Reliability of mean of standard deviations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:46:12Z http://mathoverflow.net/feeds/question/51663 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51663/reliability-of-mean-of-standard-deviations Reliability of mean of standard deviations Frank Meulenaar 2011-01-10T15:24:35Z 2011-01-20T15:52:55Z <p>(moved to <a href="http://stats.stackexchange.com/questions/6410/reliability-of-mean-of-standard-deviations" rel="nofollow">http://stats.stackexchange.com/questions/6410/reliability-of-mean-of-standard-deviations</a>)</p> <p>Hi all,</p> <p>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 :)</p> <p>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?</p> <p>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</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/51663/reliability-of-mean-of-standard-deviations/51676#51676 Answer by Carlo Beenakker for Reliability of mean of standard deviations Carlo Beenakker 2011-01-10T17:43:47Z 2011-01-10T17:43:47Z <p>you want the variance of the variance, see for example: <a href="http://www.bls.gov/osmr/pdf/st030280.pdf" rel="nofollow">http://www.bls.gov/osmr/pdf/st030280.pdf</a></p>