Damped absorbers, seen in the spectra of background quasars, are unique probes to select Hi-rich galaxies. These galaxies allow one to estimate the neutral gas mass over cosmological scales. The neutral gas mass is a possible indicator of gas consumption as star formation proceeds. The damped Ly? absorbers (DLAs; NHI≥ 2 × 1020 cm-2) and sub-DLAs (1019 ≤ NHI ≤2 × 1020 cm-2) are believed to contain a large fraction of neutral gas mass in the Universe. In Paper I of the series, we presented the results of a search for DLAs and sub-DLAs in the European Southern Observatory (ESO) Ultraviolet Visual Echelle Spectrograph (UVES) advanced data products dataset of 250 quasars. Here we use an unbiased subsample of sub-DLAs from this dataset to derive their statistical properties. We built a subset of 122 quasars ranging from 1.5 <zem <5.0, suitable for statistical analysis. The statistical sample was analyzed in conjunction with other sub-DLA samples from the literature. This resulted in a combined sample of 89 sub-DLAs over a redshift path of δz = 193. We derived the redshift evolution of the number density and the line density for sub-DLAs and compared them with the Lyman-limit systems (LLSs) and DLA measurements from the literature. The results indicate that these three classes of absorbers are evolving in the redshift interval 1.0 ≤ z ≤ 5.0. Thanks to the ESO UVES advanced data products data, we were able to determine the column density distribution, fHI(N, z), down to the sub-DLA limit. The flattening of fHI(N, z) in the sub-DLA regime is present in the observations. The redshift evolution of fHI(N, z) down to log NHI = 19.0 cm?2 is also presented, indicating that there are more sub-DLAs at high redshift than at low redshift. fHI(N, z) was also used to determine the neutral gas mass density, σg, at 1.5 <z <5.0. The complete sample shows that sub-DLAs contribute 8-20% to the total g from 1.5 <z <5.0. In agreement with previous studies, no evolution of σg was observed from low to high redshift (i.e., 1.5 <z <5.0), suggesting that star formation alone cannot explain this non-evolution and replenishment of gas and that recombination of ionized gas is needed. © 2013 ESO.