log transformation and mean question

I have question about log2 transformation and performing mean
on log2 data. I am doing analysis for ELISA data. the OD values
and the concentration values for the standards were log2
transformed before performing the lm. the OD values for samples
were log2 transformed and coefficients of lm were applied to get
the log2 concentration values. I then backtransformed these
log2 concentrations and the trouble started. when i take the
mean of log2 concentrations the value is different than the
backtransformed concentrations.

100+1000/2

[1] 600

2^( ( log2(100)+log2(1000) )/2 )

[1] 316.2278

What I am doing wrong to get the different values

Apart from the fact that I think your first line should be

  (100+1000)/2
  # [1] 550

you are doing nothing whatever wrong! The difference is an
inevitable result of the fact that, for any set of positive
numbers X = c(x1,x2,...,xn), not all equal,

  mean(log(X)) < log(mean(X))

This is because the curve of y = log(x) lies below the
tangent to the curve at any given point. If that point is
mean(X), and the tangent is y = a + b*x, then

  mean(log(X)) < mean(a + b*X) = a + b*mean(X) = log(mean(X))

since y = a + b*x is tangent to y = log(x) at x = mean(X).
This is a special case of a general result called Jensen's
Inequality.

Your second line is

  2^mean(log2(X)) < 2^log2(mean(X)) = mean(X).

where X = c(100,1000).

Ted.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <ted.harding at wlandres.net>
Fax-to-email: +44 (0)870 094 0861
Date: 12-May-11                                       Time: 17:37:45
------------------------------ XFMail ------------------------------