Least-squares problems set on high-dimensional spaces are considered. The solution is approximated employing a Monte Carlo method that takes the average of a random variable defined as the solutions of random small least-squares problems drawn as subsystems of the original problem. The conditions that ensure convergence and consistency of the method are discussed, along with an analysis of the computational cost in specific instances. The proposed algorithms generalize and improve the statistical analysis of distributed multipoles (SADM) approach put forth by Chipot et al in [6] for the derivation of distributed atomic multipoles from the quantum-mechanical electrostatic potential.