  An important question about black holes is to what extent a typical pure
state differs from the ensemble average. We show that this question can be
answered within semi-classical gravity. We focus on the quantum deviation,
which measures the fluctuations in the expectation value of an operator in an
ensemble of pure states. For a large class of ensembles and observables, these
fluctuations are calculated by a correlation function in the eternal black hole
background, which can be reliably calculated within semi-classical gravity.
This implements the idea of [arXiv:2002.02971] that wormholes can arise from
averages over states rather than theories. As an application, we calculate the
size of the long-time correlation function $\langle A(t) A(0)\rangle$.
