  Contrary to what would be predicted on the basis of Cram\'er's model
concerning the distribution of prime numbers, we develop evidence that the
distribution of $\psi(x+H)- \psi(x)$, for $0\le x\le N$, is approximately
normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^\delta \le H
\le N^{1-\delta}$.
