# BS ISO 16269-6-2014

BSI Standards Publication BS ISO 16269-6:2014 Statistical interpretation of data Part 6: Determination of statistical tolerance intervalsBS ISO 16269-6:2014 BRITISH STANDARD National foreword This British Standard is the UK implementation of ISO 16269-6:2014. It supersedes BS ISO 16269-6:2005 which is withdrawn. The UK participation in its preparation was entrusted to Technical Committee SS/2, Statistical Interpretation of Data. A list of organizations represented on this committee can be obtained on request to its secretary. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. © The British Standards Institution 2014. Published by BSI Standards Limited 2014 ISBN 978 0 580 74604 8 ICS 03.120.30 Compliance with a British Standard cannot confer immunity from legal obligations. This British Standard was published under the authority of the Standards Policy and Strategy Committee on 31 January 2014. Amendments issued since publication Date Text affectedBS ISO 16269-6:2014 © ISO 2014 Statistical interpretation of data — Part 6: Determination of statistical tolerance intervals Interprétation statistique des données — Partie 6: Détermination des intervalles statistiques de dispersion INTERNATIONAL STANDARD ISO 16269-6 Second edition 2014-01-15 Reference number ISO 16269-6:2014(E)BS ISO 16269-6:2014ISO 16269-6:2014(E)ii © ISO 2014 – All rights reserved COPYRIGHT PROTECTED DOCUMENT © ISO 2014 All rights reserved. Unless otherwise specified, no part of 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 ISO’s member body in the country of the requester. ISO copyright office Case postale 56 • CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyright@iso.org Web www.iso.org Published in SwitzerlandBS ISO 16269-6:2014ISO 16269-6:2014(E)© ISO 2014 – All rights reserved iii Contents Page Foreword iv Introduction v 1 Scope . 1 2 Normative references 1 3 T erms, definitions and s ymbols 1 3.1 Terms and definitions . 1 3.2 Symbols . 2 4 Procedures 3 4.1 Normal population with known mean and known variance . 3 4.2 Normal population with unknown mean and known variance 3 4.3 Normal population with unknown mean and unknown variance . 4 4.4 Normal populations with unknown means and unknown common variance . 4 4.5 Any continuous distribution of unknown type . 4 5 Examples . 4 5.1 Data for Examples 1 and 2 4 5.2 Example 1: One-sided statistical tolerance interval with unknown variance and unknown mean . 5 5.3 Example 2: Two-sided statistical tolerance interval under unknown mean and unknown variance 6 5.4 Data for Examples 3 and 4 6 5.5 Example 3: One-sided statistical tolerance intervals for separate populations with unknown common variance 7 5.6 Example 4: Two-sided statistical tolerance intervals for separate populations with unknown common variance 8 5.7 Example 5: Any distribution of unknown type 10 Annex A (informative) Exact k-factors for statistical tolerance intervals for the normal distribution 12 Annex B (informative) Forms for statistical tolerance intervals 17 Annex C (normative) One-sided statistical tolerance limit factors, k C (n; p; 1− α), for unknown σ .21 Annex D (normative) Two-sided statistical tolerance limit factors, k D (n; m; p; 1− α), for unknown common σ (m samples) 26 Annex E (normative) Distribution‑fr ee statistical t oler anc e int erv als 40 Annex F (informative) Computation of factors for two-sided parametric statistical tolerance intervals .42 Annex G (informative) C onstruction of a distribution‑fr ee statistical t oler anc e int erv al for an y type of distribution .44 Bibliogr aph y .46BS ISO 16269-6:2014ISO 16269-6:2014(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation 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. 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 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. www.iso.org/directives Attention 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 will be in the Introduction and/or on the ISO list of patent declarations received. www.iso.org/patents Any 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 related to conformity assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information The committee responsible for this document is ISO/TC 69, Applications of statistical methods. This second 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 intervals — 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) Introduction A statistical tolerance interval is an estimated interval, based on a sample, which can be asserted with confidence level 1 − α, 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 − α is the probability that a statistical tolerance interval constructed in the prescribed 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 α. This part of ISO 16269 describes both one-sided and two-sided statistical tolerance intervals; a one-sided interval is constructed 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 − α and a stated proportion p of the population. The confidence level of a statistical tolerance interval is well understood 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 − α of the cases in a long series of repeated random samples under identical conditions. Similarly the confidence statement of a statistical tolerance interval states that at least a proportion p of the population is contained in the interval in a proportion 1 − α 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 parameter, 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 observations 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 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 − α if the normality assumption is true. For normally distributed characteristics, the statistical tolerance interval 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 constructed. 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 specification 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 p U( I S O 3 5 3 4 -2 : 2 0 0 6 , 2 . 5 . 4) and below the lower specification limit L there is the lower fraction nonconforming p L(ISO 3534-2:2006, 2.5.5). The sum p U+ p L= p tis 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 − p t . The ideas behind statistical tolerance intervals are more widespread than is usually appreciated, for example in acceptance 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, p U, p Lor p twill be specified as an acceptable quality limit (AQL), α will be implied and the lot is accepted if there is at least an implicit 100(1−α)% confidence that the AQL is not exceeded. In statistical process management the limits U and L are fixed in advance and the fractions p U , p Land p tare either calculated, if the distribution is assumed to be known, or otherwise estimated. This is an example of a 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 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 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 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 these 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 [5]uses the term “coverage 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 factor 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 other 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, each sample being of the same size n.vi © ISO 2014 – All rights reservedBS ISO 16269-6:2014INTERNATIONAL ST ANDARD ISO 16269-6:2014(E) Statistical interpretation of data — Part 6: Determination of statistical tolerance intervals 1 Scope This part of ISO 16269 describes procedures for establishing statistical tolerance 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 lower 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 establishment of two-sided statistical tolerance intervals for more than one normal sample with common unknown variance. 2 Normative references The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition 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 probability ISO 3534-2:2006, Statistics — Vocabulary and symbols — Part 2: Applied statistics 3 T erms, d efinitions and s ymbols For the purposes of this document, the terms and definition given in ISO 3534-1, ISO 3534-2 and the following apply. 3.1 T erms and definiti ons 3.1.1 statistical tolerance interval interval determined from a random sample in such a way that one may have a specified level of confidence