Malmquist bias

A flux-limited sample preferentially includes intrinsically brighter objects at a given distance. Calibrating standard candles from that selected sample can bias inferred distances.

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.

A noise-free flux threshold truncates the distance distribution of objects with exactly identical relevant-band luminosity. It cannot preferentially select intrinsically brighter members when no luminosity scatter exists. A luminosity distribution introduces the accessible-volume weighting $L^{3/2}$ in homogeneous Euclidean space. Measurement scatter across the flux threshold and nonuniform completeness introduce additional selection effects even for identical intrinsic luminosities.

For a flux threshold $f_{\min }$ and isotropic point-source luminosity $L$, the detection radius is $r_{\max }=\sqrt{L/(4\pi f_{\min })}$. In a homogeneous Euclidean population the accessible volume is proportional to $L^{3/2}$, so the selected luminosity distribution is proportional to $\phi (L)L^{3/2}$. This assumes flux selection and inverse-square propagation, not an arbitrary surface-brightness-selected diffuse field.