Gaussian absolute-magnitude Malmquist shift ($\Delta ⟨M⟩=-0.6\ln (10){\sigma }_M^2$)

For a uniform Euclidean population, a complete flux-limited sample weights luminosity by accessible volume $L^{3/2}$. Since $L\propto 10^{-0.4M}$, an intrinsic Gaussian magnitude distribution is multiplied by $e^{-0.6\ln (10)M}$. Completing the square shifts its mean by the displayed amount without changing its variance. Extra distance bounds, extinction, evolution and other selection cuts alter this simple result.