Sufficient statistics are combinations of data in terms of which the likelihood function can be rewritten without loss of information. Depending on the data volume reduction, the use of sufficient statistics as a preliminary step in a Bayesian analysis can lead to significant increases in efficiency when sampling from posterior distributions of model parameters. Here we show that the frequency integrand of the cross-correlation statistic and its variance are approximate sufficient statistics for ground-based searches for stochastic gravitational-wave backgrounds. The sufficient statistics are approximate because one works in the weak-signal approximation and uses measured estimates of the autocorrelated power in each detector. We perform analytic and numerical calculations to show that, in this approximation, LIGO-Virgo's hybrid frequentist-Bayesian parameter estimation analysis is equivalent to a fully Bayesian analysis. This work closes a gap in the LIGO-Virgo literature and suggests directions for additional searches.