The merit of a prime gap equals $(p\_{n+1}-p\_n)/\ln p\_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very definition got rigged to make the mean of merit equal 1 in the limit. What about higher moments? Do we have theorems, standard conjectures, or even just numerical evidence concerning their values or their mere existence? 



Moments of merit
The standard conjecture is that the statistics of $(p\_{n+1}-p\_n)/\ln n$ are, in the limit, precisely an exponential distribution with parameter $1$; one can recover conjectures about individual moments from this, although evaluating the moments is the natural way to approach the conjecture. This is the natural heuristic that follows from the model that every integer $m$ is prime with probability $1/\ln m$. I believe this conjecture was proved conditionally by [Gallagher (in Mathematika in 1976)](https://mathscinet.ams.org/mathscinet-getitem?mr=409385), assuming a strong form of the prime $k$-tuples conjecture.


