What is average run length?
Average run length of a control chart can be used to assess the performance of the control chart for to determine the appropriate value of various parameters of the control chart presented hair.
How to find average run length?
ARL=1/p
P is the probability of any point exceeding the control limits.
if the process is in control and the control chart signals out of control then we say that a false alarm has occurred.
For example
considered the X mean control chart with the standard 3 variance limits. For this situation p=0.0027 is the probability that a single point falls outside the limit when the process is in control. The in control ARL for X mean control chart is
ARL=1/p=1/0.0027=370.
Even if the process remains in control we should expect on the average and out of control signal every 370 samples.
if the process is actually in control then we desire a large value of the ARL.
ARL(0)=1/Alpha
Alpha is a probability that a simple point beyond the control limit.
On the other hand the process is out of control, then a small ARL value is desirable.
A small value of the ARL,indicates that the control chart will signal out of control soon after the process has shifted.
The out of control ARL
ARL(1)=1/(1-beta)
where beta is the probability of not detecting a shift on the first sample after a shift has occurred.
Cautions in the Use of ARLs
ARLprovides valuable information for designing and evaluating control schemes, there are drawbacks to relying on the ARL, Ajay design criterion.it should be noted that run length follows a geometric distribution, since it presents The number of the samples before of success occurs. One drawback is the standard deviations of the runway length is quite large. Second because the distribution of the run length follows a geometric distribution, the mean of distribution may not be reliable estimate of the true run length.
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