# Quantification of “modality” of distribution

Does anyone know of a quantification of the modality of a distribution?

For example, exactly how "high" must the "second" peak be in order to qualify as bimodal rather than unimodal with some un-smooth regions? what about tri-modal?

I have discrete distributions in mind, but the question is general.

I am looking for a quantification because I would like to programmatically "classify" the modality (uni-, bi-, tri-, 4+) of thousands of distributions, and then tally up how many e.g. bimodal distributions I have observed.

I have invented some measures myself, but of course I would like to use an established method, if one exists.

• As the number of modes is clearly an answer, the question becomes how do you recognise a "mode" as worth flagging as such? That is no easier. I conjecture that there is no accepted answer. Experienced data analysts tend to be much more cautious than beginners, as they have a stronger sense that apparent modes can be chance fluctuations and/or side-effects of choices in binning or smoothing. I know the question is how to automate recognition, so such comments may seem to miss the point entirely, but saying that this is very difficult to do well is within what a comment can say. – Nick Cox Jun 22 '15 at 13:33
• The second peak merely needs to be higher than some value between it & the first peak to qualify the distribution as being (at least) bimodal. So are you really interested in inference from the modality of a sample to the modality of the population from which it was drawn? – Scortchi Jun 22 '15 at 13:35