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

Wolfgang Huber huber at ebi.ac.uk
Tue Dec 5 18:17:20 CET 2006

```Hi Jesper,

the asymmetry only becomes appreciable when SD(X)/X = SD(A) (btw this is
called the coefficient of variation) becomes big; say, much bigger than
0.1 (=10%). You can see this from the power series of the exponential
function. e^x = 1 + x + O(x^2), and the quadratic and higher order terms
that cause the asymmetry become non-negligible when |x|>>0.

Hope this helps
Wolfgang

> 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???
>
>
> cheers:-)
> jesper ryge
> Karolinska Institutet
> Dep. of Neuroscience
>
> _______________________________________________
> Bioconductor mailing list
> Bioconductor at stat.math.ethz.ch
> https://stat.ethz.ch/mailman/listinfo/bioconductor
> Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor

--
------------------------------------------------------------------
Wolfgang Huber  EBI/EMBL  Cambridge UK  http://www.ebi.ac.uk/huber

```