We have lots of models for overdispersed count data but we rarely see underdispersed data. But now I know what example I’ll be giving when this next comes up in class. From a book review by Theo Tait:

A number of shark species go in for oophagy, or uterine cannibalism. Sand tiger foetuses ‘eat each other in utero, acting out the harshest form of sibling rivalry imaginable’. Only two babies emerge, one from each of the mother shark’s uteruses: the survivors have eaten everything else. ‘A female sand tiger gives birth to a baby that’s already a metre long and an experienced killer,’ explains Demian Chapman, an expert on the subject.

That’s what I call underdispersion. E(y)=2, var(y)=0. Take that, M. Poisson!