1、raising standards worldwideNO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAWBSI Standards PublicationBS ISO 7870-4:2011BS 5703:2011Control chartsPart 4: Cumulative sum chartsIncorporating corrigendum February 2012BS ISO 7870-4:2011/BS 5703:2011 BRITISH STANDARDNational forewordT
2、his British Standard is the UK implementation of ISO 7870-4:2011. It supersedes BS 5703-1:2003, BS 5703-2:2003, BS 5703-3:2003, and BS 5703-4:2003, which are withdrawn.This international standard has been produced by compressing the four parts of BS 5703 into a single part of ISO 7870. The resultant
3、 standard contains everything that was already in BS 5703 but it is written in a more concise manner. The content of this international standard has been further enhanced and augmented by the inclusion of text and examples that have been contributed by other nations during the drafting phase of the
4、project.The UK participation in its preparation was entrusted to Technical Committee SS/4, Statistical Process Management.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 c
5、ontract. Users are responsible for its correct application. The British Standards Institution 2012 Published by BSI Standards Limited 2012 ISBN 978 0 580 77134 7 ICS 03.120.30 Compliance with a British Standard cannot confer immunity from legal obligations.This British Standard was published under t
6、he authority of the Standards Policy and Strategy Committee on 30 September 2011.Amendments issued since publicationDate Text affected29 February 2012 This corrigendum renumbers BS ISO 7870-4:2011 as dual numbered standard BS ISO 7870-4:2011/ BS 5703:2011Reference numberISO 7870-4:2011(E)ISO 2011INT
7、ERNATIONAL STANDARD ISO7870-4First edition2011-07-01Control charts Part 4: Cumulative sum charts Cartes de contrle Partie 4: Cartes de contrle de lajustement de processus COPYRIGHT PROTECTED DOCUMENT ISO 2011 All rights reserved. Unless otherwise specified, no part of this publication may be reprodu
8、ced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISOs member body in the country of the requester. ISO copyright office Case postale 56 CH-1211 Geneva 20 Tel. + 41 22 749
9、 01 11 Fax + 41 22 749 09 47 E-mail copyrightiso.org Web www.iso.org Published in Switzerland ii BS ISO 7870-4:2011/BS 5703:2011 ISO 2012iiiContents Page Foreword iv Introduction.v 1 Scope1 2 Normative references1 3 Terms and definitions, abbreviated terms and symbols.1 3.1 Terms and definitions .1
10、3.2 Abbreviated terms .2 3.3 Symbols3 4 Principal features of cumulative sum (cusum) charts.4 5 Basic steps in the construction of cusum charts Graphical representation5 6 Example of a cusum plot Motor voltages.5 6.1 The process .5 6.2 Simple plot of results 6 6.3 Standard control chart for individu
11、al results .7 6.4 Cusum chart Overall perspective7 6.5 Cusum chart construction8 6.6 Cusum chart interpretation 9 6.7 Manhattan diagram12 7 Fundamentals of making cusum-based decisions 12 7.1 The need for decision rules12 7.2 The basis for making decisions.13 7.3 Measuring the effectiveness of a dec
12、ision rule14 8 Types of cusum decision schemes .16 8.1 V-mask types .16 8.2 Truncated V-mask .16 8.3 Alternative design approaches 22 8.4 Semi-parabolic V-mask.23 8.5 Snub-nosed V-mask 24 8.6 Full V-mask 24 8.7 Fast initial response (FIR) cusum25 8.8 Tabular cusum .25 9 Cusum methods for process and
13、 quality control 27 9.1 The nature of the changes to be detected 27 9.2 Selecting target values .28 9.3 Cusum schemes for monitoring location .29 9.4 Cusum schemes for monitoring variation 39 9.5 Special situations 47 9.6 Cusum schemes for discrete data.49 Annex A (informative) Von Neumann method56
14、Annex B (informative) Example of tabular cusum.57 Annex C (informative) Estimation of the change point when a step change occurs.61 Bibliography63 BS ISO 7870-4:2011/BS 5703:2011 ISO 2012iv Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standa
15、rds 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 organ
16、izations, 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 drafted in accordance with the rules given i
17、n 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 Standard requires approval by at least 75 %
18、 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 7870-4 was prepared by Technical Committee ISO/TC 69, Appli
19、cations of statistical methods, Subcommittee SC 4, Applications of statistical methods in process management. This first edition of ISO 7870-4 cancels and replaces ISO/TR 7871:1997. ISO 7870 consists of the following parts, under the general title Control charts: Part 1: General guidelines Part 3: A
20、cceptance control charts Part 4: Cumulative sum charts The following part is under preparation: Part 2: Shewhart control charts Additional parts on specialized control charts and on the application of statistical process control (SPC) charts are planned. BS ISO 7870-4:2011/BS 5703:2011 ISO 2012vIntr
21、oduction This part of ISO 7870 demonstrates the versatility and usefulness of a very simple, yet powerful, pictorial method of interpreting data arranged in any meaningful sequence. These data can range from overall business figures such as turnover, profit or overheads to detailed operational data
22、such as stock outs and absenteeism to the control of individual process parameters and product characteristics. The data can either be expressed sequentially as individual values on a continuous scale (e.g. 24,60, 31,21, 18,97.), in “yes”/“no”, “good”/“bad”, “success”/“failure” format, or as summary
23、 measures (e.g. mean, range, counts of events). The method has a rather unusual name, cumulative sum, or, in short, “cusum”. This name relates to the process of subtracting a predetermined value, e.g. a target, preferred or reference value from each observation in a sequence and progressively cumula
24、ting (i.e. adding) the differences. The graph of the series of cumulative differences is known as a cusum chart. Such a simple arithmetical process has a remarkable effect on the visual interpretation of the data as will be illustrated. The cusum method is already used unwittingly by golfers through
25、out the world. By scoring a round as “plus” 4, or perhaps even “minus” 2, golfers are using the cusum method in a numerical sense. They subtract the “par” value from their actual score and add (cumulate) the resulting differences. This is the cusum method in action. However, it remains largely unkno
26、wn and hence is a grossly underused tool throughout business, industry, commerce and public service. This is probably due to cusum methods generally being presented in statistical language rather than in the language of the workplace. This part of ISO 7870 is a revision of ISO/TR 7871:1997. The inte
27、ntion of this part is, thus, to be readily comprehensible to the extensive range of prospective users and so facilitate widespread communication and understanding of the method. The method offers advantages over the more commonly found Shewhart charts in as much as the cusum method will detect a cha
28、nge of an important amount up to three times faster. Further, as in golf, when the target changes per hole, a cusum plot is unaffected, unlike a standard Shewhart chart where the control lines would require a constant adjustment. In addition to Shewhart charts, an EWMA (exponentially weighted moving
29、 average) chart, can be used. Each plotted point on an EWMA chart incorporates information from all of the previous subgroups or observations, but gives less weight to process data as they get “older” according to an exponentially decaying weight. In a similar manner to a cusum chart, an EWMA chart
30、can be sensitized to detect any size of shift in a process. This subject is discussed further in another part of this International Standard. BS ISO 7870-4:2011/BS 5703:2011 ISO 2012BS ISO 7870-4:2011/BS 5703:2011 ISO 2012INTERNATIONAL STANDARD1Control charts Part 4: Cumulative sum charts 1 Scope Th
31、is part of ISO 7870 provides statistical procedures for setting up cumulative sum (cusum) schemes for process and quality control using variables (measured) and attribute data. It describes general-purpose methods of decision-making using cumulative sum (cusum) techniques for monitoring, control and
32、 retrospective analysis. 2 Normative 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. I
33、SO 3534-1, Statistics Vocabulary and symbols Part 1: General statistical terms and terms used in probability ISO 3534-2, Statistics Vocabulary and symbols Part 2: Applied statistics 3 Terms and definitions, abbreviated terms and symbols For the purposes of this document, the terms and definitions gi
34、ven in ISO 3534-1 and ISO 3534-2 and the following apply. 3.1 Terms and definitions 3.1.1 target value value for which a departure from an average level is required to be detected NOTE 1 With a charted cusum, the deviations from the target value are cumulated. NOTE 2 Using a “V” mask, the target val
35、ue is often referred to as the reference value or the nominal control value. If so, it should be acknowledged that it is not necessarily the most desirable or preferred value, as may appear in other standards. It is simply a convenient target value for constructing a cusum chart. 3.1.2 datum value t
36、abulated cusum value from which differences are calculated NOTE The upper datum value is T + fe, for monitoring an upward shift. The lower datum value is T fe, for monitoring a downward shift. BS ISO 7870-4:2011/BS 5703:2011 ISO 20122 3.1.3 reference shift F, f tabulated cusum difference between the
37、 target value (3.1.1) and datum value (3.1.2) NOTE It is necessary to distinguish between f that relates to a standardized reference shift, and F to an observed reference shift, F = fe. 3.1.4 reference shift F, f truncated V-mask slope of the arm of the mask (tangent of the mask angle) NOTE It is ne
38、cessary to distinguish between f that relates to a standardized reference shift, and F to an observed reference shift, F = fe. 3.1.5 decision interval H, h tabulated cusum cumulative sum of deviations from a datum value (3.1.2) required to yield a signal NOTE It is necessary to distinguish between h
39、 that relates to a standardized decision interval, and H to an observed decision interval, H = he. 3.1.6 decision interval H, h truncated V-mask half-height at the datum of the mask NOTE It is necessary to distinguish between h that relates to a standardized decision interval, and H to an observed d
40、ecision interval, H = he. 3.1.7 average run length L average number of samples taken up to the point at which a signal occurs NOTE Average run length (L) is usually related to a particular process level in which case it carries an appropriate subscript, as, for example, L0, meaning the average run l
41、ength when the process is at target level, i.e. zero shift. 3.2 Abbreviated terms ARL average run length CS1 cusum scheme with a long ARL at zero shift CS2 cusum scheme with a shorter ARL at zero shift DI decision interval EWMA exponentially weighted moving average FIR fast initial response LCL lowe
42、r control limit RV reference value UCL upper control limit BS ISO 7870-4:2011/BS 5703:2011 ISO 201233.3 Symbols a scale coefficient C cusum value Crdifference in the cusum value between the lead point and the out-of-control point c4factor for estimating the within-subgroup standard deviation amount
43、of change to be detected standardized amount of change to be detected d lead distance d2factor for estimating the within-subgroup standard deviation from within-subgroup range F observed reference shift f standardized reference shift H observed decision interval h standardized decision interval J in
44、dex number size of process adjustment K cusum datum value for discrete data k number of subgroups L0average run length at zero shift Laverage run length at shift population mean value m mean count number n subgroup size p probability of “success” R mean subgroup range r number of plotted points betw
45、een the lead point and the out-of-control point process standard deviation 0within-subgroup standard deviation 0 estimated within-subgroup standard deviation BS ISO 7870-4:2011/BS 5703:2011 ISO 20124 estandard error s observed within-subgroup standard deviation s average subgroup standard deviation
46、xs realized standard error of the mean from k subgroups T target value Tmreference or target rate of occurrence Tpreference or target proportion true change point t observed change point Vavgaverage voltage avgV estimated average voltage w difference between successive subgroup mean values x individ
47、ual result x arithmetic mean value (of a subgroup) x mean of subgroup means 4 Principal features of cumulative sum (cusum) charts A cusum chart is essentially a running total of deviations from some preselected reference value. The mean of any group of consecutive values is represented visually by t
48、he current slope of the graph. The principal features of a cusum chart are the following. a) It is sensitive in detecting changes in the mean. b) Any change in the mean, and the extent of the change, is indicated visually by a change in the slope of the graph: 1) a horizontal graph indicates an “on-
49、target” or reference value; 2) a downward slope indicates a mean less than the reference or target value: the steeper the slope, the bigger the difference; 3) an upward slope indicates a mean more than the reference or target value: the steeper the slope, the bigger the difference. c) It can be used retrospectively for investigative purposes, on a running basis for control, and for prediction of performance in the immediate future. BS ISO 7870-4:2011/BS 5703:2011 ISO 20125Referring to point
