I'm looking for an intuitive explanation of why the average of numbers taken as $e^{\text{mean}(10+5+6+1)}$ is not equal to $\text{mean}(e^{10}+e^5+e^6+e^1)$.
It is no discussion about this fact, I'd like to understand it better. It seems that Jensen's inequality is related to it, but my maths understanding is not good enough to get all the essence of it.
The 2 side questions would be :
Is there a way to calculate the average of numbers at an exponential?
If I have data from a exponential process (let's say for example a qPCR measurement, where on each "cycle" you synthesise new DNA fragments copies from the already present ones + the ones of the last generation) : if I want to estimate the average of different replicates, should I use the mean of the exponential (estimated to be the real copy number) or of the CT value (which is basically a Log2 score of the first one)
Edit : I just permit myself an edit given the fact that a) I'm bad at understand maths, so please help me understanding it "intuitively" b) I'd really like to have an answer for my side questions :) Thanks !
Because $$e^{p+q}=e^pe^q\neq e^p+e^q$$ by basic indices rules.