I have data that turn out to be describable by a lognormal distribution. However, I feel like the peak value (mode) of the lognormal distribution would be more representative of the data points. Is there an analytical expression for this, and what kind of dispersion measure would be most suitable to report along with it?
Mode of lognormal distribution has closed-form solution:
As about dispersion, assuming that you are interested in variability around the mode, if I were you I'd simply use bootstrap to compute intervals over it. You could also use mean absolute deviation around the mode. However if you are interested in general measure of variability, why not simply use standard deviation?