Designation B762 − 90 (Reapproved 2016) Standard Test Method of Variables Sampling of Metallic and Inorganic Coatings1 This standard is issued under the fixed designation B762; the number immediately[.]
Designation: B762 − 90 (Reapproved 2016) Standard Test Method of Variables Sampling of Metallic and Inorganic Coatings1 This standard is issued under the fixed designation B762; the number immediately following the designation indicates the year of original adoption or, in the case of revision, the year of last revision A number in parentheses indicates the year of last reapproval A superscript epsilon (´) indicates an editorial change since the last revision or reapproval Scope B602 Test Method for Attribute Sampling of Metallic and Inorganic Coatings B697 Guide for Selection of Sampling Plans for Inspection of Electrodeposited Metallic and Inorganic Coatings 2.2 ANSI Standards:3 ANSI/ASQC Z1.9-1979 Sampling Procedures and Tables for Inspection by Variables for Percent Non-Conformance ANSI/ASQC Z1.4-1981 Sampling Procedures and Tables for Inspection by Attributes 2.3 Military Standards:4 MIL-STD-105 Sampling Procedures and Tables for Inspection by Attributes MIL-STD-414 Sampling Procedures and Tables for Inspection by Variables for Percent Defective 1.1 This test method provides sampling plans that are intended for use in the inspection of metallic and inorganic coatings on products for the purpose of deciding whether submitted lots of coated products comply with the specifications applicable to the coating 1.2 The sampling plans are variables plans In plans of this type, several articles of product are drawn from a production lot A characteristic of the coating on the drawn articles is measured The values obtained are used to estimate the number of articles in the lot that not conform to a numerical limit, for example a minimum thickness The number is compared to a maximum allowable 1.3 Variables plans can only be used when the characteristic of interest is measurable, the test method gives a numerical measure of the characteristic, and the specification places a numerical limit on the measured value It is also necessary that the variation of the characteristic from article to article in a production lot be normally distributed (see Appendix X2) Each article must be tested in the same way (for example, coating thickness must be measured at the same location, see X2.7) so that the values from article to article are comparable If one or more of these conditions are not met, a variables plan cannot be used Instead, an attributes plan must be used These are given in Test Method B602 and Guide B697 1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use Definitions 3.1 destructive test—test that destroys the tested article or makes it nonconforming to a requirement 3.2 nondestructive test—test that neither destroys the tested article nor makes it nonconforming to a requirement 3.3 inspection lot—collection of articles of the same kind that is submitted to inspection for acceptance or rejection as a group 3.4 sample—articles randomly selected from an inspection lot whose quality is used to decide whether or not the inspection lot is of acceptable quality 3.5 standard deviation—measure of dispersion equal to the square root of the mean of the squares of the deviations from the arithmetic mean of the distribution (see 9.2.6) Summary of Test Method Referenced Documents 4.1 The plans in this test method provide the same protection as the attributes plans in Tables 1, 2, and of Test Method B602 and are interchangeable with them when the conditions necessary for variables sampling exist This method has no plan comparable to Table of Test Method B602, because variables plans are subject to an excessive probability of error when the number of nonconforming articles in a lot is expected 2.1 ASTM Standards:2 This test method is under the jurisdiction of ASTM Committee B08 on Metallic and Inorganic Coatings and is the direct responsibility of Subcommittee B08.10 on Test Methods Current edition approved Nov 1, 2016 Published November 2016 Originally approved in 1986 Last previous edition approved in 2010 as B762 – 90 (2010) DOI: 10.1520/B0762-90R16 For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org For Annual Book of ASTM Standards volume information, refer to the standard’s Document Summary page on the ASTM website Available from American National Standards Institute (ANSI), 25 W 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org Available from Standardization Documents Order Desk, DODSSP, Bldg 4, Section D, 700 Robbins Ave., Philadelphia, PA 19111-5098 Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959 United States B762 − 90 (2016) TABLE Level II—Sampling Plans for Nondestructive Tests, Standard Deviation UnknownA to be approximately % or less as it is for the Table plan Also for this reason, comparable variables plans are not given for the smallest lot sizes of Tables and of Test Method B602 The plans of Table 4, and Tables and in Test Method B602 are described as Level I, Level II, and Level III respectively For consistency, Table and Table of this method are described as Level II since they are comparable to Table of Test Method B602, and Table and Table are described as Level III Inspection Lot Size 91 through 280 281 through 500 501 through 200 201 through 200 201 through 10 000 10 001 through 35 000 Over 35 000 4.2 The main advantage of a variables sampling plan over an attributes plan is that fewer articles need to be inspected to obtain the same protection For example, a sample of 12 using variables can give the same protection as a sample of 50 using attributes On the other hand, more expensive test methods may be required to yield the measurements required by variables sampling A 51 through 150 151 through 280 281 through 500 501 through 200 201 through 200 201 through 16 000 16 001 through 35 000 Over 35 000 A 4.5 Additional variables plans are given in Appendix X3 Also found there are instructions for the calculation of plans for needs that are not covered 5.1 Sampling inspection permits the estimation of the overall quality of a group of product articles through the inspection of a relatively small number of product articles drawn from the group 51 through 150 151 through 280 281 through 500 501 through 200 201 through 200 201 through 16 000 16 001 through 35 000 Over 35 000 TABLE Level II—Sampling Plans for Nondestructive Tests, Standard Deviation KnownA 91 through 280 281 through 500 501 through 200 201 through 200 201 through 10 000 10 001 through 35 000 Over 35 000 A LQL 50/50 Point AOQL 1.664 1.1 12 4.8 2.4 12 1.649 1.7 10 5.0 2.6 16 1.712 1.7 8.2 4.4 2.3 25 1.704 2.1 7.4 4.4 2.5 36 1.778 2.0 5.9 3.8 2.2 52 1.829 2.0 4.9 3.4 2.1 82 1.893 1.9 4.0 2.9 1.9 50/50 Point AOQL 16 1.663 1.0 12 4.8 2.4 29 1.649 1.7 10 5.0 2.6 40 1.713 1.7 8.2 4.3 2.2 61 1.704 2.1 7.4 4.4 2.5 92 1.778 2.0 5.9 3.8 2.2 137 1.825 2.0 4.9 3.4 2.0 223 1.893 1.9 4.0 3.0 1.9 n k AQL LQL 50/50 Point AOQL 1.432 1.8 18 7.6 3.8 10 1.411 2.7 16 7.9 4.1 14 1.470 2.8 13 7.1 3.5 23 1.492 3.3 11 6.8 3.8 30 1.551 3.2 9.4 6.0 3.5 44 1.618 3.1 7.7 5.3 3.2 66 1.680 3.0 6.4 4.6 3.0 103 1.719 3.0 5.6 4.4 2.9 The AQL, LQL, 50/50 Point, and AOQL are in percent Inspection Lot Size AQL LQL TABLE Level III—Sampling Plans for Nondestructive Tests, Standard Deviation UnknownA Significance and Use k AQL The AQL, LQL, 50/50 Point, and AOQL are in percent Inspection Lot Size 4.4 The sampling plans in Tables and of this test method are considered to be standard for nondestructive testing and will be used unless the buyer specifies otherwise Tables and will be used for destructive testing; these plans use smaller samples to reduce the cost of inspection with a resultant reduction of the ability to distinguish between conforming and nonconforming lots n k TABLE Level III—Sampling Plans for Nondestructive Tests, Standard Deviation KnownA 4.3 Generally, thickness is the only characteristic of a coating that meets the conditions of a variables plan given in 1.3 For that reason, the plans in this method are designed to be used when the specification for the characteristic in question is a minimum value, which is the usual case for coating thickness Variables plans can be used when the limit is a maximum and when there are both a minimum and a maximum Plans for these cases are given in the references Inspection Lot Size n A n k AQL LQL 50/50 Point AOQL 12 1.433 1.7 19 7.6 3.8 19 1.410 2.6 16 7.9 3.7 29 1.470 2.8 13 7.1 3.8 48 1.494 3.3 11 6.7 3.8 66 1.551 3.2 9.4 6.0 3.5 102 1.618 3.1 7.7 5.3 3.2 159 1.680 3.0 6.4 4.6 3.0 248 1.717 3.0 5.6 4.3 2.9 The AQL, LQL, 50/50 Point, and AOQL are in percent 5.2 The specification of a sampling plan provides purchasers and sellers a means of identifying the minimum quality level that is considered to be satisfactory 5.3 Because sampling plans yield estimates of the quality of a product, the results of the inspection are subject to error The AQL, LQL, 50/50 Point, and AOQL are in percent B762 − 90 (2016) TABLE Sampling Plans for Destructive Tests, Standard Deviation KnownA Inspection Lot Size 26 through 200 201 through 35 000 Over 35 000 A n k AQL LQL 50/50 Point 10 14 1.262 1.411 1.519 2.3 2.7 2.5 25 16 12 10 7.9 6.5 5.9 The use of a sampling plan involves the balancing of the costs of inspection against the consequences of accepting an undesirable number of nonconforming articles There is always a risk that a random sample will not describe correctly the characteristics of the lot from which it is drawn, and that an unacceptable lot will be accepted or an acceptable lot will be rejected The larger the sample, the smaller this risk but the larger the cost of inspection The AQL, LQL, and 50/50 Point are in percent 5.10 To understand the risks, consider that if every article in an inspection lot conforms to its requirements, every article in the sample will conform also Such lots will be accepted (Note 1) If only a few articles in an inspection lot are nonconforming, the sample probably will indicate that the lot is acceptable; but there is a small probability that the sample will indicate that the lot is unacceptable The larger the proportion of nonconforming articles in an inspection lot, the more likely it will be that the sample will indicate that the lot is unacceptable If every article in an inspection lot is nonconforming, a sample will always indicate that the lot is unacceptable TABLE Sampling Plans for Destructive Tests, Standard Deviation UnknownA Inspection Lot Size 26 through 200 201 through 35 000 Over 35 000 A n k AQL LQL 50/50 Point 19 34 1.181 1.412 1.497 2.8 2.5 2.8 27 16 12 12 7.9 6.7 The AQL, LQL, and 50/50 Point are in percent Through the selection of a sampling plan, the potential error is known and controlled NOTE 1—Throughout this method, it is assumed that no mistakes are made in sampling, measurement, and calculation 5.4 Sampling inspection is used when a decision must be made about what to with a quantity of articles This quantity may be a shipment from a supplier, articles that are ready for a subsequent manufacturing operation, or articles ready for shipment to a customer 5.11 The probability of accepting an inspection lot that contains nonconforming items is often described in terms of the Acceptable Quality Level (AQL) and the Limiting Quality Level (LQL) The AQL is the quality level that is considered to be acceptable The LQL is a quality level that is considered to be barely tolerable A sampling plan is selected that has a high probability of accepting lots of AQL quality and of rejecting lots of LQL quality In this method, the AQL given for a sampling plan is the quality level of lots (expressed as the percentage of nonconforming articles) that have a 95 % probability of being accepted The LQL is the quality level of lots that have a 10 % probability of being accepted or, in other words, a 90 % probability of being rejected The tables in this method give the AQL and LQL of each plan They also give the 50/50 point, the quality level of a lot that is just as likely to be accepted as rejected 5.5 In sampling inspection, a relatively small number of articles (the sample) is selected randomly from a larger number of articles (the inspection lot); the sample is inspected for conformance to the requirements placed on the articles Based on the results, a decision is made whether or not the lot conforms to the requirements 5.6 Since only a portion of a production lot is inspected, the quality of the uninspected articles is not known The possibility exists that some of the uninspected articles are nonconforming Therefore, basic to any sampling inspection plan is the willingness of the buyer to accept lots that contain some nonconforming articles The number of nonconforming articles in accepted lots is controlled by the size of the sample and the criteria of acceptance that are placed on the sample 5.12 The disposition of nonconforming inspection lots is beyond the scope of this method because, depending on the circumstances, lots may be returned to the supplier, kept and used, put to a different use, scrapped, reworked, or dealt with in some other way An alternative is rectifying inspection in which rejected lots are screened and used 5.7 Acceptance sampling plans are used for the following reasons: 5.7.1 When the cost of inspection is high and the consequences of accepting a nonconforming article are not serious 5.7.2 When 100 % inspection is fatiguing and boring and, therefore, likely to result in errors 5.7.3 When inspection requires a destructive test, sampling inspection must be used 5.13 In rectifying inspection, when an inspection lot is rejected, all of the articles in the lot are inspected and nonconforming ones are removed They may be replaced with conforming articles The now 100 % conforming lot is accepted With this practice, the average quality level for a series of lots taken as a whole will be better because of the addition of the 100 % conforming lots When the incoming lots are of a good quality level, the average quality level of a series of lots will be even better when the rejected lots are screened and resubmitted When incoming lots are of a poor quality level, the average quality of a series of accepted lots will again be good because many of the incoming lots will be rejected and upgraded At intermediate quality levels of incoming lots, the average quality level of a series of accepted lots will again be 5.8 In acceptance sampling by variables, the coating characteristic of each article in the sample is measured Using the arithmetic mean of these values, the standard deviation of the process, and the factor k that is found in the Tables, a number is calculated (see 9.3) If this number equals or exceeds the specified minimum, the inspection lot conforms to the requirements If it is less, the lot does not conform If the standard deviation of the process is not known, the standard deviation of the sample is calculated and used B762 − 90 (2016) 8.2 Nondestructive Tests—For nondestructive testing, the size of the sample shall be that specified for the sampling plan level that is required by the purchaser The sampling plans are given for Level II in Tables and and for Level III in Tables and If the purchaser does not specify the level, Level II shall be used The plans in Table and Table shall be used when the standard deviation of the coating process is known Tables and plans shall be used when the standard deviation is not known and must be estimated from the sample values improved, but it will not be improved as much as in either of the above cases; and there will be an intermediate quality level where the degree of improvement is the least This improved quality level is called the Average Outgoing Quality Limit (AOQL) It is the worst condition that can occur under rectifying inspection The tables give the AOQL for each plan There is no AOQL for the plans used with destructive tests because destructive tests cannot be used to screen rejected lots NOTE 2—The AOQLs given in the tables are strictly correct only when the sample is small with respect to the lot If this is not the case, the correct AOQL will be smaller than the tabulated value The correct values are obtained by multiplying the tabulated values by the following equation: sample size/lot size 8.3 Destructive Tests—For destructive testing, the size of the sample shall be that specified in Table when the standard deviation of the process is known and Table when it is not known (1) 5.14 Rectifying inspection will substantially increase the cost of inspection if the incoming lots are much worse than AQL quality 8.4 The sample shall be drawn randomly from the inspection lot, that is, in a manner that ensures each article an equal chance of being selected regardless of other considerations such as location in the inspection lot, appearance, quality, location on a fixture during coating, and chronological relationship to the other articles Random sampling procedures are given in the Appendixes 5.15 Rectifying inspection is used only when required by the purchaser Ordering Information 6.1 Unless otherwise specified by the purchaser, the sampling plans given in Tables and will be used for nondestructive testing, and the plans given in Tables and for destructive testing Calculations 9.1 Calculate the arithmetic mean of the measured characteristic by adding the values obtained for the articles and dividing the number of articles that were tested using the following equation: 6.2 When either a nondestructive or a destructive test can be used to inspect an article for conformance to a particular requirement, the purchaser should specify which test is to be used When a test is neither clearly nondestructive nor destructive, the purchaser should specify which it is considered to be n X¯ (X i51 i (2) n where: X¯ = arithmetic mean of the measured values, = measured value, Xi n = sum of the measured values, and X NOTE 3—The nature of a destructive test can be such that the tested article can be reclaimed, for example by stripping and reapplying the coating Other tests can destroy the coating in nonessential locations, in which case the article can still be functional In these instances, the purchaser needs to decide and state whether the tests are to be considered destructive or nondestructive ( i51 n 6.3 Rectifying inspection will be used only when specified by the purchaser When rectifying inspection is used, nonconforming articles will be replaced with conforming ones only when specified by the purchaser i = number of articles tested 9.2 If the standard deviation of the coating process is known, continue the calculations as directed in 9.3 The symbol for the standard deviation for the process is σ If the standard deviation for the process is not known, calculate an estimated value from the measurements obtained from the sample as directed in 9.2.1 through 9.2.6 The symbol for this estimated standard deviation is s 9.2.1 Subtract the arithmetic mean from the first measured value using the following equation: Formation of Inspection Lot 7.1 An inspection lot shall be formed from articles that are of the same kind, that have been produced to the same specification, and that have been coated by a single supplier at one time or at approximately the same time under essentially identical conditions X X¯ (3) NOTE 4—These requirements are intended to ensure that the lot is homogeneous and that variations between articles in the lot are the result only of the inherent variation of the production process (see Appendix X1) 9.2.2 Calculate the square of the difference obtained in 9.2.1 using the following equation: Sampling 9.2.3 Repeat 9.2.1 and 9.2.2 for each measured value 9.2.4 Add all of the squares obtained in 9.2.2 and 9.2.3 using the following equation: ~X 8.1 General—A sample shall be selected randomly from the inspection lot If the test method to be used is nondestructive, the sample size shall be that directed in 8.2 If the test method is destructive, the sample size shall be that directed in 8.3 ~X 2 X¯ ! ¯ ¯ ¯ X ! ~ X 2 X ! 1…1 X n X ~ (4) ! n ( ~X i51 2 X¯ ! (5) B762 − 90 (2016) 9.2.5 Divide the sum obtained in 9.2.4 by one less than the number of articles that were tested using the following equation: n ( ~ X X¯ ! X¯ kσ or, calculate the following when the standard deviation is not known: X¯ ks i i51 (6) n21 s5 ! n ( ~ X X¯ ! s5 !( i 10.1 Inspection—Each article in the sample shall be inspected as directed in the applicable coating standard 10.2 Lot Classification: 10.2.1 The number calculated in 9.3 shall be compared to the minimum number stated in the coating specification If the number in 9.3 equals or exceeds the specified minimum, the lot conforms to the requirements If it is less than the specified minimum, the lot does not conform 10.2.2 When specified by the purchaser, nonconforming lots shall be 100 % inspected, and nonconforming articles shall be removed When required by the purchaser, the nonconforming articles shall be replaced with conforming articles i n21 NOTE 5—The following equation can also be used: X (7) ~ ( X i! n21 n (10) 10 Inspection and Lot Classification 9.2.6 Calculate the square root of the value obtained in 9.2.5 using Eq This is standard deviation, s i51 (9) (8) 9.3 Using the k that is in the table and the standard deviation from 9.2, calculate the following number when the standard deviation is known: APPENDIXES (Nonmandatory Information) X1 DRAWING OF SAMPLES follows: the 85’s are rejected because they are over 80, and the second 06 is rejected because it has already appeared The sample then consists of articles numbered 31, 20, 8, 26, 53, 65, 64, 46, 22, 6, 41, and 67 X1.1 The success of acceptance sampling is totally dependent on the sample being drawn from the lot at random Random sampling means that the selection of an article for the sample is totally by chance and that every article in the lot is equally likely to be selected If the articles in the inspection lot are thoroughly mixed, such as barrel-plated articles, a sample drawn from anywhere in the lot will be random (see X2.5) Rack-plated articles cannot be sampled this way unless thorough mixing is done before sampling, otherwise a random sampling procedure must be used Methods of random sampling are described in the following paragraphs X1.4 When product items are arranged in an order without regard to quality, such as articles in a tray, a sample can be drawn by using the constant interval procedure Here, a constant interval is maintained between the items drawn for the sample For example, every 9th, 19th, or 24th unit is selected The first item drawn from the lot can be determined from the table of random numbers All other items are then drawn at a constant interval following the first item The constant interval is determined by dividing the lot size by the sample size X1.2 When random numbers are used to select a sample, each article in the lot is identified by a different number If the units have serial numbers, the serial numbers can be used The numbers of the articles that are to be inspected are selected from a table of random numbers such as Table X1.1 Other tables of random numbers can be obtained from books on statistics Some pocket calculators are designed to generate random numbers X1.5 As an example, assume that a lot of 3000 items is to be inspected In accordance with Table 3, a sample of 30 items is to be drawn The constant interval is 100 (3000 divided by 30) A random number from to 100 is selected either from a table or by another appropriate method After the first item is taken, the remaining items in the required sample are drawn by selecting every 100th item from the lot until 30 are selected X1.3 As an example, assume that a sample of 12 articles is to be selected from an inspection lot of 80 articles The articles are numbered through 80 A pencil is allowed to fall blindly at some number in Table X1.1 Starting at this point, a coin is tossed to decide whether to go up or down the column; heads, up; tails, down If the pencil falls on column 10, line 11, and the coin is tails; the decision is to read down the column until 12 numbers are chosen Take the first two digits in each group of five digits The selection of random numbers is made as X1.6 References (1 through 2)5 give additional information and procedures on random sampling X1.7 The numbers of a random sample can be generated by the following microsoft BASIC computer program: The boldface numbers in parentheses refer to the list of references at the end of the standard B762 − 90 (2016) 10 30 40 50 60 70 80 90 100 110 120 130 140 150 160 REM—Program to select random samples for testing PRINT “ENTER LOT SIZE” INPUT L PRINT “ENTER SAMPLE SIZE” INPUT S DIM A(L) FOR K = TO L A(K) = NEXT K PRINT “MATRIX ZEROED” RANDOMIZE PEEK(11) FOR K = TO L N = INT(L*RND(1) + 1) IF A(N)0 THEN 140 A(N) = K 170 180 190 200 NEXT K PRINT “MATRIX LOADED” PRINT “FOR A LOT SIZE OF ”;L:LPRINT “FOR A LOT SIZE OF” ;L PRINT “AND A SAMPLE SIZE OF” ;S:LPRINT “AND A SAMPLE SIZE OF ”;S PRINT “THE SAMPLE NUMBERS ARE:”:LPRINT “THE SAMPLE NUMBERS ARE:” FOR R = TO S M = INT(L*RND(1) + 1) IF A(M) = THEN 220 PRINT A(M);“,”;:LPRINT A(M);“,”; NEXT R PRINT “END OF SAMPLE LIST”:LPRINT “END OF SAMPLE LIST” END 205 210 220 230 240 250 260 270 TABLE X1.1 Table of Random Numbers Line 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 Column (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 10480 22368 24130 42167 37570 77921 99562 96301 89579 85475 28918 63553 09429 10365 07119 51085 02368 01011 52162 07056 48663 54164 32639 29334 02488 81525 29676 00742 05366 91921 00582 00725 69011 25976 09763 91567 17955 46503 92157 14577 98427 34914 70060 53976 76072 90725 64364 08962 95012 15664 16408 18629 73115 57491 30405 16631 96773 15011 46573 48360 93093 39975 06907 72905 91977 14342 36857 69578 40961 93969 61129 97336 12765 21382 54092 53916 97628 91245 58492 32363 27001 33062 72295 20591 57392 04213 26418 04711 69884 65795 57948 83473 42595 56349 18584 80634 62765 07523 63976 28277 54914 29515 52210 67412 00358 68379 10493 81899 81953 35101 16703 83946 35006 20206 01536 25595 22527 06243 81837 11008 56420 05463 63661 53342 88231 48235 52636 87529 71048 51821 52404 33362 46369 33787 85828 22421 05597 87637 28834 04839 68086 39064 25669 64117 87917 62797 95876 29888 73577 27958 90999 18845 94824 35605 33362 88720 39475 06990 40980 83974 33339 31662 93526 20492 04153 05520 47498 23167 23792 85900 42559 02011 85393 97265 61680 16656 42751 69994 07972 10281 53988 33276 03427 92737 85689 08178 51259 60268 94904 58586 09998 14346 74103 24200 87308 07351 96423 26432 66432 26422 94305 77341 56170 55293 88604 12908 30134 49127 49618 78171 81263 64270 82765 46473 67245 07391 29992 31926 25388 70765 38391 53381 91962 87637 49323 14422 98275 78985 81647 30995 76393 07856 06121 27756 98872 18876 17453 53060 70997 49626 88974 48237 77233 77452 89368 31273 23216 42698 09172 47070 13363 58731 19731 24878 46901 84673 44407 26766 42206 86324 18988 67917 30883 04024 20044 02304 84610 39667 01638 34476 23219 68350 58745 65831 14883 61642 10592 91132 79401 04739 99016 45021 15059 32388 05300 91646 89198 64809 16376 91782 53498 31016 20922 18103 59533 79936 69445 33488 52267 13916 16308 19885 04146 14513 06691 30168 25306 38005 00256 92420 82651 20849 40027 44048 25940 35126 88072 27354 48708 18317 86385 59931 51038 82834 47358 92477 17032 53416 82948 25774 38857 24413 31072 04542 21999 21438 13092 71060 33132 45799 52390 22164 69179 27982 15179 39440 60468 18602 71194 94595 57740 38867 56865 18663 36320 67689 47564 60756 55322 18594 83149 76988 90229 76468 94342 45834 60952 66566 89768 32832 37937 39972 74087 76222 26575 18912 28290 29880 06115 20655 09922 56873 66969 87589 94970 11398 22987 50490 59744 81249 76463 59516 83035 97662 88824 12544 22716 16815 24369 14194 53402 24830 53537 81305 70659 18738 56869 84378 62300 05859 72695 17617 93394 81056 92144 44819 29852 98736 13602 04734 26384 28728 15398 61280 14778 81536 61362 63904 22209 99547 36086 08625 82271 35797 99730 20542 58727 25417 56307 98420 40836 25832 42878 80059 83765 92351 35648 54328 81652 92350 24822 71013 41035 19792 69298 54224 62590 93965 49340 71341 49684 90655 44013 69014 25331 08158 90106 52180 30015 01511 97735 49442 01188 71585 23495 51851 59193 58151 35806 46557 50001 76797 86645 98947 45766 71500 81817 84637 40801 65424 05998 55536 18059 28168 44137 61607 04880 32427 69975 80287 39911 55657 97473 56891 02349 27195 36693 94730 18735 80780 09983 82732 35083 36207 34095 32081 57004 60672 15053 48840 60045 12566 17983 31595 20847 08272 26358 85977 53900 65255 85030 64350 46104 22178 06646 06912 41135 67658 14780 12659 96067 66134 64568 42607 93161 59920 69774 41688 84855 02008 15475 48413 49518 45585 70002 94884 88267 96189 14361 89286 69352 17247 48223 31238 06496 20286 45393 74353 38480 19687 20969 52666 30680 00849 14110 21916 63213 18425 58678 16439 01547 12234 84115 85104 29372 70960 64835 51132 94738 88916 30421 21524 17012 10367 32586 13300 92259 64760 75470 91402 43808 76038 29841 33611 34952 29080 73708 56942 25555 89656 46565 70663 19661 47363 41151 31720 35931 48373 28865 46751 59649 35090 23153 44812 68668 73817 11052 99570 19174 19655 74917 06927 81825 21069 84903 44947 11458 85590 90511 27156 20285 74461 63990 44919 01915 17752 19509 61666 15227 64161 07684 86679 87074 57102 64584 66520 42416 76655 65855 80150 54262 37888 09250 83517 53389 21246 20103 04102 88863 72828 46634 14222 57375 04110 45578 14777 22923 91754 04822 72924 12515 30429 32523 91491 91291 39615 63348 97758 01263 44394 10634 42508 05585 18593 91610 33703 30613 29975 28551 75601 05944 92747 35156 25625 99904 96909 18296 36188 50720 79666 80428 96096 34693 07844 62028 77919 12777 85963 38917 79656 36103 20562 35509 77490 46880 77775 00102 06541 60697 56228 23726 78547 62730 32261 72772 86774 35165 98931 70735 41961 60383 90700 99505 58629 16379 54613 42880 12952 32307 56941 64952 78188 90322 74952 89868 90707 40719 55157 64951 35749 58104 32812 44592 22851 18510 94953 95725 25280 98253 90449 69618 76630 88006 48501 03547 88050 73211 42791 87338 20468 18062 45709 69348 66794 97809 59583 41546 51900 81788 92277 85653 02338 98289 43040 91202 25499 44437 19746 B762 − 90 (2016) TABLE X1.1 Line 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Continued Column (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 38935 31624 78919 03931 74426 09066 42238 16153 21457 21581 55612 44657 91340 91227 50001 65390 27504 37169 11508 37449 46515 30986 63798 82486 21885 60336 43937 97656 03299 79626 85636 18039 08362 79556 92608 23982 09915 59037 42488 46764 03237 86591 38534 64202 76384 19474 33309 33278 00903 12426 08002 40742 57802 78095 66999 84979 21199 38140 05224 96131 94851 70225 30362 70331 81223 64995 84846 32906 98782 46891 63175 01221 06486 68335 14367 15656 29068 82674 25835 96306 33300 78077 86273 45430 81482 01715 14349 17403 23632 57047 43972 20795 87025 26504 29820 02050 83197 99324 46949 31935 66321 72958 83944 39117 51111 06694 85922 42416 46583 99254 92431 07408 24010 89303 05418 03574 47539 61337 60627 04142 27072 40055 05908 26695 69882 63003 55417 52667 94964 82674 53363 27889 74211 10119 95452 14267 41744 96783 89728 33732 51281 81973 27022 19924 28609 41575 89632 38351 54690 38329 58353 09785 67632 09060 53458 25560 16275 38982 17668 03129 06117 36478 16268 32534 67006 97901 62247 61657 93017 63282 61582 87288 66523 44167 47914 63445 89917 92648 20979 81959 29400 17937 05810 84463 37949 84067 72163 81406 10573 00959 19444 04052 57015 21532 44160 43218 64297 13564 86355 07100 55758 07785 65651 12143 65648 15387 17075 12293 28395 69927 34136 31204 90816 14972 65680 44133 64486 02584 17361 15665 45454 04508 65642 21840 37621 24813 60563 61023 05462 09538 39147 08619 16487 66499 53115 15765 30502 78128 50076 51674 59089 33941 92063 92237 76020 11977 46609 16764 12856 27698 02753 14186 76123 79180 36692 17349 90053 43772 00697 64758 37680 62825 52872 09552 64535 74240 15035 47075 86902 79312 43997 35216 12151 25549 64482 65536 71945 62757 97161 32305 83991 21361 64126 26445 25786 21942 26759 79924 02510 32989 53412 66227 98204 14827 00821 50842 97526 40202 88298 89534 39560 35552 75366 20801 39908 73823 88815 31355 56302 34537 42080 60397 93454 15263 14486 06878 48542 73923 49071 05422 95348 17869 86482 42865 64816 62570 29789 54990 18611 86367 25651 26113 74014 09013 38358 63863 23235 80703 43834 43092 35275 90183 76036 12918 35970 76554 72152 05607 73144 16553 86064 00033 33310 97403 16489 68876 80644 29891 91903 42627 36152 39782 13442 78662 45349 05174 92520 51202 26123 85205 71899 47348 21216 83325 99447 64708 07832 22478 11951 35071 70426 86654 04098 57306 36600 49199 86537 19124 31601 39339 91284 88662 51125 29472 67107 06116 48626 03264 25471 43942 68607 18749 45233 05184 17095 78675 11163 61796 07901 83531 88124 05155 41001 15475 20203 98442 88428 68645 00533 41574 73373 34648 99704 75647 70959 73571 55543 78406 43716 62738 63318 12614 34806 68833 88970 79375 47689 77510 95240 68995 88525 93911 89203 41867 34405 57202 94142 02330 84081 81651 66345 54339 80377 41870 59194 12535 95434 18534 08303 85076 34327 35398 17639 88732 88022 37543 76310 79725 80799 53203 06216 97548 19636 29686 33072 08930 25570 74492 97596 05974 70625 15957 43805 42786 25650 71795 14951 56087 94617 25299 73401 66938 50245 81073 58861 35909 52689 52799 12133 98227 03862 56613 72811 15152 58408 82163 09443 56148 11601 88717 93872 76536 18098 95787 04379 51132 03387 60332 85001 38818 51805 16296 52468 28725 16572 33386 05269 12682 99533 91696 82790 23772 84387 00275 93654 34971 49106 74818 81250 51275 28225 14645 21824 78095 91511 22717 55230 13261 60859 82558 34925 35503 37890 28117 71255 47625 42579 46370 25739 59846 92325 87820 46920 99378 66092 16834 34191 06004 21597 92532 73572 50501 85065 70925 07896 34925 48280 59894 52924 79860 46942 54238 83556 85762 23541 19585 50136 75928 50585 93448 47908 75567 05250 57031 85171 40129 19233 64239 88684 90730 28672 56947 X2 NORMAL DISTRIBUTION X2.1 Articles produced by a manufacturing process are never identical Minor variations in the process occur that affect the characteristics of the articles Such variations often occur at random and tend to cancel each other out Under these conditions, the articles are quite similar to each other Less often, the chance variations not cancel out as expected and some articles will differ from the typical As a result, it often is the case that most of the articles produced by a controlled process are closely grouped around an average condition, while smaller numbers deviate more from the average, and the greater the deviation the fewer articles there are Frequently this distribution of the articles can be closely described by a mathematical equation which, when plotted, gives the bellshaped curve shown in Fig X2.1 This is called a normal or Gaussian distribution variation and, thus, of little consequence X2.2 Along the horizontal, Xaxis in Fig X2.1 is plotted the numerical value of the characteristic that is being considered, for example, the thickness of the coating The area beneath the curve and above the Xaxis represents all of the articles in a production lot The arithmetic mean thickness is X¯, which is at the middle of the curve The vertical line at X¯ divides the curve in half so that half of the area is to the left, thicknesses less than the mean, and half is to the right, thicknesses greater than the mean It can be seen that if a plating thickness specification is given as a minimum value, and if the mean thickness of the lot equals the specification value, the thickness of the plating on half of the parts will be below the specification limit; that is, half of the article will be nonconforming Usually, it is required that most of the articles be conforming, which means that the mean thickness has to exceed the specified minimum The NOTE X2.1—There is also a random variation introduced by the measurement method This normally is small relative to the product B762 − 90 (2016) FIG X2.1 Normal Distribution variables plan based on the normal curve will give invalid results Therefore, statistical tests should be made of a process to confirm that the product characteristic is approximately normally distributed before a variables plan is used in the sampling inspection of the product Tests for normality are described in Ref (3) standard deviation can be used to determine by how much X2.3 If the mean thickness is one standard deviation higher than the specified minimum thickness, the thickness will be less than the specified minimum on 16 % of the articles If the mean is two standard deviations above the specification value, there will be about 2.3 % nonconforming articles in the lot Hence, if the standard deviation of the process is known, a mean thickness can be calculated that will ensure that no more than a given percentage (for example, the AQL) of a lot will be nonconforming Once a lot of articles is produced, a random sample of the articles can be inspected and their mean calculated If the mean is equal to or larger than the required mean, it is known that the percentage of nonconforming articles in the lot is no more than the AQL Actually, the mean of the sample can be different from that of the lot In variables plans, an additional factor is placed in the calculation to allow for this In processes where the standard deviation is not known and the sample standard deviation is used in the calculation, there is another potential error This is guarded against by using larger samples X2.5 The distribution of plating thickness on barrel-plated articles tends to be normal, provided good barrel plating practices are observed A production lot that consists of several barrel loads will also be normal if the loads are produced under essentially the same conditions But if two or more loads that have different average thicknesses are mixed, the mixed lot may not conform to the normal distribution X2.6 The distribution of coating thickness on products that are processed on racks may or may not be normal Tests of normality are required to determine if a variables plan can be used in these cases X2.7 The thickness of the coating must be measured at the same location on every article in the sample If this is not done, the variation in thickness that occurs naturally over the surface of a product will result in invalid values for the average and the standard deviation X2.4 It is important to remember that variables sampling plans are based on the normal curve If the product of a manufacturing process is not distributed normally, the use of a B762 − 90 (2016) X3 ADDITIONAL SAMPLING PLANS TABLE X3.1 Sampling Plans, Standard Deviation Known AQL, % LQL, % 10 15 20 25 n k 50/50 Point 18 1.943 1.740 1.600 1.493 1.396 2.6 4.1 5.5 6.8 8.1 AOQL n 1.4 2.0 2.8 3.4 4.2 51 14 5 k 50/50 Point AOQL n 1.824 1.619 1.481 1.373 1.283 3.4 5.3 6.9 8.5 10 2.1 2.8 3.5 4.3 5.0 65 23 13 10 k 50/50 Point AOQL n k 50/50 Point AOQL 1.441 1.303 1.193 1.009 7.5 9.6 12 14 4.9 5.6 6.3 6.9 142 44 23 1.144 1.034 0.940 13 15 17 9.5 9.9 11 TABLE X3.2 Sampling Plans, Standard Deviation Unknown AQL, % LQL, % 10 15 20 25 n k 50/50 Point 53 20 14 1.943 1.739 1.601 1.492 1.398 2.6 4.1 5.5 6.8 8.1 AOQL n 1.4 2.1 2.7 3.5 4.3 136 33 17 11 k 50/50 Point AOQL n 1.824 1.620 1.482 1.372 1.278 3.4 5.3 6.9 8.5 10 2.1 2.7 3.5 4.3 5.0 132 43 23 15 10 k 50/50 Point AOQL n k 50/50 Point AOQL 1.441 1.303 1.193 1.099 7.5 9.6 12 14 4.9 5.6 6.3 7.1 236 68 34 1.144 1.034 0.940 13 15 17 9.5 10 11 X3.2 The plans in Table X3.1 and Table X3.2 provide AQL’s of 1, 2, 5, and 10 % and LQL’s of 5, 10, 15, 20, and 25 % To select a plan, go to the table, find the column headed with the desired AQL, read down to the row headed by the desired LQL, and note the sample size (n) and k X3.1 Table X3.1 and Table X3.2 provide additional sampling plans that may be useful in situations where the standard plans of Tables 2-6 are unsuitable The plans of Table X3.1 are to be used when the standard deviation of the process is known The plans of Table X3.2 are to be used when the standard deviation is not known X4 CALCULATIONS OF VARIABLES SAMPLING PLANS TABLE X4.1 Values of z Used in Designing Variables Sampling Plans X4.1 The equations in this appendix can be used to calculate the sample size (n) and the k value of variables sampling plans where: X4.1.1 The coating requirement is stated as a lower limit; X4.1.2 The probability of accepting product of AQL quality is 0.95 (95 %); and X4.1.3 The probability of accepting product of LQL quality is 0.10 (10 %) X4.2 First, select the AQL and LQL values Next, go to Table X4.1 and find the values of z that correspond to the AQL and LQL For AQL’s and LQL’s that are not given in the table, thez’s can be calculated by interpolation The z corresponding to the AQL isz1, and the one corresponding to the LQL is z2 X4.3 Using z1 and z2, calculate n and k Round the value of n to the nearest whole number Use Eq X4.1 and Eq X4.2 when the standard deviation is known and Eq X4.3 and Eq X4.4 when the standard deviation is not known The equations are derived in Ref (3), Chapters 11 and 12 z 10 12.5 15 20 25 2.326 2.054 1.889 1.751 1.645 1.555 1.476 1.405 1.341 1.282 1.150 1.036 0.8416 0.6745 k5 ~ =n ~ z 1z ! 0.3633! /2 =n X4.3.2 Standard deviation unknown: k 0.4379 z 10.5621 z ! (X4.2) Use the rounded value of n to calculate k X4.3.1 Standard deviation known: η 8.564/ ~ z z AQL or LQL %Nonconforming n 4.2822 ~ 21k ! / ~ z z (X4.1) (X4.3) 2 ! (X4.4) B762 − 90 (2016) X4.4 Sample calculations are as follows: Desired AQL = % Desired LQL = 10 % Standard Deviation of Process Known Select z’s from Table X4.1 For AQL of % z1 = 2.054 For LQL of 10 % z2 = 1.282 n 8.564/ ~ 2.054 1.282! 14.4 (X4.5) Round to 14 X4.4.2 Calculate k as follows: k5 ~ =14 ~ 2.05411.282! 0.3633! /2 =14 1.619 (X4.6) X4.4.3 Note that these are the values of n and k given in Table X4.1 for an AQL of %, an LQL of 10 %, and a known standard deviation X4.4.1 Calculate n as follows: REFERENCES (1) Military Handbook MIL-HDBK-53, “Guide for Sampling Inspection.” (2) Manual on Presentation of Data and Control Chart Analysis, ASTM STP15D, ASTM, 1983 (3) Duncan, Acheson, J., Quality Control and Industrial Statistics, 4th Edition, Richard D Irwin, Inc., Homewood, IL, 1976 (4) General Services Administration Handbook FSS P4440.1, Guide for the Use of MIL-STD-105 (5) Duncan, Acheson J., “Acceptance Sampling Plans,”Standardization News, ASTM, Vol 3, No 9, September 1975, pp 8–9 (6) Duncan, Acheson, J., “An Introduction to Acceptance Sampling Plans,” ibid, pp 10–14 (7) Duncan, Acheson, J., “What Sampling Plan to Use,” ibid, pp 15–19 (8) Reynolds, John, H., “Sampling Plans for Mandatory Standards,” Standardization News, ASTM, Vol 5, No 3, March 1977, pp 8–12 (9) Dodge, H F., and Romig, H G., Sampling Inspection Tables, Single and Double Sampling, Second Edition, John Wiley and Sons, New York, NY, 1959 (10) Bowker, A H., and Goode, H P., Sampling Inspection by Variables, McGraw-Hill Book Co., New York, NY, 1952 ASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentioned in this standard Users of this standard are expressly advised that determination of the validity of any such patent rights, and the risk of infringement of such rights, are entirely their own responsibility This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years and if not revised, either reapproved or withdrawn Your comments are invited either for revision of this standard or for additional standards and should be addressed to ASTM International Headquarters Your comments will receive careful consideration at a meeting of the responsible technical committee, which you may attend If you feel that your comments have not received a fair hearing you should make your views known to the ASTM Committee on Standards, at the address shown below This standard is copyrighted by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States Individual reprints (single or multiple copies) of this standard may be obtained by contacting ASTM at the above address or at 610-832-9585 (phone), 610-832-9555 (fax), or service@astm.org (e-mail); or through the ASTM website (www.astm.org) Permission rights to photocopy the standard may also be secured from the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, Tel: (978) 646-2600; http://www.copyright.com/ 10