On Wikipedia I found about ARL:
- Even when a process is in control (that is, no special causes are present in the system), there is approximately a 0.27% probability of a point exceeding 3-sigma control limits.
- So, even an in-control process plotted on a properly constructed control chart will eventually signal the possible presence of a special cause, even though one may not have actually occurred.
- For a Shewhart control chart using 3-sigma limits, this false alarm occurs on average once every 1/0.0027 or 370.4 observations.
- Therefore, the in-control average run length (or in-control ARL) of a Shewhart chart is 370.4.
I do not know how to get the formula to produce such a result.
Can somebody give me some hints?
Thank you in advance
I dare! :) The Shewhart charts are based on the normal (Gaussian) distribution. It is assumed, that the distribution of the results is normal. (If not the calculation is incorrect).
To calculate, you must use the normal distribution charts. Assuming, that is no bias, the probability of having a result over the 3-sigma limit is
2*(1-F(3)=2*0,00135=0,0027, where F(x) is the Gaussian function , F(3) is its value, if x=3*sigma (3 standard deviations). the result must be multiplied by 2, because the deviation may occur in two directions.
0,0027 (0,27%) is the probability of an alarm in the absence of any bias. If two measurements are made the probability of a false alarm increases to 2*0.0027=0.0054.
As an average in every 1/0.0027=370.4 runs occurs a false alarm. this is the average run lenghts ARL
A simpler analogy: Rolling a dice the probability of having 1 is 1/6=0,16666. As an average each 1/0.1666=6th result will be 1, this is the ARL for dice rolling.
@ Atilla Barna Vandra
Are you joking?
ARL can be (and MUST be) for any control chart.
Why do you consider the "Normal Distribution" only?
The question is very general: for any distribution!
Massimo Sivo, I am not joking at all. You have written: "...there is approximately a 0.27% probability of a point exceeding 3-sigma control limits." suggesting that you are referring to the Gaussian (normal) distribution. The 0.27% probability refers to the probability (in the case of a normal distribution) of having a result beyond the 3-sigma (3SD) limits. This is only true in the case of the normal distribution. E.g., in the case of the triangular, uniform or sinusoidal distributions no value is beyond the 3SD limits.
If you want to define ARL more generally, ARL=1/P(alarm) (=1/0.0027=370.4 in the presented example) where P(alarm) is the probability of a (false) alarm , and defines the average number of runs between two (false) alarms. In the case of Cusum charts, ARL defines the average number of runs before exceeding the decision limit, and ARL=1/Bias(averaged), where Bias(averaged) is the average value of the bias in the time frame.
With a simpler analogy: if rolling a dice, the probability of having e.g., a 5 is 1/6=0.1666. The average number of rollings is 1/0.1666= 6. The next 5 may happen in the next rolling but also may happen only after 60 rollings. But the average is 6. Therefore, the ARL=6.
Returning to our example, and other distributions, ARL of having a result beyond the 3SD limits (3*(1/0,57) =1.71=square root of 3) is infinite (=1/0), because in the case of uniform distribution, no value is beyond the value of 1.
@ Atilla Barna Vandra
You explained why Wikipedia wrote
- Even when a process is in control (that is, no special causes are present in the system), there is approximately a 0.27% probability of a point exceeding 3-sigma control limits.
Thank you.
But what about the ARL for data Exponentially distributed?
The ARL concept should be valid for those Control Charts...
I s the 3-sigma limit still to be considered?
The "sigma" is a parameter of the Gaussian equation, therefore neither the sigma nor its estimator the SD (standard deviation) has the usual meaning in other distributions. Of course, SD can be calculated, but cannot be used for accurate predictipns.
Atilla Barna Vandra
You say:
- neither the sigma nor its estimator the SD (standard deviation) has the usual meaning in other distributions.
The contrary is TRUE...
Are you saying that we cannot compute the ARL for Control Charts with EXPONENTIALLY distributed data?
I did not say that. I said that in other disyribution you cannot calculate with the laws of the normal distributions. I have given example with the uniform distribution. Give a concrete example with exponential distribution.and show me the meaning of the sigma and SD.
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