Transcript DevStat8e_16_02

16
Quality Control
Methods
Copyright © Cengage Learning. All rights reserved.
16.2 Control Charts for Process
Location
Copyright © Cengage Learning. All rights reserved.
Control Charts for Process Location
Suppose the quality characteristic of interest is associated
with a variable whose observed values result from making
measurements.
For example, the characteristic might be resistance of
electrical wire (ohms), internal diameter of molded rubber
expansion joints (cm), or hardness of a certain alloy
(Brinell units).
One important use of control charts is to see whether some
measure of location of the variable’s distribution remains
stable over time. The most popular chart for this purpose is
the chart.
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The X Chart Based on Known
Parameter Values
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X
The
Chart Based on Known Parameter Values
Because there is uncertainty about the value of the variable
for any particular item or specimen, we denote such a
random variable (rv) by X. Assume that for an incontrol
process, X has a normal distribution with mean value  and
standard deviation .
Then if denotes the sample mean for a random sample of
size n selected at a particular time point, we know that
1. E( ) = 
2.
3.
has a normal distribution.
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X
The
Chart Based on Known Parameter Values
It follows that
where Z is a standard normal rv. It is thus highly likely that
for an in-control process, the sample mean will fall within 3
standard deviations
of the process mean .
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X
The
Chart Based on Known Parameter Values
Consider first the case in which the values of both  and 
are known. Suppose that at each of the time points
1, 2, 3, . . . , a random sample of size n is available.
Let
denote the calculated values of the
corresponding sample means. An chart results from
plotting these
over time—that is, plotting points
and so on—and then drawing horizontal
lines across the plot at
LCL = lower control limit =
UCL = upper control limit =
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X
The
Chart Based on Known Parameter Values
Such a plot is often called a 3-sigma chart. Any point
outside the control limits suggests that the process may
have been out of control at that time, so a search for
assignable causes should be initiated.
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Example 1
Once each day, three specimens of motor oil are randomly
selected from the production process, and each is analyzed
to determine viscosity.
The accompanying data
(Table 16.1) is for a
25-day period.
Viscosity Data for Example 1
Table 16.1
9
Example 1
cont’d
Extensive experience with this process suggests that when
the process is in control, viscosity of a specimen is
normally distributed with mean 10.5 and standard deviation
.18.
Thus
are
so the 3 SD control limits
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Example 1
cont’d
All points on the control chart shown in Figure 16.2 are
between the control limits, indicating stable behavior of the
process mean over this time period (the standard deviation
and range for each sample will be used in the next
subsection).
chart for the viscosity data of Example 1
Figure 16. 2
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X
Chart Based on Estimated
Parameters
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Charts Based on Estimated Parameters
X
In practice it frequently happens that values of  and  are
unknown, so they must be estimated from sample data
prior to determining the control limits.
This is especially true when a process is first subjected to a
quality control analysis. Denote the number of observations
in each sample by n, and let k represent the number of
samples available.
Typical values of n are 3, 4, 5, or 6; it is recommended that
k be at least 20. We assume that the k samples were
gathered during a period when the process was believed to
be in control. More will be said about this assumption
shortly.
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Charts Based on Estimated Parameters
X
With
denoting the k calculated sample means,
the usual estimate of  is simply the average of these
means:
There are two different commonly used methods for
estimating  : one based on the k sample standard
deviations and the other on the k sample ranges (recall that
the sample range is the difference between the largest and
smallest sample observations).
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Charts Based on Estimated Parameters
X
Prior to the wide availability of good calculators and
statistical computer software, ease of hand calculation was
of paramount consideration, so the range method
predominated.
However, in the case of a normal population distribution,
the unbiased estimator of  based on S is known to have
smaller variance than that based on the sample range.
Statisticians say that the former estimator is more efficient
than the latter. The loss in efficiency for the estimator is
slight when n is very small but becomes important for n > 4.
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Charts Based on Estimated Parameters
X
Recall that the sample standard deviation is not an
unbiased estimator for . When X1, …., Xn is a random
sample from a normal distribution, it can be shown that
E(S) = an  
where
and
denotes the gamma function. A tabulation of an for
selected n follows:
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Charts Based on Estimated Parameters
X
Let
Where S1, S2,. . ., Sk are the sample standard deviations for
the k samples. Then
Thus
So
is an unbiased estimator of .
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Charts Based on Estimated Parameters
X
Control Limits Based on the Sample Standard
Deviations
where
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Example 2
Referring to the viscosity data of Example 1, we had n = 3
and k = 25. The values of
and si(i = 1, . . . , 25) appear
in Table 16.1, from which it follows that
= 261.896/25 = 10.476 and s = 3.834/25 = .153.
With a3 = .886, we have
These limits differ a bit from previous limits based on
 = 10.5 and  = .18 because now = 10.476 and
= .173.
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Example 2
cont’d
Inspection of Table 16.1 shows that every is between
these new limits, so again no out-of-control situation is
evident.
Viscosity Data for Example 1
Table 16.1
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Charts Based on Estimated Parameters
X
To obtain an estimate of  based on the sample range,
note that if X1, . . . ,Xn form a random sample from a normal
distribution, then
R = range(X1, . . . , Xn) = max(X1, . . . , Xn) – min(X1, . . . , Xn)
= max(X1 – , . . . , Xn – ) – min(X1 – , . . . , Xn – )
=   {max(Z1, . . . , Zn) – min(Z1 , . . . , Zn)}
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Charts Based on Estimated Parameters
X
=   {max(Z1, . . . , Zn) – min(Z1 , . . . , Zn)}
Where Z1 , . . . , Zn are independent standard normal rv’s.
Thus
E(R) =   E(range of a standard normal sample)
=   bn
so that
is an unbiased estimator of .
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Charts Based on Estimated Parameters
X
Now denote the ranges for the k samples in the quality
control data set by r1, r2, . . . ,rk. The argument just given
implies that the estimate
comes from an unbiased estimator for .
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Charts Based on Estimated Parameters
X
Selected values of bn appear in the accompanying table
[their computation is based on using statistical theory and
numerical integration to determine E(min(Z1, . . . , Zn)) and
E(max(Z1 , . . . , Zn))
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Charts Based on Estimated Parameters
X
Control Limits Based on the Sample Ranges
where
sample ranges.
and r1, . . . , rk are the k individual
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Recomputing Control Limits
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Recomputing Control Limits
We have assumed that the sample data used for estimating
 and  was obtained from an in-control process. Suppose,
though, that one of the points on the resulting control chart
falls outside the control limits.
Then if an assignable cause for this out-of-control situation
can be found and verified, it is recommended that new
control limits be calculated after deleting the corresponding
sample from the data set.
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Recomputing Control Limits
Similarly, if more than one point falls outside the original
limits, new limits should be determined after eliminating any
such point for which an assignable cause can be identified
and dealt with.
It may even happen that one or more points fall outside the
new limits, in which case the deletion/recomputation
process must be repeated.
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Performance Characteristics of
Control Charts
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Performance Characteristics of Control Charts
Generally speaking, a control chart will be effective if it
gives very few out-of- control signals when the process is in
control, but shows a point outside the control limits almost
as soon as the process goes out of control.
One assessment of a chart’s effectiveness is based on the
notion of “error probabilities.” Suppose the variable of
interest is normally distributed with known  (the same
value for an in-control or out-of-control process).
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Performance Characteristics of Control Charts
In addition, consider a 3-sigma chart based on the target
value 0, with  = 0 when the process is in control. One
error probability is
 = P(a single sample gives a point outside the control
limits when  = 0)
= P( > 0 + 3/
or X < 0 – 3 /
when  = 0)
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X
Performance Characteristics of Control Charts
The standardized variable
has a
standard normal distribution when  = 0, so
 = P(Z > 3 or Z < –3) = (–3.00) + 1 – (3.00) = .0026
If 3.09 rather than 3 had been used to determine the
control limits (this is customary in Great Britain), then
 = P(Z > 3.09 or Z < –3.09) = .0020
The use of 3-sigma limits makes it highly unlikely that an
out-of-control signal will result from an in-control process.
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Performance Characteristics of Control Charts
Now suppose the process goes out of control because 
has shifted to  +  ( might be positive or negative);  is
the number of standard deviations by which  has
changed.
A second error probability is.
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Performance Characteristics of Control Charts
We now standardize by first subtracting 0 +  from each
term inside the parentheses and then dividing by
 = P(– 3 –
= (3 –
< standard normal rv < 3 –
) – (–3 –
)
)
This error probability depends on , which determines the
size of the shift, and on the sample size n.
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Performance Characteristics of Control Charts
In particular, for fixed ,  will decrease as n increases (the
larger the sample size, the more likely it is that an
out-of-control signal will result), and for fixed n, 
decreases as || increases (the larger the magnitude of a
shift, the more likely it is that an out-of-control signal will
result).
The accompanying table gives  for selected values of 
when n = 4.
It is clear that a small shift is quite likely to go undetected in
a single sample.
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Performance Characteristics of Control Charts
If 3 is replaced by 3.09 in the control limits, then 
decreases from .0026 to .002, but for any fixed n and , 
will increase.
This is just a manifestation of the inverse relationship
between the two types of error probabilities in hypothesis
testing. For example, changing 3 to 2.5 will increase  and
decrease .
The error probabilities discussed thus far are computed
under the assumption that the variable of interest is
normally distributed.
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Performance Characteristics of Control Charts
If the distribution is only slightly nonnormal, the Central
Limit Theorem effect implies that will have approximately
a normal distribution even when n is small, in which case
the stated error probabilities will be approximately correct.
This is, of course, no longer the case when the variable’s
distribution deviates considerably from normality.
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Performance Characteristics of Control Charts
A second performance assessment involves expected or
average run length needed to observe an out-of-control
signal.
When the process is in control, we should expect to
observe many samples before seeing one whose lies
outside the control limits.
On the other hand, if a process goes out of control, the
expected number of samples necessary to detect this
should be small.
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Performance Characteristics of Control Charts
Let p denote the probability that a single sample yields an
x value outside the control limits; that is,
Consider first an in-control process, so that
are
all normally distributed with mean value 0 and standard
deviation
Define an rv Y by
Y = the first i for which Xi falls outside the control limits
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Performance Characteristics of Control Charts
If we think of each sample number as a trial and an
out-of-control sample as a success, then Y is the number of
(independent) trials necessary to observe a success.
This Y has a geometric distribution, and we seen earlier
that E(Y) = 1/p. The acronym ARL (for average run length)
is often used in place of E(Y).
Because p =  for an in-control process, we have
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Performance Characteristics of Control Charts
Replacing 3 in the control limits by 3.09 gives
ARL = 1/.002 = 500.
Now suppose that, at a particular time point, the process
mean shifts to  = 0 + .
If we define Y to be the first i subsequent to the shift for
which a sample generates an out-of-control signal, it is
again true that ARL = E(Y) = 1/p, but now p = 1 – .
The accompanying table gives selected ARLs for a 3-sigma
chart when n = 4.
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Performance Characteristics of Control Charts
These results again show the chart’s effectiveness in
detecting large shifts but also its inability to quickly identify
small shifts.
When sampling is done rather infrequently, a great many
items are likely to be produced before a small shift in  is
detected. The CUSUM procedures were developed to
address this deficiency.
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Supplemental Rules for X Charts
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X
Supplemental Rules for
Charts
The inability of charts with 3-sigma limits to quickly detect
small shifts in the process mean has prompted
investigators to develop procedures that provide improved
behavior in this respect.
One approach involves introducing additional conditions
that cause an out-of-control signal to be generated.
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X
Supplemental Rules for
Charts
The following conditions were recommended by Western
Electric (then a subsidiary of AT&T). An intervention to take
corrective action is appropriate whenever one of these
conditions is satisfied:
1. Two out of three successive points fall outside 2-sigma
limits on the same side of the center line.
2. Four out of five successive points fall outside 1-sigma
limits on the same side of the center line.
3. Eight successive points fall on the same side of the
center line.
A quality control text should be consulted for a discussion
of these and other supplemental rules.
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Robust Control Charts
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Robust Control Charts
The presence of outliers in the sample data tends to reduce
the sensitivity of control-charting procedures when
parameters must be estimated.
This is because the control limits are moved outward from
the center line, making the identification of unusual points
more difficult.
We do not want the statistic whose values are plotted to be
resistant to outliers, because that would mask any
out-of-control signal.
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Robust Control Charts
For example, plotting sample medians would be less
effective than plotting
as is done on an chart.
The article “Robust Control Charts” by David M. Rocke
(Technometrics, 1989:173–184) presents a study of
procedures for which control limits are based on statistics
resistant to the effects of outliers.
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Robust Control Charts
Rocke recommends control limits calculated from the
interquartile range (IQR), which is very similar to the fourth
spread. In particular,
For a random sample from a normal distribution,
E(IQR) = kn; the values of kn are given in the
accompanying table.
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Robust Control Charts
The suggested control limits are
The values of
are plotted.
Simulations reported in the article indicated that the
performance of the chart with these limits is superior to that
of the traditional chart.
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