ISO 7870-2:2023

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

General Information

• Status: Published
• Publication Date: 13-Mar-2023
• ICS: 03.120.30 - Application of statistical methods
• Technical Committee: ISO/TC 69/SC 4 - Applications of statistical methods in product and process management
• Drafting Committee: ISO/TC 69/SC 4/WG 10 - Control charts
• Current Stage: 6060 - International Standard published
• Start Date: 14-Mar-2023
• Due Date: 05-Mar-2023
• Completion Date: 14-Mar-2023

Relations

• Revises: ISO 7870-2:2013 - Control charts - Part 2: Shewhart control charts
• Effective Date: 23-Apr-2020

Overview

ISO 7870-2:2023 - “Control charts - Part 2: Shewhart control charts” is an international standard that provides a clear, practical guide to using Shewhart control charts for Statistical Process Control (SPC). The document focuses exclusively on the Shewhart approach to monitoring process stability and identifying assignable causes of variation. It also briefly introduces compatible topics such as warning limits, trend/pattern analysis, and basic process capability considerations.

Keywords: ISO 7870-2, Shewhart control charts, control charts, statistical process control (SPC), process capability, industrial quality control

Key Topics

• Core concepts: Definitions of Shewhart control charts, control limits, action limits, warning limits, Type I and Type II errors, and what it means for a process to be in statistical control.
• Phases of control: Guidance on Phase 1 (initial analysis and set-up) and Phase 2 (ongoing monitoring).
• Chart types covered: Variables charts (X̄ & R, X̄ & s, individuals X and moving ranges R, median charts) and attribute charts (e.g., p charts for fraction nonconforming).
• Control procedure: Steps for data collection, chart construction, examining and homogenizing variation charts, and interpreting X and variation charts.
• Pattern tests and assignable causes: Recognition of unnatural patterns, tests for assignable causes, and practical notices on investigating out-of-control signals.
• Process capability & improvement: Introductory treatment of linking control to capability and improvement actions.
• Cautions & limitations: Notes on correlated data, alternative rule sets to the three-sigma rule, and contexts where other chart types might be more appropriate.

Applications

ISO 7870-2:2023 is practical for organizations and professionals who need standardized, reliable methods for process monitoring and improvement:

• Quality engineers and statisticians implementing SPC on production lines.
• Process engineers and manufacturing managers in discrete and continuous industries (automotive, electronics, chemical, pharmaceutical, food).
• Quality assurance teams and auditors assessing control systems and compliance with quality management systems.
• Suppliers and contract manufacturers establishing control plans and supplier quality monitoring.

Typical uses include real-time process monitoring, early detection of special causes, reducing waste through prevention (rather than end‑of‑line inspection), and forming the statistical basis for process improvement projects.

Related Standards (if applicable)

• Part of the ISO 7870 series - consult the ISO website for other parts covering additional chart types and broader SPC guidance.
• Prepared by ISO/TC 69 (Applications of statistical methods) - aligns with general SPC best practices used in ISO-based quality management systems.

For implementation, users should combine ISO 7870-2 guidance with organization-specific procedures, measurement systems analyses, and process capability studies.

Standards Content (Sample)

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 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
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Phone: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii
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
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
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
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
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
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
σ True process standard deviation
σ A given or prespecified value of σ

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
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
c
Average number of nonconformities for all subgroups
u Number of nonconformities per unit in a subgroup
u
Average number of nonconformities per unit
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
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 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 Because it is generally impractical to make a meaningful estimate of probability of Type 2 error
in any given situation, Shewhart control chart system is designed to control the risk (or probability) of
Type 1 error.
4.8 Process not in control
When a plotted value falls outside of either control limit, or a series of values display an unusual pattern
such as discussed in Clause 8 and Annex B, the state of statistical control can no longer be accepted.
When this occurs, an investigation is initiated to locate the assignable cause, and the process may be
stopped or adjusted. Once the assignable cause is determined and eliminated, the process is ready to
continue. As discussed in 4.3.2, on rare occasions when no assignable cause can be found and it must
be concluded that the point outside the limits represents the occurrence of a rare event, a false signal,
which has resulted in a value outside of the control limits even though the process is in control.
NOTE Point on the control line is considered as point in control.
4.9 Phase 1 of statistical process control
When a process is to be studied for the first time with the objective of bringing the process in a state of
statistical control, it is often found necessary to use historical data that has previously been obtained
from the process or to undertake to obtain new data from a series of samples before attempting to
establish the control chart. This retrospective stage during which the control chart parameters are
being established is often referred to as Phase 1. Sufficient data will need to be found in order to
obtain reliable estimates of the centre line and control limits for the control charts. The control limits
established in Phase 1 are trial control limits as they are based upon data collected when the process
may not be in control. The identification of the precise causes for signals given by the control chart at
this stage may prove to be difficult because of the lack of information about the historical operating
characteristics of the process. However, when special causes of variation can be identified and corrective
action taken, the retrospective data from the process when under the influence of the special cause
should be removed from consideration and the control chart parameters re-determined. This iterative
procedure is continued until the final trial control chart shows no signals and the control limits then
correspond to the process in control. Because some data may have to be removed from consideration
during Phase 1, some additional data may have to obtained from the process to maintain the reliability
of the parameter estimates.
4.10 Phase 2 of control charts
Once statistical control has been established, the revise control limits in Phase 1 are taken as the control
limits for the ongoing monitoring of the process. The objective now, in what is referred to as Phase 2, is
the maintenance of the process in a state of control as well as the rapid identification of special causes
that may affect the process from time to time. It should be recognized that moving from Phase 1 to
Phase 2 may be time consuming and difficult. However, it is critical, because failure to remove special
causes of variation will result in overestimation of the process variation. In this case the control chart
will have control limits that are set too wide apart resulting in a control chart that is not sufficiently
sensitive for detecting the presence of special causes.
5 Types of control charts
5.1 Types of Shewhart control charts
5.1.1 Shewhart control charts are of following two types:
a) variables control charts;
b) attribute control charts.
5.1.2 For each of these control charts, there are two distinct situations:
a) when no pre-specified process parameters values are given;
b) when pre-specified process parameters values are given.
5.2 Control charts where no pre-specified values of process parameters are given
The purpose is to identify whether the values of the statistics, which are being plotted on the control
charts for different subgroups, differ from the centre line by an amount greater than that can be
attributed to chance causes only. Control charts will be constructed using only the data collected from
samples from the process. The control charts are used for detecting those variations caused other than
by chance with the purpose being to bring the process in a state of statistical control.
5.3 Control charts with respect to given pre-specified values of process parameters
5.3.1 The purpose is to identify whether the observed values of X , s, etc., for several subgroups of n
observations each, differ from the respective given values of μ σ , etc. by amounts greater than that
0, 0
expected to be due to chance causes only. The difference between charts with given parameter values
and those where no pre-specified values are given, is the additional requirement concerning the
determination of the location of the centre and variation of the process. The pre-specified values may
be based on experience obtained by using control charts with no prior information or specified values.
They may also be based on economic values established upon consideration of the need for service and
cost of production or be nominal values designated by the product specifications.
5.3.2 Preferably, the specified values should be determined through an investigation of preliminary
data that is supposed to be typical of all future data. The specified values should be compatible with
the inherent process variability for effective functioning of the control charts. Control charts based on
such pre-specified values are used particularly during process operation to control processes and to
maintain product or service uniformity at the desired level.
5.4 Types of variables and attribute control charts
5.4.1 Variables control charts
The following control charts for variables are considered when measurements are on continuous scales:
a) average, X chart, and range, R chart, or standard deviation, s chart;
b) individuals, X chart and moving range, R chart;

c) median, X chart and range, R chart.
5.4.2 Attribute control charts
The following attribute control charts are used when items are classified as conforming and
nonconforming or number of nonconformities are counted on the items:
a) p chart for proportion of nonconforming items, when sample size is not constant;
b) np chart for number of nonconforming items when the sample size is constant.
NOTE p chart can also be used in such a case. As it involves additional calculation to find p value for each
subgroup for plotting them on p chart, and the result being the same as that of np chart; it is recommended
to use np chart when sample size is constant.
c) c chart for number of nonconformities when the sample size is constant;
NOTE u chart can also be used in such a case. As it involves additional calculation to find u value for each
subgroup for plotting them on u chart, and the result being the same as that of c chart; it is recommended to
use c chart when sample size is constant.
d) u chart for the number of nonconformities per unit when the sample size is not constant.
Figure 2 shows a process of selecting an appropriate control chart for a given situation.
a)  Variables data b)  Attribute data
Figure 2 — Types of control charts
6 Variables control charts
6.1 Usefulness of variables control charts
Control charts for variables are particularly useful for several reasons including the following:
a) Most processes, and their output, have characteristics that are measurable, hence generate
variables data, so the potential applicability is broad.
b) Variables charts are more informative than attribute charts since specific information about the
process average and variation is obtained directly.
c) Although obtaining information on variables data for one item is more costly than that for attribute
data, the subgroup sizes needed for variables data are smaller than those for attribute data, for an
equivalent monitoring efficiency. This helps to reduce the total inspection cost and to shorten the
time gap between the occurrence of a process problem and its corrective action.
d) Variables charts provide visual means to directly assess process performance regardless of the
specifications.
6.2 Assumption of normality
For all variables control charts considered in this document, it is assumed that the distribution of the
quality characteristic is normal. The factors used for computing control limits are derived using the
assumption of normality. So, departure from this assumption will affect the performance of the charts.
Since most control limits are used as empirical guides in making decisions, reasonably small departures
from normality should not be of concern. In any case, because of the central limit theorem, averages
tend to be normally distributed even when individual observations are not; this makes it reasonable for
evaluating control to assume normality for X charts, even for sample sizes as small as 4 or 5. When
dealing with individual observations for capability study purposes, the true form of the distribution is
important. Periodic checks on the continuing validity of such assumptions are advisable, particularly
for ensuring that only data from a single population are being used. It should be noted that the
distributions of the ranges and standard deviations are not normal. Although normality is necessarily
assumed in the determination of the constants for the calculation of control limits for the range or
standard deviation chart, moderate deviations from normality of the process data should not be of
major concern in the use of these charts as an empirical decision procedure.
6.3 Pair of control charts
6.3.1 As normality is assumed for variables type of data, and normal distribution has two parameters,
namely, mean and standard deviation; a pair of control charts is prepared and analysed together, one
for controlling variation of the process and the other for process mean. So, variables charts can describe
process data in terms of both process variability (spread) and process average (location). Average, X
chart is commonly used to control location and range, R chart to control inherent variability.
6.3.2 Each chart can be plotted using either estimated control limits, in which case limits are based
on the information contained in the sample data plotted on the chart, or pre-specified control limits
based on adopted specified values applicable to the statistical measures plotted on the chart.
6.3.3 The chart for spread is analysed first, since it provides the rationale and justification for the
estimation of the process standard deviation. The resulting estimate of the process standard deviation
is then be used in establishing control limits for the chart for location.
6.4 Average, X chart and range, R chart or average, X chart and standard deviation,
s chart
X and R control charts can be used when subgroup sample size is small or moderately small, usually
less than 10. X and s control charts are preferable in the case of large subgroup sample sizes (n ≥ 10),
since the range becomes increasingly less efficient in estimating the process standard deviation when
the sample size gets larger. Where software is available to calculate process limits, standard deviation
chart is preferable. Table 1 and Table 2 give the control limit formulae and the factors for each of these
variables control charts.
Table 1 — Control limit formulae for average, range and standard deviation
Statistic Estimated control limits Pre-specified control limits
Centre line U and L Centre line U and L
CL CL CL CL
μ μσ± A
X
X XA±±RXand As
0 00
R DR , DR d σ DDσσ,
R
43 20 20 10
s s Bs , Bs c σ BBσσ,
43 40 60 50
NOTE μ and σ are given values of parameters.
0 0
Table 2 — Factors for computing control chart lines
subgroup Factors for
Factors for control limits
size centre line
a a
n s chart R chart Using s Using R
X chart
A A A B B B B D D D D c d
2 3 3 4 5 6 1 2 3 4 4 2
2 2,121 1,880 2,659 0 3,267 0 2,606 0 3,686 0 3,267 0,798 1,128
3 1,732 1,023 1,954 0 2,568 0 2,276 0 4,358 0 2,575 0,886 1,693
4 1,500 0,729 1,628 0 2,266 0 2,088 0 4,698 0 2,282 0,921 2,059
5 1,342 0,577 1,427 0 2,089 0 1,964 0 4,918 0 2,114 0,940 2,326

6 1,225 0,483 1,287 0,030 1,970 0,029 1,874 0 5,079 0 2,004 0,952 2,534
7 1,134 0,419 1,182 0,118 1,882 0,113 1,806 0,205 5,204 0,076 1,924 0,959 2,704
8 1,061 0,373 1,099 0,185 1,815 0,179 1,751 0,388 5,307 0,136 1,864 0,965 2,847
9 1,000 0,337 1,032 0,239 1,761 0,232 1,707 0,547 5,394 0,184 1,816 0,969 2,970
10 0,949 0,308 0,975 0,284 1,716 0,276 1,669 0,686 5,469 0,223 1,777 0,973 3,078

11 0,905 0,285 0,927 0,321 1,679 0,313 1,637 0,811 5,535 0,256 1,744 0,975 3,173
12 0,866 0,266 0,886 0,354 1,646 0,346 1,610 0,923 5,594 0,283 1,717 0,978 3,258
13 0,832 0,249 0,850 0,382 1,618 0,374 1,585 1,025 5,647 0,307 1,693 0,979 3,336
14 0,802 0,235 0,817 0,406 1,594 0,399 1,563 1,118 5,696 0,328 1,672 0,981 3,407
15 0,775 0,223 0,789 0,428 1,572 0,421 1,544 1,203 5,740 0,347 1,653 0,982 3,472

16 0,750 0,212 0,763 0,448 1,552 0,440 1,526 1,282 5,782 0,363 1,637 0,984 3,532
17 0,728 0,203 0,739 0,466 1,534 0,458 1,511 1,356 5,820 0,378 1,622 0,985 3,588
18 0,707 0,194 0,718 0,482 1,518 0,475 1,496 1,424 5,856 0,391 1,609 0,985 3,640
19 0,688 0,187 0,698 0,497 1,503 0,490 1,483 1,489 5,889 0,404 1,596 0,986 3,689
20 0,671 0,180 0,680 0,510 1,490 0
...