Shewhart Control Charts
Let M be some measure of quality. We will also call M a control statistic. Let the
mean of M be M and the standard deviation be M. Then the central line (CL), the
upper control limit (UCL) and the lower control limit (LCL) are fixed at
UCL = M + kM
CL = M
LCL = M - kM
where k is the distance of the control limits from the central line, expressed in
standard deviation units. This configuration, known as Shewhart control chart, is
shown in Figure19. The estimation of M and M and fixing a value for k are
statistical problems. Say that M has a normal distribution. If we assume a value of 3
for k, the probability that an observed value of a statistic falling outside the control
limits is 0.0027. If k is fixed as 3.09, then the probability of fallout will be 0.002. In
order to compute the control limits, the unknown parameters M and M of the process
are estimated using a reasonable number of subgroups. Even if the original
characteristic of interest X does not follow a normal distribution, the value of k is
generally fixed as 3. Hence the control limits are known as 3-sigma limits. The central
limit theorem forms the underlying foundation for control charts and fixing a value of
3 for k is justified if M is an average of the various X values. That is, even if the
original characteristic of interest does not follow a normal distribution, the statistic
under consideration such as the mean will follow a normal distribution provided the
sample or subgroup sizes are large. In fact the central limit theorem requires the
averages be independent random variables, though not identically distributed. This is
the reason why great care must be taken in forming rational subgroups. Since the
assumption of normal distribution is not always easy to justify for the individual
characteristic values, 3-sigma control charts for individual values are to be used with
caution. If the computed value of the quality measure for a given subgroup breaches
the 3-sigma limits, action will be initiated to look for assignable causes. Hence, we
will also call the 3-sigma limits as action limits.
Figure.1 Shewhart Control Chart
The control charting procedure is actually a sequence of tests of hypothesis. If one is
using a Shewhart chart with mean as the quality measure, the chart is nothing but a
sequence of tests that the current mean is equal to the hypothesised mean (central
line). The control limits may therefore be viewed as confidence limits.
The parameters of the probability distribution of the quality characteristic may be
estimated using extensive past (or historical or retrospective) data. This may help to
hypothesise the constants of a control chart such as the mean, standard deviation etc.
Such control charts based on the hypothesised values will be called standards given
charts. We will also describe nominal or target values as standard values. But it is
necessary that the target or nominal values are consistent with the historical
performance.
Certain terms relevant to the technique of control charting are summarised below:
central line: A line on a control chart representing the long term average or a standard
(nominal or target) value of the quality measure.
control limits: These are the lines on a control chart which are used as the criteria for a
signal for action. That is, control limits help find whether the data indicate a state of
statistical control or not.
lower control limit: A (control) limit for the points below the central line.
upper control limit: A (control) limit for the points above the central line.
The Shewhart control charts can be classified as of variables or attribute type
depending on the type of quality measure M used. See the following table:
Control Chart Types
Quality
Name of
Measure M Chart
Average
chart
Type
Variables
Standard
S-chart Variables
deviation S
Range R
R-chart Variables
Proportion
p-chart Attribute
defective p
Number of
np-chart Attribute
defectives np
Number of count or
Attribute
defects c
c-chart
Defects per
u-chart Attribute
unit u
If the characteristic of interest has a standard value (such as nominal or target), then
such charts are distinguished from those charts having no standard values.
Note that the Shewhart control chart model is based on the following assumptions:
1. The control statistic is normally distributed so that the control limit constant k
can be fixed as 3.
2. The variation between subgroups is absent when the process is in control.
3. The control statistic is independently distributed (and there is no
autocorrelation in the production process).
If any of the assumptions is violated, the frequency of false alarms(chart showing an
out-of-control situation when it is actually under control) will increased. False alarms
result in unnecessary engineering investigation of the process variables, stoppage of
production etc and are expensive. Hence it is necessary to ensure that the control chart
assumptions are not grossly violated
Establishing a Control Procedure
The following are the steps for establishing control of a production process
recommended in ANSI/ASQC Standards A1 (1987) and B3 (1986). Steps 1 to 4 are
preliminary ones, steps 5 to 7 are for starting the control chart, while steps 8 to 11
relate to using a control chart during production. It should be noted that these steps are
merely guidelines, and the actual practice may vary depending on the type and nature
of the industry or process.
1. Selection of Quality Characteristic
Select the characteristic(s) affecting the performance of the product. Characteristics
selected may be features of materials or the component parts or even finished product.
(eg. the COUNT of cotton yarn produced by a spinning machine by which we mean
the number of 840 yard length in one pound of yarn).
2. Analysis of Production Process
For the quality characteristic(s) selected, study the production process in detail to
determine the kind and location of defects that are likely to give rise to special cause
of variation. Also:
Review the requirements imposed by the specification limits.
Study the relation between each production step and the selected characteristic, noting
where and how variation may arise from causes associated with raw material,
machine, human operation, etc.
Study the method of inspecting a unit, and avoid inspection errors due to faulty
gauges, human errors, etc.
Decide whether the entire output of a product forms a single process having a
common cause system or two or more distinct streams (eg different conveyor lines,
machines, shifts, operators, etc, may warrant separate controls).
Decide the earliest point in the production process at which inspection and testing can
be carried out.
3. Planning Subgrouping and Data collection
Decide what quality data should be recorded, and how to divide the data into
subgroups. Also bear in mind the following points:
Clear instructions for inspecting an individual unit and recording the result are
required. The record may be a measured value or a symbol to denote whether the unit
is conforming or not, depending on the type of chart to be used.
Decide how the observed values are to be grouped, so that the units in any subgroup
are produced under identical conditions. Subgrouping requires technical knowledge of
the production process and familiarity with production conditions. While making a
decision on subgrouping based on time, such as once during every half an hour, hour
or day or from every production lot, etc, the periodic factors associated with the
production process should not coincide with the time of sampling and such biases
should be avoided. If inspection of a unit is simple, like using a ‘go and no-go gauge’,
a sample of 4 or 5 every half hour or every hour is preferable. If sampling is done
frequently, a consecutive 4 or 5 or even 10 units may be grouped to form a subgroup.
When subgrouping is based on the order of production, use of small and frequent
subgroups is preferable to infrequent large subgroups.
Keep all records of inspection data (check sheets) in a complete form for future
reference so that one can identify the source, time, location of the trouble, etc, easily
and quickly.
4. Choice of Quality Measures
Decide the statistical measures of quality to be used for the control chart(s).
Sometimes the unit must be noted on an attribute basis and the control chart may be of
p or np type (to be discussed later).
5. Collection and Analysis of Preliminary data
Once decisions have been made on the choice of characteristic, subgroup planning
and the statistical measures used for the control chart, collect and analyse some
inspection data to establish a preliminary or trial control chart ie., central line and
control limits. It is desirable to have sample-by-sample data for at least 25 subgroups.
If past data are available, they may be directly used for subgrouping.
6. Establishment of Control Limits
From the analysis of the preliminary data, establish control limits as action limits. The
control limits are to be used as action limits for future inspection results as they are
plotted subgroup by subgroup. Use of 3-sigma limits is generally recommended,
employing the relevant formulae to compute the central line and control limits.
7. Preparing Control Chart for Use
On a suitable form or squared paper, lay out a chart with vertical scales at the left for
the statistical measures chosen and with a horizontal scale for the subgroup number,
possibly supplemented by date and sample number. Computer software also proves
handy for this purpose.
8. Plotting Points on the Control Chart and Taking Action
Decide on the general type of action that is to be taken if a point falls outside the
control limits.
The action may be on the lot of product sampled or on the production process. As
soon as the results of inspection are available from a sample, compute the value of the
statistical measure and plot a point on the chart at once. If a point falls outside control
limits, take the action that is deemed appropriate. Even though all the points fall
within the control limits, indications of trouble or change in the process may be
observed by unusual patterns such as the following:
a series of points falling close to one of the control limits
a long series of points falling above or below the central line
a series of points revealing a trend.
9. Assumption of Existence of Control
It is usually not safe to assume that a state of control exists before 25 successive
subgroup points plotted fall within the 3-sigma limits. In addition, if not more than 1
out of 35, or not more than 2 out of 100, fall outside the 3-sigma control limits, a state
of control may ordinarily be assumed to exist.
10. Review of Control Chart Standards
The standards initially adopted for constructing the control charts should usually be
reviewed after 10 to 25 points have been plotted. If special causes are located and
eliminated, it is necessary to alter the standard (limits). Even though a fairly high
degree of control is attained, it is desirable to have a definite schedule for review, such
as after every 50 or 100 points.
11. Record Keeping
It is often necessary to keep all control chart records as a permanent records of quality
history. These records are also useful in maintaining good producer and consumer
relationship.
Some of the concepts, such as computing the control limits, drawing a control chart,
etc, will explained in the following sections. Some theory behind control charting has
already been presented.
In the next section, the construction of
using an example.
and related control charts is discussed
Xbar and Related Control Charts
-chart is used to control the mean of a quality characteristic (ie the quality
measure M is the mean). This chart will be accompanied by either the range (R) chart
or standard deviation (S) chart which will monitor the increase in (within subgroup)
variability over time.
To construct both charts, it is necessary to estimate the mean () and standard
deviation () of the quality characteristic during a state of control (ie when the
process is subjected to only chance causes) using historical data. The approach
followed in SPC for estimating  and  will be explained considering the following
example.
The estimated values of  and  can be used in constructing the control charts limits.
Quick methods for computing the control limits using published control limit factors
will also be discussed in this section.
Cotton yarn is produced by a spinning machine (called a ring spinning frame). This
machine will draws slivers of cotton (which are produced by several preparatory
machines) and spins them into yarn of continuous lengths. A typical spinning mill will
have a number of spinning frames, each machine having a number of spindles with
the spinning operation done by each of the spindles. Let us consider controlling the
process of spinning yarn with respect to the quality characteristic COUNT of yarn
produced by a given spinning frame. The term COUNT means the number of 840
yard lengths in one pound of yarn. Obviously, the higher the COUNT, the thinner the
yarn. A unit for inspection purposes is fixed at a 120 yard length of yarn (called a lea).
Lea testing and measuring instruments are available which will quickly determine the
quality characteristics COUNT, strength, number of thick and thin places, etc.
Let the nominal or target COUNT be 40, ie a pound of yarn will give 40(840) = 33600
yards of length. Assume that the lower and upper specification limits for COUNT are
respectively 39.8 and 40.2.
The mill has been sampling five leas from randomly selected spindles from the
spinning frame for testing during a production shift. On some days/shifts samples
were not taken. Multiple samples were taken on a few shifts. Table.2 provides the
historical data collected by the mill. This Table indicates certain important process
conditions that were noted during the period of study.
The mill found that the input for the spinning machine, namely the yarn slivers
produced in the preparatory process, was not uniform during certain periods. Such
cases, indicated as 'input sliver problem', occurred intermittently. It took some time to
locate the sources of trouble in the preparatory stages and correct this problem.
Samples numbered 17 and 34 are associated with clear (engineering) evidence that
they represent unusual production conditions or measurement problems. These
samples must be dropped. The same is the case with subgroups associated with 'input
sliver problem' and hence the samples numbered 4, 14, and 21 are also dropped. All
cases where there is no strong technical evidence for lack of control such as sample 25
(casual operator employed) will be included in the analysis using control charts.
It is also very likely that certain unusual production conditions or special causes
would have existed during the period of data collection which are not evident in the
ordinary course of production operations. A trial control chart will be used to detect
the presence of special causes so that further technical investigation can be initiated to
locate and eliminate the sources of trouble.
TABLE.2 Retrospective Data for Trial Control Chart
Sample #
Date
Shift
Obs-1
Obs-2
Obs-3
Obs-4
Obs-5
1
28-09
1
40.029
39.928
39.998
40.053
40.004
2
28-09
2
39.984
39.995
40.089
40.034
40.080
3
28-09
3
39.971
39.966
39.996
39.957
39.918
4
29-09
1
41.009
40.895
40.990
41.041
40.997 input sliver problem
5
29-09
2
40.033
39.978
40.066
40.018
39.994
6
29-09
3
40.006
40.001
40.041
39.942
39.904
7
30-09
2
39.968
40.091
40.024
40.008
40.006
8
30-09
3
40.000
39.952
39.941
39.988
40.065
9
04-10
1
40.070
40.019
39.901
40.004
39.980
10
04-10
2
40.005
39.965
40.053
39.963
39.981
11
04-10
3
40.004
39.996
40.037
40.057
39.995
12
05-10
1
40.049
40.031
39.953
40.032
40.025
13
05-10
2
40.041
39.956
40.018
39.853
39.991
14
05-10
3
40.508
40.989
40.950
40.839
41.005 input sliver problem
15
06-10
2
40.027
39.932
39.994
39.999
40.010
16
06-10
3
40.092
40.016
40.030
40.002
40.014
17
07-10
1
40.054
*
*
*
*
18
07-10
2
40.072
39.967
40.073
39.922
39.918
19
07-10
3
40.018
39.939
39.965
40.064
39.994
20
08-10
1
40.060
40.005
40.010
39.973
40.082
21
08-10
2
39.031
38.076
38.996
39.012
39.598 input sliver problem
22
08-10
3
39.975
40.006
39.960
40.143
39.981
23
11-10
1
39.949
40.032
39.912
40.002
39.997
24
11-10
1
40.004
40.032
40.020
40.063
40.063
25
11-10
2
40.152
40.011
40.067
39.999
39.977
26
11-10
3
40.057
39.995
40.024
40.006
39.939
27
12-10
1
40.055
40.074
40.053
39.966
40.088
28
12-10
2
40.048
40.006
39.965
39.977
39.984
29
12-10
3
39.952
40.053
39.968
39.914
40.028
30
13-10
1
40.006
39.980
40.071
40.003
40.017
31
13-10
2
39.934
39.993
39.993
40.072
39.910
32
13-10
2
40.015
40.013
40.012
40.031
40.067
33
13-10
3
39.941
39.981
40.077
39.971
39.922
34
14-10
1
60.027
59.936
60.012
40.068
40.037 count mix up in lab
35
14-10
2
39.993
40.067
40.052
40.023
40.004
36
14-10
3
40.087
40.054
39.995
39.943
40.031
spinning motor
failed
casual operative
37
15-10
1
40.948
40.974
39.945
40.029
41.023
38
15-10
2
39.897
39.958
40.003
40.056
40.095
39
15-10
3
40.068
40.019
40.111
40.021
39.954
How COUNT varies during a shift is important for rational subgrouping and
effectiveness of the control charts. More studies must be done by collecting data in the
same shift at different time intervals to understand how other process variables such
as interference due to doffing, operatives, maintenance schedules, etc, affect COUNT.
They may suggest more frequent sampling during a given shift of operation, etc. With
the available data, the subgrouping is made based on shifts over time.
Let us first quickly explore the distribution of COUNT (without samples 4, 14, 17, 21
and 34) using the following plots:
Figure 2 Plots to Explore the Distribution of COUNT
Obviously the historical data appear to indicate mostly periods of stable production
(which can be modelled by a normal distribution) as well as some unstable periods
which may be associated with special causes. We will use the control charts to test
this.
In order to estimate the true mean () or standard deviation () of the quality
characteristic COUNT, the historical data will NOT be pooled. The retrospective data
may contain periods dominated by one or more special causes, and a pooled estimate
of the standard deviation can be used only when the process is in control, ie
dominated only by chance causes. The Shewhart control charts allow only variation
within a subgroup, and any extra variation between subgroups is inadmissible and is
attributed to the presence of special causes. For any given subgroup i, the usual
standard deviation Si (n-1 in divisor) or the range Ri will be used to estimate  . If
there are k (say) such subgroups, the mean of the k subgroup standard deviations (Si
values) or ranges (Ri values) will be used to estimate the process  . Similarly the
mean of the k subgroup means (
i (say) values) is used to estimate the true process
mean  . Consider the following table which gives the means, standard deviations and
ranges for the COUNT data. Note that this table omits the samples 4, 14, 17, 21 and
34 and relates to a total of 34 subgroups only. The subgroup numbers match with the
original sample numbers for easy reference.
Table 3 Table of Subgroup Means, Ranges and Standard Deviations
Old sample
number
Subgroup i
1
2
3
5
6
7
8
9
10
11
12
13
15
16
18
19
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
i
40.0024
40.0364
39.9616
40.0178
39.9788
40.0194
39.9892
39.9948
39.9934
40.0178
40.0180
39.9718
39.9924
40.0308
39.9904
39.9960
40.0260
Ri
Si
Old sample
number
Subgroup i
0.125
0.105
0.078
0.088
0.137
0.123
0.124
0.169
0.090
0.062
0.096
0.188
0.095
0.090
0.155
0.125
0.109
0.046971
0.047784
0.028343
0.034296
0.054888
0.044998
0.048915
0.061933
0.037320
0.027797
0.037417
0.073578
0.036060
0.035626
0.077378
0.048275
0.044153
22
23
24
25
26
27
28
29
30
31
32
33
35
36
37
38
39
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
i
40.0130
39.9784
40.0364
40.0412
40.0042
40.0472
39.9960
39.9830
40.0154
39.9804
40.0276
39.9784
40.0278
40.0220
40.0638
40.0018
40.0346
Ri
Si
0.183
0.120
0.059
0.175
0.118
0.122
0.083
0.139
0.091
0.162
0.055
0.155
0.074
0.144
0.329
0.198
0.157
0.074542
0.047564
0.026235
0.070279
0.043356
0.047620
0.032673
0.056727
0.033872
0.062883
0.023341
0.059923
0.031316
0.055453
0.123918
0.078305
0.058901
The overall mean of the 34 (= k) subgroups estimates  . Let us denote the grand or
overall mean as
=
where
(X double bar). That is
= (40.0024+40.0364 + ....+ 40.0018+40.0346) / 34 = 40.008
i
is the ith subgroup mean. The process standard deviation  is estimated as
=
where
/c4
is the average of the subgroup standard deviations, viz
where Si is the standard deviation of the ith subgroup and c4 is a constant that
ensures
an unbiased estimator of  . That is c4 = E(S)/. Hence, the constant c4 is
known as the unbiasing constant. c4 is purely a function of the subgroup size n and
values are given in Table in the appendix 1 to this chapter. For COUNT data,
= (0.046971 + 0.047784 + .... + 0.078305 + 0.058901) / 34 = 0.050372
giving
= 0.050372 / 0.94 = 0.0535872.
It is also possible to estimate the process sigma using ranges. That is, the estimator
will be
=
where
/d2
is the mean of the subgroup ranges given by
and d2 is the unbiasing constant for the range. That is, d2 = E(R)/ values are given in
appendix for selected subgroup sizes. For COUNT data, we find
= (0.125 + 0.105 + .... + 0.198 + 0.157) / 34 = 0.12715
yielding
= /d2= 0.12715 / 2.326 = 0.0546647. In general (ie if subgroup size n >
2), the range estimate of  is inefficient compared to the estimate based on the
standard deviation.
The control limits of an
-chart can be established using the above estimated values
of  and  . If if X ~ N( , ), then
limits for the
~ N( , /n). Hence the 3-sigma control
-chart are:
±3
Hence the control limits for the
are:
/n
-chart based on the standard deviation estimate of
,
.
For COUNT data, it is easy to see that
LCL = 40.008 - 3(0.0535872/ 5) = 39.936.
UCL = 40.008 + 3(0.0535872/ 5) = 40.080.
Hence the control limits for the
-chart based on the range estimate of are:
,
The
.
-chart control limits for the COUNT data are:
LCL = 40.008 - 3(0.0546647/ 5) = 39.935
LCL = 40.008 + 3(0.0546647/ 5) = 40.081.
It is easy to compute the control limits using the control limit formulae appearing in
Appendix. The table also gives certain constants (A2, A3, B1, B2 , B3, B4, D1, D2, D3,
D4) called control limit factors for computing control limits. For example, Appendix 1
gives the control limits (based on the
estimate of  ) as ±A2
with control
limit factor A2 = 0.577 (which is equal to 3/d2n). The control limit factors are useful
for manual computation of control limits.
The control limits will be displayed on a time sequence plot or run chart for the mean.
That is, the subgroup means
i
will be plotted against the subgroup number i with
reference lines for the control limits. The overall mean
will also be placed on the
chart to produce a central line. The following is a MINITAB
COUNT data based on the
-chart output for the
estimate of  .
Figure 4 Xbar-Chart
None of the plotted points (subgroup means
i ) crossed the control limits and hence
we call the mean COUNT to be under control. Note that the significance of subgroup
means is tested using the estimated  which is based on the variation within the
subgroups. In other words, the above chart indicates that there is no significant shift in
the mean COUNT level for the production period related to the 34 subgroups given
the amount of variation within a subgroup due to chance causes.
In order answer the question whether the variability within the subgroups is stable,
either an R-chart (range chart) or a S-chart (standard deviation chart) will be used.
These charts evaluate the variability within a process in terms of the subgroup ranges
and standard deviations. One of these two charts will accompany the
monitoring the variation within the subgroups over time.
S chart: The control limits are given by LCL = B3
line being
and UCL = B4
-chart for
with the central
. For the given subgroup size of 5, one finds the factors for control limits
B3 = 0 and B4 = 2.089. For COUNT data,
= 0.050372 and hence
LCL = 0(0.050372) = 0 and UCL = 2.089(0.050372) = 0.1052.
These control limits are then placed on a run chart of Si values with a reference central
line for = 0.050372. The following is the S chart for COUNT data drawn using
MINITAB:
Figure 5. S-Chart
breaches the UCL of 0.1052. This means that the variability within the process
represented by the 34 subgroups is not in control. Now consider the computation of
control limits for the R-chart.
R chart: The control limits are given by LCL = D3 and UCL = D4 with the
central line being . For the given subgroup size of 5, one finds the factors for
control limits D3 = 0 and D4 = 2.115. For COUNT data,
= 0.12715 and hence
LCL = 0(0.12715) = 0 and UCL = 2.115(0.12715) = 0.2689.
These control limits are then placed on a run chart of Ri values with a reference
central line for
= 0.12715. The following is the R-chart for COUNT data drawn
using MINITAB:
Figure 6 R-Chart
Again subgroup #32 signals that the variability within the process is not under control.
We will be using either
- and S-charts or
- and R-charts for monitoring the
process mean level and the variation within the process respectively. It is therefore
desirable to show the two charts on a single display as shown below:
Figure 7 Two Charts On a Single Display
From the above analysis of retrospective data, we find that the quality characteristic
COUNT is not under control. The reasons for an increase in variability for subgroup
#32 must be explored and then the charts must be revised. This is discussed in the
next section.
Revision of Control Charts
When a signal (which could possibly be a false alarm) is obtained for lack of control
from a control chart, one usually looks for the presence of special causes. In the event
of finding and eliminating a special cause, it is necessary to revise the control limits
deleting the subgroup(s) which signalled the presence of special causes. It may also
be possible that this identified special cause affected the process represented by
certain subgroups adjacent to the one that signalled its presence. Any such subgroup
which was influenced by the special cause variables must be dropped. In other words,
we identify a set of subgroups that represents a process subjected to only chance
causes.
An out-of-control point on the
-chart need not necessarily be an out-of-control
point on the R- or S-chart (and vice versa). Such a point need not be dropped from the
associated R- or S-chart; nor will it call for a revision. For a normally distributed
quality characteristic X, the mean
and the sample variance S2 are independently
distributed, and hence such an action may be justified. (In the revised charts given in
Figure 8 the out of control point #32 was dropped for both charts but this need not be
done.)
Consider the signal given by the subgroup #32 in case of the COUNT data. Assume
that a process variable was identified that caused an increase in variability for
subgroup #32. We will drop #32 and recompute the control limits without this
subgroup. If a thorough investigation reveales no trouble with the process, then the
signal given by the subgroup #32 is a false alarm and we will not recompute the
control limits and use them directly for any real time control of COUNT.
Figure 8 shows the revised control charts drawn for the COUNT data using
MINITAB. Subgroup #32 was ignored for control limit calculations but not for
display.
Figure 8 Revised Control Charts
Here all the points (other than subgroup #32) lie within the control limits (sometimes
another revision may be necessary if a further point falls outside the revised limits).
Hence, one sets the Standard values as for the mean and standard deviation as:
standard value for mean =
'
= 40.00
standard value for standard deviation = ' =
/c4 = 0 04814/ 0.94 = 0.0512
It is also possible to fix a standard value for the range of subgroup size 5 as = R'4=
d2'= (0.0512)(2.326) = 0.1191. This value sightly differs from the figure of 0.1210
displayed on the revised R-chart. This value can also be used as a standard value for
range. But estimation of the true process sigma using standard deviations is more
efficient than estimating it using ranges. While using MINITAB for standard values,
they should be input as historical mean and sigma in the appropriate windows.
Using the fixed standard values, one can draw
- and R-charts or
- and S-charts
for future use, ie for on-line or real timecontrol. That is, the production process will
be sampled for quality monitoring, ie subgroups are formed one by one. As soon as a
subgroup is formed, the point must be plotted on the control charts for the standards
known case. If a later subgroup is associated with a special cause, then it is necessary
to disregard the subgroup while revising the standards or during the periodic revision
of the chart once 50 or 100 subgroups are taken. Before any real time control of the
process takes place, it is also necessary to compare the standard values with the
constraints imposed by the specifications. This is discussed below:
By natural process limits (also called natural tolerance limits), we mean the limits
which include a stated fraction of the individuals in a population. For populations
following a normal distribution, the natural process limits are  . If placed around
the standard level, these limits identify the boundaries which will include 99.7% of
the individuals of a process that is properly centred and in a state of statistical control.
Natural process limits will be mostly used to compare the natural capability of the
process to specification limits. Consider Figure 9. Here the proportion defective
is
where
,
the cumulative standard normal distribution function.
Specification limits, as discussed earlier, are determined externally and have no
mathematical relationship with control limits. Specification limits should be
sufficiently wider than the natural process limits; otherwise, the percent defective
production will be high.
Figure 9 Natural Process and Specification Limits
Since control limits of the
chart are set at  ± 3/n the control limits will lie
within the natural process limits as can be seen from Figure 9a
Figure 9a Natural Process and Control limits
Whenever a signal is received from the control chart for lack of control, one should
see that the value falling outside the control limits does not exceed the lower or upper
specification limit. If this is so, it indicates a major shift in the process level. It is
usually desirable to have the state of the art of the process such that the natural
process limits are within the specification limits. The control limits of the
-chart
are also ideally set at a distance of
inside the natural tolerance limits.
This is to reduce the risk of accepting an unsatisfactory shift in process level near the
specification limits.
In the above discussion, we have assumed that the values of  and  are known, rather
than estimated using a large number of subgroups. If they are estimated, the interval
that includes a desired fraction of individuals will be called a statistical tolerance
interval and a constant different from 3 will be used in constructing such intervals.
This course does not cover this topic. We will discuss process capability ratios, which
are more popular than the statistical tolerance intervals, in a separate section. Before
any real time control begins, it is necessary to ensure that the process is capable of
meeting the specification requirements, ie it will avoid production of nonconforming
units.
Average Run Length (ARL) and Operating Characteristic (OC) Curves
The performance of a control chart is revealed by its ARL and OC curves.
By average run length (ARL), we mean the average number of times the process will
have been sampled before a shift in the process level is signalled by the control chart.
By operating characteristic (OC) curve, we mean a curve showing the probability of
accepting a process as a function of the process level. Recall that by false alarm we
mean the situation when a control chart gives a signal for lack of control when it is in
control. The probability of such a false alarm can be read from the OC curve.
If the process is operating at an acceptable level, the probability of accepting the
process as
in control should be high from its control chart. This is equivalent to saying that if the
process is operating at an acceptable level, then the average run length at that process
level should be very high. Similarly, if the process is operating at an unacceptable
level, the probability of accepting the process should be low and a signal for a shift in
the process level should come from the control chart with a smaller number of
subgroups drawn. That is, the ARL at unacceptable process level should be small.
If sampling is done for every h hours, then the quantity (ARL)h will be the average
time to signal (ATS). That is, the measure ATS expresses the ARL in time units. It is
also possible to express the ARL in terms of the number of units produced. That is,
the ARL can be multiplied by the number of units that will be produced during a
sampling interval so that the performance of the control chart can be evaluated in
terms of the quantity of units produced. We will be following the terminology of ARL
only, rather than the linear transformations of it.
OC curve of
Chart
Suppose that we have an
-chart configured only with action lines (that is, with only
3-sigma limits). Let 0 be the acceptable process level and 1 be the unacceptable
process level. Also let 1 = 0 +k where  is the population standard deviation
which is not affected by a shift in the average level of the process. The constant k
stands for the shift in the process level in standard deviation units. Let the signal from
the
chart for out of control be a point falling outside the control limits (upper or
lower). For the
chart, the upper and lower control limits are UCL = 0 + 3/n
and LCL = 0 - 3/n since
follows normal distribution with mean  and standard
deviation /n. Then the probability of accepting the process (ie a subgroup average
falling within the limits) at the unacceptable process level 1 is
Pa = Pr { LCL 
 UCL   =  =  +k } or
where denotes the standard cumulative normal distribution. Substituting for UCL
and LCL and simplifying one gets:
Pa =[+3-k n ] -[-3-k n].
Figure 10 shows the OC curves for a given level of (upward) shift k in standard
deviation units for various subgroup sizes. Note that large subgroup size n leads to
faster detection of shifts, ie it keeps Pa low.
Figure 10 OC Curves of Xbar-Chart for Various Subgroup Sizes
If k is zero, then there is no shift in the process level. In such a case, Pa is
[3]- [-3] = 0.99865-(1-0.99865) = 0.9973.
This means that if the process is operating at an acceptable level, the control chart
accepts it with a very high probability. Suppose that k = 2. This means that the
process has shifted upwards to an unsatisfactory level above two standard deviations
to the acceptable level. Now Pa stands for the probability of failing to pick up a shift
in the process level with a single sample after the shift has occurred. Let the subgroup
size be 5. In this case Pa will be
 [3-2 5]- [-3-2 5] = ( -1.47214)- ( -7.4721)
= 0.070492-0.000000
 0.0705,
which is a low figure. That is, the control chart will detect the shift with the very high
probability of 0.9295. In case of a shift in process average, 1- Pa is the probability of
detecting the shift with one sample (subgroup) only. If the shift is not detected with
the first subgroup, it may be detected with the second subgroup and so on. Thus we
may have a run of points falling within the control limits and then a shift may be
detected with a point falling outside the limits. One therefore has the run length (RL)
distribution (geometric) given in Table 4. Here the definition of run length includes
the last subgroup that gave a signal.
Table 4 Run Length Distribution
Run Length Probability
1
(1-Pa)
2
Pa(1-Pa)
3
Pa2(1-Pa)
4
Pa3(1-Pa)
.
.
.
.
m
.
.
.
Pam-1(1-Pa)
.
.
.
The mean and variance of the run length are:
ARL = E(RL) =  m Pam-1(1- Pa ) = 1/ (1- Pa ).
V(RL) = Pa / (1-Pa )2.
For example, let the process level shifts to two standard deviations on the upper side.
Then the average run length will be 1/ 0.0705 = 14.18. That is, the chart will detect
the shift in the process average with about 14 subgroup averages plotted on the chart.
The variance of the above RL is 185.37.
For example, to draw the OC curve of an
-chart having UCL = 11, LCL = 9,
subgroup size = 5 and  = 0.70, consider the following:
Pa = Pr (LCL 
 UCL)
= ( ZUCL) - ( ZLCL)
where
and
For the given problem '/n = 0.3130495. For assumed  values, Pa can be computed
Using  and Pa values, the OC curve is then drawn as shown in Figure 11. This OC
curve gives the Pa values when the shift is either upward or downward. Note that the
Pa is high at the central value 10 and starts to drop if the process level shifts from 10.
If one considers an unacceptable process level, say 11, the Pa is still high, which is not
desirable.
Figure 11 OC Curve of an Xbar-Chart
OC Curve of R-chart
Table 5, giving the percentage points of the distribution of relative range W = R/ 
(for normal distribution), is useful in constructing the OC curve of an R-chart.
Table 5 Percentage Points of W
w values for
Pr[W
w]
0.999
0.995
0.990
0.975
0.950
0.900
0.750
0.500
0.250
0.100
0.050
0.025
0.010
0.005
0.001
n=4 n=5 n=6 n=10
5.31
4.69
4.40
3.98
3.63
3.24
2.62
1.98
1.41
0.98
0.76
0.59
0.43
0.34
0.20
5.48
4.89
4.60
4.20
3.86
3.48
2.88
2.26
1.70
1.26
1.03
0.85
0.66
0.55
0.37
5.62
5.03
4.76
4.36
4.03
3.66
3.08
2.47
1.93
1.49
1.25
1.06
0.87
0.75
0.54
5.97
5.42
5.16
4.79
4.47
4.13
3.59
3.02
2.51
2.09
1.86
1.67
1.47
1.33
1.08
The probability that R will be less than or equal to UCL is the same as W being less
than or equal to a limit of UCL/ . This implies that  = (UCL/w). Using the above
table, w values are obtained and then the OC curve of an R-chart is drawn. For
example, let n = 4 and UCL = 2.0. A table of true R and Pa values is obtained as
shown below:
Table 6 Points for drawing OC Curve of an R-chart
w for n =
=
4
(UCL/w)
0.999 5.31
0.3766
0.995 4.69
0.4264
Pa
true R= d2(d2=
2.3059)
0.8685
0.9833
0.990
0.975
0.950
0.900
0.750
0.500
0.250
0.100
0.050
0.025
0.010
0.005
0.001
4.40
3.98
3.63
3.24
2.62
1.98
1.41
0.98
0.76
0.59
0.43
0.34
0.20
0.4545
0.5025
0.5510
0.6173
0.7634
1.0101
1.4184
2.0408
2.6316
3.3898
4.6512
5.8824
10.0000
1.0481
1.1587
1.2705
1.4234
1.7602
2.3292
3.2708
4.7059
6.0682
7.8166
10.7251
13.5641
23.0590
Charts with Probability Limits
The standard control chart factors are based on the assumption that the control statistic
is normally distributed and hence "three sigma" limits can be used. This may be hard
to justify in the case of sample variance, range etc. One therefore makes use of the
exact distribution of the control statistic, and the control limits fixed in this manner
are known as probability limits.
When working with an S-chart, one can avoid using the factors for control limits
namely B1 to. B4 and use the probability limits directly. It is known that if X ~ N( ,
), then
with n-1 d.f.
It therefore follows that
If the process sigma is estimated, then the control limits become
and
where
=
/c4is the usual unbiased estimate of sigma. For a subgroup size of 4, one
has c4 = 0.921 and 20.001 = 0.024 and 20.999 = 16.266. Once
computation of the control limits
will be straight forward.
is computed, the
Some directly use the sample variance S2 instead of S and draw an S2 chart. Note that
S2 is an unbiased estimate of 2 . For given k rational subgroups, one may compute
the mean of the sample variances
and set the control limits at
where n is the subgroup size. The control limits of the S-chart based on probability
limits are not the square root of the corresponding control limits of the S2 chart
since
is not equal to
An R-chart with probability limits may also be constructed using the percentage
distribution of the relative range W. For example, when the subgroup size is five, the
probability limits for an
R-chart having 0.002 false alarm probability are
LCL = 0.37
/d2
UCL = 5.48
/d2.
and
Supplementary Run Rules for Shewhart Control Charts
The operation of a control chart with only action limits is slow in detecting shifts of
smaller order in the process level. In control charts with only action limits, the signal
comes only if a point falls above or below the control limits. In order to improve the
performance or sensitivity of the chart, it is usual to consider warning limits which are
set at ±2 as well as at ±1 (the symbol  is used to represent the standard deviation
of the control statistic, ie M). The rules for using these limits differ according to
practice. In general, the warning limits are used in the same manner as control limits,
but a count of continuous or interrupted runs of points lying between the warning and
action limits is also taken as a signal for action. A control chart with the three zones
for warning is shown in Figure 12
Zone A : +2 to +3 and -2 to -3 limits
Zone B : +1 to +2 and -1 to -2 limits
Zone C : -1 to +1 limits.
Figure 12 Control Chart with Warning Lines
When the process is in control, the probability that a point will fall in each of the
zones can be found. If the characteristic of interest X ~ N( , ), then
~ N( , n ) or
The control limits for the
statistic
~ N( 0 ,1).
-chart are
and ±3 for the standardised
. Hence, one has the following table:
Table 7 Warning Zone Probabilities
Zone
A
B
C
beyond A
Probability of a point falling in the Zone
2[(3) - (2)] = 2[0.99865-0.97725] = 0.04280
2[(2) - (1)] = 2[0.97725-0.84134] = 0.27182
2[(1) - (0)] = 2[0.84134-0.50000] = 0.68268
2(-3) = 0.00270
The following is a list of tests (or the supplementary run rules, also called Western
Electric run rules) generally adopted for use:
Test or Rule 1: one point beyond zone A (usual signal with action
limits).
Test or Rule 2: nine points in a row in Zone C and beyond.
Test or Rule 3: six points steadily increasing or decreasing.
Test or Rule 4: fourteen points in a row alternating up and down.
Test or Rule 5: two out of three points in a row in Zone A or beyond.
Test or Rule 6: four out of five points in a row in Zone B or beyond.
Test or Rule 7: fifteen points in a row in Zone C (above and below
central line).
Test or Rule 8: Eight points in a row on both sides of central line with
none in Zone C.
The sample graphs showing the eight different rules are shown in Figure 13. The
following points on the application of the run rules are noteworthy (see Nelson
(1984)).
1. When the process is in statistical control, the probability of a false alarm is less
than five in a thousand for each of the rules.
2. If rules 1 to 4 are routinely applied then the probability of a false alarm is
about one in a hundred.
3. If the first four rules are supplemented by rules 5 and 6 when it is desirable to
have earlier warning, it will increase the probability of a false alarm to about
two in a hundred.
4. Rules 7 and 8 are diagnostic tests for stratification. These rules show when the
observations in a subgroup have been taken from two or more processes. Rule
7 reacts when the observations in the subgroup always come from both
processes. Rule 8 reacts when the subgroups are taken from one process at a
time.
Figure 13 Illustration of Run Tests
One should be careful while using several or all of the above run rules simultaneously.
If k rules with probability of false alarm i for rule i are used, then the overall false
alarm probability will be
provided the rules are assumed to be "independent". Assuming that the probabilities
of all the false alarms are equal to 0.005 (being the maximum), the total probability of
false alarm will be
 = 1-(1-0.005)8 = 0.039.
That is, nearly four out of 100 cases could be a false alarm. Hence, one should be
careful in applying a number of tests simultaneously, particularly tests 7 and 8. The
commonly used run rules are 1, 5, 6 and 8. MINITAB has a provision to perform one
or more of the above tests when the subgroup sizes are equal. (see Figure 14).
Figure 14 MINITAB Run Test Sample Output
Whenever run rules are used, no closed form expression for the OC function or Type I
error is available. Further, the run rules are not independent. For example, rules 5 and
8 cannot be assumed to be independent. In such a situation, obtaining an overall value
of Type I error as
will be approximate and may be
avoided. This is one of the reason why SPC researchers prefer to use the concept of
ARL in place of the OC curve or Type I error. Champ and Woodall (1987) have
provided a method of obtaining the exact ARLs for Shewhart charts with run rules for
a normally distributed quality characteristic. The procedure is based on Markov chain
formulation and is computationally intensive and hence not discussed here. Table 8
gives the ARL values for an
chart with certain run rule combinations and for
various levels of shifts in process level. This table is based on the method of Champ
and Woodall and produced using a SAS procedure.
Table 8 Average Run Lengths for Various Shift Levels and Run rules
Process level shift (in 
terms)
Rule 1
Rules 1 & 5
Rules 1 & 6
Rules 1, 5 & 8
0 or no shift
370.352
(369.653)
225.410
(224.226)
166.034
(163.582)
122.036
(118.347)
0.5
155.205
(154.621)
77.715
(76.501)
46.176
(43.538)
36.171
(32.335)
1.0
43.889
(43.363)
20.003
(18.824)
12.663
(10.202)
11.725
(8.491)
2.0
6.302
(5.778)
3.646
(2.633)
3.680
(1.923)
3.502
(2.199)
(Standard Deviations of Run Lengths are in brackets.)
It is noted that the in-control ARL (corresponding to shift level zero) is decreasing
with the application of run rules. The gain is that of reducing the ARL corresponding
to a shifted process level, which is the main purpose of the run rules. In practice, one
may perform a sensitivity analysis: that is, varying the process level and finding what
types of signals (false negative or false positive) are obtained from the chart.
For successful application of supplementary run rules, symmetric distribution of the
control statistic at least is necessary.
Zone Control Chart
Jaehn (1991) introduced the zone control chart with the objective of signalling at
roughly the same time as a Shewhart chart with the commonly used run rules. A zone
control chart has eight zones, four on each side of the central line with boundaries of
the central line, one and two sigma limits (see Figure 15). Scores will be assigned to
each zone in the following manner:
Zone
Score
Between central line and one sigma
0
Between one and two sigma
2
Between two and three sigma
4
Beyond three sigma
8
The score for points between the central line and one sigma can also be fixed at one
instead of zero. For the first observation of the control statistic, the score is shown in a
circle drawn in the appropriate zone. For the subsequent observations of the control
statistic, the circled scores will be cumulated. That is, the score for the latest one is
added to the previous score. This cumulation will take place only when an observation
falls on one side of the central line and the cumulation will stop if a point falls on the
other side of the central line. At this point, the score restarts on the latest actual score.
This scoring method is simple to apply and no pattern checking or counts of points are
needed to apply the supplementary run rules. Once the cumulative score reaches or
exceeds 8, the zone control chart signals action for an assignable cause.
It may be seen that the scoring method roughly corresponds to the application of run
rules, viz:
1. A point falling outside the three sigma limits,
2. Two of three consecutive points falling outside the two sigma limits on the
same side of the central line,
3. Four out of five consecutive points falling outside the one sigma limits on the
same side of the central line, and
4. Eight consecutive points falling on the same side of the central line.
This does not mean that zone control chart and the Shewhart chart with the usual tests
will give the signals at the same time. For instance, the score is reset if a point is on
the opposite side. The usual run rules 5 and 8 discussed earlier do not reset. The zone
control chart basically combines the points of different zones but not necessarily the
run rules.
Figure 15 Zone Control Chart
Davis et al. (1994) have shown that zone control charts with score zero instead of one
(for points between the central line and one sigma) outperform the Shewhart charts
whenever the shift to be detected is of the order of two standard deviations or less.
Also because of its simplicity, the zone control chart is a viable alternative to
Shewhart charts. However does not compare favourably with more complicated charts
such as CUSUM and EWMA (to be discussed later).
Fast initial response (FIR) is a technique used to provide a rapid signal in case of an
initial out-of-control situation whenever decisions are based on a cumulative type of
control statistic such as a score considered in the zone control chart. That is, the FIR
feature gives a "head start" by initialising the score at a state where a signal is very
likely after the few observations if the process is not in control. It is also possible to
introduce the FIR feature to the zone control chart; the FIR feature will be further
discussed in the section on CUSUM charting.
It is also possible to adopt a different weighting pattern for initial process conditions
and yet another weighting pattern to detect shifts of smaller order once the process
condition is stable. This practice is currently empirical and more research is required
in the area.
Control Chart for Individual Measurements (X-chart or I-chart)
The I-chart for individual observations monitors the process level in terms of a single
measurement per sample. This chart is equivalent to using an
-chart with subgroup
size equal to one. Control charts for individual measurements are used under any of
the following situations:
Technology permits 100% inspection of individual units.
Slow production rate and hence samples can accumulate.
Chemical processes and continuous process conditions such as paper production etc.
For a series of measurements X1, X2, X3, ....Xk, we compute
.
That is, the process mean  is estimated using the sample mean. The sample standard
deviation S, given by
,
can be used to estimate the true sigma through the unbiasing constant c4 (c4
corresponding to sample size k). That is, we estimate  as S/c4 and set the control
limits at
± 3S/c4
with the central line at
range given by
. It is also common to estimate  using the average moving
.
That is,  is estimated as
then set at
=
/d2 (d2 for sample size 2). The control limits are
±3
/d2
with the central line at
.
Example: Consider the series of measurements of a quality characteristic (one-at-atime data) given in Table 9
Table 9 Data for I-Chart
4.1 5.9 5.0 4.4 5.4 5.0 4.1 5.0 6.1 4.5
4.2 3.5 3.0 4.3 5.6 6.6 7.3 5.0 3.9 4.7
6.1 5.3 2.5 6.2 5.4 5.7 5.2 3.1 4.5 4.9
5.2 4.1 5.2 6.1 5.3 7.3 5.6 5.0 4.9 5.0
6.4 5.3 4.3 6.3 3.9 4.7 7.1 4.5 4.1 5.4
(Read from left to right)
It is easy to compute
= 5.0440 , S = 1.0475 giving
= S/c4 = 1.0475 / 0.9949 =
1.0528. The control limits are therefore 5.044 3(1.0528). That is, LCL = 1.8856 and
UCL = 8.2024. The following control chart is then drawn for the data in Table 9
Figure 15 I-Chart Based on the Standard Deviation Estimate of Sigma
It is easy to compute the average of the moving ranges (absolute values) as
= (1.8 + 0.9 + 0.6 + 1.0 + .... + 2.6 + 0.4 + 1.3)/49 = 1.1735
and compute
=
/d2 = 1.1735 / 1.128 = 1.0404. The control limits of the Ichart (based on the moving range estimate of sigma) are 5.044 3(1.0404), ie LCL =
1.9228 and UCL = 8.1652. The individual values are then shown on the I-chart in
Figure 16
Figures 15 and 16 both suggest that the quality characteristic under consideration is
under control.
One may also conduct the supplementary run tests (preferably using MINITAB).
Figure 16 MINITAB I-Chart Output
The assumption of normality and independence for individual measurements must be
justified for using the Shewhart chart for individual measurements. Random errors
such as measurement error are likely to be present in individual values. Further, the
moving range values may have correlated errors. These aspects should be noted
before applying the chart for individual measurements.
The normality assumption of the control statistic is more relevant in the case of a
Shewhart chart for individual values. There are several methods available for testing
the normality assumption such as regression test, chi square goodness of fit test,
distance tests, moment test, etc. MINITAB provides for three tests namely, 1) the
Anderson-Darling test, 2) the Ryan-Joiner test and 3) the Kolmogorov-Smirnov test.
To apply these tests, a value for alpha, the Type I error, needs to be chosen (usually
10% or above) and compared with the P-value that is displayed in the MINITAB
normal plot. If the displayed P-value is smaller than the alpha value chosen, one
rejects the assumption of normality.
On violation of the normality assumption, transformations need to be tried (log,
square root, square etc). Box and Cox (1964) provided maximum likelihood methods
for choosing normalising transformations. The Box-Cox power transformation is
useful for correcting non-normality as well as unstable variation in the process data.
MINITAB provides for optimal estimation of the appropriate power  of the
transformation
Y()=
Y
{ log Y
e
= 0
=0
with common transformations such as square root, etc, being particular cases as seen
below:
Value of  Transformation
2
Y2
0.5
Y
0
loge Y
-0.5
1/Y
-1
1/Y