Anions of the second and third periods\char22{}${\mathrm{B}}^{\ensuremath{-}},$ ${\mathrm{O}}^{\ensuremath{-}},$ ${\mathrm{F}}^{\ensuremath{-}},$ ${\mathrm{Al}}^{\ensuremath{-}},$ ${\mathrm{S}}^{\ensuremath{-}},$ and ${\mathrm{Cl}}^{\ensuremath{-}}$\char22{}had their total electron detachment cross sections measured, in the 0.3\char21{}1.5 a.u. velocity range, for He, Ne, and Ar targets. It is observed that, for any given target, the cross sections present similar velocity dependence and a conspicuous maximum at the same velocity, which differs for distinct targets. It was also obtained that these cross sections scale with target-independent factors, and the ratio of factors within the same group of the periodic table is also nearly group independent. Explanations for these behaviors are presented considering the presence of low- and high-velocity regions, and describing the latter by a free-electron model. The roles of the double ionization and the strong electron correlation in the detachment process are also discussed.