The distribution of protein stability effects is known to be well approximated by a Gaussian distribution from previous empirical fits. Starting from first-principles statistical mechanics, we more rigorously motivate this empirical observation by deriving per-residue-position protein stability effects to be Gaussian. Our derivation requires the number of amino acids to be large, which is satisfied by the standard set of 20 amino acids found in nature. No assumption is needed on the number of residues in close proximity in space, in contrast to previous applications of the central limit theorem to protein energetics. We support our derivation results with computational and experimental data on mutant protein stabilities across all types of protein residues.