ISO 7870-5:2014

INTERNATIONAL ISO
STANDARD 7870-5
First edition
2014-01-15
Control charts —
Part 5:
Specialized control charts
Cartes de contrôle —
Partie 5: Cartes de contrôle particulières
Reference number
©
ISO 2014
© ISO 2014
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ii © ISO 2014 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms . 4
4.1 Symbols . 4
4.2 Abbreviated terms . 5
5 Specialized control charts . 5
6 Moving average and moving range control charts . 5
6.1 Control limits . 6
6.2 Interpretation . 6
6.3 Advantages . 6
6.4 Limitations . 6
6.5 Example . 6
7 z- chart. 9
7.1 Control limits . 9
7.2 Advantages .10
7.3 Limitations .10
7.4 Example .10
8 Group control chart .10
8.1 Control limits .12
8.2 Advantages .12
8.3 Limitations .13
8.4 Example .13
9 High-low control chart .16
9.1 Control limits .16
9.2 Interpretation .17
9.3 Advantages .17
9.4 Limitations .17
9.5 Example .17
10 Trend control chart .19
10.1 Control limits .20
10.2 Advantages .20
10.3 Limitations .21
10.4 Example .21
11 Control chart for coefficient of variation .24
11.1 Control limits .24
11.2 Advantage .24
11.3 Limitation .24
11.4 Example .25
12 Control chart for non-normal data .26
12.1 Control limits .27
12.2 Example .28
13 Standardized p- chart .32
13.1 Control limits .34
13.2 Advantages and limitations .34
13.3 Example .34
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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committee has been established has the right to be represented on that committee. International
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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
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For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 69, Applications of statistical methods,
Subcommittee SC 4, Applications of statistical methods in process management.
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 control charts
— Part 5: Specialized control charts
— Part 6: EWMA control charts
iv © ISO 2014 – All rights reserved

Introduction
The Shewhart control charts as given in ISO 7870-2 aid in detection of unnatural patterns of variations
in data from repetitive processes and provide criteria for detecting a lack of statistical control.
However, there may be several special situations for variables data where Shewhart control charts
may be inadequate, insufficient or less efficient in detecting the unnatural patterns of variation of the
process, particularly where:
a) it takes considerable time to produce an item and as such sample results are available at large
intervals;
b) there are several subgroup sources that have approximately the same production rate, process
average and process capability;
c) process average is changing systematically;
d) sample size is large and sequence of production is irrelevant;
e) process does not have a constant target value.
In such situations, specialized control charts are to be used.
Similarly, special situations may be encountered in dealing with attributes data. There may be situations
when criticality of an incidence in a subgroup (nonconformity) is a matter of concern, but different
nonconformities are having different criticality. As such, all types of nonconformities cannot be treated
alike. Depending upon criticality, different ratings (weights) are required to be given to each class of
nonconformity, and accordingly demerit scores are calculated. The control limits are calculated based
on such demerit scores and accordingly control charts are plotted to exercise process control.
There may be situations when inspection by attributes is preferred to that by variables, from practical
considerations, for controlling both the location and the variability parameters of a measureable
characteristic of a process (for example, inspection by gauging). The information is also available on the
number of items less than the lower specification limits (no-go gauge) as well as the number of items
above upper specification limit (go gauge) in assembly operations. In such situation, a specialized pair
of control charts may be used.
There may also be situations when data do not follow normal distribution. Such situations of non-normal
data are quite often encountered in service industry, besides in special processes of manufacturing. In
such a situation specialized control chart is to be used.
This part of ISO 7870 has been prepared to provide guidance on the use of specialized control charts to
address above typical, unusual situations.
INTERNATIONAL STANDARD ISO 7870-5:2014(E)
Control charts —
Part 5:
Specialized control charts
1 Scope
This part of ISO 7870 establishes a guide to the use and understanding of specialized control charts in
situations where commonly used Shewhart control chart approach to the methods of statistical control
of a process may either be not applicable or less efficient in detecting unnatural patterns of variation of
the process.
The specialized control charts included in this part of ISO 7870 for variables data are:
a) moving average and moving range charts;
b) z-charts;
c) group control charts;
d) high–low control charts;
e) trend control charts;
f) control charts for coefficient of variation;
g) control charts for non-normal data.
For attributes data, specialized control charts included in this part of ISO 7870 are:
a) standardized p-charts;
b) demerit control charts;
c) control charts for inspection by gauging.
This part of ISO 7870 also provides guidance as to when each of the above control charts should be used,
their control limits, advantages and limitations. Each control chart is illustrated with an example.
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
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-2 and the following
apply.
3.1
control chart
chart on which some statistical measure of a series of samples is plotted in a particular order to steer
the process with respect to that measure and to control and reduce variation
Note 1 to entry: The particular order is usually based on time or sample number order.
Note 2 to entry: The control chart operates most effectively when the measure is a process variable which is
correlated with an ultimate product or service characteristic.
[SOURCE: ISO 3534-2:2006, 2.3.1]
3.2
Shewhart control chart
control chart with Shewhart control limits intended primarily to distinguish between the variation in
the plotted measure due to random causes and that due to special causes
[SOURCE: ISO 3534-2:2006, 2.3.2]
3.3
variables control chart
Shewhart control chart in which the measure plotted represents data on a continuous scale
[SOURCE: ISO 3534-2:2006, 2.3.6]
3.4
attributes control chart
Shewhart control chart in which the measure plotted represents countable or categorized data
[SOURCE: ISO 3534-2:2006, 2.3.7]
3.5
Xbar control chart
average control chart
variables control chart for evaluating the process level in terms of subgroup averages
[SOURCE: ISO 3534-2:2006, 2.3.12]
3.6
R chart
range control chart
variables control chart for evaluating variation in terms of subgroup ranges
Note 1 to entry: The value of the subgroup range, given by the symbol R, is the difference between the largest and
smallest observation of a subgroup.
Note 2 to entry: The average of the range values for all subgroups is denoted by the symbol R
.
[SOURCE: ISO 3534-2:2006, 2.3.18]
3.7
moving average control chart
control chart for evaluating the process level in terms of the arithmetic average of each successive n
observations
Note 1 to entry: This chart is particularly useful when only one observation per subgroup is available. Examples
are process characteristics such as temperature, pressure and time.
Note 2 to entry: The current observation replaces the oldest of the latest n + 1 observations.
Note 3 to entry: It has the disadvantage of an unweighted carry-over effect lasting n points.
[SOURCE: ISO 3534-2:2006, 2.3.14]
2 © ISO 2014 – All rights reserved

3.8
moving range control chart
variables control chart for evaluating variation in terms of the range of each successive n observations
Note 1 to entry: The current observation replaces the oldest of the latest n +1 observations.
[SOURCE: ISO 3534-2:2006, 2.3.20]
3.9
z-chart
variables control chart for evaluating the process in terms of subgroup standardized normal variates
3.10
group control chart for averages
variables control chart for evaluating the process level in terms of subgroup (with several sources)
highest and lowest averages with corresponding source identification
3.11
group control chart for ranges
variables control chart for evaluating the process variation in terms of subgroup (with several sources)
highest ranges with corresponding source identification
3.12
high – low control chart
variables control chart for evaluating the process level in terms of subgroup largest and smallest values
3.13
trend control chart
control chart for evaluating the process level with respect to the deviation of the subgroup averages
from an expected change in the process level
Note 1 to entry: The trend may be determined empirically or by regression techniques.
Note 2 to entry: A trend is an upward or downward tendency, after exclusion of the random variation and cyclical
effects, when observed values are plotted in the time order of the observations.
[SOURCE: ISO 3534-2:2006, 2.3.17]
3.14
control chart for coefficient of variation
variables control chart for evaluating variation in terms of subgroup coefficient of variation
3.15
p chart
proportion or percent categorized units control chart
attributes control chart for number of units of a given classification per total number of units in the
sample expressed either as a proportion or percent
Note 1 to entry: In the quality field, the classification usually takes the form of “nonconforming unit”.
Note 2 to entry: The “p” chart is applied particularly when the sample size is variable.
Note 3 to entry: The plotted measure can be expressed as a proportion or as a percentage.
[SOURCE: ISO 3534-2:2006, 2.3.11]
3.16
standardized p-chart
attributes control chart where proportions of given classification are expressed as standardized normal
variates
Note 1 to entry: In this chart, the centre line is zero, upper control limit is +3 and lower control limit is −3.
3.17
demerit control chart
quality score chart
multiple characteristic control chart for evaluating the process level where different weights are
apportioned to events depending on their perceived significance
[SOURCE: ISO 3534-2:2006, 2.3.23]
3.18
control chart for inspection by gauging
attributes control chart when the inspection is done by gauging and the information is available on the
number of units above upper gauge limit and below lower gauge limit
4 Symbols and abbreviated terms
4.1 Symbols
n subgroup sample size
k number of subgroups
x
individual measured value
average value of i-th subgroup
x
i
average of the subgroup average values
x
μ
true process mean value
σ
true process standard deviation value
R range
average range
R
s
sample standard deviation
s average of subgroup sample standard deviations
p
proportion or fraction of units
p
average value of the proportion or fraction of units
C centre line
L
upper control limit
U
CL
lower control limit
L
CL
average value of the variable X plotted on a control chart
X
x largest observation in a subgroup
H
smallest observation in a subgroup
x
L
4 © ISO 2014 – All rights reserved

average of largest observations for all subgroups
x
H
x average of smallest observations for all subgroups
L
z
variable that has a normal distribution with zero mean and unit standard deviation
v
coefficient of variation
v
average of coefficient of variation values
4.2 Abbreviated terms
BPO business process outsourcing
CV coefficient of variation
L lower gauge limit
GL
U upper gauge limit
GL
5 Specialized control charts
The following specialized control charts for variables have been included:
a) moving average and moving range control charts;
b) z-charts;
c) group control charts;
d) high–low control charts;
e) trend control charts;
f) control charts for coefficient of variation;
g) control charts for non-normal data.
The following specialized control charts for attributes have been included:
a) standardized p-chart;
b) demerit control chart;
c) control chart for inspection by gauging.
6 Moving average and moving range control charts
In certain cases of industrial production it takes considerable time to produce a new item or the tests
are destructive in nature. As a result, it is inconvenient to sample frequently to accumulate sample of
size n > 1. In the meantime process average or dispersion may have changed and this may incur some
appreciable loss. Under such situations subgroups, each consisting of individual observations, are used
for process monitoring.
In these situations, use of moving averages and moving ranges instead of Shewhart control charts has
been suggested. Moving averages of k subgroups (each of size one) are obtained as follows. Initially,
the values of first k subgroups are averaged. Then in the second step the value for the first subgroup
th
is dropped in favour of the value for (k+1) subgroup and an average obtained. Next, the value for the
th
second subgroup is dropped and the value for (k+2) subgroup is included and these values are averaged,
and so on. In a similar manner moving ranges are obtained. The rate of production helps to decide the
number of subgroups to be considered at a time for moving average and moving range. Additionally, the
lesser the magnitude of shift in process average and variation one wishes to detect, the higher will be
the value of k.
6.1 Control limits
6.1.1 Moving range chart
C = R
L
UD= R
CL 4
LD= R
CL 3
6.1.2 Moving average chart
C = x
L
U =+xRA
CL 2
L =−xRA
CL 2
where, R is the homogenized range. The values of A , D and D are given in Annex A for various sample
2 3 4
sizes (n) = k.
6.2 Interpretation
Unlike the case of Shewhart control charts, here successive moving averages and moving ranges are
not independent. Hence, in moving average and moving range control charts, runs on either side of
the centre line do not have the same interpretation as is given by Shewhart control chart. However, a
point beyond control limits here has the same significance as in case of Shewhart control chart. Cyclic
pattern and/or increasing or decreasing trend in the moving range chart is indicative of potential for
improvement. However, the assignable causes for the moving average chart and those for moving range
chart may be different.
6.3 Advantages
In some situations a control chart for moving average and moving range is more efficient. It gives a
warning signal earlier than with usual (,XR) charts. It is not necessary to wait until an entire new
sample is accumulated. This may be important if the product is either expensive or the rate of output is
small.
6.4 Limitations
Successive points are not independent. Since the probability of obtaining a run of any kind is much larger
with control chart for moving average or moving range as compared to the Shewhart control charts, the
traditional interpretation of runs is not valid for these control charts.
6.5 Example
The crown of the watchcase is used to adjust the time. The pin of the crown is fitted through a hole in the
watch case. The diameter of the hole has to be maintained at 0,005 ± 0,001mm. Table 1 gives the data in
6 © ISO 2014 – All rights reserved

order of production, where reaming operation is done to make the hole for the pin of crown to fit in the
watchcase. It is decided to plot control charts for the moving average and moving range by averaging
diameter values from 3 consecutive subgroups.
Table 1 — Subgroup results from diameter of the hole for the pin of crown
Sub Hole diameter Sum of 3 mov- Moving average Moving range Remarks
ing observa-
group
tions
number
1 0,003
2 0,005
3 0,001 0,009 0,0030 0,004
4 0,003 0,009 0,0030 0,004
5 0,002 0,006 0,0020 0,002
6 0,005 0,010 0,0033 0,003
7 0,006 0,013 0,0043 0,004 Shift change
8 0,003 0,014 0,0047 0,003
9 0,004 0,013 0,0043 0,003
10 0,005 0,012 0,0040 0,002
11 0,005 0,014 0,0047 0,001
12 0,006 0,016 0,0053 0,001
13 0,001 0,012 0,0040 0,005
14 0,002 0,009 0,0030 0,005 Tool changed
15 0,007 0,010 0,0033 0,006
16 0,001 0,010 0,0033 0,006
17 0,003 0,011 0,0037 0,006
18 0,004 0,008 0,0027 0,003
19 0,003 0,010 0,0033 0,001
20 0,001 0,008 0,0027 0,003
21 0,006 0,010 0,0033 0,005
22 0,005 0,012 0,0040 0,005
23 0,004 0,015 0,0050 0,002
24 0,002 0,011 0,0037 0,003
25 0,001 0,007 0,0023 0,003
Total 0,0829 0,080
6.5.1 Control limits for moving range control chart
0,080
C ==R =0,0035
L
U ==D R 2,575×0,0035=0,0090
CL 4
L ==D R 0×0,0035=0
CL 3
The above values of D and D are taken from Annex A for n = 3. As all range values are less than U ,
3 4 CL
the value of average homogenized range is taken as 0,0035 for computation of control limits for moving
average control chart.
6.5.2 Control limits for moving average control chart
0,0829
C ==x =0,0036
L
U =+xRA =+0,,0036 1023×=0,,0035 00072
CL 2
L =−xRA =−0,,0036 1023×=0,0035 0
CL 2
The value of A is taken as 1,023 from Annex A for n = 3. The control chart is plotted in Figure 1.
0.0100
0.0090
0.0080
U
CL
0.0070
0.0060
0.0050
0.0040
C
L
0.0030
0.0020
0.0010
0.0000 L
CL
0123456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Subgroup number
8 © ISO 2014 – All rights reserved
Moving average
0.01
0.009 U
CL
0.008
0.007
0.006
0.005
0.004
C
L
0.003
0.002
0.001
L
0 CL
012345678 910111213141516171819202122232425
Subgroup number
Figure 1 — Moving average and moving range control charts
6.5.3 Interpretation
Process appears to be in state of statistical control.
7 z- chart
There are situations where processes are needed to be controlled when there are large varieties of
products with different specifications, production runs are small and varying sample/lot sizes. When
there are substantial differences in the variance of these products, using the deviation from the process
target becomes problematic. There may also be situations when the process does not have a constant
target value; instead, the target value keeps on changing with time.
In such cases the commonly used charts like X,R or Xs, fail to provide a basis for viewing and validly
() ()
interpreting control chart and appropriate decision making. A suitable chart to see the pattern and take
decisions is the z-chart. The idea is to standardize the data to compensate for the different product
parameters in terms of averages and variability and transform each point to a standard normal variate
by using the transformationzx=−()μσ/ , provided the expected value of the standard deviation is
known at that time point. If process is under control then standard normal variates will lie between +3
and –3. These types of charts are referred to as z-charts.
7.1 Control limits
C =0
L
U =+3
CL
L =−3
CL
Moving range
7.2 Advantages
The z-chart has the advantage of simplified calculation, presentation and most important it facilitates
usual interpretation of Shewhart control charts to control the processes and decision-making.
7.3 Limitations
When there is no historical data, this chart may be difficult to apply because it requires past data for
estimating variability.
7.4 Example
Graphite rods used in steel manufacturing are baked in furnaces. At different time points (subgroups),
the temperature inside the furnace differs. The target values of temperature and the estimates of the
inherent standard deviation of the furnace temperature at different times are also given in Table 2.
Standard normal variate (z) values for different subgroups have been calculated and are given in Table 2.
It is then seen whether they lie between +3 and –3. If it is above +3 or below −3, then it is out of control
situation. The control chart is plotted in Figure 2.
8 Group control chart
In industrial production it may happen that the data presented for the purpose of controlling the quality
comes from a number of sources, say from multi-spindle machine with same standard output, or several
workers or several machines. In such cases, unless proper steps are taken in choosing the sample, it is
difficult for the quality engineer to single out the problem when the control chart shows lack of control.
One obvious way is to maintain a separate chart for each possible source of variation, which is rather
uneconomical and time-consuming. A group control chart, first devised with a view to controlling
the dimensions on multiple-spindle automatics, having wide applicability, provides an answer to the
problem.
The group control charts are valid only when there are enough reasons to presume that the averages of
each source of data, and also the variability of each source are uniform. Instead of maintaining a pair of
average and range charts for each possible source, (such as machine or worker) only one pair of average
and range charts is maintained. In the average chart, the highest and lowest average values are plotted
along with suitable source identifications (such as serial number of spindles/machines/workers) and the
largest range is plotted on the range chart. In the average chart, the highest values are connected by a
line, so also, the lowest values in order to avoid confusion. The underlying idea is that if corresponding to
a particular sample the highest value is below the Upper Control Limit (U ), the others are necessarily
CL
so. Similarly, if the lowest value is above the Lower Control Limit (L ), others are necessarily so. The
CL
identifying number attached to the highest value that is beyond the U or to a lowest value that is below
CL
the L at once detects the trouble-yielding source. It calls for an attention if a particular identifying
CL
number is appearing more frequently either in the high value or in low value. If high and the low values
together show a cyclic pattern for the same identifying number, this provides vital clue for attention.
Table 2 — Subgroup results from the temperature in side furnace
Subgroup Time Target Standard devia- Observed Remarks
x−μ
()
z=
number (h) value(μ) tion from past value (x)
σ
data (σ)
1 2 205 2,12 200 −2,36
2 4 210 7,07 200 −1,41
3 6 210 8,48 210 0,00
4 8 220 6,36 215 −0,79
5 10 220 7,07 215 −0,71
6 12 230 7,07 220 −1,41
10 © ISO 2014 – All rights reserved

Table 2 (continued)
Subgroup Time Target Standard devia- Observed Remarks
x−μ
()
z=
number (h) value(μ) tion from past value (x)
σ
data (σ)
7 14 230 6,36 225 −0,79
8 16 230 17,68 240 0,57
9 18 240 11,31 245 0,44
10 20 240 10,61 260 1,89
11 22 240 7,07 265 3,54 Heating system mal-
functioned
12 24 240 3,53 245 1,42
13 26 240 5,53 255 2,71
14 28 250 8,08 260 1,24
15 30 250 12,65 270 1,58
16 32 250 13,62 285 2,57
17 34 260 10,5 285 2,38
18 36 260 10,07 285 2,48
19 38 270 8,48 285 1,77
20 40 270 6,36 285 2,36
21 42 270 7,07 285 2,12
22 44 270 7,07 285 2,12
23 46 280 6,36 300 3,14
24 48 280 7,67 300 2,61
25 50 320 4,95 330 2,02
26 52 380 4,95 350 −6,06
27 54 460 5,15 430 −5,83
28 56 480 6,7 460 −2,99
29 58 550 8,1 530 −2,47
30 60 550 5,1 545 −0,98
31 62 550 4,8 555 1,04
32 64 550 5,25 550 0,00
33 66 550 4,5 545 −1,11
34 68 550 6,02 540 −1,66
35 70 550 8,07 530 −2,48
36 72 460 7,8 450 −1,28
37 74 340 10,2 350 0,98
38 76 300 8,76 310 1,14
NOTE Events such as change in raw material, shift, operator, etc. may be recorded in “Remarks” to facilitate
traceability of assignable cause at that stage.
6.00
4.00
U
CL
2.00
0.00
C
L
-2.00
L
CL
-4.00
-6.00
-8.00
1357 91113151719212325272931333537
Subgroup number
Figure 2 — z chart
8.1 Control limits
8.1.1 Group control chart for range
C = R
L
UD= R
CL 4
LD= R
CL 3
8.1.2 For group control chart for averages
k
C ==xx
L ∑ i
i=1
U =+xRA
CL 2
L =−xRA
CL 2
The values of the factors A , D , and D are given in Annex A for various sample sizes
2 3 4
8.2 Advantages
The advantages are:
a) It involves less work in plotting.
12 © ISO 2014 – All rights reserved
z value
b) A compact presentation of all information from a group of machines on a single chart makes the
interpretation easier.
c) It is easier to find out whether a particular source is giving consistently high or low values on
average or range chart. If there is no real difference among the sources, the numbers corresponding
to the various sources should occur on the charts almost equally in the long run.
8.3 Limitations
The limitations are:
a) The group control charts requires that there should be several subgroup sources that yield
approximately equal number of subgroups at approximately the same rate, such as different
spindles on one automatic machine, several identical machines, or several operators each doing
the same operation. There should be consistency among the averages or dispersions of various
subgroups which can be sustained. For example, if there are 10 machines on the same job but two
of the machines have different process capability, then the group control chart cannot be applied
for all the 10 machines and should not include these two machines. The two machines, which give
different process capability, should be treated separately.
b) Experience and skill are needed for interpretation.
c) Conventional interpretation of runs above or below the centre line is not applicable.
8.4 Example
Table 3 gives two measurements of the diameters of two pieces produced on each of six spindles of an
automatic screw machine. The values given are in units of 0,001 mm in excess of 12 mm. The highest
and lowest average values are indicated in Table 3 as H and L respectively. The highest ranges are also
indicated by H in Table 3. It is decided to plot group control chart.
8.4.1 Control limits for group control chart for range
Cm==R =09, 7 icrons=0,00097mm
L
UD==R 3,,267×0973= ,17microns=0,00317mm
CL 4
LD==R 00× ,97=0microns
CL 3
The values of D and D for sample size (n) = 2 from Annex A are 0 and 3,267 respectively. Since all range
3 4
values are less than U , the ranges are homogeneous. The average range can therefore be used for
CL
computing control limits for group control chart for average.
8.4.2 Control limits for group control chart for average
195,5
Cm==x ==54,,30icrons 00543mm
L
UA=+xR=+54,,31 88×09,,77==25micronsm0,00725 m
CL 2
LA=−xR=−54,,31 88×09,,73==61micronsm0,00361 m
CL 2
The value of A from Annex A for sample size 2 is 1,880.
As the values are in excess of 12 mm, the actual control limits for the group control chart for average are:
C = 12,0054 mm
L
U = 12,0072 mm
CL
L = 12,0036 mm
CL
Table 3 — Subgroup results from diameter measurements (microns in excess of 12 mm)
Average
Subgroup Spindle Diameter Range Remarks
number number
x
Piece 1 Piece 2 R
1 6 7 6,5 H 1
2 4 6 5,0 2 H
3 6 4 5,0 2 H
4 5 4 4,5 L 1
5 6 5 5,5 1
6 4 5 4,5 L 1
1 6 6 6,0 H 0
2 6 6 6,0 H 0
3 5 6 5,5 1
4 5 5 5,0 L 0
5 5 6 5,5 1
6 7 5 6,0 H 2 H
1 5 6 5,5 1 H
2 6 6 6,0 H 0
3 5 5 5,0 L 0
4 6 5 5,5 1 H
5 5 5 5,0 L 0
6 6 6 6,0 H 0
1 5 6 5,5 1
2 6 5 5,5 1
3 5 5 5,0 0
4 4 4 4,0 L 0
5 5 7 6,0 H 2 H
6 6 4 5,0 2 H
1 5 6 5,5 1
2 5 4 4,5 L 1
3 6 5 5,5 1
4 7 4 5,5 3 H
5 7 6 6,5 H 1
6 5 7 6,0 2
14 © ISO 2014 – All rights reserved

Table 3 (continued)
Average
Subgroup Spindle Diameter Range Remarks
number number
x
Piece 1 Piece 2
R
1 5 5 5,0 L 0
2 6 5 5,5 1
3 4 7 5,5 3 H
4 7 6 6,5 H 1
5 5 5 5,0 L 0
6 6 5 5,5 1
Total 195,5 35
The group control chart for average and range are plotted in Figure 3. In this figure, in the group
control chart for average, the highest and lowest average values along with suitable source identifying
indications (spindle number) are plotted. Similarly in the group control chart for range, the highest
range along with suitable source identifying indications of spindle number is plotted.
8.4.3 Interpretation
There is no evidence of any out of control situation
12.008
UUU
CLCLCL
12.007
(1) (5) (4)
(2,6)
(5)
(1,2,6)
12.006
C
L
12.005
(4)
(3,5) (1,5)
(4)
(2)
12.004
(4,6)
L
CL
12.003
12.002
12.001
Subgroup number
Average
0.0035
U
CL
0.003
(4)
(3)
0.0025
(5,6)
0.002 (6)
0.0015
0.001
C
L
(1,4)
0.0005
L
C
L
Subgroup number
Figure 3 — Group control chart for averages and ranges
9 High-low control chart
There may be situations when sample size is large and the sequence of production is not traceable. For
example, in batch production (e.g. zinc coating, heat treatment for annealing) the sequence of production
is lost. Also, since several batches get mixed up, the systematic variation becomes an inherent part of
further processing. In such situations, it is desirable to use a control chart for largest and smallest values
or the high-low control chart, as it is popularly called, in place of the conventional Shewhart control
charts.
9.1 Control limits
9.1.1 Average and standard deviation not known
When the values of the process average and dispersion are not known from the past data, they are
estimated with the help of the initial data collected and the control limits are computed as follows:
()xx+
HL
CM= =
L
UM=+  H R
CL 2
LM=− H R
CL 2
where Rx=−x where,x and x denote the largest (high) and smallest (low) values respectively in
HL H L
each subgroup, and x and x are the average of largest and smallest values respectively for all
H L
subgroups. The values of H are given in Annex A.
16 © ISO 2014 – All rights reserved
Range
9.1.2 Average and standard deviation known
If the values of process mean and standard deviation are known as μ and σ respectively, the control
limits are:
C = μ
L
U = μ + H σ
CL
L = μ – H σ
CL
The values of H are given in Annex A.
9.2 Interpretation
Although an upper control limit for x and a lower control limit for x are usually drawn, it is possible
H L
to have an upper and a lower control limit for each x and x separately. The control limits forx and
H L H
1 1
   
x are given by xR±−Η and xR±−Η respectively. In such a case, shift in the process is
L H  2  L  2 
2 2
   
indicated if both x and x are above the respective upper control limits or below the lower control
H L
limits. On the other hand, if x is above the relevant upper control limit and x is below the relevant
H L
lower control limit, this is enough evidence to conclude the increase in process variability.
A run of 6 or 7 points of high as well as low points closer to the centre line shows improvement in the
process. Control limits for the subsequent process may be consequentially changed. If there is a trend in
both x andx , dip or rise simultaneously, then it shows a shift in the average. Likewise, any increasing
H L
or decreasing trends and cyclic patterns should be investigated for special causes. If x and x points
H L
are very close to centre line, then either sampling method is not appropriate or the data are not authentic.
9.3 Advantages
This type of chart is extremely simple since no calculations are needed for plotting the points on control
chart. Besides, only one chart needs to be maintained in this case in place of the two conventional charts
since information concerning both process level and variation are provided on the single chart.
It has been found that under most conditions, high-low control charts are nearly as good as the XR,
()
charts for detecting lack of control
...