For the unbiased estimation of control limits several constants (like a_g, b_g, c_g, etc.) are in use. Although everyone seems to take equal constants, I haven't found formulas for calculating them (only some diffuse hints on Gamma-distribution or the difficulties of calculating the distribution of the range).

Due to my work as a statistician I was asked several times for an answer, but I could not find anyone yet. Could somebody help me with this issue?

Any suggestions and hints are appreciated :thanx: :thanx: :thanx:

Barbara

Barbara,

There was a previous thread on this topic a few years ago at
Control limit - How is table for x-bar & R control chart derived? (http://elsmar.com/Forums/showthread.php?t=2021)
It discusses several of your questions, but not specifically how c_4 is calculated.

That equation can be found at http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc32.htm
The calculation does indeed related to the gamma function (which is a generalization of the more familair factorial function, n!)

Another option is the Monte Carlo approach. You could use Excel generate 4 columns or 1000 rows of random binomial data (mean = 0, sigma = 1). Take the range of the 4 numbers for each of the 1000 rows. Take the average of the 1000 rows. This is the average range you would expect for this distribution. This number should be d2 from the table. I just tried it and the answer agreed to 2 decimal places with the table (2.056 vs 2.059), the difference being attributable to random fluctuations.

Similar simulations could obtain the other constants.


Tim F

Tim,

thank you for your helpful hints. You mentioned the random numbers from a binomial distribution with mean=0 and sd=1. Perhaps I missunderstood it, but this seems to me as a determination of the normal distribution N(0,1), not for a binomial. Am I right? And what is the connection between the "2" of d2 and the simulation?

Do you have an ASTM Manual 7 or the ANSI/ANSQ B1/B3-1996? It was mentioned in the other thread as a source for formulas and I want to know if that holds true.

Barbara

Barbara,

Yes, I meant normal, not binomial. :bonk:

No, I don't have a copy of the standard. :(

I'm not sure of the reasoning behind the names: "d2", "c4", "A2", etc. :confused:


Tim F

Tim,

:thanks: for your reply.

I tried your simulation approach: The estimated values for d seem to convergate, but rather slowly. For a basis of 4x1'000 random numbers the average range of 1000 ranges counts between 1.959 and 2.158 - more or less narrow to the true value d_4=2.059. Increasing the basic sample size up to 10'000 decreases variation, but the deviation from the true value is still given (average range between 2.015 and 2.087, see attached boxplots).

IMO the index with d stand for the amount of consecutive numbers considered in the calculation of the moving range. So the simulation approach is an estimation of d_4, as 4 columns are taken into account.

Besides this, I'm really interested in the formulas for the whole constants. Any further help appreciated :)

Barbara

I found this comment from "Ross" on another forum, regarding Shewhart Constants, and thought it might help answer your questions. I kind of think of these constants the same way as other constants in mathematics, such as Pi. The main difference is the control chart constants don't have a button on the calculator :lmao:

"The d2 and c4 factors are mathematically derived. Unfortunately, the formulas for these two factors are quite involved (the first involves an integral, the second the gamma function). Both formulas are given on page 137 in the ASTM Manual on Presentation of Data and Control Chart Analysis, STP 15D, by the American Society for Testing and Materials (Philadelphia, PA)."

I have found AJ Duncan's book Quality Control and Industrial Statistics, Irwin publishing to be very handy for these formulas. They give the references for each constant and its derivation.

I have found AJ Duncan's book Quality Control and Industrial Statistics, Irwin publishing to be very handy for these formulas. They give the references for each constant and its derivation.

:thanx: Dave and Steven for your sources :)

@Steven: Could you give me the ISBN-no. of Duncan's book, please? I've got probs in finding it. Thanks!

Barbara

:thanx: Dave and Steven for your sources :)

@Steven: Could you give me the ISBN-no. of Duncan's book, please? I've got probs in finding it. Thanks!

Barbara
I have the 5th Edition, but here is the ISBN Number 0-256-03535-0

Thanks, Steven. Unfortunately Ducan's book is not available any more :( The only edition I found is out of 1965... so I'll order the ASTM Manual :)

Barbara