Good metric to distinguish between fat tailed and narrow distribution Could anyone point me to a good metric to distinguish between the following distributions? One distribution seems to be exponential type whereas the other is fatter and sometimes also has a peak around 16Hz.
Type 1

Type 2

Given an unlabelled distribution, how to classify it as type 1 or type 2?
 A: It is unlikely that your histograms are going to be very helpful here.  To diagnose whether a distribution has "fat tails" you would usually construct a tail plot (see this related answer).  If you want to reduce that to a single metric then I would suggest conducting a weighted regression on the tails to estimate whether the tail behaviour is consistent with a power law, and if so, estimate the rate parameter in the power law.  Anything less than cubic decay indicates a "heavy tailed" distribution in the sense of having an infinite variance.
A: My eye says that Type 1 distribution has higher kurtosis (it seems more peaked while Type 2 more flat) and smaller variance. Maybe from just these values combined with the mode/mean/median you could determine the distribution type, but the other central moments could also be informative. I would recommend you to grow a decision tree using these as features and possibly than try other combinations of central moments and mode/mean/median as explanatory variables and see what works.
