Some Special Process Control Procedures
•
•
•
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X and s charts
Warning limits
Control charts for moving averages
X chart with a linear trend
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X and s Chart
• Why s chart instead of R chart?
– Range is computed with only two values, the
maximum and the minimum. However, s is
computed using all the measurements
corresponding to a sample.
– For large samples, s becomes a much better
estimator (of standard deviation) than R.
2
X and s Chart
• The control limits of X chart are
X 3

n
Note: This can also be written as X  A
Where
3
A
(also given in Appendix 3)
n
3
X and s Chart
R
• Previously, the value of  has been estimated as:
d2
s
• The value of  may also be estimated as:
c4
where, s is the sample standard deviation and
is as obtained from Appendix 3
• Control limits may be different with different
estimators of  (i.e., R and s )
c4
4
X and s Chart
• The control limits for X and s charts are:
UCL X  X  A3 s
UCL S  B4 s
LCL X  X  A3 s
LCLs  B3 s
Where, the values of A3, B3 and B4 are as obtained
from Appendix 3
5
X and s Chart
• For variable sample size, each of the X and s
obtained from an weighted average:
2
is
n1 X 1  n2 X 2  n3 X 3    nk X k
X 
n1  n2  n3    nk
s
n1  1s12  n2  1s22  n3  1s32    nk  1sk2
n1  n2  n3    nk
6
X and s Chart
• For nearly uniform samples sizes, X and s
obtained from unweighted averages:
2
may be
X1  X 2  X 3  X k
X 
k
s1  s2  s3    sk
s
k
• Following is a rule that identifies nearly uniform sample
sizes
– a rule: the largest sample is at least twice the
smallest
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X and s Chart
• For large samples:
3
c4  1, A3 
n
3
3
B3  1 
, B4  1 
2n
2n
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Warning Limits
• Two sets of limits on X charts
– Outer limits - conventional 3 sigma limits
– Inner limits - warning limits usually set at 2 sigma
• Two sets of limits may be confusing. However, they
can be very useful too:
– Two points in succession outside the same inner
limit give stronger evidence of a process shift
– Similarly, strong evidences are two points out of
three beyond one inner limit, three out of seven,
four out of ten etc.
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Problem 10.1: A certain company manufactures
electronic components for television sets. One
particular component is made to a critical length of
0.450 in. On the basis of past production experiences,
the standard deviation of this dimension is 0.010 in.
Because of the critical nature of the dimension, the
quality control group maintains warning limits on X
control chart as well as the normal 3-sigma control
limits. The X chart is based on subgroups of four
samples, and warning limits are maintained at two
standard deviations from the mean. Compute the
warning limits and the control limits for the X chart.
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Problem 10.2: For Problem 10.1
(a) What is the probability that a subgroup average will
exceed the UWL but not exceed the UCL when the
process is correctly centered?
(b) Supervisor should be notified if (i) 2 successive
subgroup averages exceed one of the warning limits or
(ii) if one subgroup average exceeds either of the
control limits. What is the probability that, when there
has been no change in the process, the supervisor will
have to be notified because of (i) or (ii)?
(c) If the process suddenly shifts to 0.460 with no change
in the standard deviation, what is the probability that 2
successive points will exceed the upper warning limit
but not exceed the upper control limit?
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Reading and Exercises
• Chapter 10 ( X and s charts):
– pp. 362-368 (Section 10.1-2)
– 10.3, 10.4, 10.5
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