Column density of molecular hydrogen vs total neutral hydrogen column density N(H I) = N(H I) + 2N(H₂) for our sample of proximate H₂ absorbers and z > 2 intervening strong H₂-bearing DLAs from Balashev et al. (2019). For a fair comparison, we removed systems that may potentially be proximate, and kept only systems with a conservative cut of c (z_abs − z_QSO)/(1 + z_QSO) <  − 10 000 km s⁻¹. Our selection of proximate H₂ systems based on strong H₂ lines directly seen in the low-resolution SDSS spectra naturally skews our sample to high H₂ columns. The horizontal blue line shows the approximate and empirical lower limit of at log N(H₂) > 18.5 for the present sample. The top histograms show the N(H) distributions for the proximate (blue) and intervening (green) systems with N(H₂) > 18.5. Finally, the different lines correspond to different averaged molecular fractions f_H2 ≡ 2N(H₂)/2N(H₂) + N(H I)).