2 Master Sampling Tables. The master samplingtables for plans based on variability known for a doublespecification limit are Tables D-3 and D-4. Table D-3 isused for normal and tightened inspection and Table D-4 forreduced inspection.

D9.3 Obtaining Sampling Plan. A sampling plan consistsof a sample size and the associated maximum allowablepercent nonconforming. The sampling plan to be applied ininspection shall be obtained from Master Table D-3 or D-4.

D9.3.1 Sample Size. The sample size n is shown in themaster table corresponding to each sample size code letterand AQL.

D9.3.2 Maximum Allowable Percent Nonconforming.The maximum allowable percent nonconforming for sam-ple estimates of percent nonconforming for the lower,upper, or both specification limits combined, correspon-ding to the sample size mentioned in paragraph D9.3.1, isshown in the column of the master table corresponding tothe applicable AQL value(s). If different AQLs areassigned to each specification limit, designate the maxi-mum allowable percent nonconforming by ML for thelower limit, and by MU for the upper limit. If one AQL isassigned to both limits combined, designate the maximumallowable percent nonconforming by M. Table D-3 isentered from the top for normal inspection and from thebottom for tightened inspection. Sampling plans forreduced inspection are provided in Table D-4.

D10. DRAWING OF SAMPLES


D11. LOT-BY-LOT ACCEPTABILITYPROCEDURES

D11.1 Acceptability Criterion. The degree of confor-mance of a quality characteristic with respect to a doublespecification limit shall be judged by the percent of non-conforming product. The percentage of nonconformingproduct is estimated by entering Table D-5 with thequality index.

PART IIDOUBLE SPECIFICATION LIMIT

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 88

D11.2 Computation of Quality Indices. The qualityindices QU = (U – X

–)v/σ and QL = (X

–– L)v/σ shall be

computed, where

U is the upper specification limit,L is the lower specification limit,v is a factor provided in Tables D-3 and D-4,X–

is the sample mean, andσ is the known variability.

D11.3 Percent Nonconforming in the Lot. The quality ofa lot shall be expressed in terms of the lot percent noncon-forming. Its estimate will be designated by pL, pU, or p. Theestimate pU indicates conformance with respect to the upperspecification limit, pL with respect to the lower specificationlimit, and p for both specification limits combined. Theestimates pL and pU shall be determined by entering TableD-5, respectively with QL and QU. The estimate p shall bedetermined by adding the corresponding estimated percentsnonconforming pL and pU found in the table.

D12. ACCEPTABILITY CRITERION ANDSUMMARY FOR OPERATION OFSAMPLING PLANS

D12.1 One AQL Value for Both Upper and LowerSpecification Limit Combined.

D12.1.1 Acceptability Criterion.4 Compare the estimatedlot percent nonconforming p = pU + pL with the maximumallowable percent nonconforming M. If p is equal to or lessthan M, the lot meets the acceptability criterion; if p isgreater than M or if QU or QL, or both are negative, then thelot does not meet the acceptability criterion.

D12.1.2 Summary of Operation of Sampling Plan. Incases where a single AQL value is established for theupper and lower specification limit combined for a singlequality characteristic, the following steps summarize theprocedures to be used:


(2) Select plan from Master Table D-3 or D-4. Obtain thesample size n, the factor v, and the maximum allowablepercent nonconforming M.



.


)v/σ andQL = (X

–– L)v/σ.

(6) Determine the estimated lot percent nonconformingp = pU + pL from Table D-5.

(7) If the estimated lot percent nonconforming p is equalto or less than the maximum allowable percent noncon-forming M, the lot meets the acceptability criterion; if p isgreater than M or if QU or QL or both are negative, then thelot does not meet the acceptability criterion.

D12.2 Different AQL Values for Upper and LowerSpecification Limit.

D12.2.1 Acceptability Criteria.5 Compare the estimated lotpercents nonconforming pL and pU with the correspondingmaximum allowable percents nonconforming ML and MU:also compare p = pL + pU with the larger of ML and MU. If pL

is equal to or less than ML, pU is equal to or less than MU, andp is equal to or less than the larger of ML and MU, the lotmeets the acceptability criteria; otherwise, the lot does notmeet the acceptability criteria. If either QL or QU or both arenegative, then the lot does not meet the acceptability criteria.

D12.2.2 Summary for Operation of Sampling Plan. Incases where a different AQL value is established for theupper and lower specification limit for a single qualitycharacteristic, the following steps summarize the proce-dures to be used:

(1) Determine the sample size code letter from Table A-2by using the lot sizes and inspection level.

(2) Select the sampling plan from Master Table D-3 or D-4. Obtain the sample size n and the factor v, corresponding

ANSI/ASQ Z1.9-2008

89

4See Example D-3A for a complete example of this procedure.5See Example D-4A for a complete example of this procedure.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 89

to the larger of the two AQL values, and also the maximumallowable percent nonconforming MU and ML, correspon-ding to the AQL values for the upper and lower specificationlimits, respectively.



.


)v/σ andQL = (X

–– L)v/σ.

(6) Determine the estimated lot percents nonconformingpU and pL, corresponding to the percents nonconforming

above the upper and below the lower specification limits.Also determine the combined percent nonconforming p =pU + pL.

(7) If all three of the following conditions:(a) pU is equal to or less than MU,(b) pL is equal to or less than ML,(c) p is equal to or less than the larger

of ML and MU,


ANSI/ASQ Z1.9-2008

90

EXAMPLE D-3A



Variability Known


Example: The specified maximum and minimum yield points for certain steel castings are 67,000 and 58,000 psi, respec-tively. A lot of 500 items is submitted for inspection. Inspection Level II, normal inspection, with AQL = 1.5%is to be used. The variability σ is known to be 3,000 psi.


1 Upper Specification Limit: U 67,000

2 Lower Specification Limit: L 58,000


4 Factor for Maximum Process Standard Deviation for an AQL of 1.5%: F σ 0.194 See Table D-6

5 Maximum Process Standard Deviation:MPSD = F σ (U – L) 1,746 0.194 (67,000 – 58,000)

6 Mandatory check: Compare σ with MPSD 3,000 < 1,746

The process fails the mandatory check that σ does not exceed the MPSD,so sampling inspection is pointless and submitted lots should not be accepted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 90

EXAMPLE D-3B



Variability Known


Example: The specified maximum and minimum yield points for certain steel castings are 70,000 and 54,000 psi, respec-tively. A lot of 500 items is submitted for inspection. Inspection Level II, normal inspection, with AQL = 1.5%is to be used. The variability σ is known to be 3,000 psi.





4 Factor for Maximum Process Standard Deviation for an AQL of 1.5%: F σ 0.194 See Table D-6

5 Maximum Process Standard Deviation: MPSD = F σ (U – L) 3,104 0.194 (70,000 – 54,000)

6 Mandatory check: Compare σ with MPSD 3,000 < 3,104

The process satisfies the mandatory check that σ does not exceed the MPSD. It follows that there is a possibility, thoughnot a certainty, that the lot is acceptable. Lot acceptability is determined by sampling. From Tables A-2 and D-3, it is seenthat a sample of size 10 is required. Suppose the yield points of the sample specimens are:

62,500; 60,500; 68,000; 59,000; 65,500;62,000; 61,000; 69,000; 58,000; 64,500;



7 Sample size: n 10 See Tables A-2 and D-3


9 Sample Mean: X–

= ∑X/n 63,000 630,000/10

10 Factor: v 1.054 See Table D-3


)v/σ 2.46 (70,000 – 63,000) 1.054/3,000


– L)v/σ 3.16% (63,000 – 54,000) 1.054/3,000

13 Est. of Lot % Ncf. Above U: pU 0.70% See Table D-5

14 Est. of Lot % Ncf. Below L: pL 0.08% See Table D-5

15 Total Est. % Ncf. in Lot: p = pU + pL 0.78% 0.697% + 0.078%

16 Max. Allowable Est. % Ncf.: M 3.63% See Table D-3

17 Acceptability Criterion: Compare p with M 0.78% < 3.63%

The lot meets the acceptability criterion, since p = pU + pL is less than M.

ANSI/ASQ Z1.9-2008

91

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 91

EXAMPLE D-4



Variability Known

Different AQL Values for Upper and Lower Specification Limits

Example: The specified maximum and minimum yield points for certain steel castings are 67,000 psi and 58,000 psi,respectively. A lot of 500 items is submitted for inspection. Inspection Level II, normal inspection withAQL = 1% for the upper and AQL = 2.5% for the lower specification limit is to be used. The variability σ isknown to be 3,000 psi. From Tables A-2 and D-3 it is seen that a sample of size 11 corresponding to the samplesize code letter, I, and the AQL value of 2.5% is required. Suppose the yield points of the sample specimens are:

62,500; 60,500; 64,000; 59,000’ 65,500;62,000; 61,000; 60,500; 68,000; 62,000; 63,000



1 Sample Size: n 11



4 Sample Mean X–

: ∑X/n 62,645 688,000/11

5 Factor: v 1.049 See Table D-3; use larger AQL




)v/σ 1.56 (67,000 – 62,645)1.049/3,000


– L)v/σ 1.59 (62,545 – 58,000)1.049/3,000

10 Est. of Lot Percent Ncf. above U: pU 5.94% See Table D-5

11 Est. of Lot Percent Ncf. below L: pL 5.59% See Table D-5


13 Max. Allowable Percent Ncf. Above U: MU 2.59% See Table D-3

14 Max. Allowable Percent Ncf. Below L: ML 5.60% See Table D-3

15 Acceptability Criteria: (a) Compare pU with MU 5.94% > 2.59% See Para. D12.2.2(7)(a)

(b) Compare pL with ML 5.59% > 5.60% See Para. D12.2.2(7)(b)

(c) Compare p with ML 11.53% > 5.60% See Para. D12.2.2(7)(c)

The lot does not meet the acceptability criteria, since 15(a), (b), and (c) are not satisfied; i.e., pU > MU, pL > ML, and p > ML.

ANSI/ASQ Z1.9-2008

92

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 92

ANSI/ASQ Z1.9-2008

93

Tabl

e D

-3M

aste

r Ta

ble

for

Nor

mal

and

Tig

hten

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n K

now

n V

aria

bilit

y(D

oubl

e Sp

ecifi

catio

n L

imit

and

For

m 2

—Si

ngle

Spe

cific

atio

n L

imit)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)


.65

v

1.41

4

1.22

5

1.15

5

1.11

8

1.08

0

1.06

9

1.03

3

1.02

3

1.01

7

1.01

2

1.00

9

1.04

9

n 2 4 4 5 7 8 16 23 30 44 5911

M 1.28

1.94

1.88

1.76

1.75

1.80

1.62

1.51

1.47

1.39

1.39

1.73

.40

v

1.41

4

1.22

5

1.11

8

1.09

5

1.06

9

1.03

8

1.02

5

1.01

9

1.01

3

1.00

9

1.05

4

n 2 3 5 6 8 14 21 27 40 5410

M

.51

0

.95

9

1.09

1.14

1.14

1.08

1.00

.98

0

.92

1

.92

0

1.14

.25

nv

1.41

4

1.22

5

1.15

5

1.09

5

1.08

0

1.04

1

1.02

7

1.02

1

1.01

4

1.01

0

1.06

1

2 3 4 6 7 13 19 25 37 49 9

M .310

.568

.681

.721

.756

.721

.673

.655

.615

.620

.760

.15

v

1.22

5

1.15

5

1.11

8

1.09

5

1.04

5

1.03

1

1.02

3

1.01

5

1.01

1

1.06

1

n 3 4 5 6 12 17 23 34 45 9

M .369

.399

.445

.478

.461

.438

.423

.397

.402

.469

.10

nv

1.15

5

1.11

8

1.09

5

1.04

9

1.03

3

1.02

4

1.01

7

1.01

2

1.06

9

4 5 6 11 16 22 31 42 8

M .290

.296

.321

.326

.308

.296

.283

.285

.330

T

1.00

.65

.40

.25

.15

.10

v

1.22

5

1.15

5

1.09

5

1.04

9

1.03

5

1.02

6

1.01

7

1.01

3

1.08

0

n 3 4 6 11 15 20 30 40 7

M .114

.161

.230

.217

.211

.207

.193

.196

.226

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

T d

enot

es p

lan

used

exc

lusi

vely

on

tight

ened

insp

ectio

n an

dpr

ovid

es s

ymbo

l for

iden

tific

atio

n of

app

ropr

iate

OC

cur

ve.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s M

val

ue. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 93

ANSI/ASQ Z1.9-2008

94

Tabl

e D

-3—

Con

tinu

edM

aste

r Ta

ble

for

Nor

mal

and

Tig

hten

ed I

nspe

ctio

n fo

r Pl

ans

Bas

ed o

n K

now

n V

aria

bilit

y(D

oubl

e Sp

ecifi

catio

n L

imit

and

For

m 2

—Si

ngle

Spe

cific

atio

n L

imit)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its (

norm

al in

spec

tion)

Acc

epta

nce

Qua

lity

Lim

its (

tight

ened

insp

ectio

n)


10.0

0

v

1.15

5

1.11

8

1.08

0

1.04

9

1.03

8

1.03

1

1.01

6

1.01

0

1.00

8

1.00

5

1.00

4

1.02

2

n 4

5

7

11

14

17

33

49

64

95

127

24

M

22.9

7

20.8

0

19.4

6

17.8

8

17.3

6

17.0

5

15.6

1

14.8

7

14.5

8

14.0

9

14.0

2

16.2

3

6.50

v

1.22

5

1.11

8

1.09

5

1.06

1

1.04

5

1.03

5

1.01

8

1.01

2

1.00

9

1.00

6

1.00

5

1.02

6

n 3

5

6

9

12

15

29

42

55

82

109

20

M

15.6

0

15.2

1

13.8

9

12.8

8

12.3

5

12.0

4

10.9

3

10.4

0

10.1

7

9.7

6

9.7

3

11.5

7

4.00

nv

1.22

5

1.15

5

1.11

8

1.06

9

1.05

4

1.04

1

1.02

1

1.01

4

1.01

1

1.00

7

1.00

5

1.02

9

3 4 5 8 10 13 25 36 48 73 9318

M

10.7

9

9.9

7

8.9

2

8.6

2

8.4

3

8.1

3

7.3

4

6.9

5

6.7

5

6.4

8

6.4

6

7.7

2

2.50

v

1.22

5

1.15

5

1.11

8

1.08

0

1.06

1

0.14

9

1.02

4

1.01

6

1.01

2

1.00

8

1.00

6

1.03

5

n 3 4 5 7 9 11 22 32 42 61 8115

M 7.56

6.99

6.05

5.83

5.68

5.60

4.98

4.68

4.55

4.35

4.34

5.34

1.50

nv

1.41

4

1.22

5

1.15

5

1.09

5

1.06

9

1.05

4

1.02

7

1.01

8

1.01

4

1.00

9

1.00

7

1.03

8

2 3 4 6 8 10 19 28 36 54 7114

M 3.00

3.85

3.87

3.77

3.68

3.63

3.28

3.05

2.99

2.82

2.82

3.43

1.00

10.0

06.

504.

002.

501.

50

v

1.41

4

1.22

5

1.15

5

1.09

5

1.08

0

1.06

1

1.03

1

1.02

1

1.01

6

1.01

0

1.00

8

1.04

5

n 2 3 4 6 7 9 17 25 33 49 6512

M 2.23

2.76

2.58

1.22

5

324

.22

1.22

5

317

.74

1.41

4 2

9.2

71.

414

26.

111.

414

23.

901.

414

22.

73

2.57

2.62

2.59

2.35

2.19

2.12

2.00

2.00

2.49

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s M

val

ue. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 94

ANSI/ASQ Z1.9-2008

95

Tabl

e D

-4M

aste

r Ta

ble

for

Red

uced

Ins

pect

ion

for

Plan

s B

ased

on

Kno

wn

Var

iabi

lity

(Dou

ble

Spec

ifica

tion

Lim

it an

d F

orm

2—

Sing

le S

peci

ficat

ion

Lim

it)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its


.65

v

1.41

4

1.41

4

1.22

5

1.15

5

1.08

0

1.06

1

1.04

9

1.03

1

1.02

1

1.09

5

n 2 2 3 4 7 9 11 17 25 6

M 2.73

2.23

2.76

2.58

2.62

2.59

2.57

2.35

2.19

2.57

.40

v

1.41

4

1.22

5

1.15

5

1.08

0

1.06

9

1.05

4

1.03

3

1.02

3

1.11

8

n 2 3 4 7 8 10 16 23 5

M 1.28

1.94

1.88

1.75

1.80

1.79

1.62

1.51

1.76

.25

nv

1.41

4

1.22

5

1.09

5

1.06

9

1.06

1

1.03

8

1.02

5

1.11

8

2 3 6 8 9 14 21 5

M

.51

0

.95

9

1.14

1.14

1.18

1.08

1.00

1.09

.15

v

1.41

4

1.22

5

1.09

5

1.08

0

1.06

9

1.04

1

1.02

7

1.15

5

n 2 3 6 7 8 13 19 4

M .310

.568

.721

.756

.791

.721

.673

.681

.10

nv

1.22

5

1.11

8

1.09

5

1.08

0

1.04

5

1.03

1

1.15

5

3 5 6 7 12 17 4

M .369

.445

.478

.507

.461

.438

.399

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s M

val

ue. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 95

ANSI/ASQ Z1.9-2008

96

Tabl

e D

-4—

Con

tinu

edM

aste

r Ta

ble

for

Red

uced

Ins

pect

ion

for

Plan

s B

ased

on

Kno

wn

Var

iabi

lity

(Dou

ble

Spec

ifica

tion

Lim

it an

d F

orm

2—

Sing

le S

peci

ficat

ion

Lim

it)

Sam

ple

Size

Cod

eL

ette

r

Acc

epta

nce

Qua

lity

Lim

its


10.0

0

v

1.22

5

1.15

5

1.09

5

1.06

9

1.03

3

1.02

6

1.02

2

1.01

3

1.00

9

1.04

5

n 4 4 6 8 16 20 24 38 5612

M

33.6

7

31.0

1

28.6

4

26.6

4

23.9

6

23.4

3

23.1

3

21.7

7

20.9

0

24.8

8

6.50

v

1.22

5

1.15

5

1.11

8

1.08

0

1.03

8

1.03

1

1.02

5

1.01

6

1.01

0

1.04

9

n 3 4 5 7 14 17 21 33 4911

M

24.2

2

22.9

7

20.8

0

19.4

6

17.3

6

17.0

5

16.7

1

15.6

1

14.8

7

17.8

8

4.00

nv

1.22

5

1.22

5

1.11

8

1.09

5

1.04

5

1.03

5

1.02

9

1.01

8

1.01

2

1.06

1

3 3 5 6 12 15 18 29 42 9

M

17.7

4

15.6

0

15.2

1

13.8

9

12.3

5

12.0

4

11.8

8

10.9

3

10.4

0

12.8

8

2.50

v

1.41

4

1.22

5

1.15

5

1.11

8

1.05

4

1.04

1

1.03

5

1.02

1

1.01

4

1.06

9

n 2 3 4 5 10 13 15 25 36 8

M

9.2

7

10.7

9

9.9

7

8.9

2

8.4

3

8.1

3

8.1

3

7.3

4

6.9

5

8.6

2

1.50

nv

1.41

4

1.22

5

1.15

5

1.11

8

1.06

1

1.04

9

1.04

1

1.02

4

1.01

6

1.08

0

2 3 4 5 9 11 13 22 32 7

M 6.11

7.56

6.99

6.05

5.68

5.60

5.58

4.98

4.68

5.83

1.00

v

1.41

4

1.41

4

1.22

5

1.15

5

1.06

9

1.05

4

1.04

5

1.02

7

1.01

8

1.09

5

n 2 2 3 4 8 10 12 19 28 6

M 3.90

3.00

3.85

3.87

3.68

3.63

3.61

3.28

3.05

3.77

All

AQ

L v

alue

s ar

e in

per

cent

non

conf

orm

ing.

Use

fir

st s

ampl

ing

plan

bel

ow a

rrow

; tha

t is,

bot

h sa

mpl

e si

ze a

s w

ell a

s M

val

ue. W

hen

sam

ple

size

equ

als

or

exce

eds

lot s

ize,

eve

ry it

em in

the

lot m

ust b

e in

spec

ted.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 96

ANSI/ASQ Z1.9-2008

97

Tabl

e D

-5Ta

ble

for

Est

imat

ing

the

Lot

Per

cent

age

Non

conf

orm

ing

for

Plan

s B

ased

on

Kno

wn

Var

iabi

lity1

QU or QL

p U or p L

QU or QL

p U or p L

QU or QL

p U or p L

QU or QL

p U or p L

.00

.01

.02

.03

.04

.05

50.0

00

49.6

01

49.2

02

48.8

03

48.4

05

48.0

06

.06

.07

.08

.09

.10

47.6

08

47.2

10

46.8

12

46.4

14

46.0

17

.11

.12

.13

.14

.15

45.6

20

45.2

24

44.8

28

44.4

33

44.0

38

.16

.17

.18

.19

.20

43.6

44

43.2

51

42.8

58

42.4

65

42.0

74

.21

.22

.23

.24

.25

41.6

83

41.2

94

40.9

05

40.5

17

40.1

29

.26

.27

.28

.29

.30

39.7

43

39.3

58

38.9

74

38.5

91

38.2

09

.31

.32

.33

.34

.35

37.8

28

37.4

48

37.0

70

36.6

93

36.3

17

.36

.37

.38

.39

.40

35.9

42

35.5

69

35.1

97

34.8

27

34.4

58

.41

.42

.43

.44

.45

34.0

90

33.7

24

33.3

60

32.9

97

32.6

36

.46

.47

.48

.49

.50

32.2

76

31.9

18

31.5

61

31.2

07

30.8

54

.51

.52

.53

.54

.55

30.5

03

30.1

53

29.8

06

29.4

60

29.1

16

.56

.57

.58

.59

.60

28.7

74

28.4

34

28.0

96

27.7

60

27.4

25

.61

.62

.63

.64

.65

27.0

93

26.7

63

26.4

35

26.1

09

25.7

85

.66

.67

.68

.69

.70

25.4

63

25.1

43

24.8

25

24.5

10

24.1

96

.71

.72

.73

.74

.75

23.8

85

23.5

76

23.2

70

22.9

65

22.6

63

.76

.77

.78

.79

.80

22.3

63

22.0

65

21.7

70

21.4

76

21.1

86

.81

.82

.83

.84

.85

20.8

97

20.6

11

20.3

27

20.0

45

19.7

66

.86

.87

.88

.89

.90

19.4

89

19.2

15

18.9

43

18.6

73

18.4

06

.91

.92

.93

.94

.95

18.1

41

17.8

79

17.6

19

17.3

61

17.1

06

.96

.97

.98

.99

1.00

16.8

53

16.6

02

16.3

54

16.1

09

15.8

66

QU or QL

p U or p L

1.01

1.02

1.03

1.04

1.05

15.6

25

15.3

86

15.1

50

14.9

17

14.6

86

1.06

1.07

1.08

1.09

1.10

14.4

57

14.2

31

14.0

07

13.7

86

13.5

67

1.11

1.12

1.13

1.14

1.15

13.3

50

13.1

36

12.9

24

12.7

14

12.5

07

1.16

1.17

1.18

1.19

1.20

12.3

02

12.1

00

11.9

00

11.7

02

11.5

07

1.21

1.22

1.23

1.24

1.25

11.3

14

11.1

23

10.9

35

10.7

49

10.5

65

QU or QL

p U or p L

1.26

1.27

1.28

1.29

1.30

10.3

83

10.2

04

10.0

27

09.8

53

09.6

80

1.31

1.32

1.33

1.34

1.35

09.5

10

09.3

42

09.1

76

09.0

12

08.8

51

1.36

1.37

1.38

1.39

1.40

08.6

91

08.5

34

08.3

79

08.2

26

08.0

76

1.41

1.42

1.43

1.44

1.45

07.9

27

07.7

80

07.6

36

07.4

93

07.3

53

1.46

1.47

1.48

1.49

1.50

07.2

14

07.0

78

06.9

44

06.8

11

06.6

81

QU or QL

p U or p L

1.51

1.52

1.53

1.54

1.55

06.5

52

06.4

26

06.3

01

06.1

78

06.0

57

1.56

1.57

1.58

1.59

1.60

05.9

38

05.8

21

05.7

05

05.5

92

05.4

80

1.61

1.62

1.63

1.64

1.65

05.3

70

05.2

62

05.1

55

05.0

50

04.9

47

1.66

1.67

1.68

1.69

1.70

04.8

46

04.7

46

04.6

48

04.5

51

04.4

57

1.71

1.72

1.73

1.74

1.75

04.3

63

04.2

72

04.1

82

04.0

93

04.0

06

QU or QL

p U or p L

1.76

1.77

1.78

1.79

1.80

03.9

20

03.8

36

03.7

54

03.6

73

03.5

93

1.81

1.82

1.83

1.84

1.85

03.5

15

03.4

38

03.3

62

03.2

88

03.2

16

1.86

1.87

1.88

1.89

1.90

03.1

44

03.0

74

03.0

05

02.9

38

02.8

72

1.91

1.92

1.93

1.94

1.95

02.8

07

02.7

43

02.6

80

02.6

19

02.5

59

1.96

1.97

1.98

1.99

2.00

02.5

00

02.4

42

02.3

85

02.3

30

02.2

75

QU or QL

p U or p L

2.01

2.02

2.03

2.04

2.05

02.2

22

02.1

69

02.1

18

02.0

68

02.0

18

2.06

2.07

2.08

2.09

2.10

01.9

70

01.9

23

01.8

76

01.8

31

01.7

86

2.11

2.12

2.13

2.14

2.15

01.7

43

01.7

00

01.6

59

01.6

18

01.5

78

2.16

2.17

2.18

2.19

2.20

01.5

39

01.5

00

01.4

63

01.4

26

01.3

90

2.21

2.22

2.23

2.24

2.25

01.3

55

01.3

21

01.2

87

01.2

55

01.2

22

QU or QL

p U or p L

2.26

2.27

2.28

2.29

2.30

01.1

91

01.1

60

01.1

30

01.1

01

01.0

72

2.31

2.32

2.33

2.34

2.35

01.0

44

01.0

17

00.9

90

00.9

64

00.9

39

2.36

2.37

2.38

2.39

2.40

00.9

14

00.8

89

00.8

66

00.8

42

00.8

20

2.41

2.42

2.43

2.44

2.45

00.7

98

00.7

76

00.7

55

00.7

34

00.7

14

2.46

2.47

2.48

2.49

2.50

00.6

95

00.6

76

00.6

57

00.6

39

00.6

21

QU or QL

p U or p L

2.51

2.52

2.53

2.54

2.55

00.6

04

00.5

87

00.5

70

00.5

54

00.5

39

2.56

2.57

2.58

2.59

2.60

00.5

23

00.5

08

00.4

94

00.4

80

00.4

66

2.61

2.62

2.63

2.64

2.65

00.4

53

00.4

40

00.4

27

00.4

15

00.4

02

2.66

2.67

2.68

2.69

2.70

00.3

91

00.3

79

00.3

68

00.3

57

00.3

47

2.71

2.72

2.73

2.74

2.75

00.3

36

00.3

26

00.3

17

00.3

07

00.2

98

QU or QL

p U or p L

2.76

2.77

2.78

2.79

2.80

00.2

89

00.2

80

00.2

72

00.2

64

00.2

56

2.81

2.82

2.83

2.84

2.85

00.2

48

00.2

40

00.2

33

00.2

26

00.2

19

2.86

2.87

2.88

2.89

2.90

00.2

12

00.2

05

00.1

99

00.1

93

00.1

87

2.91

2.92

2.93

2.94

2.95

00.1

81

00.1

75

00.1

69

00.1

64

00.1

59

2.96

2.97

2.98

2.99

3.00

00.1

54

00.1

49

00.1

44

00.1

39

00.1

35

QU or QL

p U or p L

3.01

3.02

3.03

3.04

3.05

00.1

31

00.1

26

00.1

22

00.1

18

00.1

14

3.06

3.07

3.08

3.09

3.10

00.1

11

00.1

07

00.1

03

00.1

00

00.0

97

3.11

3.12

3.13

3.14

3.15

00.0

94

00.0

90

00.0

87

00.0

84

00.0

82

3.16

3.17

3.18

3.19

3.20

00.0

79

00.0

76

00.0

74

00.0

71

00.0

69

3.21

3.22

3.23

3.24

3.25

00.0

66

00.0

64

00.0

62

00.0

60

00.0

58

QU or QL

p U or p L

3.26

3.27

3.28

3.29

3.30

00.0

56

00.0

54

00.0

52

00.0

50

00.0

48

3.31

3.32

3.33

3.34

3.35

00.0

47

00.0

45

00.0

43

00.0

42

00.0

40

3.36

3.37

3.38

3.39

3.40

00.0

39

00.0

38

00.0

36

00.0

35

00.0

34

3.41

3.42

3.43

3.44

3.45

00.0

32

00.0

31

00.0

30

00.0

29

00.0

28

3.46

3.47

3.48

3.49

3.50

00.0

27

00.0

26

00.0

25

00.0

24

00.0

23

QU or QL

p U or p L

3.51

3.52

3.53

3.54

3.55

00.0

22

00.0

22

00.0

21

00.0

20

00.0

19

3.56

3.57

3.58

3.59

3.60

00.0

19

00.0

18

00.0

17

00.0

17

00.0

16

3.61

3.62

3.63

3.64

3.65

00.0

15

00.0

15

00.0

14

00.0

14

00.0

13

3.66

3.67

3.68

3.69

3.70

00.0

13

00.0

12

00.0

12

00.0

11

00.0

11

3.71

3.72

3.73

3.74

3.75

00.0

10

00.0

10

00.0

10

00.0

09

00.0

09

QU or QL

p U or p L

3.76

3.77

3.78

3.79

3.80

00.0

08

00.0

08

00.0

08

00.0

08

00.0

07

3.81

3.82

3.83

3.84

3.85

00.0

07

00.0

07

00.0

06

00.0

06

00.0

06

3.86

3.87

3.88

3.89

3.90

00.0

06

00.0

05

00.0

05

00.0

05

00.0

05

3.91

3.92

3.93

3.94

3.95

00.0

05

00.0

04

00.0

04

00.0

04

00.0

04

3.96

3.97

3.98

3.99

4.00

00.0

04

00.0

04

00.0

03

00.0

03

00.0

03

1 Val

ues

tabu

late

d ar

e re

ad in

per

cent

.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 97

ANSI/ASQ Z1.9-2008

98

Table D-6Value of F σ for Maximum Process Standard Deviation

The MPSD may be obtained by multiplying the factor F σ by the difference between the upper specification limit U andthe lower specification limit L. The formula is MPSD = F σ (U – L).

The MPSD indicates the greatest allowable magnitude of the process standard deviation when using plans for the doublespecification limit case with known variability. If the process standard deviation is less than the MPSD, there is a possibil-ity but not a certainty that the lot will be accepted.

AQL(% Ncf)

F σ

.10

0.147

.15

0.152

.25

0.157

.40

0.165

.65

0.174

1.00

0.184

1.50

0.194

2.50

0.206

4.00

0.223

6.50

0.243

10.00

0.271

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 98

APPENDIX D

Definitions

Symbol Read Definition


X–


σ Sigma Known variability. The predetermined variability of the quality character-istic which will be used with the variability known acceptability plans.



k The acceptability constant given in Tables D-1 and D-2.

v A factor used in determining the quality indices when using the knownvariability acceptability plan. The v values are given in Tables D-3 and D-4.

QU Q sub U Quality index for use with Table D-5.

QL Q sub L Quality index for use with Table D-5.

pU p sub U Sample estimate of the lot percent nonconforming above U from Table D-5.

pL p sub L Sample estimate of the lot percent nonconforming below L from Table D-5.

p Total sample estimate of the lot percent nonconforming p = pU + pL

M Maximum allowable percent nonconforming for sample estimates givenin Tables D-3 and D-4.

MU M sub U Maximum allowable percent nonconforming above U given in Tables D-3and D-4. (For use when different AQL values for U and L are specified.)

ML M sub L Maximum allowable percent nonconforming below L given in Tables D-3and D-4. (For use when different AQL values for U and L are specified.)


p–U p bar sub U The estimated process average for an upper specification limit.




∑ Sum of Sum of.

T AQL symbol denoting plan used exclusively on tightened inspection (provides identification of appropriate OC curve).

ANSI/ASQ Z1.9-2008

99

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ANSI/ASQ Z1.9-2008

100

E1. INTRODUCTION

The original version of this variables inspection standard(ANSI/ASQC Z1.9-1972) corresponded directly to the mil-itary standard MIL-STD-414 dated 11 June 1957, just asthe attributes inspection standard ANSI/ASQC Z1.4 corre-sponded to MIL-STD-105D, dated 29 April 1963. The planscontained in these variables and attributes standards were,however, not matched. Subsequent to the promulgation ofthese standards, the International Organization forStandardization Working Group C, in June 1974, present-ed a procedure for roughly matching the MIL-STD-414and MIL-STD-105D Normal plans by a realignment of theMIL-STD-414 code letters.* It is this realignment whichwas used as a basis of the ANSI/ASQC Z1.9-1980 andsubsequent revisions.

The extent to which the plans of ANSI/ASQ Z1.9-2008match those of ANSI/ASQ Z1.4-2008 is shown in the fol-lowing tables which give:

Table 1—Matching Code Letters Table

Table 2—ANSI/ASQ Z1.9-2008 percentage points forthe 95, 50, 10th percentiles.

Table ANSI/ASQ Z1.4-2008 percentage points for the95, 50, 10th percentiles.

Table 4—Difference between ANSI/ASQ Z1.9-2008 andANSI/ASQ Z1.4-2008 percentiles.

The percentage points are the percents nonconforminghaving probability of acceptance equal to the percentilesshown. Table 4 can be employed by the user to determinethe practical significance of the difference in the operatingcharacteristics of the corresponding plans in ANSI/ASQZ1.4-2008 and ANSI/ASQ Z1.9-2008.

Procedures for switching to and from tightened or reducedinspection were taken directly from MIL-STD-105D,eliminating the use of limit numbers for reduced inspec-tion and also the procedure allowing termination ofreduced inspection without either acceptance or rejectioncriteria being met.

E2. TABLES

Tables showing the extent to which ANSI/ASQ Z1.9-2008matches ANSI/ASQ Z1.4-2008 follow.

SECTION E

APPENDIX

MATCH WITH ANSI Z1.4

*Working Group C, ISO/TC69, “Sampling by Variables,” April 1974, Draft.

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 100

ANSI/ASQ Z1.9-2008

101

E2—TablesTable 1

Matching Code Letters and ANSI/ASQ Z1.9-2008 Sample Size

3

4

5

7

10

15

20

25

35

50

75

100

150

200

B

C

D

E

F

G

H

I

J

K

L

M

N

P

B

C

D

E

F

G

H

I

K

M

N

O

P

Q

B

C

D

E

F

G

H

H

J

K

L

M

N

P

ANSI/ASQZ1.9-2008

Sample Size,Normal

Inspection,Level II

ANSI/ASQZ1.9-2008

NewCodeLetter

ANSI/ASQCZ1.9-1972

(414)Old

CodeLetter

ANSI/ASQZ1.4-2008

(105)Matched

CodeLetter

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 101

ANSI/ASQ Z1.9-2008

102

Table 2ANSI/ASQ Z1.9-2008 Percentage Points in Terms of Percent Nonconforming

Probabilityof

Acceptance

ANSI/ASQZ1.9-2008

CodeLetter .10 .15 .25 .40 .65 1.00 1.50 2.50 4.00 6.50 10.00

95.050.010.0

B1.04

16.6849.34

1.8920.3052.83

3.5225.2257.24

6.0230.9762.08

95.050.010.0

C.44

9.5234.88

.6911.2837.26

1.3214.4441.15

2.2917.9345.05

4.1322.8950.13

6.8528.6155.55

95.050.010.0

D.28

6.3425.94

.467.82

28.40

.779.71

31.24

1.3812.4734.98

2.4315.9739.25

4.3020.7544.55

7.1126.4050.32

95.050.010.0

E.11

2.89 14.42

.18 3.72

16.33

.324.83

18.60

.536.18

21.09

.837.69

23.58

1.5010.2827.43

2.6513.6631.93

4.5718.1137.28

7.4623.5343.25

95.050.010.0

F.07

1.53 7.95

.12 2.08 9.44

.21 2.79

11.15

.363.77

13.23

.574.82

15.23

.946.33

17.84

1.658.62

21.40

2.8311.6925.66

4.8415.9130.99

7.8121.0936.98

95.050.010.0

G.06 .90

4.31

.09 1.17 5.07

.15 1.57 6.13

.25 2.20 7.58

.453.099.41

.683.99

11.12

1.095.32

13.38

1.917.51

16.77

3.0910.1520.48

5.3014.2725.76

8.4119.2531.63

95.050.010.0

H.07 .76

3.16

.11 1.01 3.85

.17 1.38 4.73

.29 1.90 5.88

.492.697.46

.793.669.23

12.14.81

11.14

2.076.86

14.25

3.399.51

17.94

5.6913.4923.01

8.8818.3128.70

95.050.010.0

I.08 .68

2.55

.12

.89 3.08

.20 1.28 3.99

.32 1.73 4.93

.562.536.46

.853.397.97

1.284.479.73

2.236.54

12.81

3.619.12

16.34

5.9813.0021.24

9.2717.7426.82

95.050.010.0

J.09 .59

1.90

.13

.76 2.29

.23 1.10 3.02

.36 1.54 3.87

.602.215.10

.943.056.50

1.404.058.07

2.385.98

10.85

3.808.41

14.11

6.2112.1018.71

9.6516.8224.23

95.050.010.0

K.10 .19

1.36

.15

.65 1.70

.26

.98 2.35

.40 1.37 3.07

.641.944.03

1.022.765.33

1.493.686.72

2.515.489.23

4.047.90

12.39

6.5211.4516.72

10.0016.0021.98

95.050.010.0

L.11 .40 .97

.17

.56 1.27

.27

.82 1.74

.43 1.19 2.37

.701.743.24

1.062.434.28

1.583.345.58

2.625.027.82

4.187.29

10.70

6.8110.8414.94

10.3415.2419.95

95.050.010.0

M.12 .37 .80

.18

.51 1.05

.29

.77 1.50

.47 1.12 2.06

.741.642.86

1.122.313.81

1.663.185.01

2.734.807.11

4.317.009.84

6.9710.4513.89

10.5114.7518.73

95.050.010.0

N.13 .32 .62

.19

.46

.85

.31

.69 1.21

.48 1.00 1.68

.771.482.36

1.182.143.26

1.732.964.34

2.824.496.26

4.416.598.78

7.079.90

12.58

10.8014.2817.44

95.050.010.0

P.143.321.571

.210

.445

.763

.344

.6831.116

.5341.0001.567

.841.482.22

1.252.083.02

1.862.964.12

3.004.485.98

4.666.588.45

7.409.88

12.19

11.2214.2716.98

Acceptance Quality Limits

01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 102

ANSI/ASQ Z1.9-2008

103

Table 3ANSI/ASQ Z1.4-2008 (MIL-STD-105) Percentage Points in Terms of Percent Nonconforming

Probabilityof

Acceptance

ANSI/ASQZ1.4-2008

Code Letter .10 .15 .25 .40 .65 1.00 1.50 2.50 4.00 6.50 10.00

95.050.010.0

B1.70

20.6 53.6

95.050.010.0

C1.02

12.9 36.9

7.6331.4 58.4

95.050.010.0

D.64

8.3025.0

2.6420.0140.6

11.1 32.1 53.9

95.050.010.0

E.394

5.19 16.2

2.8112.6 26.8

6.6320.0 36.0

11.3 27.5 44.4

95.050.010.0

F.256

3.41 10.9

1.808.25

18.1

4.2213.1 24.5

7.1318.1 30.4

14.0 27.9 41.9

95.050.010.0

G.161

2.14 6.94

1.135.19

13.38

2.598.29

15.8

4.3911.4 19.7

8.5017.5 27.1

13.1 23.7 34.1

95.050.010.0

H.103

1.38 4.50

.7123.33 7.56

1.665.31

10.3

2.777.30

12.9

5.3411.3 17.8

8.2015.2 22.4

12.9 21.2 29.1

95.050.010.0

J.064.863

2.84

.4442.09 4.78

1.03 3.33 6.52

1.734.578.16

3.327.06

11.3

5.069.55

14.2

7.9113.3 18.6

11.9 18.3 24.2

95.050.010.0

K.0410.554

1.84

.2841.34 3.11

.6542.14 4.26

1.09 2.94 5.35

2.094.547.42

3.196.149.42

4.948.53

12.3

7.4011.7 16.1

11.9 17.3 22.5

95.050.010.0

L.178.839

1.95

.4091.34 2.66

.6831.84 3.34

1.31 2.84 4.64

1.993.845.89

3.095.337.70

4.627.33

10.1

7.4510.8 14.1

95.050.010.0

M.112.534

1.23

.259

.8481.69

.4331.17 2.12

.8291.80 2.94

1.26 2.43 3.74

1.963.394.89

2.944.666.39

4.736.888.95

95.050.010.0

N.071 .336 .778

.164

.5351.06

.273

.7341.34

.5231.13 1.86

.7961.53 2.35

1.23 2.13 3.08

1.852.934.03

2.984.335.64

95.050.010.0

P.102 .334 .665

.171

.459

.835

.327

.7091.16

.498

.9591.47

.7711.33 1.93

1.16 1.83 2.52

1.862.713.52

95.050.010.0

Q.109 .294 .534

.209

.454

.742

.318

.614

.942

.494

.8531.23

.7401.17 1.61

1.19 1.73 2.25

95.050.010.0

Q.131 .284 .464

.199

.384

.589

.309

.533

.770

.462

.7331.01

.7451.08 1.41


01.Z1.9(001-104).qxd 2/17/09 2:40 PM Page 103

ANSI/ASQ Z1.9-2008

104

Table 4Difference in Percentage Points—ANSI/ASQ Z1.9-2008 Minus

ANSI/ASQ Z1.4-2008 (MIL-STD-105)

Probabilityof

Acceptance

ANSI/ASQZ1.4-2008

CodeLetter .10 .15 .25 .40 .65 1.00 1.50 2.50 4.00 6.50 10.00

95.050.010.0

B.19

–.30–.77

95.050.010.0

C.30

1.544.25

–.78–2.79–2.85

95.050.010.0

D.13

1.416.24

1.66.65

3.95

–3.99–5.70–3.58

95.050.010.0

E.136.99

4.89

–.161.065.13

–2.06–1.891.28

–3.84–5.70–1.15

95.050.010.0

F.104.36

2.33

–.15.37

3.30

–1.39–1.411.16

–2.29–2.19

.59

–6.19–6.81–4.52

95.050.010.0

G.089.06 .64

–.04.13

1.78

–.68–.76.97

–1.30–1.25

.78

–3.20–3.23–1.34

–4.69–4.45–2.47

95.050.010.0

H.067.000.23

.078

.33 1.67

.45–.50.84

–.70–.441.35

–1.95–1.79

.14

–2.51–1.71

.61

–4.02–2.89–.40

95.050.010.0

I.138.06 .41

–.38–.84–.57

–.54–.76–.09

–1.73–2.18–1.46

–2.22–2.20–1.16

–3.63–3.46–2.28

95.050.010.0

J.066

–.103–.55

.156

.12

.32

–.09 –.28 –.02

–.33–.52–.09

–.94–1.08–.45

–1.26–1.14–.09

–1.70–1.20

.11

–2.25–1.48

.03

95.050.010.0

K.059

–.364–.48

.116

.03 –.04

–.014–.20 –.23

–.07 –.18 –.02

–.60–.86–.70

–.68–.66–.19

–.90–.63.09

–.88–.25.62

–1.90–1.30–.52

95.050.010.0

L.092

–.019–.21

.021–.15 –.29

.017–.10 –.10

–.25 –.41 –.36

–.41–.50–.31

–.47–.31.12

–.44–.04.60

–.64.04.84

95.050.010.0

M–.42.12.89

.068–.022–.18

.031–.078–.19

.037–.05 –.06

–.089–.16 –.08

–.14 –.12 –.07

–.30–.21.12

–.21.14.72

95.050.010.0

N.059

–.015–.158

.026–.075–.21

.037–.044–.13

–.043–.13 –.18

–.026–.05 –.01

–.05 .01 .18

–.12.03.31

.16

.16

.62

95.050.010.0

P.041

–.013–.094

.039–.014–.072

.017–.026–.044

.036

.041

.097

.069

.15

.29

.09

.25

.50

.00

.25

.60

ANSI/ASQZ1.9-2008

CodeLetter

B

C

D

E

F

G

H

I

J

K

L

M

N

P


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