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1、BRITISH STANDARD BS 5703-2: 1980 Incorporating Amendment No. 1 Guide to Data analysis and quality control using cusum techniques Part 2: Decision rules and statistical tests for cusum charts and tabulations UDC 519.244.8:658.562.012.7:519.816 Licensed Copy: sheffieldun sheffieldun, na, Fri Dec 01 13
2、:39:41 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 5703-2:1980 This British Standard, having been prepared under the direction of the Quality Management and Statistics Standards Committee, was published under the authority of the Executive Board and comes into effect on 29 February 1980 BSI 10-199
3、9 The following BSI references relate to the work on this standard: Committee reference QMS/1 Draft for comment 77/60981 DC ISBN 0 580 11046 X Cooperating organizations The Quality Management and Statistics Standards Committee, under whose direction this British Standard was prepared, consists of re
4、presentatives from the following Government department and scientific and industrial organizations: Confederation of British Industry Consumers Association Electronic Engineering Association Institute of Cost and Management Accountants Institute of Quality Assurance* Institute of Statisticians* Inst
5、itution of Electrical Engineers Institution of Production Engineers* Ministry of Defence* National Council for Quality and Reliability National Terotechnology Centre The organizations marked with an asterisk in the above list, together with the following, were directly represented on the committee e
6、ntrusted with the preparation of this British Standard: Association of Public Analysts British Paper and Board Industry Federation (PIF) Chemical Industries Association Department of Industry (National Engineering Laboratory) Economist Intelligence Unit Limited Electronic Components Industry Federat
7、ion Institute of Petroleum Ministry of Agriculture, Fisheries and Food National Coal Board Post Office Royal Statistical Society University of Birmingham University of Essex Amendments issued since publication Amd. No.Date of issueComments 3735September 1982 Indicated by a sideline in the margin Lic
8、ensed Copy: sheffieldun sheffieldun, na, Fri Dec 01 13:39:41 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 5703-2:1980 BSI 10-1999i Contents Page Cooperating organizationsInside front cover Forewordiv 0Introduction1 0.1Basis of cusum chart1 0.2Simple example of cusum chart1 0.3Monitoring or retrospe
9、ctive analysis3 1Scope4 2References4 3Symbols and terminology5 4Simple decision rules (a) Monitoring and control6 4.1Introduction6 4.2Cusum V-masks8 4.3Practical considerations concerning cusum masks11 4.4The full V-mask11 4.5Local decision lines11 4.6Average run length characteristics of basic deci
10、sion rule11 4.7A semi-parabolic mask14 4.8Monitoring non-normal variables16 5Simple decision rules (b) Retrospective analysis16 5.1Introduction16 5.2“Span” tests17 5.3Construction of decision lines17 5.4Ad hoc tests of significance17 5.5Distribution-free cusum tests18 5.6A comparison of the four pro
11、cedures18 5.7An example of application of the tests over a longer series18 5.8Automatic procedures18 6Examples of applications18 6.1Introduction18 6.2Setting up a monitoring procedure from a specification18 6.3Control of volatile matter in a synthetic resin20 6.4Ash-content in coal22 6.5Evaluation o
12、f the performance of a blending machine24 6.6Systematic patterns in fabric coating26 6.7Faults in knitted fabric28 7Extended cusum techniques29 7.1General29 7.2Cusums without charts29 7.3Application of automatic cusum procedure to package weight control32 7.4Binary data, or counts of events containi
13、ng frequent zeros33 7.5An application of cusums to binary data33 7.6Data elements of varying size or importance35 7.7Estimation of standard error for weighted data36 7.8Application of weighted cusum to machine stoppages38 7.9Monitoring vehicle fuel consumption39 Licensed Copy: sheffieldun sheffieldu
14、n, na, Fri Dec 01 13:39:41 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 5703-2:1980 ii BSI 10-1999 Page Appendix A The full V-mask44 Appendix B Local decision lines46 Appendix C Monitoring non-normal variables46 Appendix D Construction of decision lines for retrospective analysis48 Appendix E Ad ho
15、c tests of significance for retrospective analysis49 Appendix F Distribution-free cusum tests50 Appendix G A comparison of the four procedures51 Appendix H An example of application of the tests over a longer series52 Appendix J Rigorous procedure for obtaining standard errors for weighted data57 Fi
16、gure 1 Conventional chart of data from Table 13 Figure 2 Cusum chart of data from Table 13 Figure 3 Cusum chart of data from Table 1, with target value 124 Figure 4 Cusum chart of data from Table 1, with target value 15 but compressed cusum scale4 Figure 5 Some typical run length distributions8 Figu
17、re 6 General-purpose truncated V-mask9 Figure 7 Truncated V-mask applied to cusum chart: no indication of shift10 Figure 8 Truncated V-mask applied to cusum chart: indication of shift10 Figure 9 Average run length data for cusum and control charts13 Figure 10 Semi-parabolic cusum mask16 Figure 11 Ap
18、plication of span test to cusum chart16 Figure 12 Nomogram for evaluating maximum vertical height (Vmax) of cusum over span of m sample intervals19 Figure 13 Cusum chart for % volatile matter21 Figure 14 Cusum chart of ash-content data from Table 723 Figure 15 Cusum chart of data from blending exper
19、iment24 Figure 16 Conventional chart of data from blending experiment25 Figure 17 Cusum charts of resin deposition data27 Figure 18 Cusum chart for fabric faults (T = 7, = 6)28 Figure 19 Alternative presentation of decision lines29 Figure 20 Modified cusums, S1 and S230 Figure 21 Generalized cusum m
20、ask construction31 Figure 22 Cusum chart of ammunition firing trials (binary data)34 Figure 23 Weighted cusum chart for data of Table 1237 Figure 24 Weighted cusum chart of machine stoppages data40 Figure 25 Cusum charts for fuel use, 1 500 cc saloon car43 Figure 26 Full cusum V-mask44 Figure 27 Alt
21、ernative method of constructing full V-mask45 Figure 28 Full V-mask applied to cusum chart (T = 15, e = 2)45 Figure 29 Construction of local decision line47 Figure 30 Decision line constructed on cusum chart (T = 15, e = 2)47 Figure 31 Test for significance of change in cusum slope, based on decisio
22、n line48 Figure 32 Cusum chart for retrospective analysis of data in Table 2056 Figure 33 Typical span and decision line tests56 Licensed Copy: sheffieldun sheffieldun, na, Fri Dec 01 13:39:41 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 5703-2:1980 BSI 10-1999iii Page Table 1 Data for cusum plotti
23、ng2 Table 2 Average run length data for cusum and control charts12 Table 3 Effect of incorrect value of e on ARL14 Table 4 Data for construction of a semi-parabolic cusum mask15 Table 5 Batch % volatile matter21 Table 6 Average run length data for cusum scheme for % volatile matter22 Table 7 Data fo
24、r ash-content in coal22 Table 8 Percentage of soluble ingredient present in discharge from a blending machine25 Table 9 Span tests on cusum segments26 Table 10 Ad hoc significance tests on cusum segments26 Table 11 Cusum calculations for modified procedure32 Table 12 Data with varying sample sizes f
25、or weighted cusum37 Table 13 Machine stoppages data38 Table 14 Fuel consumption data, 1 500 cc saloon car39 Table 15 Further fuel consumption details42 Table 16 Percentage points for cusum score (+ 1, 1) about median (distribution-free test)50 Table 17 Scores and cumulatives of sequence 8 to 2451 Ta
26、ble 18 Scores and cumulatives of sequence 8 to 3151 Table 19 Differenceindicated as significant by various cusum tests (normally distributed variable)52 Table 20 Data and cusum calculations53 Table 21 Cusum span tests on data of Table 2055 Table 22 Tests of significance between mean levels of segmen
27、ts55 Table 23 Use of distribution-free cusum tests57 Publications referred toInside back cover x1x2 Licensed Copy: sheffieldun sheffieldun, na, Fri Dec 01 13:39:41 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 5703-2:1980 iv BSI 10-1999 Foreword It was originally intended that this British Standard
28、should form part of an overall revision of British Standards relating to statistical quality control techniques. However, cusum methods have found applications in fields beyond that of quality control in manufacturing (e.g. management, administration, medical science, marketing, commerce) and commen
29、ts on the draft standard revealed interest in a more general document. Part 1 of the standard is an attempt to meet this demand, and is intended as an introduction to the principles of cusum procedures and charting. Part 2 details decision rules, significance tests for retrospective analysis of accu
30、mulated data, and contains a variety of applications of cusum charting, some concerned mainly with data presentation, some with monitoring in quality or budgetary control, and others with retrospective analysis. Part 31) deals with applications of cusum methods to process and quality control by meas
31、urement. The techniques cover control of both average level and variability, and may be implemented using charts, tabulations or computers. They are especially appropriate when location of change points, and estimation of the corrections required to bring the process on to target are desirable featu
32、res of the control system. Part 41) deals with control of counts of defectives. Acknowledgement is made to the Department of Statistics and Computer Science, University College London, and Cambridge University Press for material reproduced, by permission, from Random Normal Deviates by H Wold publis
33、hed as Tracts for computers, No. XXV, 1954; to the authors of papers in the statistical journals whose work has been drawn upon; to the contributors of examples; to Mr P L Goldsmith for useful discussions; to ICI Fibres (Pontypool) for provision of computing facilities; and to Dr Derek Bissell (a re
34、presentative of the Institute of Statisticians) who was primarily responsible for the development work on this standard, carried out on behalf of BSI committee QMS/1/4. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsib
35、le for their correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pages i to iv, pages 1 to 58, an inside back cover and a back cover. This standard has been u
36、pdated (see copyright date) and may have had amendments incorporated. This will be indicated in the amendment table on the inside front cover. 1) In course of preparation. Licensed Copy: sheffieldun sheffieldun, na, Fri Dec 01 13:39:41 GMT+00:00 2006, Uncontrolled Copy, (c) BSI BS 5703-2:1980 BSI 10
37、-19991 0 Introduction 0.1 Basis of cusum chart. To avoid repetition of much of the material in Part 1, it is necessary to assume that the user of Part 2 is familiar with the basis of cusum charting as presented in Part 1. However, a brief recapitulation of the fundamentals is given before decision r
38、ules and statistical tests are detailed. The cumulative sum chart (hereafter referred to by the generally accepted contraction “cusum chart”) is a highly informative means of graphical presentation of data which can be ordered into a logical sequence. Frequently, though not necessarily, this sequenc
39、e corresponds to the order of observation on a time scale. However, examples are given where other, but equally logical, ordering is based on separation of the observations into groups having distinctive features. The essential principle is that a target value, T, is subtracted from each observation
40、. This target value is generally a constant but it may be a value that changes for each observation, e.g. a prediction from a forecasting model or a conditional target which may vary depending on the circumstances in which the observation is obtained. The cumulative sum of the deviations from target
41、 is formed, and this cusum (C) is plotted against the serial number of the observation. Unlike conventional plotting (where the average level of the values in the series corresponds to height above the serial number axis), the cusum method of plotting results in the representation of average by the
42、local slope of the chart. When the average corresponds to the target value, the path of the cusum lies roughly parallel to the sequence axis. When the local average of the series is greater than the target value (positive), the cusum slopes upwards (away from the sequence axis); conversely, when the
43、 local average is less than the target value (negative), the cusum slopes downwards. The greater the discrepancy between the local average and the target value, the steeper the slope of the cusum path. The result of plotting in the cusum mode is that changes in average level over different subdivisi
44、ons of the sequence of observations are clearly indicated by changes in slope of the chart. The local averages in such subdivisions can be readily calculated, either from the numerical values of the cusum (from which the chart is plotted) or directly, albeit with slight loss in accuracy in some case
45、s, from the chart itself. 0.2 Simple example of cusum chart. The above principles are best appreciated from a simple example. The calculations and plotting procedure will, at this point, be developed without mathematical symbolism; appropriate algebraic formulae were developed in Part 1. It is suppo
46、sed that the following data have been obtained, over a time sequence in the order shown, and that a target value of 15 is appropriate. For a conventional chart, as in Figure 1, the observed values are plotted against their corresponding observation numbers. There is some indication that the last doz
47、en values appear to be clustered around a different mean level from the first 20 or so. Plotting in the cusum mode gives a much clearer display than the conventional chart. The cusum (column 4 of Table 1) is plotted against the observation number, on conventional axes, using the y (“vertical”) axis
48、for the cusum and the x (“horizontal”) axis for the observation number, as in Figure 2. The chart clearly separates into three segments. From observations numbered 1 to 7 (inclusive) the cusum path is generally parallel to the observation number axis, i.e. the path, in conventional terms, is roughly
49、 horizontal. From observations 8 to 21 inclusive the path is recognizably downward (despite local irregularities such as at observations 14, 20). From observations 22 to 33, the path is upward (again with local irregularities that do not blur the overall pattern). Thus it could be inferred that a) observations 1 to 7 constitute a sample from a “population” whose mean is at or near the target value (15); b) observations 8 to 21 appear to have been sampled from a