  We present new estimates for the statistical properties of damped
Lyman-$\alpha$ absorbers (DLAs). We compute the column density distribution
function at $z>2$, the line density, $\mathrm{d}N/\mathrm{d}X$, and the neutral
hydrogen density, $\Omega_\mathrm{DLA}$. Our estimates are derived from the DLA
catalogue of Garnett 2016, which uses the SDSS-III DR12 quasar spectroscopic
survey. This catalogue provides a probability that a given spectrum contains a
DLA, allowing us to use even the noisiest data without biasing our results and
thus substantially increase our sample size. We measure a non-zero column
density distribution function at $95\%$ confidence for all column densities
$N_\mathrm{HI} < 5\times 10^{22}$ cm$^{-2}$. We make the first measurements
from SDSS of $\mathrm{d}N/\mathrm{d}X$ and $\Omega_\mathrm{DLA}$ at $z>4$. We
show that our results are insensitive to the signal-to-noise ratio of the
spectra, but that there is a residual dependence on quasar redshift for
$z<2.5$, which may be due to remaining systematics in our analysis.
