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International
Standard
ISO 7870-6
Second edition
Control charts —
2024-07
Part 6:
EWMA control charts for the
process mean
Cartes de contrôle —
Partie 6: Cartes de contrôle EWMA pour la moyenne d'un
processus
Reference number
© ISO 2024
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ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms. 2
5 EWMA for inspection by variables . 3
5.1 General .3
5.2 Weighted average explained .3
5.3 Control limits for EWMA control chart .4
5.4 Construction of EWMA control chart .5
5.5 Example .7
6 Choice of the control chart . 10
6.1 Shewhart control chart versus EWMA control chart .10
6.2 Average run length.11
6.3 Choice of parameters for EWMA control chart .11
6.3.1 Choice of λ .11
6.3.2 Choice of L . . . 12
z
6.3.3 Calculation for n . 12
6.3.4 Example . 13
7 Procedure for implementing the EWMA control chart .13
8 Sensitivity of the EWMA to non-normality . 14
9 Advantages and limitations . 14
9.1 Advantages . .14
9.2 Limitations .14
Annex A (informative) Application of the EWMA control chart .16
Annex B (informative) EWMA control chart for a proportion of nonconforming items .20
Annex C (informative) EWMA control charts for a number of nonconformities .22
Bibliography .24
iii
Foreword
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This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 4, Applications of statistical methods in product and process management.
This second edition cancels and replaces the first edition (ISO 7870-6:2016), which has been technically
revised.
A list of all parts in the ISO 7870 series can be found on the ISO website.
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iv
Introduction
Shewhart control charts are the most widespread statistical control methods used for controlling a process,
but they are slow in signalling shifts of small magnitude in the process parameters. The exponentially weighted
[13]
moving average (EWMA) control chart makes possible faster detection of small to moderate shifts.
The Shewhart control chart is simple to implement and it rapidly detects shifts of major magnitude. However,
it is fairly ineffective for detecting shifts of small or moderate magnitude. It happens quite often that the
shift of the process is slow and progressive (in case of continuous processes in particular); this shift has to
be detected very early in order to react before the process deviates seriously from its target value. There are
two possibilities for improving the effectiveness of the Shewhart control charts with respect to small and
moderate shifts.
— The simplest, but not the most economical possibility is to increase the subgroup size. This may not
always be possible due to low production rate; time consuming or too costly testing. As a result, it may
not be possible to draw samples of size more than 1.
— The second possibility is to take into account the results preceding the control under way in order to try
to detect the existence of a shift in the production process. The Shewhart control chart takes into account
only the information contained in the last sample observation and it ignores any information given by
the entire sequence of points. This feature makes the Shewhart control chart relatively insensitive to
small process shifts. Its effectiveness can be improved by taking into account the former results.
Where it is desired to detect slow, progressive shifts, it is preferable to use specific charts which take into
account the past data and which are effective with a moderate control cost. Two very effective alternatives
to the Shewhart control chart in such situations are
a) Cumulative sum (CUSUM) control chart. This chart is described in ISO 7870-4. The CUSUM control chart
reacts more sensitively than the X-bar chart to a shift of the mean value in the range of half to two
sigma. If one plots the cumulative sum of deviations of successive averages from a specified target,
even minor, permanent shifts in the process mean will eventually lead to a sizable cumulative sum of
deviations. Thus, this chart is particularly well-suited for detecting such small permanent shifts that
may go undetected when using the X-bar chart.
b) Exponentially weighted moving average (EWMA) control chart which is covered by this document.
This chart is presented like the Shewhart control chart; however, instead of placing on the chart the
successive averages of the samples, one monitors a weighted average of the current average and of the
previous averages.
EWMA control charts are generally used for detecting small shifts in the process mean. They will detect
shifts of half sigma to two sigma much faster. They are, however, slower in detecting large shifts in the
process mean. EWMA control charts can also be preferred when the subgroups are of size n = 1.
The joint use of an EWMA control chart with a small value of smoothing parameter (λ) and a Shewhart
control chart has been recommended as a means of guaranteeing fast detection of both small and large
shifts. The here considered EWMA control chart monitors only the process mean; monitoring the process
variability requires the use of some other technique including special EWMA control charts.
The numbers in all tables and figures were calculated using the R-package SPC, (Knoth 2022), which makes
use of the algorithm proposed by Crowder (1987).
The R-file containing the calculations can be downloaded on https://standards.iso.org/iso/7870/-6/ed-2/en.
v
International Standard ISO 7870-6:2024(en)
Control charts —
Part 6:
EWMA control charts for the process mean
1 Scope
)[16]
This document covers EWMA control charts, originally proposed by Roberts (1959 , as a statistical
process control technique to detect small shifts in the process mean. It makes possible the faster detection
of small to moderate shifts in the process mean. In this chart, the process mean is evaluated in terms of
exponentially weighted moving average of all previous observations or averages.
The EWMA control chart’s application is worthwhile in particular when
— production rate is slow,
— a minor or moderate shift in the process mean is vital to be detected,
— sampling and inspection procedure is complex and time consuming,
— testing is expensive, and
— it involves safety risks.
NOTE EWMA control charts are applicable for both variables and attributes data. The given examples illustrate
both types (see 5.5, Annex A, Annex B and Annex C).
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 refe
...