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Shot noise can affect the performance of free-electron lasers (FELs) by driving instabilities (e.g., the microbunching instability) or by competing with seeded density modulations. Recent papers have proposed suppressing shot noise to enhance FEL performance. In this paper we use a one-dimensional (1D) model to calculate the noise amplification from an energy modulation (e.g., electron interactions from space charge or undulator radiation) followed by a dispersive section. We show that, for a broad class of interactions, selecting the correct dispersive strength suppresses shot noise across a wide range of frequencies. The final noise level depends on the beam’s energy spread and the properties of the interaction potential. We confirm and illustrate our analytical results with 1D simulations.