The energy distribution of electrons emitted from an Al/Al2O3/Au metal-insulator-metal (MIM) electron emitter is measured. The thickness of the insulator is 5.5 nm. The energy distribution becomes narrower as the operating voltage V-d decreases since the low energy tail of the distribution is cut off by the potential barrier of the surface work function phi of the emitter. When the emitter is operated in the nonformed state, Delta E, the full width at half-maximum of the distribution, is 0.32 eV for V-d = 5.0 V, which is slightly above phi of Au (4.7 eV). As V-d increases, the high-energy tail of the distribution broadens whereas the shape of the low-energy tail remains unchanged. For a formed MIM emitter, Delta E becomes broader by 0.15-0.2 eV more than Delta E of a nonformed emitter at each V-d; thus, operation in the nonformed state is essential to obtain good monochromaticity. The spatial distribution of the work function in the emitter surface is also measured by the retarding potential method. The variation of phi, which limits the ultimately attainable monochromaticity with the near-threshold drive method, is measured as 0.05 eV. It is estimated that Delta E less than 0.2 eV could be attained by the near-threshold drive if the insulator thickness and V-d are further reduced.