ISO 7870-4:2021

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 2021
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ii
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
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
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
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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
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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.
v
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
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
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 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
Τ
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.
3.1.2
representative 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
e
is T − fσ , for monitoring a downward shift.
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
e
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 signal
Note 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 .
e
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
d factor for estimating the within-subgroup standard deviation from within-subgroup range
F observed reference shift
f 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
δ
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
R
σ process standard deviation
σ within-subgroup standard deviation
estimated within-subgroup standard deviation
σˆ
σ standard error
e
s observed within-subgroup standard deviation
s
average subgroup standard deviation
s realized standard error of the mean from m subgroups
x
T target value
τ true change point
τˆ
estimated change point
x individual result
x
arithmetic mean value (of a subgroup)
mean of subgroup means
x
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.
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;
e
or
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.
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
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.
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 — Tabular arrangement for calculating CUSUM values
from a sequence of individual values
(1) (2) (3) (4) (5) (6) (7) (8)
Motor no. Voltage Voltage −10 CUSUM Motor no. Voltage Voltage −10 CUSUM
1 9 −1 −1 21 3 −7 +11
2 16 +6 +5 22 9 −1 +10
3 11 +1 +6 23 7 −3 +7
4 12 +2 +8 24 14 +4 +11
5 16 +6 +14 25 2 −8 +3
6 7 −3 +11 26 6 −4 −1
7 13 +3 +14 27 4 −6 −7
8 12 +2 +16 28 12 +2 −5
9 13 +3 +19 29 8 −2 −7
10 11 +1 +20 30 8 −2 −9
11 12 +2 +22 31 12 +2 −7
12 8 −2 +20 32 6 −4 −11
13 8 −2 +18 33 14 +4 −7
14 11 +1 +19 34 13 +3 -4
15 14 +4 +23 35 12 +2 −2
16 8 −2 +21 36 14 +4 +2
17 6 −4 +17 37 13 +3 +5
18 14 +4 +21 38 10 0 +5
19 4 −6 +15 39 13 +3 +8
20 13 +3 +18 40 13 +3 +11
7 Fundamentals of making CUSUM-based decisions
7.1 Need for decision rules
Decision rules are needed to rationalize the interpretation of a CUSUM chart. When an appropriate
decision rule so indicates, some action is taken, depending on the nature of the process. Typical actions
are:
a) for in-process control: adjustment of process conditions;
b) in an improvement context: investigation of the underlying cause of the change; and
c) in a forecasting mode: analysis of and, if necessary, adjustment to the forecasting model or
its parameters.
7.2 Basis for making decisions
Establishing the base criteria against which decisions are to be made is obviously an essential
prerequisite.
To provide an effective basis for detecting a signal, a suitable quantitative measure of ‘noise’ in the
system is required. What represents noise, and what represents a signal, is determined by the
monitoring strategy adopted, such as how many observations to take, and how frequently, and how to
constitute a sample or a subgroup. Also, the measure used to quantify variation can affect the issue.
It is usual to measure inherent variation by means of a statistical measure termed either of the
following.
a) Standard deviation: where individual observations are the basis for plotting CUSUMs.
The individual observations for calculation of the standard deviation are often taken from a
homogeneous segment of the process data. This performance then becomes the more onerous
criterion from which to judge. Any variation greater than this inherent variation is taken to arise
from special causes indicating a shift in the mean of the series or a change in the natural magnitude
of the variability, or both.
b) Standard error: where some function of a subgroup of observations, such as mean, median or range,
forms the basis for CUSUM plotting.
The concept of subgrouping is that variation within a subgroup is made up of common causes
with all special causes of variation occurring between subgroups. The primary role of the CUSUM
chart is then to distinguish between common and special cause variation. Hence, the choice of
subgroup is of vital importance. For example, making up each subgroup of four consecutively from
a high-speed production process each hour, as opposed to one measurement taken every quarter
of an hour to make up a subgroup of four every hour, gives very different variabilities on which to
base a decision. The standard error would be minuscule in the first instance compared with the
second. One CUSUM chart would be set up with consecutive part variation as the basis for decision-
making as opposed to 15 min to 15 min variation for the other chart.
However, the prerequisite that stability should exist over a sufficient period to establish reliable
quantitative measures, such as standard deviation or standard error, is too restrictive for some
potential areas of application of the CUSUM method.
For instance, observations of a continuous process can exhibit small unimportant variations in the
average level. It is required that it is against these variations that systematic or sustained changes
should be judged. Illustrations are:
a) an industrial process is controlled by a thermostat or other automatic control device;
b) the quality of raw material input can be subject to minor variations without violating specification;
and
c) in monitoring a patient's response to treatment, there might be minor metabolic changes connected
with meals, hospital or domestic routine, etc., but any effect of treatment should be judged against
the overall typical variation.
On the other hand, samples can comprise output or observations from several sources (administrative
regions, plants, machines and operators). As such, there can be too much local variation to provide
a realistic basis for assessing whether the overall average shifts. Because of this factor, data arising
from a combination of sources should be treated with caution, as any local peculiarities within each
contributing source might be overlooked. Moreover, variation between the sources might mask any
changes occurring over the whole system as time progresses.
One of the important assumptions in CUSUM procedures is that the process standard deviation σ is
stable. Therefore, before constructing the CUSUM procedure, any process should be assessed to see if it
is in a state of statistical control (by using the R-chart, s-chart or moving range chart) so that a reliable
estimate of σ can be obtained.
Serial correlation between observations can also manifest itself — namely, one observation might
have some influence over the next. An illustration of negative serial correlation is the use of successive
gauge readings to estimate the use of a bulk material, where an overestimate on one occasion tends to
produce an underestimate on the next reading. Another example is where overordering in one month
is compensated by underordering in the subsequent month. Positive serial correlation is likely in some
industrial processes where one batch of material might partially mix with preceding and succeeding
batches.
7.3 Measuring the effectiveness of a decision rule
7.3.1 Basic concepts
The ideal performance of a decision rule is for real changes of at least a prespecified magnitude to be
detected immediately and for a process with no real changes to be allowed to continue indefinitely
without giving rise to false alarms. In real life, this is not attainable. A simple and convenient measure
of actual effectiveness of a decision rule is the average run length (ARL).
The ARL is the expected value of the number of samples (usually denoted as run length, RL) taken up to
that which gives rise to a decision that a real change is present.
If no real change is present, the ideal value of the ARL is infinity. A practical objective in such a situation
is to make the ARL large. Conversely, when a real change is present, the ideal value of the ARL is 1, in
which case the change is detected when the next sample is taken. The choice of the ARL is a compromise
between these two conflicting requirements. Making an incorrect decision to act when the process has
not changed gives rise to ‘overcontrol’. This, in effect, increases variability. Not taking appropriate action
when the process has changed gives rise to ‘under control’. This also, in effect, increases variability and
results in increasing the cost of production.
The actual RL itself is subject to statistical variation. Sometimes one can be fortunate in obtaining no
false alarms over a long run, or in detecting a change very quickly. Occasionally, an unfortunate run of
samples can generate false alarms or mask a real change so that it does not yield a signal. The actual
pattern of such variation deserves attention occasionally. Generally, however, the ARL is looked upon as
a reasonable measure of effectiveness of a decision rule. The aim is summarized in Table 2.
Table 2 — ARL patterns and process conditions
True process condition Required CUSUM response Ideal response
At or near target Long ARL (few false alarms) ARL → infinity
Substantial departure from target Short ARL (rapid detection) ARL is 1
7.3.2 Example of calculation of ARL
The ARL concept is not particular to CUSUMs. Take a standard Shewhart control chart with control
limits set at ± 3 standard deviations from the centreline. It is well-known that about 0,135 % of the
observations are expected, on average, to fall beyond each of these limits when the process average
is on the centreline or target value. This can readily be translated into an average run length, ARL, by
calculating 1/0,001 35 = 741. In other words, one expects, on average, to see a value beyond the upper
control limit only once in every 741 observation intervals.
Hence the need, in practice, to design a control system that ensures a high ARL when the process is
running at the target value.
When two-sided limits are considered, with the process mean still on target, the ARL is halved resulting
in 1/(0,001 35 + 0,001 35) = 370.
Suppose that the process mean shifts one standard deviation towards the upper control limit. The
expectation is then that roughly 2,28 % lie above the upper control limit. The ARL in respect of the
UCL then becomes 1/0,022 8 = 44 for this single-sided limit. In other words, on average, it takes 44
observation intervals to signal a shift in the mean of one standard deviation.
When two-sided limits are considered, only 0,003 2 % is expected below the LCL as the process mean is
four standard deviations away from the LCL. As 1/(0,000 032 + 0,022 8) does not materially affect the
ARL calculated for a single limit, for a one standard deviation shift in the mean, the ARL for a double-
sided limit is approximately the same as for a single-sided one, namely 44.
Summarizing:
— with the mean at the target value → ARL for a two-sided limit is half that of a single-sided limit;
— as the shift in the mean increases → ARL for a two-sided limit approaches that of a single-sided limit.
In practice, other signalling rules such as the addition of warning limits, runs above and below the
mean, and so on, secure more rapid detection of shift but at the expense of an increase in spurious
signals when the process is on target. The Shewhart chart is very attractive and popular because of
its extreme simplicity and its effectiveness in detecting isolated special causes which give rise to large
shifts.
However, it is recognized that it has an inherent limitation in signalling other than large shifts even if
they persist without seriously prejudicing the extent of false alarms. This indicates a role for quite a
different method to achieve a more rapid detection of shift while retaining long ARLs when on target.
The CUSUM method is well suited to this.
8 Types of CUSUM decision schemes
8.1 V-mask
Plotting the cumulated deviations from target T and applying a V-mask constitutes one way of executing
the CUSUM control chart.
8.1.1 Configuration and dimensions
A V-mask is illustrated in Figure 2. The apex O is placed in distance d from the last CUSUM value which
is placed consequently at A. The V-mask is then fully specified by either giving the distance H (= 5σ ,
e
for example) to the limbs or by setting the angle between the horizontal line and the upper limb.
Key
A recent CUSUM value
H decision interval
d distance between apex and recent CUSUM value
O apex of V-mask
Figure 2 — Configuration and dimensions of a V-mask
8.1.2 Application of the V-mask
The mask is used by placing point A (see Figure 2) on a selected plotted value on the CUSUM chart, with
the datum line aligned horizontally on the chart. In an ongoing control situation, this selected plotted
value is usually the most recent point.
If the path of the CUSUM lies within the sloping arms of the mask (or their extensions), no significant
shift in mean is indicated up to that plotted value. In a control situation, the process is then said to
be in a state of statistical control with respect to the target value. If, however, the path of the CUSUM
wanders outside the sloping arms of the mask, a significant departure from the target value is signalled.
In process management, the process is then said to be out-of-control.
Figure 3 illustrates an ‘in-control’ situation where no significant departure from the target value is
detected, and two ‘out-of-control’ situations, one where there is a significant decrease in value indicated
and the other where a significant increase is revealed. A standard deviation of 0,2 is used in the three
illustrations in Figure 3. The target value used to construct the CUSUM charts is equal to the target
mean for the process.
The current situation is determined by offering up the V-mask to the CUSUM chart progressively as
data points accumulate.
a) No significant change in process mean with respect to CUSUM target value
b) Significant decrease in process mean with respect to CUSUM target value
c) Significant increase in process mean with respect to CUSUM target value
Key
X observation value
Y1 CUSUM 1
Y2 CUSUM 2
Y3 CUSUM 3
Figure 3 — Illustrations of use of the V-mask to detect significant change in process mean
Although Figure 3 a) indicates a process mean less than the CUSUM target value, the V-mask does not
yet register this change as a significant departure.
Figure 3 b) indicates that the process mean is significantly less than the target value. While the
significant departure is not detected until observation 10, from a visual perspective the process mean
appears to be running low from as early as observation 1. By noting the slope of the line through the
observation points, an assessment of the actual mean of the process can be made. This provides both
a guide to the magnitude of the correction required to restore the process to its target value, and a
diagnostic indicator to pinpoint what happened at observation 1 to set the process on this low level in
the first place.
Figure 3 c) indicates that the process mean is significantly greater than the target value. This was
not registered as significant until observation 14. The process appeared to be running lower than the
target value until observation 6, but this was not sufficient to trigger an out-of-control condition. Then,
following observation 6, the level changed to a higher value than that targeted. By measuring the line
slope up to, and from, observation 6, together with its origins, both a corrective tool and a diagnostic
aid are provided.
When only an upper or lower specification lim
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