1、BSI Standards PublicationBS ISO 16269-6:2014Statistical interpretation ofdataPart 6: Determination of statistical toleranceintervalsBS ISO 16269-6:2014 BRITISH STANDARDNational forewordThis British Standard is the UK implementation of ISO 16269-6:2014.It supersedes BS ISO 16269-6:2005 which is withd
2、rawn.The UK participation in its preparation was entrusted to TechnicalCommittee SS/2, Statistical Interpretation of Data.A list of organizations represented on this committee can beobtained on request to its secretary.This publication does not purport to include all the necessaryprovisions of a con
3、tract. Users are responsible for its correctapplication. The British Standards Institution 2014. Published by BSI StandardsLimited 2014ISBN 978 0 580 74604 8ICS 03.120.30Compliance with a British Standard cannot confer immunity fromlegal obligations.This British Standard was published under the auth
4、ority of theStandards Policy and Strategy Committee on 31 January 2014.Amendments issued since publicationDate Text affectedBS ISO 16269-6:2014 ISO 2014Statistical interpretation of data Part 6: Determination of statistical tolerance intervalsInterprtation statistique des donnes Partie 6: Dterminati
5、on des intervalles statistiques de dispersionINTERNATIONAL STANDARDISO16269-6Second edition2014-01-15Reference numberISO 16269-6:2014(E)BS ISO 16269-6:2014ISO 16269-6:2014(E)ii ISO 2014 All rights reservedCOPYRIGHT PROTECTED DOCUMENT ISO 2014All rights reserved. Unless otherwise specified, no part o
6、f this publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below or ISOs member body
7、in the country of the requester.ISO copyright officeCase postale 56 CH-1211 Geneva 20Tel. + 41 22 749 01 11Fax + 41 22 749 09 47E-mail copyrightiso.orgWeb www.iso.orgPublished in SwitzerlandBS ISO 16269-6:2014ISO 16269-6:2014(E) ISO 2014 All rights reserved iiiContents PageForeword ivIntroduction v1
8、 Scope . 12 Normative references 13 Terms, definitions and symbols 13.1 Terms and definitions . 13.2 Symbols . 24 Procedures 34.1 Normal population with known mean and known variance . 34.2 Normal population with unknown mean and known variance 34.3 Normal population with unknown mean and unknown va
9、riance . 44.4 Normal populations with unknown means and unknown common variance . 44.5 Any continuous distribution of unknown type . 45 Examples . 45.1 Data for Examples 1 and 2 45.2 Example 1: One-sided statistical tolerance interval with unknown variance and unknown mean . 55.3 Example 2: Two-side
10、d statistical tolerance interval under unknown mean and unknown variance 65.4 Data for Examples 3 and 4 65.5 Example 3: One-sided statistical tolerance intervals for separate populations with unknown common variance 75.6 Example 4: Two-sided statistical tolerance intervals for separate populations w
11、ith unknown common variance 85.7 Example 5: Any distribution of unknown type 10Annex A (informative) Exact k-factors for statistical tolerance intervals for the normal distribution 12Annex B (informative) Forms for statistical tolerance intervals 17Annex C (normative) One-sided statistical tolerance
12、 limit factors, kC(n; p; 1), for unknown .21Annex D (normative) Two-sided statistical tolerance limit factors, kD(n; m; p; 1), for unknown common (m samples) 26Annex E (normative) Distributionfree statistical tolerance intervals 40Annex F (informative) Computation of factors for two-sided parametric
13、 statistical tolerance intervals .42Annex G (informative) Construction of a distributionfree statistical tolerance interval for any type of distribution .44Bibliography .46BS ISO 16269-6:2014ISO 16269-6:2014(E)ForewordISO (the International Organization for Standardization) is a worldwide federation
14、 of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee.
15、International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.The procedures used to develop this document and those
16、 intended for its further maintenance are described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the different types of ISO documents should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2. ww
17、w.iso.org/directivesAttention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document wi
18、ll be in the Introduction and/or on the ISO list of patent declarations received. www.iso.org/patentsAny trade name used in this document is information given for the convenience of users and does not constitute an endorsement.For an explanation on the meaning of ISO specific terms and expressions r
19、elated to conformity assessment, as well as information about ISOs adherence to the WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary informationThe committee responsible for this document is ISO/TC 69, Applications of statistical methods.This se
20、cond edition cancels and replaces the first edition (ISO 16269:2005), which has been technically revised.ISO 16269 consists of the following parts, under the general title Statistical interpretation of data: Part 4: Detection and treatment of outliers Part 6: Determination of statistical tolerance i
21、ntervals Part 7: Median Estimation and confidence intervals Part 8: Determination of prediction intervalsiv ISO 2014 All rights reservedBS ISO 16269-6:2014ISO 16269-6:2014(E)IntroductionA statistical tolerance interval is an estimated interval, based on a sample, which can be asserted with confidenc
22、e level 1 , for example 0,95, to contain at least a specified proportion p of the items in the population. The limits of a statistical tolerance interval are called statistical tolerance limits. The confidence level 1 is the probability that a statistical tolerance interval constructed in the prescr
23、ibed manner will contain at least a proportion p of the population. Conversely, the probability that this interval will contain less than the proportion p of the population is . This part of ISO 16269 describes both one-sided and two-sided statistical tolerance intervals; a one-sided interval is con
24、structed with an upper or a lower limit while a two-sided interval is constructed with both an upper and a lower limit.A statistical tolerance interval depends on a confidence level 1 and a stated proportion p of the population. The confidence level of a statistical tolerance interval is well unders
25、tood from a confidence interval for a parameter. The confidence statement of a confidence interval is that the confidence interval contains the true value of the parameter a proportion 1 of the cases in a long series of repeated random samples under identical conditions. Similarly the confidence sta
26、tement of a statistical tolerance interval states that at least a proportion p of the population is contained in the interval in a proportion 1 of the cases of a long series of repeated random samples under identical conditions. So if we think of the stated proportion of p of the population as a par
27、ameter, the idea behind statistical tolerance intervals is similar to the idea behind confidence intervals.Statistical tolerance intervals are functions of the observations of the sample, i.e. statistics, and they will generally take different values for different samples. It is necessary that the o
28、bservations be independent for the procedures provided in this part of ISO 16269 to be valid.Two types of statistical tolerance interval are provided in this part of ISO 16269, parametric and distribution-free. The parametric approach is based on the assumption that the characteristic being studied
29、in the population has a normal distribution; hence the confidence that the calculated statistical tolerance interval contains at least a proportion p of the population can only be taken to be 1 if the normality assumption is true. For normally distributed characteristics, the statistical tolerance i
30、nterval is determined using one of the Forms A, B, or C given in Annex B.Parametric methods for distributions other than the normal are not considered in this part of ISO 16269. If departure from normality is suspected in the population, distribution-free statistical tolerance intervals may be const
31、ructed. The procedure for the determination of a statistical tolerance interval for any continuous distribution is provided in Form D of Annex B.The statistical tolerance limits discussed in this part of ISO 16269 can be used to compare the natural capability of a process with one or two given speci
32、fication limits, either an upper one U or a lower one L or both in statistical process management.Above the upper specification limit U there is the upper fraction nonconforming pU(ISO 3534-2:2006, 2.5.4) and below the lower specification limit L there is the lower fraction nonconforming pL(ISO 3534
33、-2:2006, 2.5.5). The sum pU+ pL= ptis called the total fraction nonconforming. (ISO 3534-2:2006, 2.5.6). Between the specification limits U and L there is the fraction conforming 1 pt.The ideas behind statistical tolerance intervals are more widespread than is usually appreciated, for example in acc
34、eptance sampling by variables and in statistical process management, as will be pointed out in the next two paragraphs.In acceptance sampling by variables, the limits U and/or L will be known, pU, pLor ptwill be specified as an acceptable quality limit (AQL), will be implied and the lot is accepted
35、if there is at least an implicit 100(1)% confidence that the AQL is not exceeded.In statistical process management the limits U and L are fixed in advance and the fractions pU, pLand ptare either calculated, if the distribution is assumed to be known, or otherwise estimated. This is an example of a
36、quality control application, but there are many other applications of statistical tolerance intervals given in textbooks such as Hahn and Meeker.13 ISO 2014 All rights reserved vBS ISO 16269-6:2014ISO 16269-6:2014(E)In contrast, for the statistical tolerance intervals considered in this part of ISO
37、16269, the confidence level for the interval estimator and the proportion of the distribution within the interval (corresponding to the fraction conforming mentioned above) are fixed in advance, and the limits are estimated. These limits may be compared with U and L. Hence the appropriateness of the
38、 given specification limits U and L can be compared with the actual properties of the process. The one-sided statistical tolerance intervals are used when only either the upper specification limit U or the lower specification limit L is relevant, while the two-sided intervals are used when both the
39、upper and the lower specification limits are considered simultaneously.The terminology with regard to these different limits and intervals has been confusing, as the “specification limits” were earlier also called “tolerance limits” (see the terminology standard ISO 3534-2:1993, 1.4.3, where both th
40、ese terms as well as the term “limiting values” were all used as synonyms for this concept). In the latest revision of ISO 3534-2:2006, 3.1.3, only the term specification limits have been kept for this concept. Furthermore, the Guide for the expression of uncertainty in measurement 5uses the term “c
41、overage factor” defined as a “numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded uncertainty”. This use of “coverage” differs from the use of the term in this part of ISO 16269.The first edition of this standard gave extensive tables of the fact
42、or k for one-sided and two-sided tolerance intervals when the mean is unknown but the standard deviation is known. In this second edition of the standard those tables are omitted. Instead, exact k-factors are given in Annex A when one of the parameters of the normal distribution is unknown and the o
43、ther parameter is known.The first edition of this standard considered statistical tolerance intervals based only on a single sample of size n. This edition considers statistical tolerance intervals for m populations with the same standard deviation, based on samples from each of the m populations, e
44、ach sample being of the same size n.vi ISO 2014 All rights reservedBS ISO 16269-6:2014INTERNATIONAL STANDARD ISO 16269-6:2014(E)Statistical interpretation of data Part 6: Determination of statistical tolerance intervals1 ScopeThis part of ISO 16269 describes procedures for establishing statistical t
45、olerance intervals that include at least a specified proportion of the population with a specified confidence level. Both one-sided and two-sided statistical tolerance intervals are provided, a one-sided interval having either an upper or a lower limit while a two-sided interval has both upper and l
46、ower limits. Two methods are provided, a parametric method for the case where the characteristic being studied has a normal distribution and a distribution-free method for the case where nothing is known about the distribution except that it is continuous. There is also a procedure for the establish
47、ment of two-sided statistical tolerance intervals for more than one normal sample with common unknown variance.2 Normative referencesThe following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edi
48、tion cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.ISO 3534-1:2006, Statistics Vocabulary and symbols Part 1: General statistical terms and terms used in probabilityISO 3534-2:2006, Statistics Vocabulary and symbols Part 2: Ap
49、plied statistics3 Terms, definitions and symbolsFor the purposes of this document, the terms and definition given in ISO 3534-1, ISO 3534-2 and the following apply.3.1 Terms and definitions3.1.1statistical tolerance intervalinterval determined from a random sample in such a way that one may have a specified level of confidence that the interval covers at least a specified proportion of the sampled populationSOURCE: ISO 3534-1:2006, 1.26Note 1 to entry: The confidence level in this context is the long-ru
