# Mode of LogNormal distribution?

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?

## 1 Answer

Mode of lognormal distribution has closed-form solution:

$$\exp(\mu-\sigma^2)$$

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?

• Bootstrapping in order to describe the dispersion of a dataset sounds like overkill.
– whuber
Oct 3 '16 at 21:12
• @whuber agree, I was thinking about confidence intervals around the mode. I don't have better ideas at the moment - I would be happy to hear about better ones.
– Tim
Oct 3 '16 at 21:20
• It's natural to describe the logarithms of the data in such cases: after all, we have been told the logs will be approximately normally distributed, so we can be confident that their mean and standard deviation will give an accurate picture of them. Furthermore, there are simple and well-known "analytical expressions" for these statistics!
– whuber
Oct 3 '16 at 21:21
• @whuber right, but as I understand the question it's about variability around the mode.
– Tim
Oct 3 '16 at 21:28
• Ah, interesting interpretation! You might want to ask the OP whether that was the intention.
– whuber
Oct 3 '16 at 21:30