Gravitational waves (GWs) create correlations in the arrival times of pulses

from different pulsars. The expected correlation $\mu(\gamma)$ as a function of

the angle $\gamma$ between the directions to two pulsars was calculated by

Hellings and Downs for an isotropic and unpolarized GW background, and several

pulsar timing array (PTA) collaborations are working to observe these. We ask:

given a set of noise-free observations, are they consistent with that

expectation? To answer this, we calculate the expected variance

$\sigma^2(\gamma)$ in the correlation for a single GW point source, as pulsar

pairs with fixed separation angle $\gamma$ are swept around the sky. We then

use this to derive simple analytic expressions for the variance produced by a

set of discrete point sources uniformly scattered in space for two cases of

interest: (1) point sources radiating GWs at the same frequency, generating

confusion noise, and (2) point sources radiating GWs at distinct

non-overlapping frequencies. By averaging over all pulsar sky positions at

fixed separation angle $\gamma$, we show how this variance may be cleanly split

into cosmic variance and pulsar variance, also demonstrating that measurements

of the variance can provide information about the nature of GW sources. In a

series of technical appendices, we calculate the mean and variance of the

Hellings-Downs correlation for an arbitrary (polarized) point source, quantify

the impact of neglecting pulsar terms, and calculate the pulsar and cosmic

variance for a Gaussian ensemble. The mean and variance of the Gaussian

ensemble may also be obtained from the discrete-source confusion noise model in

the limit of a high density of weak sources.