ISO 7870-2:2013

INTERNATIONAL ISO
STANDARD 7870-2
First edition
2013-04-01
Control charts —
Part 2:
Shewhart control charts
Cartes de contrôle —
Partie 2: Cartes de contrôle de Shewhart
Reference number
©
ISO 2013
© ISO 2013
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ii © ISO 2013 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 General . 1
3.2 Symbols . 1
4 Nature of Shewhart control charts . 3
5 Types of control charts. 5
5.1 Control charts where no pre-specified values are given . 5
5.2 Control charts with respect to given pre-specified values . 6
5.3 Types of variables and attributes control charts . 6
6 Variables control charts . 7
6.1 Mean ( X ) chart and range (R) chart or mean ( X ) chart and standard deviation (s) chart 8
6.2 Control chart for individuals (X) and control chart for moving ranges (R ) . 9
m

6.3 Control charts for medians ( X ).10
7 Control procedure and interpretation for variables control charts .11
7.1 Collect preliminary data .11
7.2 Examine the s (or R) chart .11
7.3 Remove assignable causes and revise the chart . .11
7.4 Examine the X chart .12
7.5 Ongoing monitoring of process .12
8 Pattern tests for assignable causes of variation .12
9 Process control, process capability, and process improvement.13
10 Attributes control charts .15
11 Preliminary considerations before starting a control chart .17
11.1 Choice of critical to quality (CTQ) characteristics describing the process to control .17
11.2 Analysis of the process .17
11.3 Choice of rational subgroups .17
11.4 Frequency and size of subgroups . .18
11.5 Preliminary data collection .18
11.6 Out of control action plan .18
12 Steps in the construction of control charts .18
12.1 Determine data collection strategy .19
12.2 Data collection and computation .20
12.3 Plotting X chart and R chart .20
13 Caution with Shewhart control charts .20
13.1 General caution .21
13.2 Correlated data .22
13.3 Use of alternative rule to the three-sigma rule .22
Annex A (informative) Illustrative examples .24
Annex B (informative) Practical notices on the pattern tests for assignable causes of variation .42
Bibliography .44
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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International
Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies
casting a vote.
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.
ISO 7870-2 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 4, Applications of statistical methods in process management.
This first edition cancels and replaces ISO 8258:1991, which has been technically revised.
ISO 7870 consists of the following parts, under the general title Control charts:
— Part 1: General guidelines
— Part 2: Shewhart control charts
— Part 3: Acceptance control charts
— Part 4: Cumulative sum charts
— Part 5: Specialized control charts
EWMA control charts will from the subject of a future Part 6.
iv © ISO 2013 – All rights reserved

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 proposed the control chart as a graphical means of applying the statistical
principles of significance to the control of a process. Control chart theory recognizes two kinds of
variability. The first kind is random variability due to “chance causes” (also known as “common/natural/
random/inherent/uncontrollable 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 which 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 statistical control, or simply “in control”, when 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.
Statistical process control is a methodology for establishing and maintaining a process at an acceptable
and stable level so as to ensure conformity of products and services to specified requirements. The
major statistical tool used to do this is the control chart, which is a graphical 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 called process capability.
The control chart method helps first to evaluate whether or not a process has attained, or continues in,
a state of statistical control. When in such a state the process is deemed to be stable and predictable and
further analysis as to the ability of the process to satisfy the requirements of the customer can then 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.
INTERNATIONAL STANDARD ISO 7870-2:2013(E)
Control charts —
Part 2:
Shewhart control charts
1 Scope
This International Standard establishes a guide to the use and understanding of the Shewhart control
chart approach to the methods for statistical control of a process.
This International Standard is limited to the treatment of statistical process control methods using
only the Shewhart system of charts. Some supplementary material that is consistent with the Shewhart
approach, such as the use of warning limits, analysis of trend patterns and process capability is briefly
introduced. There are, however, several other types of control chart procedures, a general description
of which can be found in ISO 7870-1.
2 Normative references
The following referenced documents, in whole or in part, are normatively referenced in this document
and are indispensable for its application. 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 16269-4, Statistical interpretation of data — Part 4: Detection and treatment of outliers
ISO 5479, Statistical interpretation of data — Tests for departure from the normal distribution
ISO 22514 (all parts), Statistical methods in process management — Capability and performance
3 Terms, definitions and symbols
3.1 General
For the purposes of this document, the terms and definitions given in ISO 3534-2:2006 apply.
3.2 Symbols
NOTE The ISO/IEC Directives makes 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.
In cases of long established practice where a symbol and/or abbreviated term means different things in different
applications, it is necessary to use a field limiter, thus 〈  〉, to distinguish between them. This avoids the alienation
of practitioners by the creation of unfamiliar abbreviated terms and symbols in their particular field that are
unlike all related texts, operational manuals and dedicated software programs. An example is the abbreviated
term ‘R’ and symbol ‘R’ which means different things in metrology from that in acceptance sampling and statistical
process control. The abbreviated term ‘R’ is differentiated thus:
R 〈metrology〉 reproducibility limit
R 〈SPC and acceptance sampling〉 range
For the purposes of this document, the following symbols apply.
n
Subgroup size; the number of sample observations per subgroup
k Number of subgroups
Lower specification limit
L
L Lower control limit
CL
Upper specification limit
U
U Upper control limit
CL
X Measured quality characteristic (individual values are expressed as (X , X , X ,.). Sometimes the symbol
1 2 3
Y is used instead of X
(X bar) Subgroup average
X
(X double bar) Average of the subgroup averages
X
μ True process mean value
σ True process standard deviation value
σ A given value of σ
Median of a subgroup

X
Average of the subgroup medians

X
R Subgroup range: difference between the largest observation and smallest observation of a subgroup
Average of the R values for all subgroups
R
R Moving range: the absolute value of the difference between two successive values
m
|X – X |,|X – X |, etc.
1 2 2 3
Average of the (n − 1) R values in a set of n observed values
m
R
m
s Sample standard deviation obtained from values within a subgroup:
()XX−
∑ i
s=
n−1
s Average of the subgroup sample standard deviations
σˆ Estimated process standard deviation value
p Proportion or fraction of units in a subgroup with a given classification
p Average value of the proportion or fraction
np Number of units with a given classification in a subgroup
2 © ISO 2013 – All rights reserved

p A given value of p
np A given value of np (for a given p )
0 0
c Number of incidences in a subgroup
c A given value of c
Average value of the c values for all subgroups
c
u Number of incidences per unit in a subgroup
Average value of the u values
u
u A given value of u
4 Nature of Shewhart control charts
A Shewhart control chart is a graph that is used to display a statistical measure obtained from either
variables or attribute data. The control chart requires data from rational subgroups 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 subgroup characteristics
are derived, such as the subgroup average, X , and the subgroup range, R, the standard deviation, s, or a
countable characteristic such as the proportion of units with a given classification.
A Shewhart control chart is a plot of the values of a given subgroup characteristic versus the subgroup
number. It consists of a centre line (CL) located at a reference value of the plotted characteristic. In
establishing whether or not a state of statistical control exists, the reference value is usually the average
of the statistical measure being considered. For process control, the reference value may be the long-
term value of the characteristic as stated in the product specifications; a value of the characteristic
being plotted based on past experience with the process when in a state of statistical control, or based
upon implied 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
Upper control limit (U )
CL
Centre line (CL)
Lower control limit (L )
CL
12 34 56 7
Subgroup number
Figure 1 — Outline of a control chart
The control limits on the Shewhart charts are placed at a distance of 3 sigma on each side of the centre
line, where sigma is the known or estimated standard deviation of the population. Shewhart chose to
use 3 sigma limits on the basis that it made economic sense with respect to balancing the cost of looking
for process problems when such problems do not exist and failing to look for problems when the process
is not performing as it should. 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 plotting statistic is approximately
normally distributed 3 sigma limits indicate that approximately 99,7 % 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 % risk, or an average of three times in a thousand, of a plotted
point being outside of either 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 sigma limits instead of 3 sigma
limits depends on costs of investigation and taking appropriate action vis-à-vis consequences of not
taking action.
It should be noted that some practitioners prefer to use the factor 3,09 instead of 3 to provide a nominal
probability value of 0,2 % or an average of one spurious observation per thousand, but Shewhart selected
3 so as not to lead to attempts to consider exact probabilities. Similarly, some practitioners use actual
probability values for the charts based on non-normal distributions such as for ranges and fraction
nonconforming. Again, the Shewhart control chart used ±3 sigma limits in view of the emphasis on
empirical interpretation.
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 sigma control limits are sometimes called the “action limits”.
Many times it is advantageous to mark 2 sigma limits on the chart also. Then, any sample value falling
beyond the 2 sigma limits can serve as a warning of an impending out-of-control situation. As such,
the 2 sigma limit lines are sometimes called “warning limits”. While no action is required as a result
of such a warning been given on the control chart, some users may wish to immediately select another
subgroup of the same size to determine if corrective action is indicated.
When assessing the status of a process using control charts, two types of errors are possible. The first
occurs when the process involved is actually in a state of control but a plotted point falls outside the control
limits due to chance. As a result, the chart has given a 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.
The second error occurs when the process involved is not in control but the plotted point falls within
the control limits due to chance. 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.
Because it is generally impractical to make a meaningful estimate of the risk and of the cost of the
second type of error in any given situation, the Shewhart control chart system is designed to control
the first of these errors. When normality is assumed and 3 sigma control limits are used, the size of this
first 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.
In fact the choice of k sigma limits instead of 3 sigma limits depends on costs of investigation and taking
appropriate action vis-à-vis consequences of not taking action.
When a process is in statistical control, the control chart provides a method, which in some senses
is analogous to continually testing a statistical null hypothesis that the process has not changed and
remains in statistical control. Because, in Phase 1, there is often uncertainty about such matters as the
probability distribution of the characteristic of interest, randomness, and the specific departures of the
process characteristic from the target value that may be of concern are not usually defined in advance,
the Shewhart control chart should not be considered to be a test of hypothesis in the purest sense.
Walter Shewhart emphasized the empirical usefulness of the control chart for recognizing departures
from an “in-control” process and de-emphasized making probabilistic interpretations.
4 © ISO 2013 – All rights reserved

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, 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
above, on rare occasions no assignable cause can be found and it must be concluded that the point outside
the limits represents the occurrence of a very rare event, a random cause, which has resulted in a value
outside of the control limits even though the process is in control.
When a process is to be studied for the first time with the objective of bringing the process into 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 trial control chart shows no signals and the process may then be considered to
be in control and hence is stable and predictable. Because some data may have to be removed from
consideration during Phase 1, the user may have to obtain additional data from the process to maintain
the reliability of the parameter estimates.
Once statistical control has been established, the final trial control chart centre line and control limits
identified in Phase 1 are taken as the control chart parameters 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 might prove to be both time consuming and
difficult. It is crucial, however, since the failure to remove special causes of variation will result in the
process variation being overestimated. 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.
Details for the procedure to establish control charts for a process will be discussed below.
5 Types of control charts
Shewhart control charts are basically of two types: variables control charts and attributes control
charts. For each of the control charts, there are two distinct situations:
a) when no pre-specified process parameter values are given;
b) when pre-specified process parameters values are given.
The pre-specified process values may be specified requirements or target values, or estimated values of
the parameters that have been determined over the long term from data when the process is in control.
5.1 Control charts where no pre-specified values are given
The purpose here is to discover whether observed values of the plotted characteristics, such as X , R or
any other statistic, vary among themselves by an amount greater than that which can be attributed to
chance alone. 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 into a state of statistical control.
5.2 Control charts with respect to given pre-specified values
The purpose here 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 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.
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.
5.3 Types of variables and attributes control charts
The following control charts are considered:
a) Variables control charts, used when the measurements are on a continuous scale:
1) average ( X ) chart and range (R) or standard deviation (s) chart;
2) individuals (X) and moving range (R );
m

3) median ( X ) chart and range (R) chart.
b) Attributes control charts, used when the measurements are countable or categorized data:
1) p chart for number of units of a given classification per total number of units in the sample
expressed as a proportion or percentage;
2) np chart for number of units of a given classification where the sample size is constant;
3) c chart for number of incidences where the opportunity for occurrence is fixed;
4) u chart for the number of incidences per unit where the opportunity is variable.
Figure 2 shows a process of selecting an appropriate control chart for a given situation.
6 © ISO 2013 – All rights reserved

VARIABLES ATTRIBUTE
DATA DATA
DEFECTIVES DEFECTS
SAMPLE SIZE
NO
n = 1
?
YES
SAMPLE SIZE SAMPLE SIZE
n ≥ 10
CONSTANT CONSTANT
NO NO NO
? ?
YES YES
YES
MEASUREMENTS MEASUREMENTS
IN FRACTIONS IN FRACTIONS
YES YES
? ?
NO NO
¯¯ ¯¯
X R X S INDIVIDUALS p np c u
CHART CHART CHART CHART CHART CHART CHART
Figure 2 — Types of control charts
6 Variables control charts
Variables control charts, or charts for variables data, and especially their most customary forms, the X
and R charts represent the classic application of control charting to process control.
Control charts for variables are particularly useful for several reasons:
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 attributes charts since specific information about
the process mean and variance are obtained directly. Variables charts will often signal a process
problem before the process has produced nonconforming items.
c) Although obtaining one item of measured data is generally more costly than obtaining one item of
go/no go data, the subgroup sizes needed for variables are almost always much smaller than those
for attributes, for an equivalent monitoring efficiency. This helps to reduce the total inspection
cost in some cases and to shorten the time gap between the occurrence of a process problem and
corrective action.
d) These charts will provide visual means to directly assess process performance regardless of the
specifications. A close look at variables charts, along with review of histograms at appropriate
intervals, will often lead to ideas or suggestions as to how to improve the process.
For all variables control chart applications considered in this International Standard, it is assumed that
the distribution of the quality characteristic is normal (Gaussian) and departures from this assumption
will affect the performance of the charts. The factors used for computing control limits were derived
using the assumption of normality. Since most control limits are used as empirical guides in making
decisions, reasonably small departures from normality should not cause 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.
Variables charts can describe process data in terms of both spread (process variability) and location
(process average). Because of this, control charts for variables are almost always prepared and analysed
in pairs – one chart for location and another for spread. The chart for spread is usually 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 may then be used in establishing control limits for
the chart for location.
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. The subscript “0” is
used in Tables 1 and 3 to designate the specified values, such as μ for the specified process mean or σ
0 0
for the specified process standard deviation.
Following are the most commonly used variables control charts.
6.1 Mean ( X ) chart and range (R) chart or mean ( X ) chart and standard deviation (s)
chart
X and R charts can be used when subgroup sample size is small or moderately small, usually less than 10.
X and s charts are preferable particularly in the case of large subgroup sample sizes (n ≥ 10), since the range
becomes increasingly less efficient at estimating the process standard deviation as the sample size gets
larger. Where electronic devices are available to calculate process limits, standard deviation is preferable.
Tables 1 and 2 give the control limit formulae and the factors for each of these variables control charts.
Table 1 — Control limit formulae for Shewhart variables control charts
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 XA±±Ro rX As
X
0 00
d σ DDσσ,
DR, DR
R
R 20 20 10
s Bs , Bs c σ BBσσ,
s
43 40 60 50
NOTE μ and σ are pre-specified values.
0 0
8 © ISO 2013 – All rights reserved

Table 2 — Factors for computing control chart lines
Obser- Factors for
Factors for control limits
vations centre line
in sub-
groups
Using Using
of size X Chart s Chart R Chart *
s * R *
n
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 – 3,267 – 2,606 – 3,686 – 3,267 0,7979 1,128
3 1,732 1,023 1,954 – 2,568 – 2,276 – 4,358 – 2,575 0,8862 1,693
4 1,500 0,729 1,628 – 2,266 – 2,088 – 4,698 – 2,282 0,9213 2,059
5 1,342 0,577 1,427 – 2,089 – 1,964 – 4,918 – 2,114 0,9400 2,326
6 1,225 0,483 1,287 0,030 1,970 0,029 1,874 – 5,079 – 2,004 0,9515 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 0,9594 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 0,9650 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 0,9693 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 0,9727 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 0,9754 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 0,9776 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 0,9794 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 0,9810 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 0,9823 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 0,9835 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 0,9845 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 0,9854 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 0,9862 3,689
20 0,671 0,180 0,680 0,510 1,490 0,504 1,470 1,549 5,921 0,415 1,585 0,9869 3,735
21 0,655 0,173 0,663 0,523 1,477 0,516 1,459 1,606 5,951 0,425 1,575 0,9876 3,778
22 0,640 0,167 0,647 0,534 1,466 0,528 1,448 1,660 5,979 0,435 1,567 0,9882 3,819
23 0,626 0,162 0,633 0,545 1,455 0,539 1,438 1,711 6,006 0,443 1,557 0,9887 3,858
24 0,612 0,157 0,619 0,555 1,445 0,549 1,429 1,759 6,032 0,452 1,548 0,9892 3,895
25 0,600 0,153 0,606 0,565 1,435 0,559 1,420 1,805 6,056 0,459 1,541 0,9896 3,931
* Not recommended for sample size n > 10.
6.2 Control chart for individuals (X) and control chart for moving ranges (R )
m
In some process control situations, it is either impossible, impractical, or it does not make sense to select
rational subgroups. It is then necessary to assess process control based on individual readings using X
and R charts.
m
In the case of control charts for individuals, since there are no rational subgroups to provide an estimate
of variability, control limits are based on a measure of variation obtained from moving ranges of two
consecutive observations. A moving range is the absolute value of the difference between successive
pairs of measurements in a series; i.e. the absolute value of the difference between the first and second
measurements, then between the second and third, and so on. From the moving ranges, the average
moving range R is calculated and used for the construction of control charts. Also, from the entire
m
collection of data, the overall average X is calculated. Table 3 gives the control limit formulae for control
charts for individuals and for control charts for moving ranges.
Some caution should be exercised with respect to control charts for individuals:
a) The charts for individuals are not as sensitive to process changes as charts based on subgroups.
b) Care shall be taken in the interpretation of charts for individuals if the process distribution is not normal.
c) Charts for individuals isolate process variability from an average of consecutive differences between
observations. Thus, it is implied that the data are time-ordered, and that no significant changes have
occurred in the process in between the collection of any two consecutive individuals. It would be
ill advised, for example, to gather data from two discontinuous campaigns of production of a batch
chemical product and to calculate a moving range between the last batch of the first campaign and
the first batch of the next campaign, if the production line has been stopped in between.
Table 3 — Control limit formulae for control charts for individuals
Statistic Estimated control limits Pre-specified control limits
Centre line U and L Central line U and L
CL CL CL CL
Individual, X
μ μσ±3
X XR±2,660
0 00
m
1,128σ 3, 686σ   0
R 3,267R   0
Moving Range, R 0 0
m m
m
NOTE 1 μ and σ are pre-specified values
0 0
NOTE 2 R denotes the average of moving ranges of 2 observations.
m

6.3 Control charts for medians ( X )
Median charts are alternatives to X charts for the control of a process location when it is desired to
reduce the influence of the extreme values in a subgroup. This might be the case for subgroups made of
many automated measurements of highly variable samples such as when measuring tensile strength.
Median charts are easy to use and do not require as many calculations, particularly for subgroups of
small size containing an odd number of observations. This can increase shop floor acceptance of the
control chart approach even more so when individual values in the subgroup are plotted together with
their median on the same chart. The chart then also shows the spread of process output and gives an
ongoing picture of the process variation. It should be noted that the
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