Astm stp 15c 1962

ASTM MANUAL 0~ QUALITY CONTROL OF MATERIALS @ Reg U S Pat OiL Prepared by ASTM COMMITTEE E-11 On Quality Control of Materials Part Presentation of Data Part 2rePresenting ± Limits of Uncertainty of an Observed Average Part S Control Chart Method of Analysis and Presentation of Data Special ff'echnical Publication 15-C January, I95I Price: $2.50; to Members,$2.00 Published by the AMERICAN SOCIETY FOR TESTING MATERIALS x916 Race St.,Philadelphia 3, Pa Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reprodu N O T E - - T h e Society is not responsible, as a body, for the statements and opinions advanced in this publication Copyrighted, 1951 by the AMERICAN SOCIETY:FORTESTING MATERIAL8 Printed in Baltimore, U.S.A First Printing, March, 1951 Second Printing, May, 1951 Third Printing August, 1952 Fourth Printing, September, 1954 Fifth Printing, September, 1956 Sixth Printing, December, 1957 Seventh Printing, July, 1960 Eighth Printing, December, 1962 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized PREFACE This Manual on the Quality Control of Materials was prepared by ASTM Technical Committee E-1I on Quality Control of Materials to make available to the ASTM membership, and others, information regarding statistical methods and quality control methods and to make recommendations for their application in engineering work of the Society The quality control methods considered herein are those methods that have been developed on a statistical basis to control the quality of product through the proper relation of specification, productio n, and inspection as parts of a continuing process This Manual consists of three Parts dealing particularly with the analysis and presentation of data It constitutes a revision and a replacement of the ASTM Manual on Presentation of Data whose main section and two supplements were first published respectively in 1933 and 1935 This early work was done with the ready cooperation of the Joint Committee on the Development of Applications of Statistics in Engineering and Manufacturing (sponsored by the American Society for Testing Materials and the American Society of Mechanical Engineers) and especially of the Chairman of the Joint Committee, W A Shewhart Over the past 15 years this material has gone through a number of minor modifications and reprintings and has become a standard of reference over wide areas in both industrial and academic fields Its nomenclature and symbolism were adopted in 1941 and 1942 in the American War Standards on Quality Control (Zl.1, Z1.2 and Z1.3) of the American Standards Association, and its Supplement B was reproduced as an appendix with one of these Standards The purposes for which the Society was founded the promotion of knowledge of the materials of engineering, and the standardization of specifications and the methods of testing involve at every turn the collection, analysis, interpretation and presentation of quantitative data Such data form an important part of the source material used in arriving at new knowledge, and in selecting standards of quality and methods of testing that are adequate, satisfactory, and economic, from the standpoints of the producer and the consumer Broadly, the three general objects of gathering engineering data are to discover: (1) physical constants and frequency distributions, (2) relationships both functional and statistical between two or more variables, and (3) causes of observed phenomena Under these general headings, the following more specific objectives in the work of the American Society for Testing Materials may be cited: iii Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized iv PRXFAC~ (a) to discover the distributions of quality characteristics of materials which serve as a basis for setting economic standards of quality, for comparing the relative merits of two or more materials for a particular use, for controlling quality at desired levels, for predicting what variations in quality may be expected in subsequently produced material; to discover the distributions of the errors of measurement for particular test methods, which serve as a basis for comparing the relative merits of two or more methods of testing, for specifying the precision and accuracy of standard tests, for setting up economical testing and sampling procedures; (b) to discover the relationship between two or more properties of a material, such as density and tensile strength; and (c) to discover physical causes of the behavior of materials under particular service conditions; to discover the causes of nonconformance with specified standards in order to make possible the elimination of assignable causes and the attainment of economic control of quality Problems falling in the above categories can be treated advantageously by the application of statistical methods and quality control methods The present Manual limits itself to several of the items mentioned under Ca) above Part discusses frequency distributions, simple statistical measures, and the presentation, in concise form, of the essential information contained in a single set of n observations Part discusses the problem of expressing 4- limits of uncertainty of an observed average of a single set of n observations, together with some working rules for rounding-off observed results to an appropriate number of significant figures Part discusses the control chart method for the analysis of observational data obtained from a series of samples, and for detecting lack of statistical control of quality This Manual is the first major revision of the earlier work The original Manual and the two supplements were prepared by the Manual Committee of the former Subcommittee IX on Interpretation and Presentation oI Data, of Committee E-1 on Methods of Testing The personnel of the Manual Committee was as follows: Messrs H F Dodge, chairman (193246), W C Chancellor (1934-37), J T MacKenzie (1932 45), R F Passano (1939 46), H G Romig (1938-46), R T Webster (1932-44), A E R Westman (1932-34) Changes and additions have been made in line with comments and suggestions received from many sources Since the last modification of the earlier work, the American Society for Quality Control has been organized (1946) and has assumed a responsible and recognized position in the field of quality control Its cooperation in the present revision is hereby acknowledged The list of members of Committee E-11 appearing in this Manual shows the personnel of the committee as of the date of publication During the preparation of the three parts of the Manual the following were also active members of the committee: Messrs C W Churchman, H F Hebley, J C Hintermaier, R F Passano, A I Peterson, T S Taylor, John Tucker, Jr Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions aut PREFACE V Additional subject material is under consideration by the committee for inclusion in this Manual as additional Parts January, 1951 In this fifth printing of the Manual there has been included in the Appendix the Tentative Recommended Practice for Choice of Sample Size to Estimate the Average Quality of a Lot or Process (ASTM Designation: E 122) This recommended practice was prepared by Dr W Edwards Deming and Miss Mary N Torrey and represents in part work done by Task Group No of Committee E-11, which consists of A G Scroggie, chairman, C A Bicking, W Edwards Deming, H F Dodge, and S B Littauer September, 1956 In this sixth printing of the Manual corrections have been made of the typographical errors on pp 61, 62, 65, and 69 December, 1957 This seventh printing of the Manual includes several minor additions and revisions The changes in Part include revised values in Tables I (c) and II (c) (and corresponding values elsewhere in the Manual where referred to); also an addition to Section Sections 20, 21, and 28 were modified to include formulas for s and s2 In Part 3, Section was expanded, and in the Example Sections 31, 32, and 33 the paragraph on Results was revised in Examples 2, 3, 4, 8, 13, 16, 21, and 22 The Appendix was expanded to include a List of Some Related Publications on Quality Control and Statistics and a Table giving a comparison of the symbols used in the Manual and those used in statistical texts These changes were prepared by an Ad Hoc Committee on Modification of ASTM Manual The personnel of this committee is as follows: H F Dodge, chairman, Simon Collier, R H Ede, R J Hader, and E G Olds July, 1960 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized M E M B E R S H I P OF C O M M I T T E E E-11 ON Q U A L I T Y C O N T R O L O F M A T E R I A L S DECEMBER, 1962 *C A Bicking, Chairman, Quality Control Manager, Carborundum Co., Niagara Falls, N Y *W P Goepfert, Vice-Chairman, Chief, Statistical Analysis Section, Metallurgical Div., Aluminum Company of America, Pittsburgh, Pa *A J Duncan, Secretar%Associate Professor, The Johns Hopkins University, Baltimore, Md D H W Allan, American Iron and Steel Inst., New York, N Y O P Beckwith, Quality Control Director, Ludlow Corp., Needham Heights, Mass J N Berrettoni, Professor of Statistics, Western Reserve University, Cleveland, Ohio *S Collier, Consultant, 10552{ Wilshire Blvd., Los Angeles 24, Calif D A Cue, Quality Manager, Hoover Ball and Bearing Co., Ann Arbor, Mich W Edwards Deming, Graduate School of Business Administration, New York University, N Y H F Dodge, Professor of Applied and Mathematical Statistics, Rutgers, The State University, New Brunswick, N J F E Grubbs, Chief, Weapon Systems Lab., Ballistic Research Labs., Aberdeen Proving Ground, Md E C Harrington, Jr., Monsanto Chemical Co., Springfield, Mass J S Hunter, Associate Professor of Chemical Engineering, Princeton University, Princeton, N J Gerald Lieberman, Stanford University, Stanford, Calif John Mandel, National Bureau of Standards, Washington, D C C L Matz, 6455 N Albany Ave., Chicago 45, Ill R B Murphy, Bell Telephone Laboratories, Inc., New York, N Y F G Norris, Metallurgical Engineer, Wheeling Steel Corp., Steubenville, Ohio *P S Olmstead, Statistical Consultant, Bell Telephone Laboratories, Inc., Whippany, N J *W R Pabst, Jr., Quality Control Div., Bureau of Ordnance, Navy Dept., Washington, D C J B Pringle, Staff Engineer, Quality Analysis, Bell Telephone Company of Canada, Montreal, P.Q., Canada L E Simon, (Honorary Member), 1761 Pine Tree Road, Winter Park, Fla R J Sobatzki, Quality Control Superintendent, Rohm & Haas Co., Philadelphia, Pa *Louis Tanner, Chief Chemist, U S Customs Laboratory, Boston, Mass Grant Wernimont, Staff Assistant, Color Control Dept., Eastman Kodak Co., Rochester, N Y * Member of Advisory Committee vi Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions autho CONTENTS PART PRESENTATION OF DATA PAGE Summary Introduction SECTION Purpose T y p e of D a t a Considered Homogeneous D a t a Typical Examples of Physical D a t a ! Ungrouped Frequency Distributions Ungrouped Frequency Distributions Remarks 5 Grouped Frequency Distributions 10 11 12 13 14 15 Introduction Definitions Choice of Cell Boundaries N u m b e r of Ceils M e t h o d s of Classifying Observations Cumulative Frequency Distribution Tabular Presentation Graphical Presentation Remarks 5 6 9 11 Functions of a Frequency Distribution 16 17 18 19 20 21 22 23 Introduction Relative Frequency Average (Arithmetic Mean) Other Measures of Central Tendency Standard Deviation Other Measures of Dispersion Skewness k Remarks 11 12 13 13 14 15 15 16 Methods of Computing X, c~, and k 24 Computation of Average and Standard Deviation 25 Short M e t h o d of Computation When ~ is Large 26 Remarks 16 19 20 vii Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions autho viii CONTENTS Amount of Information Contained in p, X, a and k SECTION PAGE 27 28 29 30 31 32 33 34 35 Introduction T h e Problem Several Values of Relative Frequency, p Single Value of Relative Frequency, p Average, ~ , Only Average, :~, a n d S t a n d a r d Deviation, a Average, X', S t a n d a r d Deviation, a, a n d Skewness, k Use of Coefficient of Variation I n s t e a d of S t a n d a r d D e v i a t i o n General C o m m e n t on Observed Frequency Distributions of a Series of A.S.T.M Observations 36 S u m m a r y 20 21 21 21 22 23 25 26 27 28 Essential Information 37 38 39 40 41 Introduction W h a t Functions of the D a t a Contain the Essential I n f o r m a t i o n Presenting X Only Versus Presenting X a n d g Observed Relationships Summary 29 29 30 31 32 Presentation of Relevant Information 42 I n t r o d u c t i o n 43 R e l e v a n t I n f o r m a t i o n 44 Evidence of Control 33 33 34 Recommendations 45 R e c o m m e n d a t i o n s for Presentation of D a t a 35 Supplements A Glossary of Symbols Used in P a r t B General References for P a r t 36 37 PART PRESENTING • LIMITS OF UNCERTAINTY AVERAGE OF AN OBSERVED Purpose T h e Problem Theoretical Background C o m p u t a t i o n of Limits Experimental Illustration P r e s e n t a t i o n of D a t a N u m b e r of Places to be R e t a i n e d in C o m p u t a t i o n a n d Presentation General C o m m e n t s on the Use of Confidence Limits 41 41 42 42 45 46 47 49 Supplements A Glossary of Symbols Used in P a r t B General References for P a r t 50 51 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproduc TABLES OF SQUARES SQUARES AND SQUARE AND SQUARE ROOTS ROOTS 125 1101-1400 (Continued) Square Square Root No 1 1 565 567 570 572 575 001 504 009 516 025 35.3695 35.3836 35.3977 35.4119 35.4260 1326 1327 1328 1329 1330 1 | 1 276 929 584 241 900 36.4143 36.4280 36.4417 30.4555 36.4692 1256 1257 1258 1259 1260 1 1 577 580 582 585 587 536 049 564 081 600 35.4401 35.4542 35.4683 35.4824 35.4965 1331 1332 1333 1334 1335 771 561 774 224 770 889 779 556 782 225 36.4829 36.4966 36.5103 36.5240 36.5377 596 :34.4384 969 34.4529 344 34.4674 721 100 34.4964 1261 1262 1263 1264 1265 1 1 590 592 595 597 600 121 644 169 696 225 35.5106 35.5246 35.5387 35.5528 35.5668 1336 1337 1338 1339 1340 784 787 569 790 244 921 795 600 30.5513 36.5650 36.5787 36.5923 36.6060 481 864 249 636 025 34.5109 134.5254 134.5398 J34.5543 34.5688 1266 692 756 87 11 75 28 28 49 1269 610 361 1270 612 900 35.5809 35.5949 35.6090 35.6230 35.6371 1341 1342 1343 1344 1345 798 281 800 964 549 806 3 809 025 36.6197 36.6333 36.6470 36.6606 36.6742 430 432 435 437 440 416 809 204 601 000 34.5832 34.5977 34.6121 34.6266 34.6410 1271 1272 1273 1274 1275 1 1 442 444 447 449 452 1206 1207 1208 1209 1210 1 1 33.7046 33.7194 33.7343 33.7491 33.7639 1211 1212 1213 1214 1215 881 304164 306 449 308 736 311 33.7787 33.7935 33.8083 33.8231 33.8378 1 1 313 315 317 320 322 316 609 904 201 500 1151 1152 1153 1154 1155 I 1 1 324 327 329 331 334 1156 1157 1158 1159 ~,160 1 1 1161 [162 1163 1164 1165 Square Square Root No 1 1 382 385 387 390 392 97fi 329 684 041 400 34.2929 34.3074 34.3220 34.3366 34.3511 1251 1252 1253 1254 1255 1181 1182 1183 1184 1185 1 1 394 397 399 401 404 761 124 489 !34.3948 856 34.4093 225 34.4238 33.3317 33.3467 33.3617 33.3766 33.3916 1186 1187 1188 1189 1190 1 1 406 408 411 413 416 456 689 924 161 400 33.4066 33,4215 33.4365 33.4515 33.4664 1191 1192 1193 1194 1195 1 I 1 418 420 423 425 428 256 258 261 263 265 641 884 129 376 625 33.4813 33.4963 33.5112 33.5261 33,5410 1196 1197 1198 1199 1200 1 1 1 1 1 267 270 272 274 276 876 129 384 641 900 33.5559 33.5708 33.5857 33.6006 33.6155 1201 1202 1203 1204 1205 1131 1132 1133 1134 1135 1 1 279 281 283 285 288 161 424 689 956 225 33.6303 33.6452 33.6601 33.6749 33.6898 1136 1137 1138 1139 1140 1 1 290 292 295 297 299 496 769 044 321 600 1141 1142 1143 1144 1145 1 1 1146 1147 1148 1149 1150 Square Square Root No 212 214 216 218 221 201 404 609 816 025 33.1813 33.1964 33.2114 33.2265 33.2415 1176 1177 1178 1179 1180 1106 223 236 1107 l 225 449 1108 I 227 664 1109 229 881 1110 232 100 33.2566 33.2716 33.2866 33,3017 33.3167 1111 1112 1113 1114 1115 1 1 234 236 238 240 243 321 544 769 996 225 1116 1117 1118 I]19 1120 I 1 1 245 247 249 252 254 1121 1122 1123 1124 1125 1 1 1128 1127 1128 1129 1130 ~NTO 1101 1102 1103 1104 1105 1 1 1 1 1 Square 758 760 763 766 768 8guare Root 615 617 620 623 625 441 984 529 076 625 35.6511 35.6651 35.6791 35.6931 35.7071 1346 1347 1348 1349 1350 1 1 811 814 817 819 822 716 409 104 801 500 36.6879 36.7015 36.7151 36.7287 36.7423 401 5 804 34.6699 209 34.6843 616 34.6987 34.7131 1276 [1 628 1277 630 1278 633 1279 635 1280 638 176 729 284 841 400 35.7211 35,7351 35.7491 35.7631 35.7771 1351 1352 1353 1354 1355 1 1 825 827 830 833 836 201 904 609 316 025 36.7560 36.7696 36.7831 36.7967 36.8103 454 456 459 461 464 436 849 264 681 100 34,7275 34.7419 34.7563 34.7707 34.7851 1281 1282 1283 128411 1285]1 640 643 646 648 651 961 524 089 656 225 35.7911 35.8050 35.8190 35.8329 35.8469 1356 1357 1358 1359 1360 1 1 838 841 844 840 849 736 449 164 881 600 36.8239 36.8375 36.8511 36.8640 36.8782 1 1 466 468 471 473 476 521 944 369 796 225 !34.7994 134.8138 134.8281 134.8425 134.8569 128611 128711 1288]1 128911 1290 653 656 658 661 664 796 369 944 521 100 35.8608 35.8748 35.8887 35.9026 35.9166 1361 1362 1363 1364 1365 1 1 852 855 857 860 863 321 044 769 496 225 36.8917 36.9053 36.9188 36.9324 36.9459 1216 1217 1218 1219 1220 1 I 1 478 481 483 485 488 656 !34.8712 089 '34.8855 524 34.8999 961 400 ,34.9285 1291 666 1292 I1 669 I 671 1294 674 1295 677 681 264 349 436 025 35.9305 35.9444 35.9583 35,9722 35.9361 1366 1367 1368 1369 1370 865 956 36.9594 808 689 36.9730 871 424 8 161 0 0 876 900 37.0135 33.8526 33.8674 33.8821 33.8969 33.9116 1221 1222 1223 1224 1225 1 1 490 493 495 498 500 841 284 729 176 625 34.9428 34.9571 34.9714 34.9857 35.0000 129611 1297 129811 1299J1 0 I1 679 682 684 687 690 616 209 804 401 000 36.0000 36.0139 36.0278 36.0416 36.0555 1371 1372 1373 1374 1375 1 1 879 882 885 887 890 641 384 129 876 625 37.0540 37.0675 37.0810 S01 104 409 716 025 33.9264 33,9411 33.9559 33.9706 33.9853 1226 1227 1228 1229 1230 1 1 503 505 507 510 512 076 529 984 441 900 35,0143 35.0286 35.0428 35.0571 35.0714 1301'1 1302 1 3 ]1 1304 1305 692 695 697 700 703 601 204 809 416 025 36.0694 36.0832 36.0971 36.1109 36.1248 1376 1377 1378 1379 1380 1 1 893 896 898 901 904 376 129 884 641 400 37.0945 37.1080 37.1214 37.1349 37.1484 336 338 340 343 345 336 649 964 281 600 34.0000 34.0147 34.0294 34.0441 34.0588 1231 1232 1233 1234 1235 1 1 515 517 520 522 525 361 135.0856 824 135.0999 289 35.1141 756 35.1283 225 35.1426 705 708 710 713 716 636 249 864 481 100 36.1386 36.1525 36.1663 36.1801 36.1939 1381 1382 1383 1384 1385 1 1 907 909 912 915 918 161 924 689 456 225 37.1618 37.1753 37.1887 37.2022 37.2156 1 1 347 350 352 354 357 921 244 569 896 225 34.0735 34.0881 34.1028 34.1174 34.1321 1236 1237 1238 1239 1240 1 1 527 530 532 535 537 696 35.1568 169 644 35.1852 121 9 600 35.2136 130611 130711 130811 1309 1310 I 1311 ] 1312 ] 1313 1314 131511 718 721 723 726 729 721 344 969 596 225 36.2077 36.2215 36.2353 36.2491 36.2629 1386 1387 1388 1389 1390 1 1 920 923 926 929 932 996 769 544 321 100 37.2290 37.2424 37.2569 37.2693 37.2827 1166 1167 1158 1169 1170 1 1 359 556 36l 889 364 224 6 561 368 900 34.1467 34.1614 34.1760 34.1906 34.2053 1241 1242 1243 1244 1245 I 1 1 540 542 545 547 550 081 2 564 35.2420 049 35,2562 536 135.2704 025 :35.2846 731 1317 734 1318 737 I1 1320 I 742 856 489 124 761 400 36.2767 36.2905 30.3043 36.3180 36.3318 1391 1392 1393 1394 1395 1 1 934 937 940 943 946 881 664 449 236 025 37.2961 37,3095 37.3229 37.3363 37.3497 1171 1172 1173 1174 1175 1 1 371 373 375 378 330 34.2199 34.2345 34.2491 34.2637 34.2783 1246 1247 1248 1249 1250 1 1 I 552 555 557 560 562 516 35.2987 009 35 3129 504 i35,3270 001 135.3412 500 5 1321 1322 1323 1324 1325 041 684 329 976 625 36.3456 36.3593 36.3731 36.3868 36.4005 1396 1397 1398 1399 1400 1 1 948 951 054 957 960 816 609 404 201 000 37.8631 37.3765 37.3898 37.4032 37.0270 37.0405 l 241 584 929 276 625 1 1 745 747 750 752 755 37.4166 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions aut 126 ASTM ~[ANUAL ON QUALITY CONTROL OF MATERIALS 1401-1700 "NO Square SQu.RE8 AND SQUARE ROOTS (Continued) Sguare Root 1~To Square Square S•uare oot Sfiuare Root NO Square Square Root ~o 601 704 411 809 2414916 025 39.3827 39.3954 39.4081 39.4208 39.4335 1626 643 1627 647 129 1628 650 1629 653 641 6 900 40.3237 40.3861 40.3485 40.3609 40,3733 1401 1402 1403 1404 971 1405 974 801 9 604 4 3 409 37.4566 216:37.4700 025 37.4833 1476 1477 1478,2 14792 1480 178 181 184 187 190 576 529 484 441 400 38,4187 38.4318 38.4448 38.4578 38.4708 1551 1552 1553 1554 1555 1406 1407 1408 1409 1410 1 1 976 979 982 985 988 83fl 649 464 281 100 37.4967 37.5100 37.5233 37.5366 37.5500 1481 1482 1483 1484 1485 193 196 199 202 205 3611 324! 289 I 256 225 38.4838 38.4968 38.5097 38.5227 38.5357 !1556 1557 i1558 1559 11560 2 2 421 424 427 430 433 136 249 364 481 600 39.4462 39.4588 39.4715 39.4842 39.4968 1631 1632 1633 1634 1635 2 2 660 663 666 669 673 161 424 689 956 225 40.3856 40.3980 40.4104 40.4228 40.4351 1411 1412 1413 1414 1415 I 1 990 993 996 999 002 921 744 569 396 225 37.5633 37,5766 37.5899 37.6032 37.6165 1486 2 1487 211 1488 2 1489 2 1490 2 196 169 144 121 100 38.5487 38.5616 38.5746 38.5876 38.6005 1561 1562 1563 1564 1565 2 2 436 439 442 446 449 721 844 969 096 225 39.5095 39.5221 39.5348 39.5474 39.5601 1636 1637 1638 1639 1640 2 2 676 679 683 686 689 496 769 044 321 600 40.4475 40.4599 40.4722 40.4846 40.4969 1416 1417 1418 1419 1420 2 2 005 007 010 013 016 056 889 724 561 400 37.6298 37.6431 37.6563 37.6696 37.6829 1491 1492 1493 1494 1495 2 2 223 226 229 232 235 081 064 049 036 025 38.6135 38.6264 38.6394 38.6523 38.6652 1566 1567 1568 1569 1570 2 2 452 455 458 461 464 356 489 624 761 900 39.5727 39.5854 39.5980 39.6106 39.6232 1641 1642 1643 1644 1645 2 2 692 696 699 702 706 88] 164 449 736 025 40.5093 40.5215 40.5339 40.5463 40.5586 1421 1422 1423 1424 1425 2 2 019 241 022 024 929 027 7 030 625 37.6962 37.7094 37.7227 37.7359 37.7492 1496 1497 1498 1499 1500 2 2 238 016 241009 244004 001 250 000 38.6782 38.6911 38.7040 38.7169 38.7298 1571 1572 1573 1574 468041 471 184 474 329 477 476 1573 480 625 39.6358 39.6485 09.6611 39.6737 39.6863 1646 709 1647 712 609 1648 715 1649 201 1650 2 0 40.5709 40.5832 40.5956 40.6079 40.6202 1426 1427 1428 1429 1430 2 2 033 036 039 042 044 476 329 184 041 900 37.7624 37.7757 37.7889 37.8021 37.8153 1501 2 001 1502 2 0 1503 2 009 1504 2 6 1505 2 025 38.7427 38.7556 38.7685 38.7814 38.7943 1576 488 776 1577!2486929 15781 0 1579 ~2 241 1580 496 400 39.6989 39.7115 39.7240 29.7366 39.7492 1651 801 1652 729 104 1653 732 1654 735 1655 40.6325 40.6448 40.6571 40,6694 40.6817 1431 1432 1433 1434 1435 2047 050 053 056 059 761 624 489 356 225 37.8286 37.8418 37.8550 37.8682 37.8814 1506 1507 1508 1509 1510 2 2 268 271 274 277 280 036 049 064 08l 100 38.8072 38.8201 38.8330 38.8458 38.8587 1581 9 1582 15832 606 1584 1585 512 661 724 889 056 225 39.7618 39.7744 39.7869 39.7996 39.8121 1656 1657 165~ 1659 1660 2 2 742 745 748 752 755 336 649 964 281 600 40.~940 40.7~q3 40.718~ 40,7308 40.7431 1436 1437 1438 1439 1440 2 2 062 064 067 070 073 096 969 844 721 37.894611511 37.907811512 37.921011513 , 1514 600 ! 5 2 2 283 286 289 292 295 121 144 169 195 225 38.8716 38.8844 38.8973 38.9102 38.9230 1586 1587 1588 1589 1590 615396 569 2521744 921 10O 39.8246 39.8372 39.8497 39.8023 39.8748 1661 1662 1663 1664 1665 2 2 758 762 765 768 772 921 244 569 896 225 40.7554 40.7676 40.7799 40.7922 40.8044 1441 1442 1443 1444 1445 2 2 076 079 082 085 088 481 364 249 136 025 37.96051516 7 1517 8 1518 0 0 1519 1520 2 2 298 301 304 307 310 256 289 324 361 400 38.9358 38.9487 38.9615 38.9744 38.9872 1591 1592 1593 1594 1595 2 2 281 464 649 836 025 39.8873 39.8999 39.9124 39.9249 39.9375 1666 7 1667 7 1668 1669 1670 788 556 889 224 561 900 40.8167 40.8289 40.8412 40.8534 1446 1447 1448 1449 1450 2 2 090 093 096 099 102 916 809 704 601 500 38.02631521 313 1522 1523 319 1524 322 1525 325 441 484 529 625 39.0000 39.0128 39.0256 39.0384 39.0512 1596 547 1597 5 1598 553 1599 556 801 1600 0 0 39.9500 39,9625 39.9750 39.9875 40.0000 1671 792 1672 12 795 1673 12 1674 802 1675 241 584 929 276 625 40.8779 40.8901 40.9023 40.9148 40.9268 1526 328 676 1527 331 1528 3 784 1529 337 841 1530 0 39.0640 39.0768 39.0896 39.1024 39,1152 1601 563 201 1602 566 4 976 329 684 041 400 40.9390 40.9512 40,9634 40.9756 40.9878 761 124 489 856 225 41.0000 41.0122 41.0244 41.0866 41.0488 596 969 344 721 100 41.0609 41.0731 41.0858 41.0974 41.1096 2 2 2 2 I 576 1451 105 401 1452 108 1453 111 114 116 1455 117 38~920 38.1051 38.1182 38.1314 38.1445 1456 1457 1458 1459 1460 2 2 119 122 125 128 131 936 849 764 681 600 38.1576 38.1707 38.1838 38.1969 38.2099 1531 11532 i 1533 11534 ! 1535 2 2 343 347 350 353 356 1461 1462 1463 1464 1465 2 2 134 137 140 143 146 621 444 369 296 225 38.2230 38.2361 38.2492 38.2623 88.2753 , 1536 ~1537 11538 11539 1540 2 2 359 362 365 368 371 6 149 156 1497 152 1468 : 5 1469 157 961 ,2 160 900 38.2884 38.3014 38.3145 38.3275 38.3406 1541 1542 1543 1544 1545 374 681 377 764 2380849 383 936 387 025 1471 163 841 : 1472 166 38.3667 1473 !2 169 729 38.3707 1474 22 172 676 38.3927 1475 175 8 1546 1547 116 393 209 531 534 537 540 544 1603 1604 1605 569 609 572 816 576 025 40.0125 40.0250 40.0375 40.0500 40.0625 1676 1677 1678 1679 1680 961 , 024 39.1408 089 5 156 139.1663 2 ! 39.1791 1606 1607 1608 1609 1610 2 2 236 449 664 881 100 40.0749 40.0874 40.0999 40.1123 40.1248 I 39 1918 369 39.2046 444 39.2173 521 39.2301 600 39.2428 1611 321 1612 598 544 1613 601 769 1614 604 996 1615 608 2 40.1373 40.1497 40.1622 40.1746 40.1871 1681 1682 16831 1684 835 168512 839 I 1686 842 1687 845 1688!2849 1689 I 852 856 39.2556 39.2683 39.2810 39.2938 39.3065 1616 611 456 1617 614 689 1618 1619 621 161 1620 624 400 40.1995 40.2119 40.2244 40.2388 40.2492 1691 1692 1693 1694 1695 2 2 859 862 866 869 873 481 864 249 636 025 41.1218 41.1339 41.1461 41.1582 41.1704 39.3192 39.3319 1621 627 1622 1623 1624 1625 ~ , 40.2616 40.2741 40.3865 40.2989 40.3113 1696 1697 1698 16ffi} 1700 2 2 $ 416 809 883 204 886601 890000 41.1825 41.1947 41.2068 41.2189 4L.2311 1548 396 304 30.3446 1549 399 401 s9.357a 5 : 402 0 0 579 582 585 588 592 641 884 129 376 625 2 2 808 812 815 819 822 40.8656 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions autho Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized o o o ~ ~ODG OP OP ~g o I-J ,q 0 o L I S T O F SOME R E L A T E D P U B L I C A T I O N S ON Q U A L I T Y CONTROL AND STATISTICS Texts: C A Bennett and N L Franklin, Statistical Analysis in Chemistry and the Chemical Industry, John Wiley & Sons, Inc., New York, 1954 *Irving W Burr, Engineering Statistics and Quality Control, McGraw-Hill Book Co., Inc., New York, 1953 H Cramer, Mathematical Methods and Statistics, Princeton University Press, Princeton, 1946 Wilfred J Dixon and Frank J Massey, Jr., Introduction to Statistical Analysis, 2d Ed., McGrawHill Book Co., Inc., New York, 1957 *Acheson J Duncan, Quality Control and Industrial Statistics, Rev Ed., Richard D Irwin, Inc., Hanewood, Ill., 1959 William Feller, An Introduction to Probability Theory and Its Application, 2d Ed., Vol I, John Wiley & Sons, Inc., New York, 1957 *E L Grant, Statistical Quality Control, 2d Ed., McGraw-Hill Book Co., Inc., New York, t952 A Held, Statistical Theory with Engineering Applications, John Wiley & Sons, Inc., New York, 1952 P G Hoel, Introduction to Mathematical Statistics, 2d Ed., John Wiley & Sons, Inc., New York, 1959 *J M Juran, Quality-Control Handbook, McGraw-Hill Book Co., Inc., New York, 1951 A M Mood, Introduction to the Theory of Statistics, McGraw-Hill Book Co., Inc., New York, 1950 M J Moroney, Facts/tom Figures, 3d Ed., Penguin Books Inc., Baltimore, 1956 W A Shewhart, Economic Control of Quality of Manufactured Product, D Van Nostrand Co., Inc., New York, 1931 *W A Shewhart, Statistical Method from the Viewpoint of Quality Control, Graduate School of the U S Dept of Agriculture, Washington, 1939 *Leslie E Simon, An Engineer's Manual of Statistical Methods, John Wiley & Sons, Inc., New York, 1941 L H C Tippett, Technological Applications of Statistics, John Wiley & Sons, Inc., New York, 1950 Journals: Applied Statistics Annals of Math Star Biometrika *Industrial Quality Control Jap ]our Star Applic for Eng & Sci Jour Am Slat Assn Jour Royal Star Soc. B *Technometrics Pamphlets: *ASA Standard Z1.1-1958, "Guide for Quality Control" (ASQC Std B1-1958) *ASA Standard Z1.2-1958, "Control Chart Method of Analyzing Data" (ASQC Std B2-1958) *ASA Standard Z1.3-1958, "Control Chart Method of Controlling Quality During Production" (ASQC Std B3-1958) *ASQC General Publications * With ~pecial reference to quality control 128 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized COMPARISON OF SYMBOLS TABLE OF S O M E SYMBOLS U S E D THOSE USED I N T H E ~V[ANUAL A N D IN STATISTICAL TEXTS N O T E - - T h e M a n u a l uses t h e p r i m e n o t a t i o n f o r u n i v e r s e p a r a m e t e r s (or s t a n d a r d v a l u e s ) , while statistical texts are inclined to use Greek letters Term An observed value Universe mean or average S a m p l e size, or n u m b e r of o b s e r v a tions Symbol Used in the Manual X (or x) X X' n (or N ) n _~ (or 2) Sample mean or average Universe standard deviation Symbol Commonly Used in Statistical Texts Ort Or Ort2 0-2 Sample standard deviation Universe variance v n ~-i ) Sample variance a S o m e a u t h o r i t i e s feel t h e t e r m " s a m p l e s t a n d a r d d e v i a t i o n " for s a n d t h e t e r m " s a m p l e v a r i a n c e " for s to be m i s a p p l i e d In a n y case s ~ is t h e u n b i a s e d e s t i m a t e of t h e u n i v e r s e v a r i a n c e 129 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions au STP15 C - E B / J a n 1951 Recommended Practice fog CHOICE OF SAMPLE SIZE TO ESTIMATE THE AVERAGE QUALITY OF A I,OT OR PROCESS' | ASTM Designation: E 122- 58 ADOPTED, 19583 This Recommended Practice of the American Society for Tcsting Materials is issued under the fixed designation E 122; the final number indicates the year of original adoption or, in the case of revision, the year of last revision NOTE. Note of Section (a) was formerly Example All subsequent notes and examples were accordingly renumbered editorially in July, 1958 Scope This recommended practice presents simple methods for calculating how many units to include in a sample in order to estimate, with a prescribed precision, the average of some characteristic for all the units of a lot of material, or the average produced by a process Empirical Knowledge Needed (a) Some empirical knowledge of the problem is necessary as follows: (1) The standard deviation or, if that is not possible, (2) The range or spread of the characteristic, from its lowest to its highest value and, if possible, some knowledge of the shape of the distribution of the characteristic; for instance, whether most of the values lie at one end of the range, or are mostly in the middle, or run Under tile standardization procedure of the Society, this recommended practice is under tile jurisdiction of the ASTM Committee E-11 oll Quality Control of Materials Prior to adoption, this recommended practice was published as tentative from 1956 to 1958 rather uniformly from one end to the other (b) If tile aim is to estimate the fraction defective, then each unit has a value of or (not defective or defective), and the standard deviation, as well as the shape, of the distribution depends only on p', the fraction defective of the lot or process (c) Sketchy knowledge is sufficient to start on, although more knowledge permits greater economy in the sample Rarely will there be difficulty in acquiring enough information to compute the required size of sample with sufficient assurance beforehand to meet the desired precision within acceptable limits A sample that is bigger than the equations indicate is used in actual practice when the empirical knowledge is only sketchy to start with, and if the desired precision is critical The extra insurance is the price of incomplete knowledge (d) In any case, even when starting with sketchy knowledge, the precision of the estimate made from a random sample 130 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 P-40-~ Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized CHOICE OF SAMPLE SIZE TO ESTIMATE AVERAGE QUALITY (E 122 -58) m a y itself be estimated from the sample This estimation of the precision reached b y the first sample makes it possible to fix more economically the sample size for the next sample of a similar material I n other words, information concerning the process, and the material produced thereby, accumulates and should be used Precision Desired T h e a p p r o x i m a t e precision desired for the estimate m u s t be prescribed T h a t is, it m u s t be decided what maxim u m difference, E, can be tolerated between the estimate to be made from the sample and the result t h a t would be obtained b y testing every unit in the universe E q u a t i o n s for C a l c u l a t i n g S a m p l e S i z e (a) T h e equation for the size n of the sample is as follows: (1) is "practically certain" t h a t the sampling error will not exceed E There are occasions, however, where a lesser degree of certainty is desired, which a smaller factor provides (Note 3) NOTE 2. In the sampling of a lot of material that has a highly skewed distribution in the characteristic measured, the factor will give a different probability, possibly as great as parts in 1000 If there is anxiety about the effect of skewness, there are two things which can be done: (1) Probe the material with a view to discovering, for example, extra-high values, or possibly spotty runs of abnormal character, in order to approximate roughly the amount of the skewness, for use with statistical theory and adjustment of the sample size if necessary (2) Search the lot for abnormal material and segregate it for separate treatment NOTE & For example, the factor gives a probability of about 45 parts in 1000 that the sampling error will exceed E Although the distributions met in practice may not be normal, the following table (based on the normal distribution) indicates approximate probabilities: where r Factor 2.58 1.96 1.64 NOTE 1. Some simple methods are given E = the m a x i m u m allowable difference between the estimate to be made from the sample and the result of testing (by the same methods) all the units in the universe = a factor corresponding to a probability of a b o u t parts in 1000 (Note 2) t h a t the difference between the sample estimate and the result of testing (by the same methods) all the units in the universe is greater than E The choice of the factor is recommended for general use W i t h the factor 3, and with a universe s t a n d a r d deviation equal to the advance estimate, it Probability _- the advance estimate of the standa r d deviation of the lot or process (Note 1) later to show how to reduce the empirical knowledge to the numerical value ,r' 131 in 1000 45 in 1000 in 100 in I00 (1 in 20) 10 in 100 (1 in 10) (b) I t is sometimes convenient to use E q in another form: namely, n (2) where: v' (coefficient of variation in per cent) = 100 ~r'/X', the advance estimate of the coefficient of variation of the m a t e r i a l , expressed in per cent = 100 E/X', the allowable sampling error expressed as a per cent of X', and X ~ = the expected value of the characteristic being measured There are some materials for which r varies a p p r o x i m a t e l y with X', in which e Po40-~ Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions a 132 CHOICE OF SAmeLE SIZE XO ESTI~AT~ AVEP,AGE QrJhr.ITY (E 122 - 58) case v' remains approximately constant from large to small values of X' If the relative error, e, is to be the same for all values of X', then everything on the right-hand side of Eq is a constant; hence n is also a constant, which means that the same sample si _ze,n, would be required for all sizes of X' (c) If the problem is to estimate the fraction defective, then (a') ~ is replaced b y p'(1 - p'), so that Eq becomes: n = p'(t p') (3) where: p' = the advance estimate of the fraction defective If p' is small, so that p'n is less than 4, then 3.25 should be used in place of in Eq to compensate for the skewness of the p distribution for small values of p' (d) When the average of a particular lot of limited size is wanted, and an estimate of the average for the process is not part of the problem, the required sample size is less than Eqs 1, 2, and indicate The sample size for estimating the average of the finite lot will be: nL = n (4) where: n = the value computed from Eqs 1, 2, or 3, and N the lot size This reduction in size is usually of little importance unless n is 10 per cent or more of N R e d u c t i o n of Empirical K n o w l e d g e to a N u m e r i c a l V a l u e of a' (Data for Previous S a m p l e s A v a i l a b l e ) (a) This section illustrates the use of the equations in Section when there are data for previous samples (b) For Equation / - - C o m p u t e the standard deviation ~ (corrected for sample size) for several samples, and use the average of them, if they are not too dissimilar, for an advance estimate of a t" NOTE - - A simple way to compute the overall ~ for a lot is to arrange the observed values in a r a n d o m order, and then average the ranges of successive groups of 4, 5, 8, or 10 observed values (Theory shows t h a t the o p t i m u m subgroup size is 8.) If R is the average of these ranges, then R/d2 is an estimate of ~r The accompanying table shows some selected valuesaof d, Group Size di 1.13 10 2.06 2.33 2.85 3.08 Example - - U s e of ~: Problem. To compute the sample size needed to estimate the average transverse strength of a lot of bricks when the desired value of E is 50 psi Solution. F~om the data of three previous lots, the values of standard deviation were found to be 215, 192, and 202 psi, based on samples of 100 bricks The average of these three standard deviations is 204 psi, whence Eq gives: = ( X 204~ n \ -~ ] = (12.2)+ = 148.8 = 149 bricks for the required size of sample to give a maximum sampling error of 50 psi (c) For Equation 2. If a' varies approximately with X ' for the characteristic of the material to be measured, compute both the average, X, and the standard deviation, a (corrected for sample size), s for several samples (unless they are already available) An average of the several values of v = a / X , if they are not too dissimilar, m a y be used as an advance estimate of v' Example 2. Use of v: Problem. To compute the sample size needed to estimate the average abrasion resistance of a material when the desired value of e is 10 per cent a See t h e A S T M M a u u a l on Quality Control of Materials, STP 15-C, p 63, for values of c2for correcting a when n is less t h a n 25, as well as for values of d~ P-40-|7 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions auth CHOICE OF SAMPLE SIZE TO E S T I m t T E AVERAGE QUALITY(E Solution. There are no d a t a from previous samples of this same material, b u t data for six samples of similar materials show a wide range of resistance However, the values of standard deviation are approximately proportional to the observed averages, as shown in the following table: 122- 58) 133 Example 3. Use of p: Problem. To compute the sample size needed to e s t i m a t e the fraction defective in a lot of alloy steel track bolts and n u t s when the desired value of E is 0.04 S o l u t i o n - - T h e following d a t a from four previous lots were used for an advance estimate of p': Coefli- a = clent of ObR Vadserved 3ample AverSize age (3;08 '-) 'ation, v, Cycle.* in per Lot No Sample Size Lot No No of Fraction Defective 040 0.100 044 O 032 Defectives cent t 5 90 40 13.0 190 1OO 32.5 140 45.5 350 450 220 71.4 1OO0 360 116.9 3550 2090 678 fi lO 10 L 10 1O 1O IO 14 17 13 16 12 19 75 100 00 125 * Values of standard d e v i a t i o n (corrected for sample size) m a y be used instead of the e s t i m a t e s made from the range, if t h e y are preferred or already available = (3 • ,52 , -i() ] = (4.6)' = 21.2 = 22 specimens for the required size of sample to give a m a x i m u m sampling error of 10 per cent of the expected value If a m a x i m u m allowable error of per cent were needed, the required sample size would be 85 specimens T h e data supplied b y the prescribed sample will be useful for the next investigation of similar material (d) For Equation & Compute the value of fraction defective, p, for each sample If the values are not too dissimilar, use the average of them for an advance estimate of p' (If the sample sizes vary, use a weighted average g= total n u m b e r of defectives in all samples total number of units in all samples instead of a simple average of the p values.) If the values are quite dissimilar, decide whether to use some of them to obtain an advance estimate of p' 21 21 = 390 n (&)' 0.054 (0.054) (0.946) T h e use of the average of the observed values of v as an advance estimate of v' in Eq gives: n 4 390 Total 15.2 Avg lO X 0.0511 0.0016 - 287.4 = 288 If the desired value of E were 0.01, the required sample size would be 4600 It would be smaller if Eq applies Reduction of Empirical Knowledge to a Numerical Value of ~' (No Data from Previous Samples of the Same or Like Material Available) (a) This section illustrates the use of the equations in Section when there are no actual observed values for the computation of a (b) For Equation / - - F r o m past experience, estimate what the smallest and largest values of the characteristic are likely to be If this is not known, obtain this information from some other source Try to picture how the other observed values are probably distributed A few simple observations and questions concerning the past behavior of the process, the usual procedure of blending, mixing, stacking, storing, etc., and concerning the aging of material and the usual pracP-40-37 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 134 CHOICE Or SAMPLE SIZE TO ESTIMATE AVERAGE QUALITY (E 122 Olstributl on, Standard Deviation: o b-o 3.5 n= ~ -(14.7)*- 216.1 217 bricks The difference between 217 and 149 bricks (found in Example 1) is the price of sketchy knowledge (c) For Equation 2. While the estimation of the coefficient of variation of Triangular o b-o 4.2 58) of values of transverse strength for a lot of bricks has been about 1200 psi The values were heaped up in the middle of this range, but not necessarily normally distributed The isosceles triangle in Fig appears to be most appropriate; the advance estimate of a' is 1200/4.9 = 245 psi Then: rice of withdrawing the material (last in, first out; or last in, last out) will usually elicit sufficient information to distinguish between one triangular distribution and ~nother in Fig I n case of doubt, or in ease the desired precision E is a critical matter, the rectangular distribution m a y be used The price of the extra protection afforded by the rectangular distribution is a larger sample size, owing to the larger standard deviation thereof At the worst, if the isosceles triangle is used when the ~ther triangle or the rectangle is a better description, then the standard error of the result is larger by no more than 40 per cent, as shown by comparing the RectangulQr - b b-o 4.2 o Normal b a b-o 4.9 b b-o 6.0 FIG 1. Some Types of Distributions and Their Standard Deviations formulas for the standard deviations given in Fig The sizes of subsequent samples m a y then be adjusted upward, if necessary NOTE 5. The standard deviation of the normal distribution in Fig is a safe assumption for materials with a good history of control, in which case an advance estimate of a' would usually be available The standard deviation estimated from one of the formulas of Fig m a y be used as an advance estimate of a t in Eq This method of advance estimation is in constant use and is often preferable to doubtful observed values Example 4. Use of ~ from Fig 1: Problem (Same as Example/). To compute the sample size needed to estimate the average transverse strength of a lot of bricks when the desired value of E is 50 psi Solution. From past experience the range a universe by use of Fig is possible, it is not recommended I n general, the knowledge that the use of v', instead of a', is preferable would be obtained from the analysis of actual data, in which case the methods of Section apply (d) For Equation - - F r o m past experience, estimate approximately the range within which the fraction defective is likely to lie Turn to Fig and read off the value of a = p(1 p) for the middle of the possible range of p, and use it in Eq I n case the desired precision is a critical matter, use the largest value of a within the possible range of P C o n s i d e r a t i o n of Cost (a) After the required size of sample to meet a prescribed precision is computed from Eqs 1, 2, or 3, the next step is to compute the cost of testing this Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized P-40-36 CHOICE OF SAMPLE SIZE TO ESTr~urATE AVERAGE QUALITY 122 - 58) (E 135 nite and willful effort to produce disorder The only universally acceptable definition of a random selection is by the use of random numbers, which are in effect the guarantee of thorough stirring of the sampling units in a lot (b) In the use of random numbers, the material must first be broken up in some manner into "sampling units." Moreover, each sampling unit is identifiable by a serial number, actual or by rule size of sample If the cost is too great, it may be possible to relax the required precision (or the equivalent, which is to accept an increase in the probability (Section 4) that the sampling error may exceed the maximum allowable error, E) and to reduce the size of the sample to meet the allowable cost (b) As an alternative to Eq 1, which gives n in terms of a prescribed precision, this equation may be solved for E I~][[i~(g~iI~F~t~t~tT[[ji]~1[i~i~[l~i~i[~tr~L~i]~t~i~)1~}iI~i~3~t)BIE~Lf] llltllll[llllJII I I I t t i l l I L L .I I.I.I I L t.(.] I ] [[~ltllll~lllllltlll I F I.I.t ll[ll I .I.J [ l .r.l.l.l.l.t.t.l t t l .l i L .I.I.L .L .l.t.].l l L I .L ~ .T .I I .l l L .I.l.i.t.l.l.t.l l l l .l l ~ l ., E i [ .~ i [ ~ .F ~ , ,L, , ,5 , ~ .i .~ t ~ ,g ,[ ~l,l,l l`,i,L[.l l~[ l [[.l lil.t zl.].l~.l l.l1l.l.t~l .L .] .] ~ ~ fItill L ~ .[ ~ .L .[ ~ .t .~ .i ~ j ~ i i ~ .I ? ~ i ~ .L .E .~ .j .~ [ i ~ ] [ [ ] ~ [ ~ I ~ i ~ [ [ ] L [ l ~ l ~ ] l l i ] j ~ r ~ I ] i ~ l ~ [ ~ t ~ ] ~ F i ~ [ ~ i ~ i ~ I ~ [ ~ f ~ L i ~ ] J i :r[TIII[LI]IIIII~IF[ [ [ ~ l [ [ i ~ ] ~ ] ~ ] i ~ r ~ ~ J J j J j ~ ] l ~ [ [ ~ ( ~ f ~ i ~ L ~ i ~ i i ~ i ~ i ~ ( ~ i i ~ ] i ~ L t ~ [ ~ i [ ~ ] ~ ] ~ i L ~ t ~ ~ i ~ J ~ j [ ~ t [ L ~ ~ J ~ ] ] L [ L [ ~ t ~ f [ [ r ~ ] ~ [ ~ ] ~ i ~ l i L [ ~ r ~ [ ~ r ~ ` ~ F ~ [ ~ t ~ i [ ( ~ { [ ~ i ~ L t I ~ i t ~ t k ~ ] ] ~ t ( ~ i , ~ ~ ] ~ [ [ ~ I j j ~ ~ j ~ i ] i j ~ ) ) i ~ T ~ ] j ~ j ~ ) i ~ t i ] ~ i ~ i i i ] [ ~ 1 ] ~ t i ~ i ~ i ~ i i i i i ~ ~ ~ t J ~ ] ~ i ~ L T ~ ~ ] ] r I T ~ ] ~ j ~ ~ ~lLllllllltliiJlilll[ll]l j ~ t I I.I I l.l l l.].l l.J.l [.l.l l.].l J ].l ~ , , , ~ ~ j ] [ ~ r ~ ~ t ~ [ ~ j ~ L T ~ i ~ ] ~ ~ r ] ~ i ~ f ~ f i ~ i ~ ~ ] [ ~ i ~ ~ [ ~ ] ) ~ ~ ] P ] ] ] ~ ~ illlilllll ]]lll)llllltllllllOlllF[~llllfl~llllll]lllllll~tllll J ].I.I I ].I.] I I.I.] J l.I.[ I.L I.I.T.~.I.) ] I.I.L.I I ~ J ] ~ ] ~ L ~ i j ~ L L i ~ I i i j [ ~ j ~ J i ~ i ~ ] ~ [ ~ T ~ L ~ I ~ i ~ i ~ i ~ I~ ,~tll~lll;'LlJllll'l.Illlli]~l ~*~ELLL]III k)jJ[jlil~ljL[~ l ~ ] ~ J ~ [ ~ ~ i ~ [ y t ~ [ ~ ~ M ~ [ [ i ~ ( ~ i ~ j ~ L ~ J ) j L [ ~ J ) j ~ i ~ j ~ F ~ ~ [ ~ ] ~ L i ~ i i ~ i j ~ j ~ j ~ i ~ u: 'LU'~ b~,'l , ll[lll ~ r ~ i i ~ L ~ L i ] ~ [ ~ ~ r ~ l ~ ] ~ L [ ~ ) ~ I f ~ j ~ j ] ~ L IJrlllllllllll[~iilllll~l]llL]llllll~lllL?i]lllllllllllll ~ i i 1 ~ t ~ ] ] j ~ ] ~ f t [ ~ i ~ i ~ F [ ~ i i ~ L ~ F ~ i i ~ ] j ~ i ~ j ~ L ~ j i ~ i [ ~ L j ] ~ % j ~ r ~ i J ] i ~ J J L ~ ] j i j i i i T ] ~ i ~ i ] ~ j ~ j L ~ i ~ ~ i ~ i ~ ~ i ~ F ~ i i ~ i ~ j ~ F ~ f ~ i ~ L ~ ] j ] ~ [ r ~ i [ [ ~ i I ] ] ~ T ~ L j ] ] ~ ] ~ ~ I ] ~ ] J L f ~ j ~ j j ~ i ~ ] ~ ] ~ L [ ~ i ~ i ~ ( i ~ L i ~ i r [ ~ t i t i ~ A ~ [ ( ~ i "IjILI'I T ~ I ~ j ~ j [ ~ ] j j ~ F ~ i j ~ i ~ j [ ~ J ) ~ ] ~ i i ~ L ~ ] ~ ] ~ i ~ J J ~ i J j ~ f ] ~ t ~ J ~ i ~ [ ~ L ~ ~ L ~ ~ rlltlt.~:lI,l~JlljlllllllJ ~ [ [ [ ~ ] ~ J ~ ] ~ i ~ I ~ ~ ~ N ~ j ~ I:lll]ll j ~ [ ] ~ ] ~ I ] ~ j ~ ] ~ I ~ ] [ ~ ] ~ ] ~ ] ~ ~ [ ~ ] ~ [ ] ] ~ ( ~ [ ~ [ ~ ~ [ [ N [ ~ [ ~ i ~ J ( [ ~ IIIIFIII[IH?.]~ ] I]]jll I11IIII1111141111111 tl IIl[llll]lllll]lll$11111lTl[[lll(llllll[llllll]ll[[Nl[ll[]ll]lll] Ikt],J~lllJI m JJ[I]lJ Ill]]l, I)~Jll~ IIJ]l]~]]~lllJ41JBdlJl(~]~]]ll]]]lllFJlilillliLP~iJ]lilllJlJllJl[i}[l~)lJlll~ tl ]t ~ ] ] ~ i ~ ] ~ ~ J ~ M ~ ] ~ ~ ] ~ ] ~ ~ *llll+l ]+llllll+]i ll]l]llt ~ + ~ I V ~ III]IIIIIIIII? ll]lllllll llll['lJ+~llli]llllilJlllll+illlLllll[llllllll11~ +l]llllllillll1111111111Flllllllliltttmlllil411 IJ]l]J |TT]IT~III~IIII[III , , , ~ t r ~ } ].1.[.i ! 1 I t l l l l l t t l l l l l l l : ~ j .] .j ~ .] .~ .i ~ .j .i ~ .i .~ .1 I .~ .i ~ ~ i .r ~ i .i i .~ .j j .~ .] .~ ~ i .~ .t ~ r .~ .F .L .~ .i + ~ ] ~ .[ ~ ] .~ .] .1 ] .~ ] .~ .[ ~ ] .~ .] .~ ] illlllI[llll l .l l .l l .l .I.l .l : .l l .l ~ , ~ l l ] ~ ~ ] ~ i I ~ ~ ILlllJIJlll [ l ~ m I i ~ ] ~ [ ~ j ~ J ~ :::::::::::::::::::::::: ~ I[ .~ [ i .i .~ .f .~ .i ~ .~ .r L .~ .j i ~ .[ ~ i .] .~ .1 .~ .1 ~ ~ r ~ ~111 iiidl [ ~ ] ~ j ] ~ r [ i ~ i i ~ i ~ [ ~ i ~ i [ ~ i i ~ ] ~ [ ~ i i ~ r [ i i i F ~ i ~ ~ i ~ [ ~ i ~ i ~ i j i i ~ ~ L ~ ] ~ ] ~ ] ~ i [ ] ~ [ ~ ] i i ~ ] ~ j ] ~ f L ~ i ~ t ~ j [ i i ~ j j i ~ i ) ~ r ~ F i ~ r ~ F ~ i 1 i ~ P ~ D ~ J ~ i ~ ] ~ i ~ T ( ~ L ~ i ~ I)1111 II ~ i ~ i i f ~ i ~ L ~ ] [ j ~ ~ i ~ ~ ~ i ~ ] i ~ ] ~ i T ~ i ~ [ ] ~ [ i i ~ ] ~ L [ ~ ~ L l ~ L ~ ~ r ~ i ~ J i [![!!!!l::l;;:::;l~;t~,~;[illl;il Illi]Ilt]IILIII+(~ Illllllllq ]jlllllllllilllllllll]lllllll{lll1111111111111llllllllllllllllllljl I~l[l+lllllllllll I j i L ~ j ~ i j ~ ] ~ ] ~ ] ] ~ j ~ j j ~ i ] j % ~ j ~ ] ~ i } i ~ i I ~ ] ~ i ] ~ r [ i ) b ~ i ~ } ] J i ~ i ~ [ i ~ f i i ~ i i ~ j ~ J ~ ) ~ i i ~ ~ i ~ ~ i ~ ] ~ [ ~ ] i ~ k t i i ~ i ? ~ ] [ j ~ ] i ~ i ] ~ f ~ J ~ i J ] j ] ~ i ~ ] ~ ] i ~ i ~ ) ~ i ~ j j ~ P i ~ ~ i i ~ i ~ i ~ ] j j j ~ F t i ~ ] ~ ~ i i m i ~j~]LLI~iii11~i~]j~1~L~)~]lj]]~j~i11jij~ii]]j]]~i~j~i~j]~ii]j~L~]~ii]1[i~i~3]~f~111~1~i~r~r~ ltii]]D~]lllll]Itll]11[lllll111111111t111t:::lltll~[It:11[111111111]:: lillll1111,,1111111411111LllllllllllllT]lllI1~[~ I , , I i]j(]~J~fI~iL~i)~ii~iPi~ii~]i*j~j[~i]][~iJ]]i~1*~H~i]]]i]]ij~iii]11]1]~[i~Liii]~Li]~[i~i]j]iiii*~fji~i l i 1 t l ~ + ~ o llli] i.l.l l.l ] 1 L r l.l l l.l l l l l J']"l l'i"l t 'l l l l Z l.l l .l.+ ~ .l.l .l.l.i .l.} .l l .1 .1 .1 .1 .1 ] .T l .l.l .l.l ] .1 .1 .1 .1 J l l l t l l t l l l l [ ] ~ ~ + ~ t ] L ~ ~ [ ~ 1 ~ ] ~ i ~ o.I ~ ~ + ~ ~ ] ~ Itl1111+It lllltl1111~[1111tl~[+! ] ~ i ~ ] ~ ~ i ~ i ~ o.~' F~a 2. Values I.O o f a , o r a~, C o r r e s p o n d i n g in terms of n, thus discovering what precision is possible for a given allowable cost The same may be done for Eqs and (c) I t is necessary to specify either the desired allowable error, E, or the allowable cost; otherwise there is no proper size of sample Selection of the Sample (a) In order to make any estimate for a lot or for a process, on the basis of a sample, it is necessary to select the units iu the sample "at random." Randomness is not just accident or lack of direction; it is the product of a deft- t o V a l u e s o f p For packaged articles, a rule is easy; the package contains a certain number of articles in definite layers, arranged in a particular way, and it is easy to devise some system for numbering the articles In the case of bulk material like ore, or coal, or a barrel of bolts or nuts, the problem of defining usable sampling units must take place at an earlier stage of manufacture (c) I t is not the purpose here to discuss the handling of materials, nor to find ways by which one can with surety discover the way to a satisfactory type of sampling unit Instead, the aim is to assume that a suitable sampling unit has P-40-39 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorize 136 CHOICE OF SAMPLE SIZE TO ESTIMATE AVERAGE been defined, and then to answer the question of how many to draw Estimation of the Precision from the Results of the Sample (a) Equation I is a prediction and is used to compute the required size of sample However, after the sample has been tested, the actual value of the maximum sampling error, E, may be estimated One procedure for computing the value of the standard deviation of the sample is that given in Section Then the estimate of the maximum difference between the sample estimate and the result of testing (by the same methods) all the units in the universe is: 3R E , = d ~ V ~ (5) where: prior information The following table indicates the sampling error in ~ for samples of size n drawn from a normal population whose standard deviation is # The following values~ for the ratio ~/~' will be exceeded by chance alone about times in 100: Sample Size 1,378 1,301 1,257 10 15 20 25 30 ~176176 50 100 200 1,228 1,207 1,191 1.152 1,110 079 (c) In estimating a fraction defective, one should remember that the estimate is subject to sampling error, the maximum of which will be: ~,,, = ~ / p ( - p) E,,t = the sample estimate of E, and n = the total sample size (b) When the sample is not apportioned by strata as described in Section 10, an equivalent estimate of the maximum sampling error is: 3~ &" = ~ n - (6) where: O" ~ QUALITY (E 122 - 58) , ,ven otUe ASTM Manual on Quality Control of Materials, Part 1.4 NOTE - - I f n is large, either estimate will be reliable If n is small, either estimate will be subject to a wide sampling error, and may not be as reliable as the advance estimate made from a See p 16 of the ASTM Manual, T P 15-C (7) "V/p(1 p) may be read from Fig if p is small, so that pn is less than 4, then 3.25 should be used in place of in Eq 7, to compensate for the skewness of the p distribution for small values of p S a m p l i n g b y S u b - l o t s or b y S t r a t a 10 It is advisable, and sometimes easier, to apportion the sample by strata (sub-lots, layers, sheets, or other natural divisions), as theory shows that such a plan will occasionally show gains in precision It is important, in the use of Eq for this kind of sampling, to average the ranges of the strata, as otherwise Eq will overestimate the sampling error 5F E Croxton and D J Cowden, Indus trial Quality Control, Vol 3, July, 1946, pp 18-2L P-O-2~ Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions au American Society for Testing and Materials 1916RaceStreet ~ Philadelphia 3,Pa Application for Membership The undersigned hereby applies for membership (which includes subscription(s) to Materials Research & Standards) in the American Society for Testing and Materials in the class of: [] Sustaining [] [] IndustriM Personal [] [] Institutional Associate and if elected to membership agrees to be governed by the Charter and By-laws of the Society and to further its objectives as laid down therein NAME (Firm, organization or person) OFFICIAL REPRESENTATIVE (If sustaining, industrial or institutional membership, indicate the name and title of individual who will exercise membership privileges.) TITLE AND DEFT NAME OF O R G A N I Z A T I O N (If personal or associate membership, indicate the name of organization with which applicant is a~liated.) ADDRESS CITY NATURE OF BUSINESS ADDRESS FOR MAIL ZONE STATE (if other than above) CITY ZONE STATE DATE OF BIRTH GRADUATE OF, OR ATTENDED (Name of college or university) YEAR Degree, or Course SIGNATURE Proposed by (Two Members) A list of A S T M members in your locality will be furnished on request 1962 E-11 QCM 137 Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 C-54-20 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproduc AMERICAN SOCIETY FOR TESTING A N D MATERIALS EXTRACT FROM CHARTER T h e name of the proposed corporation is the " A m e r i c a n Society for T e s t i n g and M a t e r i a l s " The corporation is formed for the p r o m o t i o n of knowledge of t h e m a t e r i a l s of engineering, and the s t a n d a r d i z a t i o n of specifications and the m e t h o d s of testing EXTRACT FROM BY-LAWS ARTICLE I Members and ]'heir Election SECTION The corporate m e m b e r s h i p of the Society shall consist of Personal Members, I n s t i t u t i o n a l Members, I n d u s t r i a l Members, Sustaining Members, Associate Members, and H o n o r a r y Members elected from a corporate "grade of membership I n addition there shall be S t u d e n t Members, and Honorary Members elected from nonmembers of tile Society The rights of m e m b e r ship of I n s t i t u t i o n a l , I n d u s t r i a l , and Sustaining Members shall be exercised b y the individual who is designated as the official r e p r e s e n t a t i v e of t h a t membership SECTION A Personal M e m b e r shall be a person meeting the qualifications established b y the B o a r d of Directors for this classification SECTION An I n s t i t u t i o n a l M e m b e r shall be a public library; e d u c a t i o n a l i n s t i t u t i o n ; a non-profit professional, scientific or technical society; governm e n t d e p a r t m e n t or agency at the federal, state, city, c o u n t y or township level; or separate divisions thereof meeting the qualifications e s t a b l i s h e d b y t h e B o a r d of Directors for this classification SECTION An I n d u s t r i a l M e m b e r shall be a plant, firm, corporation, p a r t nership, or other business enterprise, or separate divisions thereof; t r a d e association, or research i n s t i t u t e meeting the qualifications established b y the B o a r d of Directors for this classification SECTION A Sustaining M e m b e r shall be a person, plant, firm, corporation, society, d e p a r t m e n t of g o v e r n m e n t or other organization, or separate divisions thereof, electing to give greater s u p p o r t to the Society's activities t h r o u g h t h e p a y m e n t of larger dues SECTION An Associate M e m b e r shall be a person less t h a n t h i r t y years of age He shall have the same rights and privileges as a Personal Member, except t h a t he shall not be eligible for office An Associate M e m b e r shall not rem a i n in this category b e y o n d the end of the calendar year in which his t h i r t i e t h b i r t h d a y occurs ARTICLE V Meetings SECTION The Society shall meet annually, for the t r a n s a c t i o n of its business, at a time and place fixed by the Board of Directors Twenty-five corporate members shall c o n s t i t u t e a quorum SECTION Special business meetings of the Society m a y be called at a n y time and place at t h e discretion of the Board of Directors, or shall be called b y the President, upon the w r i t t e n request of a t least one per cent of the Corporate Membership ARTICLE VIII Dues SECTION The m e m b e r s h i p year shall commence on the first day of J a n u ary The a n n u a l dues*, payable in advance, shall be as follows: For Personal Members, $18; for I n s t i t u t i o n a l Members, $25; for I n d u s t r i a l Members, $75; for Sustaining Members, $200; for Associate Members, $10; for S t u d e n t Members, $3 H o n o r a r y Members shall not be subject to dues SECTION The e n t r a n c e fees, payable on admission to the Society, shall be $10 for Personal Members, I n s t i t u t i o n a l Members, I n d u s t r i a l Members and Sustaining Members, and $5 for Associate Members S t u d e n t Members shall pay no e n t r a n c e fee T h e r e shall be no fee for t r a n s f e r from one class of membership to another SECTION Any person elected after six m o n t h s of any m e m b e r s h i p year shall have expired, m a y pay only one-half of the a m o u n t of dues for t h a t year *NOTE~:)I the annual dues $5.00 is for subscription to )/[&TERIALSRESEARCH& STANDARDS Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 138 C-5~29 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authoriz THIS PUBLICATION is one of many issued by the American Society for Testing Materials in connection with its work of promoting knowledge of the properties of materials and developing standard specifications and tests for materials Much of the data result from the voluntary contributions of many of the country's leading technical authorities from industry, scientific agencies, and government Over the years the Society has published many technical symposiums, reports, and special books These may consist of a series of technical papers, reports by the ASTM technical committees, or compilations of data developed in special Society groups with many organizations cooperating A list of ASTM publications and information on the work of the Society will be furnished on request Copyright by ASTM Int'l (all rights reserved); Fri Dec 11 20:03:48 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authoriz