# ISO 7870-6-2016

© ISO 2016 Control charts — Part 6: EWMA control charts Cartes de contrôle — Partie 6: Cartes de contrôle de EWMA INTERNATIONAL STANDARD ISO 7870-6 First edition 2016-02-15 Reference number ISO 7870-6:2016(E) ISO 7870-6:2016(E)ii © ISO 2016 – All rights reserved COPYRIGHT PROTECTED DOCUMENT © ISO 2016, Published in Switzerland 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 Ch. de Blandonnet 8 • CP 401 CH-1214 Vernier, Geneva, Switzerland Tel. +41 22 749 01 11 Fax +41 22 749 09 47 copyright@iso.org www.iso.org ISO 7870-6:2016(E)Foreword iv Introduction v 1 Scope . 1 2 Normative references 1 3 Symbols and abbreviated terms . 1 4 EWMA for inspection by variables . 2 4.1 General . 2 4.2 Weighted average explained 3 4.3 Control limits for EWMA control chart 4 4.4 Construction of EWMA control chart 4 4.5 Example . 6 5 Choice of the control chart . 9 5.1 Shewhart control chart versus EWMA control chart . 9 5.2 Average run length .10 5.3 Choice of parameters for EWMA control chart 10 5.3.1 Choice of λ . 10 5.3.2 Choice of L z11 5.3.3 Calculation for n. 11 5.3.4 Example 12 6 Procedure for implementing the EWMA control chart 12 7 Sensitivity of the EWMA to non-normality .13 8 Advantages and limitations 13 8.1 Advantages 13 8.2 Limitations .13 Annex A (informative) Application of the EWMA control chart 14 Annex B (normative) EWMA control chart for controlling a proportion of nonconforming units .18 Annex C (normative) EWMA control charts for a number of nonconformities .20 Annex D (informative) Control chart effectiveness 22 Bibliography .26 © ISO 2016 – All rights reserved iii Contents Page ISO 7870-6:2016(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 (see 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 (see 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, Subcommittee SC 4, Applications of statistical methods in process management. ISO 7870 consists of the following parts, under the general title Control charts: — Part 1: General guidelines — Part 2: Shewhart control charts — Part 3: Acceptance control charts — Part 4: Cumulative sum charts — Part 5: Specialized control charts — Part 6: EWMA control charts A future part on charting techniques for short runs and small mixed batches is planned.iv © ISO 2016 – All rights reserved ISO 7870-6:2016(E) Introduction Shewhart control charts are the most widespread statistical control methods used for controlling a process, but they are slow in signalling shifts of small magnitude in the process parameters. The exponentially weighted moving average [10](EWMA) control chart makes possible faster detection of small to moderate shifts. The Shewhart control chart is simple to implement and it rapidly detects shifts of major magnitude. However, it is fairly ineffective for detecting shifts of small or moderate magnitude. It happens quite often that the shift of the process is slow and progressive (in case of continuous processes in particular); this shift has to be detected very early in order to react before the process deviates seriously from its target value. There are two possibilities for improving the effectiveness of the Shewhart control charts with respect to small and moderate shifts. — The simplest, but not the most economical possibility is to increase the subgroup size. This may not always be possible due to low production rate; time consuming or too costly testing. As a result, it may not be possible to draw samples of size more than 1 or 2. — The second possibility is to take into account the results preceding the control under way in order to try to detect the existence of a shift in the production process. The Shewhart control chart takes into account only the information contained in the last sample observation and it ignores any information given by the entire sequence of points. This feature makes the Shewhart control chart relatively insensitive to small process shifts. Its effectiveness may be improved by taking into account the former results. Where it is desired to detect slow, progressive shifts, it is preferable to use specific charts which take into account the past data and which are effective with a moderate control cost. Two very effective alternatives to the Shewhart control chart in such situations are a) Cumulative Sum (CUSUM) control chart. This chart is described in ISO 7870-4. The CUSUM control chart reacts more sensitively than the X-bar chart to a shift of the mean value in the range of half or two sigma. If one plots the cumulative sum of deviations of successive sample means from a specified target, even minor, permanent shifts in the process mean will eventually lead to a sizable cumulative sum of deviations. Thus, this chart is particularly well-suited for detecting such small permanent shifts that may go undetected when using the X-bar chart. b) Exponentially Weighted Moving Average (EWMA) control chart which is covered by this document. This chart is presented like the Shewhart control chart; however, instead of placing on the chart the successive averages of the samples, one monitors a weighted average of the current average and of the previous averages. EWMA control charts are generally used for detecting small shifts in the process mean. They will detect shifts of half sigma to two sigma much faster. They are, however, slower in detecting large shifts in the process mean. EWMA control charts may also be preferred when the subgroups are of size n = 1. The joint use of an EWMA control chart with a small value of lambda and a Shewhart control chart has been recommended as a means of guaranteeing fast detection of both small and large shifts. The EWMA control chart monitors only the process mean; monitoring the process variability requires the use of some other technique.© ISO 2016 – All rights reserved v Control charts — Part 6: EWMA control charts 1 Scope This International Standard covers EWMA control charts as a statistical process control technique to detect small shifts in the process mean. It makes possible the faster detection of small to moderate shifts in the process average. In this chart, the process average is evaluated in terms of exponentially weighted moving average of all prior sample means. EWMA weights samples in geometrically decreasing order so that the most recent samples are weighted most highly while the most distant samples contribute very little depending upon the smoothing parameter (λ). NOTE 1 The basic objective is the same as that of the Shewhart control chart described in ISO 7870-2. The Shewhart control chart’s application is worthwhile in the rare situations when — production rate is slow, — sampling and inspection procedure is complex and time consuming, — testing is expensive, and — it involves safety risks. NOTE 2 Variables control charts can be constructed for individual observations taken from the production line, rather than samples of observations. This is sometimes necessary when testing samples of multiple observations would be too expensive, inconvenient, or impossible. For example, the number of customer complaints or product returns may only be available on a monthly basis; yet, one would like to chart those numbers to detect quality problems. Another common application of these charts occurs in cases when automated testing devices inspect every single unit that is produced. In that case, one is often primarily interested in detecting small shifts in the product quality (for example, gradual deterioration of quality due to machine wear). 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 7870-1, Control charts — Part 1: General guidelines ISO 7870-2, Control charts — Part 2: Shewhart control charts ISO 7870-4, Control charts — Part 4: Cumulative sum charts 3 Symbols and abbreviated terms μ 0 Target value for the average of the process U μ , L μ Upper rejectable value of the average, lower rejectable value of the average Mean of the sample i N Number of units in a sample (sample size) INTERNATIONAL ST ANDARD ISO 7870-6:2016(E) © ISO 2016 – All rights reserved 1 ISO 7870-6:2016(E) z i EWMA value placed on the control chart z 0 Initial value of z i λ Value of the smoothing parameter L z Parameter used to establish the control limit for z i(expressed in number of standard deviations of z) s Estimator of the standard deviation σ σ True standard deviation of the distribution of x σ 0 True standard deviation of binomial distribution for P = p 0 σ x Standard deviation of the averages of n individual observations; σσ = / σ z Standard deviation of z iwhen i tends towards infinity δ Drift related to the average expressed in number of standard deviations δ 1 Maximum acceptable drift of the average, expressed in number of standard deviations p Proportion of nonconforming units of the process p 0 Target value for the proportion of nonconforming units of the process p 1 Upper refusable value of the proportion of nonconforming units p i Proportion of nonconforming units in the ith sample c Average number of nonconformities c 0 Target value for the average number of nonconformities c 1 Refusable average of nonconformities c i Number of nonconforming units in the ith sample U CL Upper control limit value for the EWMA control chart L CL Lower control limit value for the EWMA control chart. If L CLis negative, then it is taken as zero ARL Average Run Length ARL 0 Average Run Length of the process in control ARL 1 Average Run Length of the process with setting drift CL Centre line of the control limit MAXRL Maximum Run Length (5 % overrun probability), expressed as an integer 4 EWMA for inspection by variables 4.1 General An EWMA control chart plots geometric moving averages of past and current data in which the values being averaged are assigned weights that decrease exponentially from the present into the past. Consequently, the average values are influenced more by recent process performance. The exponentially weighted moving average is defined as Formula (1): z i= λx i+ (1 - λ) z i-1(1) NOTE 1 When the EWMA control chart is used with rational subgroups of size n 1 then x ii s s i m p l y replaced with . Where 0 λ 1 is a constant and the starting value (required with the first sample at i = 1) is the process target, so that z 0= μ 0 . NOTE 2 μ 0can be estimated by the average of preliminary data. The EWMA control chart becomes an X chart for λ = 1.2 © ISO 2016 – All rights reserved ISO 7870-6:2016(E) 4.2 Weighted average explained To demonstrate that the EWMA is a weighted average of all previous sample means, the right-hand side of Formula (1) in 4.1 can be substituted with z i-1to obtain Formula (2): λλ λλ λλλλ (2) Continuing to substitute recursively for z i-j , where j = 2, 3, ., we obtain Formula (3): λλ λ (3) For i = 1, z 1= λx 1+ (1 – λ)μ 0 . The weights, λ(1 – λ) j , decrease geometrically with the age of the sample mean. Furthermore, the weights sum to unity, since λλ λ λ λ λ (4) If λ = 0,2, then the weight assigned to the current sample mean is 0,2 and the weights given to the preceding means are 0,16; 0,128; 0,102 4 and so forth. These weights are shown in Figure 1. Because these weights decline geometrically, the EWMA is sometimes called a geometric moving average (GMA). Key X age of sample mean (EWMA λ = 0,2) Y weights λ(1-λ) j Figure 1 — Weights of past sample means Since the EWMA value can be viewed as a weighted average of all past and current observations, it is very insensitive to the normality assumption. It is, therefore an ideal control chart to use with individual observations.© ISO 2016 – All rights reserved 3 ISO 7870-6:2016(E) 4.3 Control limits for EWMA control chart If the observations x iare independent random variables with variance σ 2 , then the variance of z iis represented by Formula (5): (5) Therefore, the EWMA control chart would be constructed by plotting z iversus the sample number i (or time). The centre line and control limits for the EWMA control chart are as follows: Centre line = μ 0 (6) (7) The factor L zis the width of the control limits and its value depends upon the confidence level. In the case of X - R charts, 3σ limits are plotted for 99,73 % (±3σ) confidence. Similarly, on EWMA control chart, this confidence level