Assume that we have objects (e.g. particles) whose properties (e.g. diameter) follow a log-normal distribution that can be described by a geometric mean $\mu_g$ and a geometric standard deviation $\sigma_g$.
Now consider that we want to measure $\mu_g$ and $\sigma_g$, by sampling $n$ samples from the distribution (e.g. measuring the diameters of $n$ individual particles). Consider further that the utilized measurement technique is imperfect, i.e. we don't receive the true value $A_i$ from a measurement, but instead we receive $A_i'=A_i+\Delta A_i$.
If we define that the errors are random, then a reasonable assumption for $A_i'$ could be that it is normally distributed around $A_i$ with a certain standard deviation $\sigma_{measurement}$.
We then calculate the measured properties of the initial log-normal distribution, based on the imperfect measurements, so we receive
$\mu_g'=\mu_g+\Delta \mu_g= \sqrt[n]{(A_1+\Delta A_1) (A_2+\Delta A_2) \cdots (A_n+\Delta A_n)}$
and
$\sigma_g'=\sigma_g+\Delta \sigma_g=\exp{\sqrt{ \sum_{i=1}^n \left(\ln { {A_i+\Delta A_i} \over \mu_g' } \right)^2 \over n }}$.
Questions:
- Can we make general statements about $\Delta \mu_g$ and $\Delta \sigma_g$? E.g. $\Delta \mu_g<0$ or $\Delta \sigma_g>0$, or something along these lines?
- How does the sample size $n$ play into this? Often, we would assume that errors will approach 0 for $n\rightarrow\infty$. Is this also the case in this scenario?
Note: I simulated the process, and obviously I'm willing to share the results, but I'd like to have an unbiased discussion first.
Thanks a lot!