In a given set of manufacturing data, you can determine the control limits for 3-sigma confidence by the following equation: CL = Mean (X) +/- 3 * Standard deviation (population, sigma) / sqrt(# of samples). Now, if you don't have the population standard deviation, there's another way of calculating it: CL = X +/- (A2)*Range (R).
I'm trying to figure out how A2 (A constant from the given table) is calculated. What I have so far is a few things:
1) My google-fu has only demonstrated how to apply the process, not how the table is derived - too many results on "how to do it" makes it very difficult to actually find the derivation. Some mention that a derivation exists, but doesn't give the HOW.
2) X+/-3 *Sigma/(sqrt(n)) = X +/- (A2)*R 3*Sigma/(sqrt(n))=A2*R Given that A2 changes based off of N, but not at the rate of the sqrt(n), and that calculating a population standard deviation requires a lot more than we have, I've come up with a few guesses, and one sure-thing
Sure thing: This is an estimation, but a close enough one that it doesn't matter. Guess: There's a way to estimate population standard deviation based off of the range and the medium, that gets more and more accurate as we add more samples. Given that the standard deviation calculation requires more precise numbers.
I'm having trouble with the last step, putting it all together. I read the similar questions, but they don't quite get there. It'd also be nice to know with what confidence we have on the chart given.