Statistics question for the process engineers

[zahc]

I need a general mathematical way to calculate the error distribution and thus the process capability of the result of multiple combined processes, not having the luxury of being able to measure the final combined process to empirically determine its process capability, but knowing only the error distributions of those sub-processes. The answer is probably something as simple as "average the means and sum the error distributions" but statistics is never an intuitive.

One example:
Suppose I need to dispense 1000ml of chemical. I have the option to use a 1l dispensing system. I have measured/tested/observed this dispensing system, and found that after a large number of dispensations, the system delivers, as expected, a distribution of volumes, say with a mean of 1 liter and a standard deviation of 10ml. The distribution appears normal, and is modeled with a normal distribution, not surprising since the error is random. It is easy to calculate a 95% confidence interval, and understand the error of that dispensing process, and use common statistical methods to characterize its process capability. This is all statistical process control 101. All is well with the cosmos.

Now, consider that I also have available a very similar dispensing system that dispenses 100ml. I have also  observed this system, and I know it dispenses with a mean of 100ml and a standard deviation of 1ml. I can use this system 10 times in a row to dispense 1 liter. Suppose I am not able to monitor this 10-injection "combination injection" itself, but I do know a lot about the process capability of the 100ml dispensing system itself.

I want to predict, in advance, the distribution and process capability of the ten-dispensation combined process, so that I can compare it with the 1 liter dispenser. How can I predict the distribution of the combined process? Do the ten dispensations simply add up the same as a 1 liter dispensation with a standard deviation 10 times as large as each individual dispensation? Or do the errors in each 100ml dispensation "average out" so that the ten 100ml dispensations is actually better? Or is the ten 100ml system actually worse than the 1 liter system?

Note, testing and measuring the combined process is not an option here because reasons.

Example 2:
I have 10 different thin-film deposition processes *of different thicknesses*. The mean thickness of each film is different by design and the standard deviation of each is different too. I know a lot about the process capability of each of them. Every day at lunch some R&D guy comes up to me and asks for a different combination of films in a different order. How can I predict the thickness process capability of an arbitrary film stack?

[cordex]

On your first question, assuming the same 1% error margin for both the 1L and the 100ml dispensers, that the error margin would not be impacted by - say - rapid repeat use, and that the error could be 1% above or below the intended amount, the 100ml dispenser would be expected to give you a significantly more accurate measurement.  

A quick modeling shows that you'll on average see about half the extreme error when dispensing 100ml 10 times at 1% error than 1000ml 1 time at 1% error.  I saw a spread of between 990L-1010 using the 1L dispenser and 994.62L-1004.74L.  This was dispensing 500L using each technique.

The standard deviation of error for the small batch was about 1/3rd for the large batch.

Not sure what you're asking on the second question.