We present a new orbit determination method, termed statistical ranging, applicable to poorly observed single-apparition asteroids having two or more observations. We examine the Bayesian a posteriori probability density of the orbital elements using a Monte Carlo technique that maps an unbiased hypervolume in element phase space. We select two observations randomly from the full set, and after introducing random deviations to the observations and generating random topocentric ranges (distances), compute a sample orbit from the Cartesian positions available. Repeating the procedure allows us to obtain a large set of orbital elements compatible with the observations. To distinguish between realistic and unrealistic orbit solutions, we can incorporate the three-or four-dimensional distribution of semimajor axes, eccentricities, inclinations, and, in some cases, the absolute magnitudes of multi-apparition asteroids as a priori information in the Bayesian technique. The application of the method to both near-Earth and main-belt asteroids clearly indicates that there may exist a broad variety of orbits, all of which are compatible with the observations. The resulting distributions of the orbital elements allow recovery attempts for lost asteroids and linkage to existing observations of single-apparition asteroids. The Bayesian approach can prove useful in constraining the broad distribution of orbits for problematic newly discovered asteroids.