We demonstrate that the self-assembly of spherical nanoparticles (NPs), grafted isotropically with polymeric ligands, into anisotropic structures is a manifestation of the fluctuations inherent in small number statistics. Computer simulations show that the organization of ligand atoms around an individual NP is not spatially isotropic for small numbers of grafts and ligand monomers. This inherent, spatially asymmetric ligand distribution causes the effective, two-body inter-NP potential to have a strong orientational dependence, which reproduces the anisotropic assembly observed ubiquitously for these systems. In contrast, ignoring this angular dependence does not permit us to capture NP self-assembly. This idea of fluctuation-driven behavior should be broadly relevant, and, for example, it should be important for the assembly of ligand-decorated quantum dots into arrays.