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 <[log in to unmask]>
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
<[log in to unmask] <mailto:[log in to unmask]>>
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: [log in to unmask]
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