Sedimentary magnetizations are fundamental to palaeomagnetism, but the mechanisms that control remanence acquisition remain poorly constrained. Observed sedimentary natural remanent magnetizations are often orders of magnitude smaller than the saturation remanent magnetization of the same sediment, which indicates inefficient remanence acquisition. We present a statistical model, based on the von Mises-Fisher distribution, in which magnetic particle reorientations towards an ambient field are considered, without representing the physics of the magnetization acquisition process. The results provide insights into the nature of sedimentary magnetizations. Specifically, an assemblage of randomly oriented magnetic particles can acquire a high-fidelity palaeomagnetic signal with only small rotations (in some cases <1°) of particles towards the ambient field direction. This demonstrates that the action of a geomagnetic torque on individual magnetic mineral particle orientation may be minor, and that a weak directional bias on an assemblage of particles could be responsible for the typically observed inefficiency of sedimentary remanence acquisition. Additionally, we demonstrate that weak fields produce sedimentary magnetizations with larger directional uncertainties. For natural sediments, however, these uncertainties appear to be small enough to allow reliable recording of directional geomagnetic field behaviour during periods with weak fields, such as palaeomagnetic reversals and excursions.