[phenixbb] question on f/sigf
Edward A. Berry
BerryE at upstate.edu
Fri Jul 13 15:58:46 PDT 2012
Thanks for asking! not often I can come up
with the answer to a math question.
Of course I answered the inverse- why sigmaI/I is 2 x sigmaF/F
but close enough.
Edward A. Berry wrote:
> Edward A. Berry wrote:
>> fn1 at rice.edu wrote:
>>> Hi All,
>>> I am wondering why f/sigf is always about twice of i/sigi. Is there
>>> any mathematics behind this relation? Maybe it is not directly related
>>> to phenix, but i sincerely hope someone could help me with it.
>> I'm not sure if that is exactly the case, and it may depend on the
>> the distribution of errors and the size of the error, but in the
>> limit of small errors it should be a good approximatin.
>> Start with some calculus:
>> d/dx(x^2) = 2x
>> dx^2 = 2xdx (the change in x^2 is 2*x times the change in x
>> but we like to express as a fraction, or percent error,
>> so divide both sides by x^2
>> dx^2/x^2 = 2 dx/x the percentage change in x^2 is twice
>> the percentage change in x (for small dx)
>> Now say <F> is X, and dX is the distance from that to
>> one of the measurements. each of the measurements will be twice as far
>> from the mean values when expressed as I as when expressed as F.
>> Then run that through the root-mean-square math for calculating sigma,
>> and see if it doesn't come out twice as large for I as for F.
> Better, use chain rule for propagation of error- sigma(F(I)) = Sigma(I)*dF/dI
>> Or take an example: F is 100, which was derived from I=10000
>> a second measure is F=101 (1% different),
>> which was derived form I=101^2 = 10201 (2.01% different)
>>> Thank you in advance!
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