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Defining
Control Limits
In order to define the control
limits, we need:
- an ample history
of the process to define the level of common cause variation, and
- a basis for determining
how wide to set the control limits.
How many subgroups are
necessary to define a process? There are two issues to be resolved. The
first issue concerns the process. In order to distinguish between "special
causes" and "common causes", you must have enough subgroups
to define the "common cause" operating level of your process.
This implies that all types of common causes must be included in the data.
The second issue deals with statistics. The statistical "constants"
used to define control chart limits (such as d2 or c4) are actually variables,
and only approach constants when the number of subgroups is "large".
For a subgroup size of five, for instance, the d2 value approaches a constant
at about twenty-five subgroups (Duncan, 1986). When a limited number of
subgroups are available, use Short Run Techniques.
In order to define the
expected limits for a given set of process data, we can either attempt to
characterize the distribution , assume
Normality, or assume that the distribution makes little difference. There
are several techniques for fitting distributions to data, which are discussed
in Curve Fitting . For the X-bar
Charts, there is some statistical arguments for assuming Normality of the
plotted subgroup averages. The Central Limit Theorem holds that, regardless
of the underlying distribution of the observations, the distribution of
the average of large samples will be approximately Normal. Research using
computer simulations has verified this, demonstrating that the Normal Distribution
will provide for a good approximation to subgroup averages and that large
subgroups may be as small as four or five observations, so long as the underlying
distribution is not very skewed or bounded.
There is some contention
within the Quality community that the distribution of both the underlying
process and the subgroup averages is irrelevant to the understanding and
use of control charts. The debate itself might be viewed as rather esoteric,
since both sides would draw similar broad conclusions: the control chart,
particularly the X-bar chart, is a useful tool for detecting shifts in a
process. The pertinence of the debate, however, is in the details, and has
particular impact when applied to other control charts, including the Individual-X
chart and the more recently developed CuSum and EWMA charts.
The argument against the
use of probability models to define the control limits includes the following
remarks:
- Shewhart did not
rely upon the Normal Distribution in his development of the control
chart; instead, he used empirical (experimental) data, and generated
limits that worked for his process.
- Since the control
chart is not based on a distinct probability model, it is not necessary
to fit a distribution or make any assumptions about the process or its
data. The control limits that are calculated using the Shewhart equations
will always provide control limits that are robust to any differences
in the underlying distribution of the process.
- If you say that
the X-bar chart relies upon the Normal Distribution, you rely upon the
Central Limit Theorem. But the Central Limit Theorem would not apply
to the subgroup range or sigma calculation anyway, so how do you define
limits for the subgroup ranges (or sigma)?
- The control limits
are set in the "tail areas" of the distribution anyway, so
that any attempt to fit a distribution will be subject to errors in
these regions.
The argument for the use
of probability models to define the control limits notes the following:
- If control charts
defined by Shewhart were based entirely on empirical data, and not based
on any theory that would have broader implications for all processes,
they would be useful for only Shewhart's processes. This is not the
case; the control charts are based upon mathematical (or more precisely,
statistical) theory that transcends particular processes.
- The control limits
are determined mathematically, and the formula used for computation
is a direct application of Normal probability theory. Although this
mathematical model could be based on empirical evidence only, it is
not coincidence that the model perfectly applies to Normally distributed
statistics, and applies much less so as the statistic looks less Normal.
Consider how to estimate the control limits on an X-Bar chart:
- two parameters
are calculated: the overall average and the average within subgroup
standard deviation. Neither of these calculations demands that the
observations be Normally distributed; however, the Normal Distribution
is the only distribution perfectly described by only these two parameters.
- one parameter
is tabulated: the factor (either d2 or c4) used to convert the average
within subgroup variation to the expected variation of the process
observations, based on the subgroup size. The estimates of the d2
or c4 factors are derived based upon the assumption of Normality
of the observations.
- one parameters
is defined: the number of standard deviations at which to place
the control limits (usually 3). The placement of the control limits
at
 3
standard deviations from the center line is appropriate only for
a Normal distribution, or distributions whose shape is similar to
a Normal Distribution. Other distributions may respond to this signal
significantly more frequently even though the process hasn't
changed or significantly less frequently when the process
has changed. Given the intent of a control chart to minimize
false alarms, this is not desirable. See Tampering
.
The Western Electric Run
Tests, in fact, make use of the probability models to determine when the
pattern of groups in the control chart are non-random. Without knowing that
the subgroup averages should be Normally distributed on the X-bar chart,
you could not apply the Western Electric Run Tests; they would have no meaning
without an understanding of the probability model that is their basis.
Similarly, the argument
against using 2-sigma limits due to their impact on tampering would have
little meaning without an understanding of the underlying distribution of
the plotted subgroups. See Tampering
.
- It is true that
the Central Limit Theorem does not apply to the subgroup range or sigma
statistics. But what does that prove? Perhaps that the distribution
of the Range or Sigma is not sensitive to the assumption of Normality
of the observations?
- Curve fitting to
define Distributions, like any modeling technique, is subject to error,
and statistical error is likely to be higher where there is less data,
such as in tail regions of distributions. But there are techniques for
dealing with this situation. See also Curve
Fitting .
What are the implications
of this debate?
- If we use the X-bar
chart, little. Both sides agree that the X-bar chart is a very useful
tool, they just disagree why it is useful. As mentioned above, there
would also be a question as to the validity of Run Tests in the absence
of the probability model.
- If we use the Individual-X
chart, or try to estimate process capability, we must either assume
that the distribution does not matter, or fit a distribution. We can
easily compare a fitted curve to the Shewhart calculations to see which
best describes the process behavior. Note that the Shewhart calculations
exactly coincide with the calculations for the Normal distribution,
as pointed out above. See Curve Fitting
.
- The EWMA control
chart may have a couple of interesting uses, depending on your point
of view:
- When we are
forced to use subgroups of size one due to Rational Subgroup considerations,
the EWMA chart does not require that we fit a distribution to the
data. Instead, it plots exponentially-weighted moving averages,
which allows the use of Normal control limits via the Central Limit
Theorem. If we don't think that fitting a distribution is needed
to define control limits for individual observations, then this
use of the EWMA chart is not so interesting.
- A mathematical
understanding of the EWMA statistic would allow proof that the EWMA
control chart can be designed to be more sensitive to small process
shifts. This knowledge would be useful for detecting small process
shifts (shifts of approximately.5 to 1.5 sigma units) that would
otherwise be lumped into "common cause variation" using
the standard control limits. Note that this sensitivity is gained
without an increase in false alarms (See Tampering
). Those who do not believe in the distribution as the basis for
the control limits also would not accept the argument that this
chart is more sensitive, or even that this chart has any valid uses.
Instead, their contention would be that this chart has the possibility
of promoting tampering, since it responds to "special causes"
not detected through the standard Shewhart calculations.
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