  We present a systematic study of the impact of a dust bias on samples of
damped Lyman-$\alpha$ absorbers (DLAs). This bias arises as an effect of the
magnitude and colour criteria utilized in the Sloan Digital Sky Survey (SDSS)
quasar target selection up until data release 7 (DR7). The bias has previously
been quantified assuming only a contribution from the dust obscuration. In this
work, we apply the full set of magnitude and colour criteria used up until SDSS
DR7 in order to quantify the full impact of dust biasing against dusty and
metal-rich DLAs. We apply the quasar target selection algorithm on a modelled
population of intrinsic colours, and by exploring the parameter space
consisting of redshift, ($z_{\rm QSO}$ and $z_{\rm abs}$), optical extinction,
and HI column density, we demonstrate how the selection probability depends on
these variables. We quantify the dust bias on the following properties derived
for DLAs at z=3: the incidence rate, the mass density of neutral hydrogen and
metals, the average metallicity. We find that all quantities are significantly
affected. When considering all uncertainties, the mass density of neutral
hydrogen is underestimated by 10 to 50%, and the mass density in metals is
underestimated by 30 to 200%. Lastly, we find that the bias depends on
redshift. At redshift z=2.2, the mass density of neutral hydrogen and metals
might be underestimated by up to a factor of 2 and 5, respectively.
Characterizing such a bias is crucial in order to accurately interpret and
model the properties and metallicity evolution of absorption-selected galaxies.
