Dear all, just to show what I mean where the problem is: I've produced 1 million Gaussian random numbers with a mean of 1 and a standard deviation of 1. I attach a plot showing both the Gaussian itself, and the distribution of the numbers obtained by taking the inverse. The latter looks quite non-Gaussian to me.
According to the formulas in Wikipedia and John Taylor's book the "inverse distribution" should also have a mean of 1 and a sigma of 1. But this is not the case. Its standard deviation is 437 (a different random number would give a different number here!), due to its long tails that arise from those values of the original distribution that are close to 0. The Wikipedia's "error propagation" article in its "Caveats and Warnings" paragraph calls this a Cauchy distribution. This is clearly an example where the first-order approximation breaks down, and common sense tells me that this happens because we may divide by numbers close to zero. And it shows that it might be useful to think about error propagation, and not blindly apply the formulas. thanks, Kay -------- Original Message -------- Subject: Re: Question about the statistical analysis-might be a bit off topic Date: Tue, 7 Jun 2011 11:28:50 +0100 From: Ian Tickle <[email protected]> Kay, the usual propagation-of-uncertainty formulae are based on a first-order approximation of the Taylor series expansion, i.e. assuming that 2nd and higher order terms in the series are can be neglected. This is clearly not the case if B is small relative to its uncertainty: you would need to include higher order terms. See the 'Caveats and Warnings' section in the Wikipedia article that Bernhard quoted. Cheers -- Ian On Tue, Jun 7, 2011 at 8:59 AM, Kay Diederichs <[email protected] <mailto:[email protected]>> wrote: what I'm missing in those formulas, and in the Wikipedia, is a discussion of the prerequisites - it seems to me that, roughly speaking, if the standard deviation of B is as large or larger than the absolute value of the mean of B, then we might divide by 0 when calculating A/B . This should influence the standard deviation of the calculated A/B, I think, and seems not to be captured by the formulas cited so far. best, Kay Am 20:59, schrieb James Stroud: The short answer can be found in item 2 in this link: http://science.widener.edu/svb/stats/error.html The long answer is "I highly recommend Error Analysis by John Taylor:" http://science.widener.edu/svb/stats/error.html If you can find the first edition (which can fit in your pocket) then consider yourself lucky. Later editions suffer book bloat. James On Jun 4, 2011, at 10:44 AM, capricy gao wrote: If means and standard deviations of A and B are known, how to estimate the variance of A/B? Thanks. -- Kay Diederichs http://strucbio.biologie.uni-konstanz.de email: [email protected] <mailto:[email protected]> �Tel +49 7531 88 4049 Fax 3183 Fachbereich Biologie, Universit�t Konstanz, Box 647, D-78457 Konstanz This e-mail is digitally signed. If your e-mail client does not have the necessary capabilities, just ignore the attached signature "smime.p7s".
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