1、BRITISH STANDARD BS ISO 16269-8:2004 Statistical interpretation of data Part 8: Determination of prediction intervals ICS 03.120.30 BS ISO 16269-8:2004 This British Standard was published under the authority of the Standards Policy and Strategy Committee on 30 September 2004 BSI 30 September 2004 IS
2、BN 0 580 44530 5 National foreword This British Standard reproduces verbatim ISO 16269-8:2004 and implements it as the UK national standard. The UK participation in its preparation was entrusted to Technical Committee SS/2, Statistical interpretation of data, which has the responsibility to: A list
3、of organizations represented on this committee can be obtained on request to its secretary. Cross-references The British Standards which implement international publications referred to in this document may be found in the BSI Catalogue under the section entitled “International Standards Corresponde
4、nce Index”, or by using the “Search” facility of the BSI Electronic Catalogue or of British Standards Online. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. Compliance with a British Standard does not of its
5、elf confer immunity from legal obligations. aid enquirers to understand the text; present to the responsible international/European committee any enquiries on the interpretation, or proposals for change, and keep the UK interests informed; monitor related international and European developments and
6、promulgate them in the UK. Summary of pages This document comprises a front cover, an inside front cover, the ISO title page, pages ii to vi, pages 1 to 108, an inside back cover and a back cover. The BSI copyright notice displayed in this document indicates when the document was last issued. Amendm
7、ents issued since publication Amd. No. Date Comments Reference number ISO 16269-8:2004(E) OSI 4002INTERNATIONAL STANDARD ISO 16269-8 First edition 2004-09-15 Statistical interpretation of data Part 8: Determination of prediction intervals Interprtation statistique des donnes Partie 8: Dtermination d
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13、004IS-96261 O8:(4002E) I SO 4002 All irthgs ersedevr iiiContents Page Foreword. v Introduction . vi 1 Scope 1 2 Normative references . 1 3 Terms, definitions and symbols 2 3.1 Terms and definitions. 2 3.2 Symbols . 2 4 Prediction intervals. 3 4.1 General. 3 4.2 Comparison with other types of statist
14、ical interval 4 4.2.1 Choice of type of interval . 4 4.2.2 Comparison with a statistical tolerance interval . 4 4.2.3 Comparison with a confidence interval for the mean . 4 5 Prediction intervals for all observations in a further sample from a normally distributed population with unknown population
15、standard deviation 4 5.1 One-sided intervals. 4 5.2 Symmetric two-sided intervals 5 5.3 Prediction intervals for non-normally distributed populations that can be transformed to normality 5 5.4 Determination of a suitable initial sample size, n, for a given maximum value of the prediction interval fa
16、ctor, k 6 5.5 Determination of the confidence level corresponding to a given prediction interval . 6 6 Prediction intervals for all observations in a further sample from a normally distributed population with known population standard deviation 6 6.1 One-sided intervals. 6 6.2 Symmetric two-sided in
17、tervals 7 6.3 Prediction intervals for non-normally distributed populations that can be transformed to normality 7 6.4 Determination of a suitable initial sample size, n, for a given value of k. 7 6.5 Determination of the confidence level corresponding to a given prediction interval . 8 7 Prediction
18、 intervals for the mean of a further sample from a normally distributed population 8 8 Distribution-free prediction intervals 8 8.1 General. 8 8.2 One-sided intervals. 8 8.3 Two-sided intervals. 9 Annex A (normative) Tables of one-sided prediction interval factors, k, for unknown population standard
19、 deviation 13 Annex B (normative) Tables of two-sided prediction interval factors, k, for unknown population standard deviation 31 Annex C (normative) Tables of one-sided prediction interval factors, k, for known population standard deviation. 49 Annex D (normative) Tables of two-sided prediction in
20、terval factors, k, for known population standard deviation. 67 BSISO162698:2004IS-96261 O8:(4002E) iv I SO 4002 All irthgs ersedevrAnnex E (normative) Tables of sample sizes for one-sided distribution-free prediction intervals .85 Annex F (normative) Tables of sample sizes for two-sided distribution
21、-free prediction intervals91 Annex G (normative) Interpolating in the tables .97 Annex H (informative) Statistical theory underlying the tables .101 Bibliography108 BSISO162698:2004IS-96261 O8:(4002E) I SO 4002 All irthgs ersedevr vForeword ISO (the International Organization for Standardization) is
22、 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 represen
23、ted 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. International Standards are dr
24、afted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International S
25、tandard requires approval by at least 75 % of the member bodies casting a vote. 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. ISO 16269-8 was prepa
26、red by Technical Committee ISO/TC 69, Application of statistical methods. ISO 16269 consists of the following parts, under the general title Statistical interpretation of data: Part 6: Determination of statistical tolerance intervals Part 7: Median Estimation and confidence intervals Part 8: Determi
27、nation of prediction intervals BSISO162698:2004IS-96261 O8:(4002E) vi I SO 4002 All irthgs ersedevrIntroduction Prediction intervals are of value wherever it is desired or required to predict the results of a future sample of a given number of discrete items from the results of an earlier sample of
28、items produced under identical conditions. They are of particular use to engineers who need to be able to set limits on the performance of a relatively small number of manufactured items. This is of increasing importance with the recent shift towards small-scale production in some industries. Despit
29、e the first review article on prediction intervals and their applications being published as long ago as 1973, there is still a surprising lack of awareness of their value, perhaps due in part to the inaccessibility of the research work for the potential user, and also partly due to confusion with c
30、onfidence intervals and statistical tolerance intervals. The purpose of this part of ISO 16269 is therefore twofold: to clarify the differences between prediction intervals, confidence intervals and statistical tolerance intervals; to provide procedures for some of the more useful types of predictio
31、n interval, supported by extensive, newly-computed tables. For information on prediction intervals that are outside the scope of this part of ISO 16269, the reader is referred to the Bibliography. BSISO162698:2004INTENRATIONAL TSANDADR IS-96261 O8:(4002E)I SO 4002 All irthgs ersedevr 1Statistical in
32、terpretation of data Part 8: Determination of prediction intervals 1 Scope This part of ISO 16269 specifies methods of determining prediction intervals for a single continuously distributed variable. These are ranges of values of the variable, derived from a random sample of size n, for which a pred
33、iction relating to a further randomly selected sample of size m from the same population may be made with a specified confidence. Three different types of population are considered, namely: a) normally distributed with unknown standard deviation; b) normally distributed with known standard deviation
34、; c) continuous but of unknown form. For each of these three types of population, two methods are presented, one for one-sided prediction intervals and one for symmetric two-sided prediction intervals. In all cases, there is a choice from among six confidence levels. The methods presented for cases
35、a) and b) may also be used for non-normally distributed populations that can be transformed to normality. For cases a) and b) the tables presented in this part of ISO 16269 are restricted to prediction intervals containing all the further m sampled values of the variable. For case c) the tables rela
36、te to prediction intervals that contain at least m r of the next m values, where r takes values from 0 to 10 or 0 to m 1, whichever range is smaller. For normally distributed populations a procedure is also provided for calculating prediction intervals for the mean of m further observations. 2 Norma
37、tive references The following referenced documents are indispensable for the application of this document. 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, Statistics Vocabulary
38、 and symbols Part 1: Probability and general statistical terms ISO 3534-2, Statistics Vocabulary and symbols Part 2: Statistical quality control BSISO162698:2004IS-96261 O8:(4002E) 2 I SO 4002 All irthgs ersedevr3 Terms, definitions and symbols 3.1 Terms and definitions For the purposes of this docu
39、ment, the terms and definitions given in ISO 3534-1 and ISO 3534-2 and the following apply. 3.1.1 prediction interval interval determined from a random sample from a population in such a way that one may have a specified level of confidence that no fewer than a given number of values in a further ra
40、ndom sample of a given size from the same population will fall NOTE In this context, the confidence level is the long-run proportion of intervals constructed in this manner that will have this property. 3.1.2 order statistics sample values identified by their position after ranking in non-decreasing
41、 order of magnitude NOTE The sample values in order of selection are denoted in this part of ISO 16269 by x 1 , x 2 , , x n . After arranging in non-decreasing order, they are denoted by x 1 , x 2 , , x n , where x 1u x 2u u x n . The word “non-decreasing” is used in preference to “increasing” to in
42、clude the case where two or more values are equal, at least to within measurement error. Sample values that are equal to one another are assigned distinct, contiguous integer subscripts in square brackets when represented as order statistics. 3.2 Symbols a lower limit to the values of the variable i
43、n the population nominal maximum probability that more than r observations from the further random sample of size m will lie outside the prediction interval b upper limit to the values of the variable in the population C confidence level expressed as a percentage: C = 100 (1 ) k prediction interval
44、factor m size of further random sample to which the prediction applies n size of random sample from which the prediction interval is derived s sample standard deviation: ()() 2 1 1 n i i sx xn = = r specified maximum number of observations from the further random sample of size m that will not lie i
45、n the prediction interval T 1lower prediction limit T 2upper prediction limit x iith observation in a random sample x iith order statistic BSISO162698:2004IS-96261 O8:(4002E) I SO 4002 All irthgs ersedevr 3x sample mean: 1 n i i xx n = = 4 Prediction intervals 4.1 General A two-sided prediction inte
46、rval is an interval of the form (T 1 , T 2 ), where T 1 T 2 ; T 1and T 2are derived from a random sample of size n and are called the lower and upper prediction limits, respectively. If a and b are respectively the lower and upper limits of the variable in the population, a one-sided prediction inte
47、rval will be of the form (T 1 , b) or (a, T 2 ). NOTE 1 For practical purposes a is often taken to be zero for variables that cannot be negative, and b is often taken to be infinity for variables with no natural upper limit. NOTE 2 Sometimes a population is treated as normal for the purpose of deter
48、mining a prediction interval, even when it has a finite limit. This may seem incongruous, as the normal distribution ranges from minus infinity to plus infinity. However, in practice, many populations with a finite limit are closely approximated by a normal distribution. The practical meaning of a p
49、rediction interval relating to individual sample values is that the experimenter claims that a further random sample of m values from the same population will have at most r values not lying in the interval, while admitting a small nominal probability that this assertion may be wrong. The nominal probability that an interval constructed in such a way satisfies the claim is called the confidence l
