How to evaluate the process capability of a data set that is non-normal (cannot be transformed and does not fit any known distribution)?

#1
The data that I have been trying to perform capability analysis on are non-normal. I have tried to transform it using Box-Cox and Johnson Trans, but minitab was not able to fit the data into either of the transformations. I had also run the data through minitab to see if it fits any other known non-normal distribution (such as Weibull, 2-parameter exponential, etc.) and the results show no fit. Now I don't know how to perform capability (Cpk or Ppk) analysis on these data or where to go from here.

Does anyone know how to approach this or have an idea on what I should do?

Below are some data sets that I am working on that has this particular problem. Also, the results of the Goodness of Fit Test performed in Minitab for these data are also included.
set1.PNG

Set 1:
0.9150,0.9160,0.9180,0.9140,0.9160,0.9160,0.9150,0.9160,0.9150,0.9150,0.9180,0.9150,0.9160,0.9150,0.9150,0.9140
(The LSL is 0.906 and USL is 0.950)

set2.PNG

Set 2:
0.1298,0.1299,0.1294,0.1296,0.1297,0.1298,0.1298,0.1298,0.1295,0.1298,0.1298,0.1298,0.1298,0.1293,0.1297,0.1299
(The LSL is 0.126 and USL is 0.130)

Thanks,
Response(s) are appreciated.
 
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outdoorsNW

Quite Involved in Discussions
#2
Can you collect more data? With a sample size of 16 (assuming I am counting correctly) the random noise is large relative to the data set. Cpk and similar are not reliable with such a small data set. Usually 30 points is considered the minimum with 125 or more preferred. I would spend more time trying to collect more data than worrying about which distribution fits the data. Once you have more data, the correct distribution may be easy to determine.
 

Miner

Forum Moderator
Staff member
Admin
#3
Set 1 is non normal due to the chunkiness of the measurement system. Having a finer measurement resolution would probably resolve this issue.

Set 2 might be a mixture of different process streams. Can you tell us more about the process and the measurement?
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#4
To add to the excellent advice above: You should spend more time understanding the process and collecting data that reflects that process’s inherent variation - including plotting the data in time sequence than you do in tryin got torture the data with ‘statistical surgery’.
What is the process? How does it vary? Waht are the sources of potential varitation? Do different lots of material effect it? Different equipment? Different operations? Are there multiple fixtures, cavities, fill heads, etc.? Different operators? Set up process? Is the measurement system effective? This will help you understand what subgrouping scheme and sampling frequency you should use.

Process capability studies are not a matter of grabbing some data and pouring it into some statistical software...
 
#5
Thanks for all the advice. I have a follow-up question, is there a way to justify our process capability if we know that there is no way to normalize the data (and the data fits no known distribution) but we know for a fact that everything produced is well within the desired USL, LSL? As you can see from the data I provided, the process is stable, everything is within range. We also have data for 20 different runs and the data results are all relatively the same, with over half of the set data coming out non-normal. It seems that the data fails normality only because the data is just repeats of 4/5 numbers like you mentioned above about the chunkiness, however this means there is not a lot of variabilities and this is what we are seeking for in a stable process, so we know that the process we have is stable, the process we have right now does produce a tight range of outcomes that meets the specification.

For example, if I have a process that produces 3 results 0.012, 0.013 and 0.014 repeatedly (assuming increasing sample size and methods of measurement would not change this) and my specifications (LSL andUSL) are 0.005 and 0.020, it will fail the normality test and show to fit no known distribution, but it is a stable process. However I cannot justify/evaluated the process capability with Cpk or Ppk because of the distribution requirement it needs. So in this case, what should I do?
 
#6
Can you collect more data? With a sample size of 16 (assuming I am counting correctly) the random noise is large relative to the data set. Cpk and similar are not reliable with such a small data set. Usually 30 points is considered the minimum with 125 or more preferred. I would spend more time trying to collect more data than worrying about which distribution fits the data. Once you have more data, the correct distribution may be easy to determine.
Thanks for all the advice, I really appreciate it. I have a follow-up question, is there a way to justify our process capability if we know that there is no way to normalize the data (and the data fits no known distribution) but we know for a fact that everything produced is well within the desired USL, LSL? As you can see from the data I provided, the process is stable, everything is within range. We also have data for 20 different runs and the data results are all relatively the same, with over half of the set data coming out non-normal. It seems that the data fails normality only because the data is just repeats of 4/5 numbers like you mentioned above about the chunkiness, however this means there is not a lot of variabilities and this is what we are seeking for in a stable process, so we know that the process we have is stable, the process we have right now does produce a tight range of outcomes that meets the specification.

For example, if I have a process that produces 3 results 0.012, 0.013 and 0.014 repeatedly (assuming increasing sample size and methods of measurement would not change this) and my specifications (LSL andUSL) are 0.005 and 0.020, it will fail the normality test and show to fit no known distribution, but it is a stable process. However, I cannot justify/evaluated the process capability with Cpk or Ppk because of the distribution requirement needs. So in this case, what should I do?
 
#7
Set 1 is non normal due to the chunkiness of the measurement system. Having a finer measurement resolution would probably resolve this issue.

Set 2 might be a mixture of different process streams. Can you tell us more about the process and the measurement?

Thanks for all the advice, I really appreciate it. I have a follow-up question, is there a way to justify our process capability if we know that there is no way to normalize the data (and the data fits no known distribution) but we know for a fact that everything produced is well within the desired USL, LSL? As you can see from the data I provided, the process is stable, everything is within range. We also have data for 20 different runs and the data results are all relatively the same, with over half of the set data coming out non-normal. It seems that the data fails normality only because the data is just repeats of 4/5 numbers like you mentioned above about the chunkiness, however this means there is not a lot of variabilities and this is what we are seeking for in a stable process, so we know that the process we have is stable, the process we have right now does produce a tight range of outcomes that meets the specification.

For example, if I have a process that produces 3 results 0.012, 0.013 and 0.014 repeatedly (assuming increasing sample size and methods of measurement would not change this) and my specifications (LSL andUSL) are 0.005 and 0.020, it will fail the normality test and show to fit no known distribution, but it is a stable process. However, I cannot justify/evaluated the process capability with Cpk or Ppk because of the distribution requirement needs. So in this case, what should I do?
 
#8
To add to the excellent advice above: You should spend more time understanding the process and collecting data that reflects that process’s inherent variation - including plotting the data in time sequence than you do in tryin got torture the data with ‘statistical surgery’.
What is the process? How does it vary? Waht are the sources of potential varitation? Do different lots of material effect it? Different equipment? Different operations? Are there multiple fixtures, cavities, fill heads, etc.? Different operators? Set up process? Is the measurement system effective? This will help you understand what subgrouping scheme and sampling frequency you should use.

Process capability studies are not a matter of grabbing some data and pouring it into some statistical software...
Thanks for all the advice, I really appreciate it. I have a follow-up question, is there a way to justify our process capability if we know that there is no way to normalize the data (and the data fits no known distribution) but we know for a fact that everything produced is well within the desired USL, LSL? As you can see from the data I provided, the process is stable, everything is within range. We also have data for 20 different runs and the data results are all relatively the same, with over half of the set data coming out non-normal. It seems that the data fails normality only because the data is just repeats of 4/5 numbers like you mentioned above about the chunkiness, however this means there is not a lot of variabilities and this is what we are seeking for in a stable process, so we know that the process we have is stable, the process we have right now does produce a tight range of outcomes that meets the specification.

For example, if I have a process that produces 3 results 0.012, 0.013 and 0.014 repeatedly (assuming increasing sample size and methods of measurement would not change this) and my specifications (LSL andUSL) are 0.005 and 0.020, it will fail the normality test and show to fit no known distribution, but it is a stable process. However, I cannot justify/evaluated the process capability with Cpk or Ppk because of the distribution requirement needs. So in this case, what should I do?
 

John Predmore

Quite Involved in Discussions
#9
I notice your sample 2 is centered around .1298 (mode) and the USL is .130 . Are you sorting out parts which measure above .130 ? Even if you aren't physically sorting, is there some barrier in your process which prevents manufacture of parts over .130 ? When a bell-shaped distribution is truncated, it will appear skewed, and thus non-normal.
 
#10
I notice your sample 2 is centered around .1298 (mode) and the USL is .130 . Are you sorting out parts which measure above .130 ? Even if you aren't physically sorting, is there some barrier in your process which prevents manufacture of parts over .130 ? When a bell-shaped distribution is truncated, it will appear skewed, and thus non-normal.
No, no physical sorting was happening, it is probably a barrier in the process, but our spec is
The data that I have been trying to perform capability analysis on are non-normal. I have tried to transform it using Box-Cox and Johnson Trans, but minitab was not able to fit the data into either of the transformations. I had also run the data through minitab to see if it fits any other known non-normal distribution (such as Weibull, 2-parameter exponential, etc.) and the results show no fit. Now I don't know how to perform capability (Cpk or Ppk) analysis on these data or where to go from here.

Does anyone know how to approach this or have an idea on what I should do?

Below are some data sets that I am working on that has this particular problem. Also, the results of the Goodness of Fit Test performed in Minitab for these data are also included.
View attachment 25693
Set 1:
0.9150,0.9160,0.9180,0.9140,0.9160,0.9160,0.9150,0.9160,0.9150,0.9150,0.9180,0.9150,0.9160,0.9150,0.9150,0.9140
(The LSL is 0.906 and USL is 0.950)

View attachment 25694
Set 2:
0.1298,0.1299,0.1294,0.1296,0.1297,0.1298,0.1298,0.1298,0.1295,0.1298,0.1298,0.1298,0.1298,0.1293,0.1297,0.1299
(The LSL is 0.126 and USL is 0.130)

Thanks,
Response(s) are appreciated.
Update:
LSL and USL is .906 and .950 for set 1
LSL and USL is .128 and .132 for set 2
 
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