Conceptual Questions on SPC - Operating characteristic curve of a control chart

T

tahashamim

#1
Dear all,

Can someone please help me in the following three issues,

1- What information is provided by the operating characteristic curve of a control chart?

2- Is the average run length performance of a control chart a more meaningful measure of performance than the type I and type II error probabilities? What information does ARL convey that the statisitical error probabilities do not?

Also, what do we mean by "false alarm" in relation to ARL. I thought a point outside 3 sigma control limit will require investigation - it could be false or true! What is the benefit of saying in advance that through ARL (if there is no shift in process mean) on an average it would take 370 samples before it gives a "false alarm". Can there be no point outside the limit before 370 point plot?

3- Why lower control limit for R chart is usually "0"?

I am going through so many books, but could not find an answer that I can understand. I need to have conceptual understanding for the above three.

Thanking you all in advance.
your friend,
taha.
 
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Steve Prevette

Deming Disciple
Staff member
Super Moderator
#2
Dear all,

Can someone please help me in the following three issues,

I am going through so many books, but could not find an answer that I can understand. I need to have conceptual understanding for the above three.

Thanking you all in advance.
your friend,
taha.
1- What information is provided by the operating characteristic curve of a control chart?

First, an operating characteristic curve in general tells you how a certain statistical test performs. For example, if I say I will fail a run of product if I find two defects in a sample of 50, I can plot the probability of getting 2 defects out of fifty when the "real" defect rate is 1%, 2%, 3%, 4%, etc. (I will use "real" in the sense of what an omniscient entity would see of the whole population, which we humans cannot see)

An operating characteristic curve for a control chart is a more difficult issue. I can say if I have a normally distributed variable with mean of 10 and standard deviation of 5, what happens as I vary the "real" mean? The Cunix bead demonstrator does this. However, control charting is not dependent upon the normal distribution. The "real" distribution may be uniform, Weibull, triangular, or something else. Any operating characteristic curve depends upon the assumption of the distribution.

In the worst case, the Tchybecheve Inequality tells us that up to 1/9th of the data may be outside of three standard deviations.

Dr. Deming also cautioned us not to treat SPC as an exercise in probabilities.

2- Is the average run length performance of a control chart a more meaningful measure of performance than the type I and type II error probabilities? What information does ARL convey that the statisitical error probabilities do not?

In the context of the discussion above, not much. If you could determine the Type I and Type II error AND the expected costs related to both, that would be far superior to the ARL. The ARL provides some inference of the ability to detect a certain change, but not much more.

3- Why lower control limit for R chart is usually "0"?

It doesn't have to be. But many times the spread of the Range is relatively large compared to the average Range. Thus, when you attempt to subtract three standard deviations from the average, you get a negative number. Since ranges cannot be negative, the LCL is either ignored, or set at zero by default

4- Also, what do we mean by "false alarm" in relation to ARL. I thought a point outside 3 sigma control limit will require investigation - it could be false or true! What is the benefit of saying in advance that through ARL (if there is no shift in process mean) on an average it would take 370 samples before it gives a "false alarm". Can there be no point outside the limit before 370 point plot?

There is always the possibilty of false alarm with any detection method. Take the smoke alarm in your house. When is the last time it went off? Likely it was a false alarm!

Dr. Deming pointed out there is no way to eliminate false alarms unless you turn off the detection scheme altogether. Go take the batteries out of your smoke detector if you want to ensure no more false alarms!

The 370 number is based upon the normal distribution curve. If the data plotted are indeed normal, and you have estimated the mean and variance perfectly, then I expect 1 point outside of 3 standard deviations every 370 trials. BUT I may get a 3 standard deviation result on the very next trial! Or it could take 1,000 trials! On the average it is 370. BUT only in the most perfect situation. Tchebychev's inequality states it may be on the average in as few as 11 trials.

The application here is we make the operational definition just like the smoke detector in your house. We train our children that if they hear the alarm - EXIT THE HOUSE! Then investigate. Reenter the house if it is safe, call 911 if you have a fire. We treat SPC the same way. Believe the indication. Then investigate. Yes, it may be a false alarm. But Dr. Shewhart established the limits such that we minimize the total costs of false alarms and failures to detect.
 
T

tahashamim

#3
Hi Steve,

Thank you for taking time to write the reponses in detail. Indeed you have been of great help. In fact this forum is much better than other forum (no need to mention names).

Once again thank you and thanks to all who will respond to this post.

Regards,
Taha.
1- What information is provided by the operating characteristic curve of a control chart?

First, an operating characteristic curve in general tells you how a certain statistical test performs. For example, if I say I will fail a run of product if I find two defects in a sample of 50, I can plot the probability of getting 2 defects out of fifty when the "real" defect rate is 1%, 2%, 3%, 4%, etc. (I will use "real" in the sense of what an omniscient entity would see of the whole population, which we humans cannot see)

An operating characteristic curve for a control chart is a more difficult issue. I can say if I have a normally distributed variable with mean of 10 and standard deviation of 5, what happens as I vary the "real" mean? The Cunix bead demonstrator does this. However, control charting is not dependent upon the normal distribution. The "real" distribution may be uniform, Weibull, triangular, or something else. Any operating characteristic curve depends upon the assumption of the distribution.

In the worst case, the Tchybecheve Inequality tells us that up to 1/9th of the data may be outside of three standard deviations.

Dr. Deming also cautioned us not to treat SPC as an exercise in probabilities.

2- Is the average run length performance of a control chart a more meaningful measure of performance than the type I and type II error probabilities? What information does ARL convey that the statisitical error probabilities do not?

In the context of the discussion above, not much. If you could determine the Type I and Type II error AND the expected costs related to both, that would be far superior to the ARL. The ARL provides some inference of the ability to detect a certain change, but not much more.

3- Why lower control limit for R chart is usually "0"?

It doesn't have to be. But many times the spread of the Range is relatively large compared to the average Range. Thus, when you attempt to subtract three standard deviations from the average, you get a negative number. Since ranges cannot be negative, the LCL is either ignored, or set at zero by default

4- Also, what do we mean by "false alarm" in relation to ARL. I thought a point outside 3 sigma control limit will require investigation - it could be false or true! What is the benefit of saying in advance that through ARL (if there is no shift in process mean) on an average it would take 370 samples before it gives a "false alarm". Can there be no point outside the limit before 370 point plot?

There is always the possibilty of false alarm with any detection method. Take the smoke alarm in your house. When is the last time it went off? Likely it was a false alarm!

Dr. Deming pointed out there is no way to eliminate false alarms unless you turn off the detection scheme altogether. Go take the batteries out of your smoke detector if you want to ensure no more false alarms!

The 370 number is based upon the normal distribution curve. If the data plotted are indeed normal, and you have estimated the mean and variance perfectly, then I expect 1 point outside of 3 standard deviations every 370 trials. BUT I may get a 3 standard deviation result on the very next trial! Or it could take 1,000 trials! On the average it is 370. BUT only in the most perfect situation. Tchebychev's inequality states it may be on the average in as few as 11 trials.

The application here is we make the operational definition just like the smoke detector in your house. We train our children that if they hear the alarm - EXIT THE HOUSE! Then investigate. Reenter the house if it is safe, call 911 if you have a fire. We treat SPC the same way. Believe the indication. Then investigate. Yes, it may be a false alarm. But Dr. Shewhart established the limits such that we minimize the total costs of false alarms and failures to detect.
 
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