[BioC] standard deviation from log to normal scale???

Jesper Ryge jesper.ryge at ki.se
Mon Dec 4 19:49:31 CET 2006

Here is just a lil "trivia" statistic question for the forum:-) I  
apologize for my clumsy annotation but i hope I get the question  
through anyhow:-)

For ratios I take it its normal procedure to calculate the average as  
the geometric mean. That is easiest done with log transformed values,  
giving something like

<R> = mean(Ratio=a/b)=2^ (<log2(a)> - <log2(b)> + - SD)

Its the SD thats giving me a little headache as i go back to normal  
(un-transformed, i use ** for annotating normal) values. SD in log  
space is not symmetric in normal space, so SD** != 2^(SD)? or?

To illustrate my clumsy annotation, if : < log2(R) > + - SD(log2(R))  
=  4+-1 in log space it becomes (2^4- 2^3) and 2^4+2^5 ~ 16-8 and 16  
+ 32. so SD** is not symmetric.

I found 2 suggestions in the litterature that doesnt seem to account  
for this asymmetry. One was giving the standard deviation of the  
geometric mean to be SD**=2^(SD) just as i reasoned was inappropriate?

Another suggestion I found was for propagating errors for exponential  
transformation :

X = e^A, 	SD(X)/X = SD(A)

So should i do SD**(X) = mean(X) * SD(A)  ---  X ~ Ratio and A ~ log 
(R) ???? again i dont see how this solved the asymmetric SD from the  
log space???

Maybe i missed something basic with log-normal distributions, in any  
case any help will be highly appreciated:-)  I have the feeling its  
rather trivial but I would really like to know how to put  
(assymetric?) error bars on my (normal scale) ratios correctly. This  
goes for both Affymterix summary ratios  and RT-PCR ratios. What's  
the correct procedure???

jesper ryge
Karolinska Institutet
Dep. of Neuroscience

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