MGT 6421
Quality Management II
Introduction to Control Charts
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Control charts are the basic tools of statistical process control. They are
simple to construct and use, and are very useful at leading to process
improvement.
•
We know that all processes exhibit variation from the target, but how much
variation should we tolerate before we adjust the process? In the language
of statistical process control, how far can the process deviate from target
and still be "in control?"
•
First we will define some terms:
Common Causes of Variation are those which are due only to random or
chance causes. This variation is expected and therefore predictable.
Special Causes of Variation are those which come from influences outside
those ordinarily encountered by the system.
Introduction to Control Charts - 1
MGT 6421
Quality Management II
•
In control, or under statistical control, means that the variable we are
measuring is centered on the mean, and its variability is due only to common
causes.
•
Our goal with statistical process control is to differentiate between special
and common causes. If we can eliminate special causes, the process will be
under statistical control, and will be improved. In addition, the process will
be predictable, and further improvements will be possible by finding ways to
reduce common causes of variation.
•
The fundamental relationships in developing control charts are as follows:
•
The center (or central) line is the mean of the process.
•
The upper control limit (UCL) is the mean plus 3 standard deviations.
•
The lower control limit (LCL) is the mean minus 3 standard deviations.
So the limits are µ
± 3σ.
Introduction to Control Charts - 2
MGT 6421
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Quality Management II
The reason for using 3 standard deviations is based on the assumption that
the process output is normally distributed. If the process is in control, there
is a very small probability of an observation falling outside of the control
limits. This probability increases as the process deviates from the in-control
state.
5
Number of standard deviations from the mean
4
UCL
3
2
1
center
line
0
-1
-2
-3
LCL
-4
-5
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MGT 6421
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Quality Management II
There are two main categories of control charts:
1. Control charts for attributes, which are used when we are counting the
number of nonconformities or nonconforming items (i.e., when the variables
of interest are discrete).
2. Control charts for variables, which are used when we are making actual
measurements (i.e., when the variables of interest are continuous).
Introduction to Control Charts - 4
MGT 6421
Quality Management II
Control Chart Decision Variables
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There are several variables that we have under our control when we design
control charts. These variables include the control limits, the sample size,
and the sampling interval.
•
We have already discussed the control limits to some degree. The criterion
we have discussed uses statistical techniques. There are other criteria that
might be considered. One obvious one is economics. Another might be to
take into account the actual distribution rather than assuming a normal
approximation.
•
The choice of sampling frequency is often dictated by the process itself.
There are natrual times at which it is relatively easy to measure the process.
Rules of thumb of every hour or every day are often employed. In
traditional control chart theory, the sampling interval is assumed to be fixed.
•
The question of sample size also depends upon the physical process, and
usually takes into account questions of economics and statistical theory. On
the economics size, we need to keep the sample size low enough that
sampling costs do not become too high. On the statistical side, we need the
sample size to be large enough to better detect shifts in the process (which
can be translated into an economic problem), and to meet the assumptions
of the normal distribution approximation. Again, rules of thumbs have been
developed.
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MGT 6421
Quality Management II
Rational Sampling
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Sampling for control chart construction is quite different from sampling for
traditional statistical analysis. With control charts, we are working with what
Deming called analytic studies (he called traditional statistical studies
enumerative studies). The main difference is that we are collecting and
analyzing data over time, rather than at one fixed interval.
•
As a result, we need a sampling method that will allow us to detect changes
over time. Hence, we want the within-group variation to be small, and the
between-group variation to be large. It is the between-group variation that
we see on the control chart.
•
The type of sampling that is used in control chart construction is called
“rational sampling,” and the resulting samples are called “rational samples”
or “rational subgroups.” These rational subgroups are formed by inspecting
several (n) units in a row taken from the process at a particular point in time.
Then, after the sampling interval has passed, another n units in a row are
taken from the process.
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MGT 6421
Quality Management II
Out-of-Control Patterns
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Observations lying outside of 3 standard deviations are not the only type of
aberration which suggests an out-of-control condition. Any deviation from
randomness or normality suggests that the process may be out of control.
•
If the process output is normally distributed and independent, we expect
most of the points to be near the center line, some points near the control
limits, no points outside the control limits, and no runs or trends.
•
•
Other rules have been developed to identify these other types of out-ofcontrol patterns. Some of these rules are summarized on the following page.
In order to determine the zones mentioned, I suggest the following:
a. First obtain the standard error by taking the difference between the UCL
and the center line, and divide by 3.
b. Then the zone C boundaries are the center line ± one standard error and
the zone B boundaries are the center line ± two standard errors.
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MGT 6421
Quality Management II
From Nelson, L. S. (1984), "The Shewhart Control Chart--Tests for Special Causes," Journal of
Quality Technology, Vol. 16, No. 4, 237-239.
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Quality Management II
Out-of-Control Conditions
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Now we want to talk about their possible causes of the patterns discussed
above.
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The different rules discussed above can help identify certain kinds of out-ofcontrol conditions. We will identify what rules are associated with what
condition.
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Shifts
A shift indicates a change in the mean of the variable being monitored.
Shifts can be of several types, including sudden shifts and gradual shifts.
Points outside the control limits or points on one side of the centerline
indicate sudden shifts. What might cause such a shift?
•
A trend on a control chart will signal a gradual shift.
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MGT 6421
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Quality Management II
Cycles:
Cycles are repeated patterns. Depending on the agreement between the
cycle frequency and the sampling frequency, they can appear as several
points on one side of the centerline, points alternating back and forth, or just
a repeating pattern (not necessarily covered by any of the rules that we have
discussed).
•
Wild Patterns:
There are two types of these: freaks and bunching. Freaks are single points
that are very different from any other points. Points outside the control
limits usually detect these. Often they are caused by recording errors.
Bunching is similar, but affects more than one point. Normally the points
that are affected are close to each other.
•
Instability Patterns
Instability is characterized by wide swings up and down, with many points
being outside of the control limits. These patterns can be caused by
disturbances that create wide multimodal distributions in a process variable.
One example of such a cause is over adjustment of a machine.
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MGT 6421
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Quality Management II
Mixtures:
Very few points near the centerline or too many points hugging the centerline
often characterize mixtures. Mixtures might also be indicated by a “sawtooth” pattern. Mixture patterns indicate the presence of two or more
distributions. An obvious example of this is if we are using two or more
suppliers. If we sample separately from the two suppliers, we are likely to
see either the saw-tooth pattern or too many points near or beyond the
control limits.
•
If we combine the supplies from the two samples, then the expected pattern
would be too many points near the centerline. This is called stratification.
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Example:
Below are some simulated data of two suppliers. The parts from the
suppliers are mixed to form batches of 100.
Supplier 1
n=50, p=.8
38
41
36
39
39
34
40
39
39
40
37
32
36
37
45
38.13
0.76
Supplier 2
n=50, p=.01
0
0
0
0
2
1
2
0
0
0
0
0
0
0
0
0.33
0.0067
Introduction to Control Charts - 11
Mixture (n=100)
38
41
36
39
41
35
42
39
39
40
37
32
36
37
45
38.467
0.38467
MGT 6421
Quality Management II
High defect np chart
50
45
40
35
30
25
20
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Mixture np chart
55
50
45
40
35
30
25
20
1
2
3
4
5
6
7
8
9
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MGT 6421
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Quality Management II
Control Chart Following:
If observations come from a normal distribution, the sample variability
(measured by the standard deviation or the range) and the mean are
independent. In other types of distributions, however, this may not be the
case. Hence if the patterns on the X and R (or s) charts follow each other,
it may be an indication that we are sampling from a skewed distribution.
Introduction to Control Charts - 13