Nanoparticles with hydrophobic capping ligands and amphiphilic phospholipids are both found to self-assemble into monolayer films when deposited on the air/water interface. By separately measuring the anisotropic stress response of these films under uniaxial compression, we obtain both the 2D compressive and shear moduli of a range of different thin nanoparticle and phospholipid films. The compressive moduli of both nanoparticle and lipid films in the solid phase are on the same order of magnitude, whereas the shear moduli of the lipid films are found to be significantly lower. Additionally, the moduli of the nanoparticle films depended substantially on the polydispersity of the constituent particles-broader size distribution lowered the stiffness of the nanoparticle film.