ISO 7870-2:2023
(Main)Control charts — Part 2: Shewhart control charts
Control charts — Part 2: Shewhart control charts
This document establishes a guide to the use and understanding of Shewhart control chart approach to the methods for statistical control of a process. This document is limited to the treatment of statistical process control methods using only Shewhart system of charts. Some supplementary material that is consistent with Shewhart approach, such as the use of warning limits, analysis of trend patterns and process capability is briefly introduced. However, there are several other types of control charts which can be used in different situations.
Cartes de contrôle — Partie 2: Cartes de contrôle de Shewhart
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Buy Standard
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INTERNATIONAL ISO
STANDARD 7870-2
Second edition
2023-03
Control charts —
Part 2:
Shewhart control charts
Cartes de contrôle —
Partie 2: Cartes de contrôle de Shewhart
Reference number
ISO 7870-2:2023(E)
© ISO 2023
---------------------- Page: 1 ----------------------
ISO 7870-2:2023(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2023
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 2023 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 7870-2:2023(E)
Contents Page
Foreword .v
Introduction . vi
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
3.1 General presence . 1
3.2 Symbols . 1
3.2.1 For the purposes of this document, the following symbols apply . 1
4 Concepts of Shewhart control charts .3
4.1 Shewhart control chart . 3
4.2 Control limits . 3
4.3 Process in statistical control . 3
4.4 Action limits . 4
4.5 Warning limits . 4
4.6 Type 1 error . . 4
4.7 Type 2 error . . 4
4.8 Process not in control . 4
4.9 Phase 1 of statistical process control . 5
4.10 Phase 2 of control charts . 5
5 Types of control charts . 5
5.1 Types of Shewhart control charts . 5
5.2 Control charts where no pre-specified values of process parameters are given. 5
5.3 Control charts with respect to given pre-specified values of process parameters . 6
5.4 Types of variables and attribute control charts . 6
5.4.1 Variables control charts . 6
5.4.2 Attribute control charts . 6
6 Variables control charts. 7
6.1 Usefulness of variables control charts . 7
6.2 Assumption of normality . 7
6.3 Pair of control charts . 8
6.4 Average, X chart and range, R chart or average, X chart and standard deviation,
s chart . 8
6.5 Control chart for individuals, X, and moving ranges, R . 9
m
6.6 Control charts for medians, X . 10
7 Control procedure and interpretation for variables control charts .11
7.1 Underlying principle . 11
7.2 Collect preliminary data . 11
7.3 Examine s (or R) chart . . 11
7.4 Homogenization for s (or R) chart . 11
7.5 Homogenization for X chart .12
7.6 Ongoing monitoring of process .12
8 Unnatural pattern and tests for assignable causes of variation .12
8.1 Natural pattern .12
8.2 Unnatural patterns .13
8.2.1 General .13
8.2.2 Lack of control in the average chart only .13
8.2.3 Lack of control in the variation chart only .13
8.2.4 Lack of control in both average and variation charts . 14
8.2.5 Depiction of unnatural patterns . 14
9 Process control, process capability, and process improvement .15
iii
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ISO 7870-2:2023(E)
9.1 Process control . 15
9.2 Process capability and improvement . 16
10 Attribute control charts .18
10.1 Attribute data . 18
10.2 Distributions . 18
10.3 Subgroup size . 18
10.4 Control chart for fraction nonconforming (p chart) . 19
11 Preliminary considerations before starting a control chart .19
11.1 Choice of critical to quality (CTQ) characteristics describing the process to control . 19
11.2 Analysis of the process . 19
11.3 Choice of rational subgroup . 20
11.4 Frequency and size of subgroups . 20
11.5 Preliminary data collection . 21
11.6 Out of control action plan . 21
12 Steps in the construction of control charts .21
12.1 Typical format of a standard control chart form . 21
12.2 Determine data collection strategy . 22
12.3 Data collection and computation . 23
12.4 Plotting X chart and R chart .23
13 Caution with Shewhart control charts .24
13.1 General caution . 24
13.2 Correlated data .26
13.3 Use of alternative rules to the three-σ rule . 26
Annex A (informative) Illustrative examples .27
Annex B (informative) Practical notices on the pattern tests for assignable causes
of variation .46
Bibliography .48
iv
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ISO 7870-2:2023(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 cancels and replaces the first edition (ISO 7870-2:2013), which has been technically
revised.
The main changes are as follows:
— various clauses have been modified for better understanding;
— some examples for control charts have been modified;
— new examples for control charts have been included.
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
© ISO 2023 – All rights reserved
---------------------- Page: 5 ----------------------
ISO 7870-2:2023(E)
Introduction
A traditional approach to manufacturing has been to depend on production to make the product and
on quality control to inspect the final product and screen out items not meeting specifications. This
strategy of detection is often wasteful and uneconomical because it involves after-the-event inspection
when the wasteful production has already occurred. Instead, it is much more effective to institute a
strategy of prevention to avoid waste by not producing unusable output in the first place. This can be
accomplished by gathering process information and analysing it so that timely action can be taken on
the process itself.
Dr. Walter Shewhart in 1924 developed the control chart method for controlling the quality during
production. Control chart theory recognizes two kinds of variability. The first kind is random
variability (also known as natural/inherent/uncontrollable variation) arising due to causes known as
chance/common/random causes. This is due to the wide variety of causes that are consistently present
and not readily identifiable, each of which constitutes a very small component of the total variability
but none of them contributes any significant amount. Nevertheless, the sum of the contributions of
all of these unidentifiable random causes is measurable and is assumed to be inherent to the process.
The elimination or correction of common causes may well require a decision to allocate resources to
fundamentally change the process and system.
The second kind of variability represents a real change in the process. Such a change can be attributed
to some identifiable causes that are not an inherent part of the process and which can, at least
theoretically, be eliminated. These identifiable causes are referred to as “assignable causes” (also
known as special/unnatural/systematic/controllable causes) of variation. They may be attributable
to such matters as the lack of uniformity in material, a broken tool, workmanship or procedures, the
irregular performance of equipment, or environmental changes.
A process is said to be in a state of statistical control, or simply “in control”, if the process variability
results only from random causes. Once this level of variation is determined, any deviation from this
level is assumed to be the result of assignable causes that should be identified and eliminated.
The major statistical tool used to do this is the control chart, which is a method of presenting and
comparing information based on a sequence of observations representing the current state of a process
against limits established after consideration of inherent process variability. The control chart method
helps first to evaluate whether a process has attained, or continues in, a state of statistical control.
When the process is deemed to be stable and predictable, then further analysis regarding the ability
of the process to satisfy the requirements of the customer may be conducted. The control chart also
can be used to provide a continuous record of a quality characteristic of the process output while
process activity is ongoing. Control charts aid in the detection of unnatural patterns of variation in data
resulting from repetitive processes and provide criteria for detecting a lack of statistical control. The
use of a control chart and its careful analysis leads to a better understanding of the process and will
often result in the identification of ways to make valuable improvements.
vi
© ISO 2023 – All rights reserved
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INTERNATIONAL STANDARD ISO 7870-2:2023(E)
Control charts —
Part 2:
Shewhart control charts
1 Scope
This document establishes a guide to the use and understanding of Shewhart control chart approach to
the methods for statistical control of a process.
This document is limited to the treatment of statistical process control methods using only Shewhart
system of charts. Some supplementary material that is consistent with Shewhart approach, such as the
use of warning limits, analysis of trend patterns and process capability is briefly introduced. However,
there are several other types of control charts which can be used in different situations.
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 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms and definitions
3.1 General presence
For the purposes of this document, the terms and definitions given in ISO 3534-2 apply.
ISO and IEC maintain terminology 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/
3.2 Symbols
NOTE The ISO/IEC Directives make it necessary to depart from common SPC usage in respect to the
differentiation between abbreviated terms and symbols. In ISO standards an abbreviated term and its symbol
can differ in appearance in two ways: by font and by layout. To distinguish between abbreviated terms and
symbols, abbreviated terms are given in Cambria upright and symbols in Cambria or Greek italics, as applicable.
Whereas abbreviated terms can contain multiple letters, symbols consist only of a single letter. For example,
the conventional abbreviation of upper control limit, UCL, is valid but its symbol in equations becomes U . The
CL
reason for this is to avoid misinterpretation of compound letters as an indication of multiplication.
3.2.1 For the purposes of this document, the following symbols apply
n Subgroup size; the number of sample observations per subgroup
k Number of subgroups
L Lower specification limit
1
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ISO 7870-2:2023(E)
L Lower control limit
CL
th
L Lower control limit for i subgroup
CLi
CL Centre line
U Upper control limit
CL
th
U Upper control limit for i subgroup
CLi
X Measured quality characteristic (individual values are expressed as (X , X , X ,.).
1 2 3
Sometimes the symbol Y is used instead of X
X (X bar) Subgroup average
(X double bar) Average of the subgroup averages
X
μ True process mean
μ A given or prespecified value of μ
0
σ True process standard deviation
σ A given or prespecified value of σ
0
Median of a subgroup
X
Average of the subgroup medians
X
R Subgroup range
R Average of subgroup ranges
R Subgroup moving range
m
R Average moving range
m
s Subgroup sample standard deviation
s
Average of subgroup sample standard deviations
p Proportion of nonconforming items in a subgroup
p
Average proportion of nonconforming items for all subgroups
np Number of nonconforming items in a subgroup
p A given value of p
0
np A given value of np (for a given p )
0 0
c Number of nonconformities in a subgroup
c A given value of c
0
c
Average number of nonconformities for all subgroups
u Number of nonconformities per unit in a subgroup
u
Average number of nonconformities per unit
2
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ISO 7870-2:2023(E)
4 Concepts of Shewhart control charts
4.1 Shewhart control chart
A Shewhart control chart is a chart that is used to display a statistical measure (also called ‘statistic’)
obtained from either variables or attribute data. The control chart requires data from rational
subgroups (see 11.3) to be taken at approximately regular intervals from the process. The intervals
may be defined in terms of time (for example hourly) or quantity (every lot). Usually, the data are
obtained from the process in the form of samples or subgroups consisting of the same process
characteristic, product or service with the same measurable units and the same subgroup size. From
each subgroup, one or more statistical measures are calculated, such as average, X , range, R, standard
deviation, s, proportion of nonconforming items p, and number of nonconformities, c.
4.2 Control limits
Shewhart control chart is a chart on which some statistical measure of the values in each subgroup is
plotted against subgroup number. It consists of centre line, CL, which is usually the average value of the
statistical measure being considered or may be based on past experience, when the process is in state
of statistical control. It may also be based on product or service target values. The control chart has
two statistically determined limit lines, one on either side of the centre line, which are called the upper
control limit, U , and the lower control limit, L , (see Figure 1).
CL CL
Key
X subgroup number
Y statistic
CL centre line
L lower control limit
CL
U upper control limit
CL
Figure 1 — Outline of a control chart
4.3 Process in statistical control
4.3.1 The upper and lower control limits on the control chart, on each side of the centre line, are
typically placed at a distance of three times the standard deviation of the statistic (3 σ) being plotted.
If large number of observations from a process in statistical control are studied in form of frequency
distribution, it often shows a bell shaped symmetrical pattern, which is well represented as normal
distribution.
4.3.2 Placing the limits too close to the centre line will result in many searches for non-existing
problems and yet placing the limits too far apart will increase the risk of not detecting process
problems when they do exist. Under an assumption that the plotted statistic is approximately normally
distributed 3 σ limits indicate that approximately 99,73 % of the values of the statistic will be included
within the control limits, provided the process is in statistical control. Interpreted another way, there
3
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ISO 7870-2:2023(E)
is a 0,27 % probability, or about three out of thousand plotted points will be out of the upper or lower
control limit when the process is in control. The word “approximately” is used because deviations from
underlying assumptions such as the distributional form of the data will affect the probability values.
In fact, the choice of k σ limits, instead of 3 σ limits, depends on costs of investigation and taking
appropriate action vis-à-vis consequences of not taking action.
4.4 Action limits
The possibility that a violation of the limits is really a chance event rather than a real signal is considered
so small that when a point appears outside of the limits, action should be taken. Since action is required
at this point, the 3 σ control limits are sometimes called the “action limits”.
4.5 Warning limits
Sometimes it is advantageous to mark 2 σ limits on the chart also. Then, any sample value falling
beyond the 2 σ limits can serve as a warning of an impending out-of-control situation. As such, the 2 σ
limits are sometimes called “warning limits”. While no action is required as a result of such a warning
on the control chart, some users may wish to immediately select another subgroup of the same size to
determine if corrective action is needed.
4.6 Type 1 error
When assessing the status of a process using control charts, two types of errors are possible. The first
occurs when the process is actually in a state of control but a plotted point falls outside the control
limits due to chance (Type 1 error). As a result, the chart has given a false signal resulting in an incorrect
conclusion that the process is out of control. A cost is then incurred in an attempt to find the cause of a
non-existent problem.
If normality is assumed and 3 σ control limits are used, the probability of Type 1 error is 0,27 %. In
other words, this error will happen only about 3 times in 1 000 samples when the process is in control.
4.7 Type 2 e
...
2022-11-07
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line: 0 pt, Tab stops: Not at 21.6 pt
ISO/FDIS 7870-2:2022(E)
Style Definition: Heading 2: Font: Bold, Tab stops: Not
at 18 pt
ISO TC 69/SC 4/WG 10
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DATE: 2022-xx
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Secretariat: DIN
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Control charts — Part 2: Shewhart control charts
Style Definition: AMEND Heading 1 Unnumbered:
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Cartes de contrôle — Partie 2: Cartes de contrôle de Shewhart
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Formatted: French (Switzerland)
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ISO/FDIS 7870-2:2022(E)
© ISO 2022
Commented [eXtyles1]: The reference is to a withdrawn
standard which has been replaced
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no
ISO/IEC 2022, Information technology — Character code
part of this publication may be reproduced or utilized otherwise in any form or by any means,
structure and extension techniques
electronic or mechanical, including photocopying, or posting on the internet or an intranet, without
Formatted: Pattern: Clear
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.
Formatted: Pattern: Clear
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.orgwww.iso.org
Published in Switzerland
ii © ISO 2022 – All rights reserved
---------------------- Page: 2 ----------------------
ISO/FDIS 7870-2:2022(E)
Contents
Foreword . iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
3.1 General presence . 1
3.2 Symbols . 1
3.2.1 For the purposes of this document, the following symbols apply . 1
4 Concepts of Shewhart control charts . 3
4.1 Shewhart control chart . 3
4.2 Control limits . 3
4.3 Process in statistical control . 3
4.4 Action limits . 4
4.5 Warning limits . 4
4.6 Type 1 error . 4
4.7 Type 2 error . 4
4.8 Process not in control . 5
4.9 Phase 1 of control charts . 5
4.10 Phase 2 of control charts . 5
5 Types of control charts . 5
5.1 Types of Shewhart control charts . 5
5.2 Control charts where no pre-specified values of process parameters are given . 6
5.3 Control charts with respect to given pre-specified values of process parameters . 6
5.4 Types of variables and attribute control charts . 6
5.4.1 Variables control charts . 6
5.4.2 Attribute control charts . 6
6 Variables control charts . 7
6.1 Usefulness of variables control charts . 7
6.2 Assumption of normality . 8
6.3 Pair of control charts . 8
6.4 Average, X chart and range, R chart or average, X chart and standard deviation,
s chart . 8
6.5 Control chart for individuals, X, and moving ranges, R . 10
m
6.6 Control charts for medians, X . 11
7 Control procedure and interpretation for variables control charts . 11
7.1 Underlying Principle . 11
7.2 Collect preliminary data . 12
7.3 Examine s (or R) chart . 12
7.4 Homogenization for s (or R) chart . 12
7.5 Homogenization for X chart . 12
7.6 Ongoing monitoring of process . 13
8 Unnatural pattern and tests for assignable causes of variation . 13
8.1 Natural pattern . 13
8.2 Unnatural patterns . 13
8.2.1 General . 13
© ISO 2022 – All rights reserved iii
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ISO/FDIS 7870-2:2022(E)
8.2.2 Lack of control in the average chart only . 14
8.2.3 Lack of control in the variation chart only . 14
8.2.4 Lack of control in both average and variation charts . 14
8.2.5 Depiction of unnatural patterns . 15
9 Process control, process capability, and process improvement . 16
9.1 Process control . 16
9.2 Process capability and improvement . 17
10 Attribute control charts . 19
10.1 Attribute data . 19
10.2 Distributions . 19
10.3 Subgroup size . 19
10.4 Control chart for fraction nonconforming (p chart) . 20
11 Preliminary considerations before starting a control chart . 20
11.1 Choice of critical to quality (CTQ) characteristics describing the process to control 20
11.2 Analysis of the process . 20
11.3 Choice of rational subgroup . 21
11.4 Frequency and size of subgroups . 22
11.5 Preliminary data collection . 22
11.6 Out of control action plan . 22
12 Steps in the construction of control charts . 23
12.1 Typical format of a standard control chart form . 23
12.2 Determine data collection strategy. 24
12.3 Data collection and computation . 25
12.4 Plotting X chart and R chart . 26
13 Caution with Shewhart control charts . 26
13.1 General caution . 26
13.2 Correlated data . 28
13.3 Use of alternative rules to the three-σ rule . 28
Annex A (informative) Illustrative examples . 30
Annex B (informative) Practical notices on the pattern tests for assignable causes
of variation . 51
Bibliography . 53
iv © ISO 2022 – All rights reserved
---------------------- Page: 4 ----------------------
ISO/FDIS 7870-2:2022(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/directiveswww.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/patentswww.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.htmlwww.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 cancels and replaces the first edition (ISO 7870-2:2013), which has been technically
Formatted: Pattern: Clear
revised. The main changes are as follows:
Formatted: Pattern: Clear
Formatted: Pattern: Clear
The main changes are as follows:a)
Formatted: Pattern: Clear
— various clauses have been modified for better understanding;
Formatted: List Continue 1, Adjust space between Latin
and Asian text, Adjust space between Asian text and
b)— some examples for control charts have been modified;
numbers, Tab stops: Not at 19.85 pt + 39.7 pt + 59.55
pt + 79.4 pt + 99.25 pt + 119.05 pt + 138.9 pt +
c)— new examples for control charts have been included.
158.75 pt + 178.6 pt + 198.45 pt
A list of all parts in the ISO 7870series7870 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.htmlwww.iso.org/members.html.
© ISO 2022 – All rights reserved v
---------------------- Page: 5 ----------------------
ISO/FDIS 7870-2:2022(E)
Introduction
A traditional approach to manufacturing has been to depend on production to make the product and on
quality control to inspect the final product and screen out items not meeting specifications. This strategy
of detection is often wasteful and uneconomical because it involves after-the-event inspection when the
wasteful production has already occurred. Instead, it is much more effective to institute a strategy of
prevention to avoid waste by not producing unusable output in the first place. This can be accomplished
by gathering process information and analysing it so that timely action can be taken on the process itself.
Dr. Walter Shewhart in 1924 developed the control chart method for controlling the quality during
production. Control chart theory recognizes two kinds of variability. The first kind is random variability
(also known as natural/inherent/uncontrollable variation) arising due to causes known as
chance/common/random causes. This is due to the wide variety of causes that are consistently present
and not readily identifiable, each of which constitutes a very small component of the total variability but
none of them contributes any significant amount. Nevertheless, the sum of the contributions of all of these
unidentifiable random causes is measurable and is assumed to be inherent to the process. The elimination
or correction of common causes may well require a decision to allocate resources to fundamentally
change the process and system.
The second kind of variability represents a real change in the process. Such a change can be attributed to
some identifiable causes that are not an inherent part of the process and which can, at least theoretically,
be eliminated. These identifiable causes are referred to as “assignable causes” (also known as
special/unnatural/systematic/controllable causes) of variation. They may be attributable to such
matters as the lack of uniformity in material, a broken tool, workmanship or procedures, the irregular
performance of equipment, or environmental changes.
A process is said to be in a state of statistical control, or simply “in control”, if the process variability
results only from random causes. Once this level of variation is determined, any deviation from this level
is assumed to be the result of assignable causes that should be identified and eliminated.
The major statistical tool used to do this is the control chart, which is a method of presenting and
comparing information based on a sequence of observations representing the current state of a process
against limits established after consideration of inherent process variability. The control chart method
helps first to evaluate whether a process has attained, or continues in, a state of statistical control. When
the process is deemed to be stable and predictable, then further analysis regarding the ability of the
process to satisfy the requirements of the customer may be conducted. The control chart also can be used
to provide a continuous record of a quality characteristic of the process output while process activity is
ongoing. Control charts aid in the detection of unnatural patterns of variation in data resulting from
repetitive processes and provide criteria for detecting a lack of statistical control. The use of a control
chart and its careful analysis leads to a better understanding of the process and will often result in the
identification of ways to make valuable improvements.
vi © ISO 2022 – All rights reserved
---------------------- Page: 6 ----------------------
FINAL DRAFT INTERNATIONAL STANDARD ISO/FDIS 7870-2:2022(E)
Control charts — Part 2: Shewhart control charts
1 Scope
This document establishes a guide to the use and understanding of Shewhart control chart approach to
the methods for statistical control of a process.
This document is limited to the treatment of statistical process control methods using only Shewhart
system of charts. Some supplementary material that is consistent with Shewhart approach, such as the
use of warning limits, analysis of trend patterns and process capability is briefly introduced. However,
there are several other types of control charts which can be used in different situations.
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 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
ISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms and definitions
3.1 General presence
For the purposes of this document, the terms and definitions given in ISO 3534-2 apply.
Formatted: Pattern: Clear
ISO and IEC maintain terminology databases for use in standardization at the following addresses: Formatted: Pattern: Clear
Formatted: Pattern: Clear
— ISO Online browsing platform: available at https://www.iso.org/obphttps://www.iso.org/obp
Formatted: English (United States)
— IEC Electropedia: available at https://www.electropedia.org/https://www.electropedia.org/
Formatted: Adjust space between Latin and Asian text,
Adjust space between Asian text and numbers, Tab
3.2 Symbols
stops: Not at 19.85 pt + 39.7 pt + 59.55 pt + 79.4 pt
+ 99.25 pt + 119.05 pt + 138.9 pt + 158.75 pt +
NOTE The ISO/IEC Directives makesmake it necessary to depart from common SPC usage in respect to the
178.6 pt + 198.45 pt
differentiation between abbreviated terms and symbols. In ISO standards an abbreviated term and its symbol can
Formatted: Hyperlink, English (United States)
differ in appearance in two ways: by font and by layout. To distinguish between abbreviated terms and symbols,
abbreviated terms are given in Cambria upright and symbols in Cambria or Greek italics, as applicable. Whereas
Formatted: English (United States)
abbreviated terms can contain multiple letters, symbols consist only of a single letter. For example, the conventional
Formatted: Hyperlink, English (United States)
abbreviation of upper control limit, UCL, is valid but its symbol in equations becomes U . The reason for this is to
CL
avoid misinterpretation of compound letters as an indication of multiplication.
3.2.1 For the purposes of this document, the following symbols apply
n Subgroup size; the number of sample observations per subgroup
© ISO 2022 – All rights reserved 1
---------------------- Page: 7 ----------------------
ISO/FDIS 7870-2:2022(E)
k Number of subgroups
L Lower specification limit
L Lower control limit
CL
th
LCLi Lower control limit for i-thi subgroup
Formatted: Font: Italic
CL Centre Lineline
U Upper control limit
CL
th
U Upper control limit for i-thi subgroup
CLi
Formatted: Font: Italic
X Measured quality characteristic (individual values are expressed as (X , X , X ,.).
1 2 3
Sometimes the symbol Y is used instead of X
Formatted: Font: Italic
(X bar) Subgroup average Formatted: Font: Italic
X
(X double bar) Average of the subgroup averages
X
μ True process mean
μ0 A given or prespecified value of μ
σ True process standard deviation
σ A given or prespecified value of σ
0
Median of a subgroup
X
Average of the subgroup medians
X
R Subgroup range
R Average of subgroup ranges
R Subgroup moving range
m
Average moving range
R
m
s Subgroup sample standard deviation
s
Average of subgroup sample standard deviations
p Proportion of nonconforming items in a subgroup
p
Average proportion of nonconforming items for all subgroups
np Number of nonconforming items in a subgroup
p A given value of p
0
np0 A given value of np (for a given p0)
c Number of nonconformities in a subgroup
c A given value of c
0
c
Average number of nonconformities for all subgroups
u Number of nonconformities per item in a subgroup
u
Average number of nonconformities per item for all subgroups
2 © ISO 2022 – All rights reserved
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ISO/FDIS 7870-2:2022(E)
4 Concepts of Shewhart control charts
4.1 Shewhart control chart
A Shewhart control chart is a chart that is used to display a statistical measure (also called ‘statistic’)
obtained from either variables or attribute data. The control chart requires data from rational subgroups
(see Clause 11.3) to be taken at approximately regular intervals from the process. The intervals may be
Formatted: Pattern: Clear
defined in terms of time (for example hourly) or quantity (every lot). Usually, the data are obtained from
the process in the form of samples or subgroups consisting of the same process characteristic, product or
service with the same measurable units and the same subgroup size. From each subgroup, one or more
statistical measures are calculated, such as average, X , range, R, standard deviation, s, proportion of
nonconforming items p, and number of nonconformities, c.
4.2 Control limits
Shewhart control chart is a chart on which some statistical measure of the values in each subgroup is
plotted against subgroup number. It consists of centre line (, CL),, which is usually the average value of
the statistical measure being considered or may be based on past experience, when the process is in state
of statistical control. It may also be based on product or service target values. The control chart has two
statistically determined limit lines, one on either side of the centre line, which are called the upper control
limit (, U ), and the lower control limit (, L ), (see Figure 1).
CL CL Formatted: Pattern: Clear
7870-2_ed2fig1.EPS
Key
X subgroup number
Y statistic
CL centre line
L lower control limit
CL
U upper control limit
CL
Figure 1 — Outline of a control chart
4.3 Process in statistical control
4.3.1 The upper and lower control limits on the control chart, on each side of the centre line, are
typically placed at a distance of 3 times the standard deviation of the statistic (3 sigmaσ) being plotted. If
large number of observations from a process in statistical control are studied in form of frequency
distribution, it often shows a bell shaped symmetrical pattern, which is well represented as normal
distribution.
© ISO 2022 – All rights reserved 3
---------------------- Page: 9 ----------------------
ISO/FDIS 7870-2:2022(E)
4.3.2 Placing the limits too close to the centre line will result in many searches for non-existing
problems and yet placing the limits too far apart will increase the risk of not detecting process problems
when they do exist. Under an assumption that the plotted statistic is approximately normally distributed
3 sigmaσ limits indicate that approximately 99,73 % of the values of the statistic will be included within
the control limits, provided the process is in statistical control. Interpreted another way, there is
approximately a 0,3 % probability, or about three out of thousand plotted points will be out of the upper
or lower control limit when the process is in control. The word “approximately” is used because
deviations from underlying assumptions such as the distributional form of the data will affect the
probability values. In fact, the choice of k sigmaσ limits, instead of 3 sigmaσ limits, depends on costs of
investigation and taking appropriate action vis-à-vis consequences of not taking action.
4.4 Action limits
The possibility that a violation of the limits is really a chance event rather than a real signal is considered
so small that when a point appears outside of the limits, action should be taken. Since action is required
at this point, the 3 sigmaσ control limits are sometimes called the “action limits”.
4.5 Warning limits
Sometimes it is advantageous to mark 2 sigmaσ limits on the chart also. Then, any sample value falling
beyond the 2 sigmaσ limits can serve as a warning of an impending out-of-control situation. As such, the
2 sigmaσ limits are sometimes called “warning limits”. While no action is required as a result of such a
warning on the control chart, some users may wish to immediately select another subgroup of the same
size to determine if corrective action is needed.
4.6 Type 1 error
When assessing the status of a process using control charts, two types of errors are possible. The first
occurs when the process is actually in a state of control but a plotted point falls outside the control limits
due to chance (Type 1 error). As a result, the chart has given a false signal resulting in an incorrect
conclusion that the process is out of control. A cost is then incurred in an attempt to find the cause of a
non-existent problem.
If normality is assumed and 3 sigmaσ control limits are used, the probability of Type 1 error is 0,3 %. In
other words, this error will happen only about 3 times in 1 000 samples when the process is in control.
4.7 Type 2 error
4.7.1 The second error occurs when the process involved is not in control but the plotted point falls
within the control limits due to chance (Type 2 error). In this case, the chart provides no signal and it is
incorrectly concluded that the process is in statistical control. There may also be a substantial cost
associated with failing to detect that a change in the process location or variability has occurred, the result
of which might be the production of nonconforming output. The risk of this type of error occurring is a
function of three things: the width of the control limits, the sample size, and the degree to which the
process is out of control. In general, because the magnitude of the change in the process cannot be known,
little can be determined about the actual size of the risk of this error.
4.7.2 Beca
...
FINAL
INTERNATIONAL ISO/FDIS
DRAFT
STANDARD 7870-2
ISO/TC 69/SC 4
Control charts —
Secretariat: DIN
Voting begins on:
Part 2:
2022-12-07
Shewhart control charts
Voting terminates on:
2023-02-01
Cartes de contrôle —
Partie 2: Cartes de contrôle de Shewhart
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BEING ACCEPTABLE FOR INDUSTRIAL, TECHNO-
ISO/FDIS 7870-2:2022(E)
LOGICAL, COMMERCIAL AND USER PURPOSES,
DRAFT INTERNATIONAL STANDARDS MAY ON
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LIGHT OF THEIR POTENTIAL TO BECOME STAN-
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NATIONAL REGULATIONS. © ISO 2022
---------------------- Page: 1 ----------------------
ISO/FDIS 7870-2:2022(E)
FINAL
INTERNATIONAL ISO/FDIS
DRAFT
STANDARD 7870-2
ISO/TC 69/SC 4
Control charts —
Secretariat: DIN
Voting begins on:
Part 2:
Shewhart control charts
Voting terminates on:
Cartes de contrôle —
Partie 2: Cartes de contrôle de Shewhart
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BEING ACCEPTABLE FOR INDUSTRIAL, TECHNO-
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LOGICAL, COMMERCIAL AND USER PURPOSES,
DRAFT INTERNATIONAL STANDARDS MAY ON
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OCCASION HAVE TO BE CONSIDERED IN THE
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ii
© ISO 2022 – All rights reserved
NATIONAL REGULATIONS. © ISO 2022
---------------------- Page: 2 ----------------------
ISO/FDIS 7870-2:2022(E)
Contents Page
Foreword .v
Introduction . vi
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
3.1 General presence . 1
3.2 Symbols . 1
3.2.1 For the purposes of this document, the following symbols apply . 1
4 Concepts of Shewhart control charts .3
4.1 Shewhart control chart . 3
4.2 Control limits . 3
4.3 Process in statistical control . 3
4.4 Action limits . 4
4.5 Warning limits . 4
4.6 Type 1 error . . 4
4.7 Type 2 error . . 4
4.8 Process not in control . 4
4.9 Phase 1 of control charts . 5
4.10 Phase 2 of control charts . 5
5 Types of control charts . 5
5.1 Types of Shewhart control charts . 5
5.2 Control charts where no pre-specified values of process parameters are given. 5
5.3 Control charts with respect to given pre-specified values of process parameters . 6
5.4 Types of variables and attribute control charts . 6
5.4.1 Variables control charts . 6
5.4.2 Attribute control charts . 6
6 Variables control charts. 7
6.1 Usefulness of variables control charts . 7
6.2 Assumption of normality . 7
6.3 Pair of control charts . 8
6.4 Average, X chart and range, R chart or average, X chart and standard deviation,
s chart . 8
6.5 Control chart for individuals, X, and moving ranges, R . 9
m
6.6 Control charts for medians, X . 10
7 Control procedure and interpretation for variables control charts .11
7.1 Underlying Principle . 11
7.2 Collect preliminary data . 11
7.3 Examine s (or R) chart . . 11
7.4 Homogenization for s (or R) chart . 11
7.5 Homogenization for X chart .12
7.6 Ongoing monitoring of process .12
8 Unnatural pattern and tests for assignable causes of variation .12
8.1 Natural pattern .12
8.2 Unnatural patterns .13
8.2.1 General .13
8.2.2 Lack of control in the average chart only .13
8.2.3 Lack of control in the variation chart only .13
8.2.4 Lack of control in both average and variation charts . 14
8.2.5 Depiction of unnatural patterns . 14
9 Process control, process capability, and process improvement .15
iii
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ISO/FDIS 7870-2:2022(E)
9.1 Process control . 15
9.2 Process capability and improvement . 16
10 Attribute control charts .18
10.1 Attribute data . 18
10.2 Distributions . 18
10.3 Subgroup size . 18
10.4 Control chart for fraction nonconforming (p chart) . 19
11 Preliminary considerations before starting a control chart .19
11.1 Choice of critical to quality (CTQ) characteristics describing the process to control . 19
11.2 Analysis of the process . 19
11.3 Choice of rational subgroup . 20
11.4 Frequency and size of subgroups . 20
11.5 Preliminary data collection . 21
11.6 Out of control action plan . 21
12 Steps in the construction of control charts .21
12.1 Typical format of a standard control chart form . 21
12.2 Determine data collection strategy . 22
12.3 Data collection and computation . 23
12.4 Plotting X chart and R chart .23
13 Caution with Shewhart control charts .24
13.1 General caution . 24
13.2 Correlated data .26
13.3 Use of alternative rules to the three-σ rule . 26
Annex A (informative) Illustrative examples .27
Annex B (informative) Practical notices on the pattern tests for assignable causes
of variation .46
Bibliography .48
iv
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ISO/FDIS 7870-2:2022(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 cancels and replaces the first edition (ISO 7870-2:2013), which has been technically
revised.
The main changes are as follows:
— various clauses have been modified for better understanding;
— some examples for control charts have been modified;
— new examples for control charts have been included.
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
© ISO 2022 – All rights reserved
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ISO/FDIS 7870-2:2022(E)
Introduction
A traditional approach to manufacturing has been to depend on production to make the product and
on quality control to inspect the final product and screen out items not meeting specifications. This
strategy of detection is often wasteful and uneconomical because it involves after-the-event inspection
when the wasteful production has already occurred. Instead, it is much more effective to institute a
strategy of prevention to avoid waste by not producing unusable output in the first place. This can be
accomplished by gathering process information and analysing it so that timely action can be taken on
the process itself.
Dr. Walter Shewhart in 1924 developed the control chart method for controlling the quality during
production. Control chart theory recognizes two kinds of variability. The first kind is random
variability (also known as natural/inherent/uncontrollable variation) arising due to causes known as
chance/common/random causes. This is due to the wide variety of causes that are consistently present
and not readily identifiable, each of which constitutes a very small component of the total variability
but none of them contributes any significant amount. Nevertheless, the sum of the contributions of
all of these unidentifiable random causes is measurable and is assumed to be inherent to the process.
The elimination or correction of common causes may well require a decision to allocate resources to
fundamentally change the process and system.
The second kind of variability represents a real change in the process. Such a change can be attributed
to some identifiable causes that are not an inherent part of the process and which can, at least
theoretically, be eliminated. These identifiable causes are referred to as “assignable causes” (also
known as special/unnatural/systematic/controllable causes) of variation. They may be attributable
to such matters as the lack of uniformity in material, a broken tool, workmanship or procedures, the
irregular performance of equipment, or environmental changes.
A process is said to be in a state of statistical control, or simply “in control”, if the process variability
results only from random causes. Once this level of variation is determined, any deviation from this
level is assumed to be the result of assignable causes that should be identified and eliminated.
The major statistical tool used to do this is the control chart, which is a method of presenting and
comparing information based on a sequence of observations representing the current state of a process
against limits established after consideration of inherent process variability. The control chart method
helps first to evaluate whether a process has attained, or continues in, a state of statistical control.
When the process is deemed to be stable and predictable, then further analysis regarding the ability
of the process to satisfy the requirements of the customer may be conducted. The control chart also
can be used to provide a continuous record of a quality characteristic of the process output while
process activity is ongoing. Control charts aid in the detection of unnatural patterns of variation in data
resulting from repetitive processes and provide criteria for detecting a lack of statistical control. The
use of a control chart and its careful analysis leads to a better understanding of the process and will
often result in the identification of ways to make valuable improvements.
vi
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FINAL DRAFT INTERNATIONAL STANDARD ISO/FDIS 7870-2:2022(E)
Control charts —
Part 2:
Shewhart control charts
1 Scope
This document establishes a guide to the use and understanding of Shewhart control chart approach to
the methods for statistical control of a process.
This document is limited to the treatment of statistical process control methods using only Shewhart
system of charts. Some supplementary material that is consistent with Shewhart approach, such as the
use of warning limits, analysis of trend patterns and process capability is briefly introduced. However,
there are several other types of control charts which can be used in different situations.
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 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms and definitions
3.1 General presence
For the purposes of this document, the terms and definitions given in ISO 3534-2 apply.
ISO and IEC maintain terminology 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/
3.2 Symbols
NOTE The ISO/IEC Directives make it necessary to depart from common SPC usage in respect to the
differentiation between abbreviated terms and symbols. In ISO standards an abbreviated term and its symbol
can differ in appearance in two ways: by font and by layout. To distinguish between abbreviated terms and
symbols, abbreviated terms are given in Cambria upright and symbols in Cambria or Greek italics, as applicable.
Whereas abbreviated terms can contain multiple letters, symbols consist only of a single letter. For example,
the conventional abbreviation of upper control limit, UCL, is valid but its symbol in equations becomes U . The
CL
reason for this is to avoid misinterpretation of compound letters as an indication of multiplication.
3.2.1 For the purposes of this document, the following symbols apply
n Subgroup size; the number of sample observations per subgroup
k Number of subgroups
L Lower specification limit
1
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---------------------- Page: 7 ----------------------
ISO/FDIS 7870-2:2022(E)
L Lower control limit
CL
th
L Lower control limit for i subgroup
CLi
CL Centre line
U Upper control limit
CL
th
U Upper control limit for i subgroup
CLi
X Measured quality characteristic (individual values are expressed as (X , X , X ,.).
1 2 3
Sometimes the symbol Y is used instead of X
X (X bar) Subgroup average
(X double bar) Average of the subgroup averages
X
μ True process mean
μ A given or prespecified value of μ
0
σ True process standard deviation
σ A given or prespecified value of σ
0
Median of a subgroup
X
Average of the subgroup medians
X
R Subgroup range
R Average of subgroup ranges
R Subgroup moving range
m
R Average moving range
m
s Subgroup sample standard deviation
s
Average of subgroup sample standard deviations
p Proportion of nonconforming items in a subgroup
p
Average proportion of nonconforming items for all subgroups
np Number of nonconforming items in a subgroup
p A given value of p
0
np A given value of np (for a given p )
0 0
c Number of nonconformities in a subgroup
c A given value of c
0
c
Average number of nonconformities for all subgroups
u Number of nonconformities per item in a subgroup
u
Average number of nonconformities per item for all subgroups
2
© ISO 2022 – All rights reserved
---------------------- Page: 8 ----------------------
ISO/FDIS 7870-2:2022(E)
4 Concepts of Shewhart control charts
4.1 Shewhart control chart
A Shewhart control chart is a chart that is used to display a statistical measure (also called ‘statistic’)
obtained from either variables or attribute data. The control chart requires data from rational
subgroups (see 11.3) to be taken at approximately regular intervals from the process. The intervals
may be defined in terms of time (for example hourly) or quantity (every lot). Usually, the data are
obtained from the process in the form of samples or subgroups consisting of the same process
characteristic, product or service with the same measurable units and the same subgroup size. From
each subgroup, one or more statistical measures are calculated, such as average, X , range, R, standard
deviation, s, proportion of nonconforming items p, and number of nonconformities, c.
4.2 Control limits
Shewhart control chart is a chart on which some statistical measure of the values in each subgroup is
plotted against subgroup number. It consists of centre line, CL, which is usually the average value of the
statistical measure being considered or may be based on past experience, when the process is in state
of statistical control. It may also be based on product or service target values. The control chart has
two statistically determined limit lines, one on either side of the centre line, which are called the upper
control limit, U , and the lower control limit, L , (see Figure 1).
CL CL
Key
X subgroup number
Y statistic
CL centre line
L lower control limit
CL
U upper control limit
CL
Figure 1 — Outline of a control chart
4.3 Process in statistical control
4.3.1 The upper and lower control limits on the control chart, on each side of the centre line, are
typically placed at a distance of 3 times the standard deviation of the statistic (3 σ) being plotted. If
large number of observations from a process in statistical control are studied in form of frequency
distribution, it often shows a bell shaped symmetrical pattern, which is well represented as normal
distribution.
4.3.2 Placing the limits too close to the centre line will result in many searches for non-existing
problems and yet placing the limits too far apart will increase the risk of not detecting process
problems when they do exist. Under an assumption that the plotted statistic is approximately normally
distributed 3 σ limits indicate that approximately 99,73 % of the values of the statistic will be included
within the control limits, provided the process is in statistical control. Interpreted another way, there
3
© ISO 2022 – All rights reserved
---------------------- Page: 9 ----------------------
ISO/FDIS 7870-2:2022(E)
is approximately a 0,3 % probability, or about three out of thousand plotted points will be out of the
upper or lower control limit when the process is in control. The word “approximately” is used because
deviations from underlying assumptions such as the distributional form of the data will affect the
probability values. In fact, the choice of k σ limits, instead of 3 σ limits, depends on costs of investigation
and taking appropriate action vis-à-vis consequences of not taking action.
4.4 Action limits
The possib
...
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