(moved to http://stats.stackexchange.com/questions/6410/reliability-of-mean-of-standard-deviations)

Hi all,

I've a question which probably is going to show my ignorance about statistics :). I have a large set of machines that produce iron bars of certain lengths. For each machine, I have ran experiments and have a list of lengths. From those I can calculate a mean and sample standard deviation. I don't really care about their means and I am mainly focused on the variation. Therefore, I basically only record a sample standard deviation per machine. I think the results of each machine follow a normal distribution. So far so good :)

I now want to combine these variations into a single number. Therefore, I calculate the quadratic average of each machine variation, let's call it X. In the next step, I also would like to give an estimate for the spread around X. What is this number called and what's the best way to calculate it?

Edit: I'll try to clarify with an example. Suppose I measure 3 machines and find that they produce M1: 100 +/- 7 M2: 120 +/- 8 M3: 130 +/- 9

where the numbers behind the +/-'s are the sample standard deviations of observed values on that single machine. As said before, I don't care about the means but only in the spread, so I define {X_1, X_2, X_3} = {7,8,9}. Their quadratic average is X = RMS(X_i) = $\sqrt{194}$ and I think of X as an indication of the average spread of a machine in my park.

Suppose that I would have found {X_1, X_2, X_3} = {3,8,11}. Their quadratic average is the same $\sqrt{194}$, but the spread around it is obviously bigger. My confidence in the correctness of $\sqrt{194}$ as the average spread of a machine should therefore be lower (I'd like to test some more machines, for instance) and I would like to express this in a number.