# ISO 7870-4:2021

(Main)## Control charts — Part 4: Cumulative sum charts

## Control charts — Part 4: Cumulative sum charts

This document describes 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 retrospective analysis.

## Cartes de contrôle — Partie 4: Cartes de contrôle à somme cumulée

### General Information

### Relations

### Standards Content (Sample)

INTERNATIONAL ISO

STANDARD 7870-4

Second edition

2021-09

Control charts —

Part 4:

Cumulative sum charts

Cartes de contrôle —

Partie 4: Cartes de contrôle à somme cumulée

Reference number

ISO 7870-4:2021(E)

© ISO 2021

---------------------- Page: 1 ----------------------

ISO 7870-4:2021(E)

COPYRIGHT PROTECTED DOCUMENT

© ISO 2021

All rights reserved. Unless otherwise specified, or required in the context of its implementation, 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

CP 401 • Ch. de Blandonnet 8

CH-1214 Vernier, Geneva

Phone: +41 22 749 01 11

Email: copyright@iso.org

Website: www.iso.org

Published in Switzerland

© ISO 2021 – All rights reserved

---------------------- Page: 2 ----------------------

ISO 7870-4:2021(E)

Contents Page

Foreword ..........................................................................................................................................................................................................................................v

Introduction .............................................................................................................................................................................................................................. vi

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ..................................................................................................................................................................................... 1

3 Terms and definitions, abbreviated terms and symbols .......................................................................................... 1

3.1 Terms and definitions ...................................................................................................................................................................... 1

3.2 Abbreviated terms .............................................................................................................................................................................. 2

3.3 Symbols ......................................................................................................................................................................................................... 2

4 Principal features of cumulative sum (CUSUM) charts .............................................................................................. 3

5 Basic steps in the construction of CUSUM charts — Graphical representation ...............................4

6 Example of a CUSUM plot — Motor voltages ........................................................................................................................... 5

6.1 Process ........................................................................................................................................................................................................... 5

6.2 Simple plot of results ........................................................................................................................................... .............................. 5

6.3 Standard control chart for individual results ............................................................................................................. 6

6.4 CUSUM chart construction .......................................................................................................................................................... 7

7 Fundamentals of making CUSUM-based decisions .......................................................................................................... 8

7.1 Need for decision rules .................................................................................................................................................................... 8

7.2 Basis for making decisions .......................................................................................................................................................... 8

7.3 Measuring the effectiveness of a decision rule .......................................................................................................... 9

7.3.1 Basic concepts ....................................................................................................................................................................... 9

7.3.2 Example of calculation of ARL ............................................................................................................................. 10

8 Types of CUSUM decision schemes .................................................................................................................................................10

8.1 V-mask ......................................................................................................................................................................................................... 10

8.1.1 Configuration and dimensions............................................................................................................................ 10

8.1.2 Application of the V-mask ........................................................................................................................................ 11

8.1.3 Average run lengths ..................................................................................................................................................... 14

8.1.4 General comments on average run lengths ............................................................................................. 15

8.2 Fast-initial response (FIR) CUSUM .................................................................................................................................... 16

8.3 Tabular CUSUM ................................................................................................................................................................................... 16

8.3.1 Rationale ................................................................................................................................................................................. 16

8.3.2 Deployment .......................................................................................................................................................................... 17

9 CUSUM methods for process and quality control ...........................................................................................................19

9.1 Nature of the changes to be detected .............................................................................................................................. 19

9.1.1 Size of the changes to be detected ................................................................................................................... 19

9.1.2 ‘Step’ changes ..................................................................................................................................................................... 19

9.1.3 Drifting ..................................................................................................................................................................................... 19

9.1.4 Cyclic .......................................................................................................................................................................................... 19

9.1.5 Hunting .................................................................................................................................................................................... 19

9.2 Selecting target values ................................................................................................................................................................. 19

9.2.1 General ..................................................................................................................................................................................... 19

9.2.2 Standard (given) value as target ....................................................................................................................... 20

9.2.3 Performance-based target ...................................................................................................................................... 20

9.3 CUSUM schemes for monitoring location ......... ............................................................................................................ 20

9.3.1 Standard schemes...........................................................................................................................................................20

9.3.2 Standard schemes — Limitations .................................................................................................................... 27

9.3.3 ‘Tailored’ CUSUM schemes ...................................................................................................................................... 27

9.4 CUSUM schemes for monitoring variation ..................................................................................................................28

9.4.1 General .....................................................................................................................................................................................28

9.4.2 CUSUM schemes for subgroup ranges ..........................................................................................................29

9.4.3 CUSUM schemes for subgroup standard deviations ........................................................................ 32

9.5 Special situations ..............................................................................................................................................................................36

iii© ISO 2021 – All rights reserved

---------------------- Page: 3 ----------------------

ISO 7870-4:2021(E)

9.5.1 Large between-subgroup variation ................................................................................................................36

9.5.2 ‘One-at-a-time’ data .......................................................................................................................................................36

9.5.3 Serial dependence between observations ................................................................................................36

9.5.4 Outliers..................................................................................................................................................................................... 37

9.6 CUSUM schemes for discrete data ......................................................................................................................................38

9.6.1 Event count — Poisson data ..................................................................................................................................38

9.6.2 Two classes data — Binomial data ..................................................................................................................40

Annex A (informative) Example of tabular CUSUM .......................................................................................................................... 44

Annex B (informative) Estimation of the change point when a step change occurs ......................................48

Bibliography .............................................................................................................................................................................................................................50

© ISO 2021 – All rights reserved---------------------- Page: 4 ----------------------

ISO 7870-4:2021(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 of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to

the World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see

www.iso.org/iso/foreword.html.This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,

Subcommittee SC 4, Applications of statistical methods in process management.This second edition of ISO 7870-4 cancels and replaces the first edition (ISO 7870-4: 2011), which has

been technical revised.The main changes compared to the previous edition are as follows:

— Manhattan diagram removed (former 6.7);

— V-mask types in Types of CUSUM decision schemes reduced to one V-mask;

— von Neumann method removed (former Annex A).

A list of all parts in the ISO 7870 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www.iso.org/members.html.© ISO 2021 – All rights reserved

---------------------- Page: 5 ----------------------

ISO 7870-4:2021(E)

Introduction

This document 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 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 measures (e.g. mean, range,

counts of events).The method has a rather unusual name, cumulative sum, or 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 cumulating (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.The CUSUM method is already used unwittingly by golfers throughout 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 unknown 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.

The intention of this document 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 detects a change 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 require constant adjustment.In addition to Shewhart charts, an EWMA (exponentially weighted moving average) chart can be used.

Each plotted point on an EWMA chart incorporates information from all 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 can be sensitized to detect any

size of shift in a process. This subject is discussed further in 7870-6.© ISO 2021 – All rights reserved

---------------------- Page: 6 ----------------------

INTERNATIONAL STANDARD ISO 7870-4:2021(E)

Control charts —

Part 4:

Cumulative sum charts

1 Scope

This document describes 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

retrospective analysis.2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements 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 and symbols — Part 1: General statistical terms and terms used in

probabilityISO 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 given in ISO 3534-1 and ISO 3534-2 and

the following apply.ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp— IEC Electropedia: available at https:// www .electropedia .org/

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 to entry: With a charted CUSUM, the deviations from the target value are cumulated.

Note 2 to entry: Using a ‘V’ mask, the target value is often referred to as the reference value or the nominal

control value. If so, it needs be acknowledged that it is not necessarily the most desirable or preferred value, as

can appear in other standards. It is simply a convenient target value for constructing a CUSUM chart.

3.1.2representative out of control value

〈tabulated CUSUM〉 value which controls the sensitivity of the procedure

Note 1 to entry: The upper out of control value is T + fσ , for monitoring an upward shift. The lower control value

is T − fσ , for monitoring a downward shift.© ISO 2021 – All rights reserved

---------------------- Page: 7 ----------------------

ISO 7870-4:2021(E)

3.1.3

reference shift

F, f

〈tabulated CUSUM〉 difference between the target value (3.1.1) and the representative out of control value

(3.1.2)Note 1 to entry: It is necessary to distinguish between f, that relates to a standardized reference shift, and F,

that relates to an observed reference shift; F = fσ . It plays a crucial role for constructing the tabular form of the

CUSUM chart.3.1.4

decision interval

H, h

〈tabulated CUSUM〉 cumulative sum of deviations from a representative out of control value (3.1.2)

required to yield a signalNote 1 to entry: It is necessary to distinguish between h, that relates to a standardized decision interval, and H,

that relates to an observed decision interval; H = hσ .3.1.5

average run length

ARL

average number of samples taken up to the point at which a signal occurs

Note 1 to entry: The average run length (ARL) is usually related to a process level, in which case it carries an

appropriate subscript as, for example, ARL , meaning the average run length 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

FIR fast initial response

LCL lower control limit

RL run length

SPC statistical process control

UCL upper control limit

3.3 Symbols

a scale coefficient

C CUSUM value

c factor for estimating the within-subgroup standard deviation

δ amount of change to be detected

Δ standardized amount of change to be detected

d lead distance

© ISO 2021 – All rights reserved

---------------------- Page: 8 ----------------------

ISO 7870-4:2021(E)

d factor for estimating the within-subgroup standard deviation from within-subgroup range

F observed reference shiftf standardized reference shift

K reference value, equal to sum of target T and observed reference shift F

H observed decision interval

h standardized decision interval

ARL average run length at δ shift

ARL average run length at no shift μ population mean value

n subgroup size

m number of subgroups within a preliminary study

p probability of ‘success’

mean subgroup range

σ process standard deviation

σ within-subgroup standard deviation

estimated within-subgroup standard deviation

σ standard error

s observed within-subgroup standard deviation

average subgroup standard deviation

s realized standard error of the mean from m subgroups

T target value

τ true change point

estimated change point

x individual result

arithmetic mean value (of a subgroup)

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 the 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.

© ISO 2021 – All rights reserved

---------------------- Page: 9 ----------------------

ISO 7870-4:2021(E)

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-target’ or at 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.Referring to point b) above, a CUSUM chart has the capacity to clearly indicate points of change; these

are clearly indicated by the change in gradient of the CUSUM plot. This has enormous benefit for process

management: to be able to quickly and accurately pinpoint the moment when a process altered so that

the appropriate corrective action can be taken.A further very useful feature of a CUSUM system is that it can be handled without plotting, i.e. in tabular

form. This is very helpful if the system is to be used to monitor a highly technical process, e.g. plastic

film manufacture, where the number of process parameters and product characteristics is large. Data

from such a process can be captured automatically, downloaded into CUSUM software to produce an

automated CUSUM analysis. A process manager can then be alerted to changes on many characteristics

on a simultaneous basis. Annex A contains an example of the method.5 Basic steps in the construction of CUSUM charts — Graphical representation

The following steps are used to set up a CUSUM chart for individual values.

Step 1 — Choose a reference, target, control or preferred value. The average of past results generally

provides good discrimination.Step 2 — Tabulate the results in a meaningful (e.g. chronological) sequence. Subtract the reference

value from each result.Step 3 — Progressively sum the values obtained in Step 2. These sums are then plotted as a CUSUM

chart.Step 4 — For reasonable discrimination, without undue sensitivity, the following options are

recommended:a) choose a convenient plotting interval for the horizontal axis and make the same interval on the

vertical axis equal to 2σ (or 2σ if a CUSUM of means is to be charted), rounding off as appropriate;

b) where it is required to detect a known change, say δ, choose a vertical scale such that the ratio of

the scale unit on the vertical scale divided by the scale unit on the horizontal scale is between δ and

2δ, rounding off as appropriate.NOTE The scale selection is visually very important since an inappropriate scale gives either the impression

of impending disaster due to the volatile nature of the plot, or a view that nothing is changing. The schemes

described in a) and b) above can give a scale that shows changes in a reasonable manner, neither too sensitive nor

too suppressed.© ISO 2021 – All rights reserved

---------------------- Page: 10 ----------------------

ISO 7870-4:2021(E)

6 Example of a CUSUM plot — Motor voltages

6.1 Process

Suppose a set of 40 values in chronological sequence is obtained of a characteristic. These happen to be

voltages, taken in order of production, on fractional horsepower motors at an early stage of production.

But they can be any individual values taken in a meaningful sequence and expressed on a continuous

scale. These are now shown:9, 16, 11, 12, 16, 7, 13, 12, 13, 11, 12, 8, 8, 11, 14, 8, 6, 14, 4, 13, 3, 9, 7, 14, 2, 6, 4, 12, 8, 8, 12, 6, 14, 13, 12,

14, 13, 10, 13, 13.The reference or target voltage value is 10 V.

6.2 Simple plot of results

To gain a better understanding of the underlying behaviour of the process, by determining patterns and

trends, a standard approach is simply to plot these values in their natural order as shown in Figure 1 a).

Apart from indicating a general drop away in the middle portion from a high start and with an equally

high finish, Figure 1 a) is not very revealing because of the extremely noisy, or spiky, data throughout.

a) Simple plot of motor voltages© ISO 2021 – All rights reserved

---------------------- Page: 11 ----------------------

ISO 7870-4:2021(E)

b) Standard control chart for individuals

c) CUSUM chart

Key

X motor number

Y1 voltage

Y2 cumulative sum

Figure 1 — Motor voltage example

6.3 Standard control chart for individual results

The next level of sophistication is to establish a standard control chart for individuals as in Figure 1 b).

Figure 1 b) is even less revealing than the previous figure. It is, in fact, quite misleading. The standard

statistical process control criterion to test for process stability and control is just “no points lying above

the upper control limit (UCL) or below the lower control limit (LCL)”. All points reside within these

limits. Hence, one can be led to the conclusion that this is a stable process, one that is ‘in control’ around

its overall average value of about 10 V, which is the target value. Further standard analysis would

reveal that although the process is stable, it is not capable of meeting the specification requirements.

© ISO 2021 – All rights reserved---------------------- Page: 12 ----------------------

ISO 7870-4:2021(E)

However, this analysis would not in itself provide any further clues as to why it is incapable of meeting

the requirements.The reason for the inability of the standard control chart for individuals to be of value is that the

control limits are based on actual process performance and not on desired or specified requirements.

Consequently, if the process naturally exhibits a large variation the control limits are correspondingly

wide. What is required is a method that is better at indicating patterns and trends, or even pinpointing

points of change, to help determine and remove primary sources of variation.NOTE By using additional tools, such as an individual and moving range chart, the practitioner can study

other process variation issues.6.4 CUSUM chart construction

The construction of a CUSUM chart using individual values, as in this example, is based on the very

simple steps given in Clause 5.Step 1 — Choose a target value, T. The preferred or reference value is given as 10 V.

Step 2 — Tabulate the results (voltages) in production sequence against motor number as in Table 1,

column 2 (and 6). Subtract the reference value of 10 from each result as in Table 1, column 3 (and 7).

Step 3 — Progressively sum the values of Table 1, column 3 (and 7) in column 4 (and 8). Plot column 4

(and 8) against the observation (motor) number as in Figure 1 c), taking note of the scale comments in

Steps 4 and 5.Table 1 — Tabu

**...**

FINAL

INTERNATIONAL ISO/FDIS

DRAFT

STANDARD 7870-4

ISO/TC 69/SC 4

Control charts —

Secretariat: DIN

Voting begins on:

Part 4:

2021-06-18

Cumulative sum charts

Voting terminates on:

2021-08-13

Cartes de contrôle —

Partie 4: Cartes de contrôle de l'ajustement de processus

RECIPIENTS OF THIS DRAFT ARE INVITED TO

SUBMIT, WITH THEIR COMMENTS, NOTIFICATION

OF ANY RELEVANT PATENT RIGHTS OF WHICH

THEY ARE AWARE AND TO PROVIDE SUPPOR TING

DOCUMENTATION.

IN ADDITION TO THEIR EVALUATION AS

Reference number

BEING ACCEPTABLE FOR INDUSTRIAL, TECHNO-

ISO/FDIS 7870-4:2021(E)

LOGICAL, COMMERCIAL AND USER PURPOSES,

DRAFT INTERNATIONAL STANDARDS MAY ON

OCCASION HAVE TO BE CONSIDERED IN THE

LIGHT OF THEIR POTENTIAL TO BECOME STAN-

DARDS TO WHICH REFERENCE MAY BE MADE IN

NATIONAL REGULATIONS. ISO 2021

---------------------- Page: 1 ----------------------

ISO/FDIS 7870-4:2021(E)

COPYRIGHT PROTECTED DOCUMENT

© ISO 2021

All rights reserved. Unless otherwise specified, or required in the context of its implementation, 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

CP 401 • Ch. de Blandonnet 8

CH-1214 Vernier, Geneva

Phone: +41 22 749 01 11

Email: copyright@iso.org

Website: www.iso.org

Published in Switzerland

ii © ISO 2021 – All rights reserved

---------------------- Page: 2 ----------------------

ISO/FDIS 7870-4:2021(E)

Contents Page

Foreword ..........................................................................................................................................................................................................................................v

Introduction ................................................................................................................................................................................................................................vi

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions, abbreviated terms and symbols ............................................................................................ 1

3.1 Terms and definitions ....................................................................................................................................................................... 1

3.2 Abbreviated terms ............................................................................................................................................................................... 2

3.3 Symbols ......................................................................................................................................................................................................... 2

4 Principal features of cumulative sum (CUSUM) charts ................................................................................................. 3

5 Basic steps in the construction of CUSUM charts — Graphical representation ..................................4

6 Example of a CUSUM plot — Motor voltages ............................................................................................................................ 5

6.1 Process ........................................................................................................................................................................................................... 5

6.2 Simple plot of results ......................................................................................................................................................................... 5

6.3 Standard control chart for individual results ............................................................................................................... 6

6.4 CUSUM chart construction ............................................................................................................................................................ 7

7 Fundamentals of making CUSUM-based decisions ........................................................................................................... 8

7.1 Need for decision rules .................................................................................................................................................................... 8

7.2 Basis for making decisions ........................................................................................................................................................... 8

7.3 Measuring the effectiveness of a decision rule ............................................................................................................ 9

7.3.1 Basic concepts .................................................................................................................................................................... 9

7.3.2 Example of calculation of ARL ...........................................................................................................................10

8 Types of CUSUM decision schemes .................................................................................................................................................10

8.1 V-mask .........................................................................................................................................................................................................10

8.1.1 Configuration and dimensions ..........................................................................................................................10

8.1.2 Application of the V-mask .....................................................................................................................................11

8.1.3 Average run lengths ...................................................................................................................................................14

8.1.4 General comments on average run lengths ...........................................................................................15

8.2 Fast-initial response (FIR) CUSUM .....................................................................................................................................16

8.3 Tabular CUSUM ....................................................................................................................................................................................16

8.3.1 Rationale ..............................................................................................................................................................................16

8.3.2 Deployment .......................................................................................................................................................................17

9 CUSUM methods for process and quality control ............................................................................................................19

9.1 Nature of the changes to be detected ...............................................................................................................................19

9.1.1 Size of the changes to be detected .................................................................................................................19

9.1.2 ‘Step’ changes ...................................................................................................................................................................19

9.1.3 Drifting ..................................................................................................................................................................................19

9.1.4 Cyclic........................................................................................................................................................................................19

9.1.5 Hunting..................................................................................................................................................................................19

9.2 Selecting target values ...................................................................................................................................................................19

9.2.1 General...................................................................................................................................................................................19

9.2.2 Standard (given) value as target .....................................................................................................................20

9.2.3 Performance-based target ....................................................................................................................................20

9.3 CUSUM schemes for monitoring location .....................................................................................................................20

9.3.1 Standard schemes ........................................................................................................................................................20

9.3.2 Standard schemes — Limitations ..................................................................................................................27

9.3.3 ‘Tailored’ CUSUM schemes ...................................................................................................................................27

9.4 CUSUM schemes for monitoring variation ...................................................................................................................28

9.4.1 General...................................................................................................................................................................................28

9.4.2 CUSUM schemes for subgroup ranges ........................................................................................................29

9.4.3 CUSUM schemes for subgroup standard deviations ......................................................................32

9.5 Special situations ...............................................................................................................................................................................36

© ISO 2021 – All rights reserved iii---------------------- Page: 3 ----------------------

ISO/FDIS 7870-4:2021(E)

9.5.1 Large between-subgroup variation ..............................................................................................................36

9.5.2 ‘One-at-a-time’ data ....................................................................................................................................................36

9.5.3 Serial dependence between observations ..............................................................................................36

9.5.4 Outliers ..................................................................................................................................................................................37

9.6 CUSUM schemes for discrete data .......................................................................................................................................38

9.6.1 Event count — Poisson data ...............................................................................................................................38

9.6.2 Two classes data — Binomial data ................................................................................................................40

Annex A (informative) Example of tabular CUSUM .............................................................................................................................44

Annex B (informative) Estimation of the change point when a step change occurs ........................................48

Bibliography .............................................................................................................................................................................................................................50

iv © ISO 2021 – All rights reserved---------------------- Page: 4 ----------------------

ISO/FDIS 7870-4:2021(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 of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/

iso/ foreword .html.This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,

Subcommittee SC 4, Applications of statistical methods in process management.This second edition of ISO 7870-4 cancels and replaces the first edition (ISO 7870-4: 2011), which has

been technical revised.The main changes compared to the previous edition are as follows:

— Manhattan diagram removed (former 6.7);

— V-mask types in Types of CUSUM decision schemes reduced to one V-mask;

— von Neumann method removed (former Annex A).

A list of all parts in the ISO 7870 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/ members .html.© ISO 2021 – All rights reserved v

---------------------- Page: 5 ----------------------

ISO/FDIS 7870-4:2021(E)

Introduction

This document 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 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 measures (e.g. mean, range,

counts of events).The method has a rather unusual name, cumulative sum, or 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 cumulating (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.The CUSUM method is already used unwittingly by golfers throughout 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 unknown 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.

The intention of this document 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 detects a change 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 require constant adjustment.In addition to Shewhart charts, an EWMA (exponentially weighted moving average) chart can be used.

Each plotted point on an EWMA chart incorporates information from all 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 can be sensitized to detect any

size of shift in a process. This subject is discussed further in 7870-6.vi © ISO 2021 – All rights reserved

---------------------- Page: 6 ----------------------

FINAL DRAFT INTERNATIONAL STANDARD ISO/FDIS 7870-4:2021(E)

Control charts —

Part 4:

Cumulative sum charts

1 Scope

This document describes 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

retrospective analysis.2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements 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 and symbols — Part 1: General statistical terms and terms used in

probabilityISO 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 given in ISO 3534-1 and ISO 3534-2 and

the following apply.ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp— IEC Electropedia: available at https:// www .electropedia .org/

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 to entry: With a charted CUSUM, the deviations from the target value are cumulated.

Note 2 to entry: Using a ‘V’ mask, the target value is often referred to as the reference value or the nominal

control value. If so, it needs be acknowledged that it is not necessarily the most desirable or preferred value, as

can appear in other standards. It is simply a convenient target value for constructing a CUSUM chart.

3.1.2representative out of control value

〈tabulated CUSUM〉 value which controls the sensitivity of the procedure

Note 1 to entry: The upper out of control value is T + fσ , for monitoring an upward shift. The lower control value

is T − fσ , for monitoring a downward shift.© ISO 2021 – All rights reserved 1

---------------------- Page: 7 ----------------------

ISO/FDIS 7870-4:2021(E)

3.1.3

reference shift

F, f

〈tabulated CUSUM〉 difference between the target value (3.1.1) and the representative out of control value

(3.1.2)Note 1 to entry: It is necessary to distinguish between f, that relates to a standardized reference shift, and F,

that relates to an observed reference shift; F = fσ . It plays a crucial role for constructing the tabular form of the

CUSUM chart.3.1.4

decision interval

H, h

〈tabulated CUSUM〉 cumulative sum of deviations from a representative out of control value (3.1.2)

required to yield a signalNote 1 to entry: It is necessary to distinguish between h, that relates to a standardized decision interval, and H,

that relates to an observed decision interval; H = hσ .3.1.5

average run length

ARL

average number of samples taken up to the point at which a signal occurs

Note 1 to entry: The average run length (ARL) is usually related to a process level, in which case it carries an

appropriate subscript as, for example, ARL , meaning the average run length 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

FIR fast initial response

LCL lower control limit

RL run length

SPC statistical process control

UCL upper control limit

3.3 Symbols

a scale coefficient

C CUSUM value

c factor for estimating the within-subgroup standard deviation

δ amount of change to be detected

Δ standardized amount of change to be detected

d lead distance

2 © ISO 2021 – All rights reserved

---------------------- Page: 8 ----------------------

ISO/FDIS 7870-4:2021(E)

d factor for estimating the within-subgroup standard deviation from within-subgroup range

F observed reference shiftf standardized reference shift

K reference value, equal to sum of target T and observed reference shift F

H observed decision interval

h standardized decision interval

ARL average run length at δ shift

ARL average run length at no shift μ population mean value

n subgroup size

m number of subgroups within a preliminary study

p probability of ‘success’

mean subgroup range

σ process standard deviation

σ within-subgroup standard deviation

estimated within-subgroup standard deviation

σ standard error

s observed within-subgroup standard deviation

average subgroup standard deviation

s realized standard error of the mean from m subgroups

T target value

τ true change point

estimated change point

x individual result

arithmetic mean value (of a subgroup)

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 the 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.

© ISO 2021 – All rights reserved 3

---------------------- Page: 9 ----------------------

ISO/FDIS 7870-4:2021(E)

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-target’ or at 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.Referring to point b) above, a CUSUM chart has the capacity to clearly indicate points of change; these

are clearly indicated by the change in gradient of the CUSUM plot. This has enormous benefit for process

management: to be able to quickly and accurately pinpoint the moment when a process altered so that

the appropriate corrective action can be taken.A further very useful feature of a CUSUM system is that it can be handled without plotting, i.e. in tabular

form. This is very helpful if the system is to be used to monitor a highly technical process, e.g. plastic

film manufacture, where the number of process parameters and product characteristics is large. Data

from such a process can be captured automatically, downloaded into CUSUM software to produce an

automated CUSUM analysis. A process manager can then be alerted to changes on many characteristics

on a simultaneous basis. Annex A contains an example of the method.5 Basic steps in the construction of CUSUM charts — Graphical representation

The following steps are used to set up a CUSUM chart for individual values.

Step 1 — Choose a reference, target, control or preferred value. The average of past results generally

provides good discrimination.Step 2 — Tabulate the results in a meaningful (e.g. chronological) sequence. Subtract the reference

value from each result.Step 3 — Progressively sum the values obtained in Step 2. These sums are then plotted as a CUSUM

chart.Step 4 — For reasonable discrimination, without undue sensitivity, the following options are

recommended:a) choose a convenient plotting interval for the horizontal axis and make the same interval on the

vertical axis equal to 2σ (or 2σ if a CUSUM of means is to be charted), rounding off as appropriate;

b) where it is required to detect a known change, say δ, choose a vertical scale such that the ratio of

the scale unit on the vertical scale divided by the scale unit on the horizontal scale is between δ and

2δ, rounding off as appropriate.NOTE The scale selection is visually very important since an inappropriate scale gives either the impression

of impending disaster due to the volatile nature of the plot, or a view that nothing is changing. The schemes

described in a) and b) above can give a scale that shows changes in a reasonable manner, neither too sensitive nor

too suppressed.4 © ISO 2021 – All rights reserved

---------------------- Page: 10 ----------------------

ISO/FDIS 7870-4:2021(E)

6 Example of a CUSUM plot — Motor voltages

6.1 Process

Suppose a set of 40 values in chronological sequence is obtained of a characteristic. These happen to be

voltages, taken in order of production, on fractional horsepower motors at an early stage of production.

But they can be any individual values taken in a meaningful sequence and expressed on a continuous

scale. These are now shown:9, 16, 11, 12, 16, 7, 13, 12, 13, 11, 12, 8, 8, 11, 14, 8, 6, 14, 4, 13, 3, 9, 7, 14, 2, 6, 4, 12, 8, 8, 12, 6, 14, 13, 12,

14, 13, 10, 13, 13.The reference or target voltage value is 10 V.

6.2 Simple plot of results

To gain a better understanding of the underlying behaviour of the process, by determining patterns and

trends, a standard approach is simply to plot these values in their natural order as shown in Figure 1 a).

Apart from indicating a general drop away in the middle portion from a high start and with an equally

high finish, Figure 1 a) is not very revealing because of the extremely noisy, or spiky, data throughout.

a) Simple plot of motor voltages© ISO 2021 – All rights reserved 5

---------------------- Page: 11 ----------------------

ISO/FDIS 7870-4:2021(E)

b) Standard control chart for individuals

c) CUSUM chart

Key

X motor number

Y1 voltage

Y2 cumulative sum

Figure 1 — Motor voltage example

6.3 Standard control chart for individual results

The next level of sophistication is to establish a standard control chart for individuals as in Figure 1 b).

Figure 1 b) is even less revealing than the previous figure. It is, in fact, quite misleading. The standard

statistical process control criterion to test for process stability and control is just “no points lying above

the upper control limit (UCL) or below the lower control limit (LCL)”. All points reside within these

limits. Hence, one can be led to the conclusion that this is a stable process, one that is ‘in control’ around

its overall average value of about 10 V, which is the target value. Further standard analysis would

reveal that although the process is stable, it is not capable of meeting the specification requirements.

6 © ISO 2021 – All rights reserved---------------------- Page: 12 ----------------------

ISO/FDIS 7870-4:2021(E)

However, this analysis would not in itself provide any further clues as to why it is incapable of meeting

the requirements.The reason for the inability of the standard control chart for individuals to be of value is that the

control limits are based on actual process performance and not on desired or specified requirements.

Consequently, if the process naturally exhibits a large variation the control limits are correspondingly

wide. What is required is a method that is better at indicating patterns and trends, or even pinpointing

points of change, to help determine and remove primary sources of variation.NOTE By using additional tools, such as an individual and moving range chart, the practitioner can study

other process variation issues.6.4 CUSUM chart construction

The construction of a CUSUM cha

**...**

## Questions, Comments and Discussion

## Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.