Differential column density distribution at 1.9 < z < 3.2 of our quasar sample using the high-order Lyman fit. Black and grey data points show the results from our quasar sample. The grey data points below log  N_H i < 12.75 mark the column densities that are affected by incompleteness. The grey data points above log  N_H i > 12.75 represent the results from the Lyα-only fit. The dashed line represents a theoretical prediction at z ~ 3 by Altay et al. (2011). The vertical error bars indicate 1σ Poisson errors, while the x-axis error bars show the N_H i range covered by each data point. The filled circles at log  N_H i > 20.1 and at 19.0 < log  N_H i < 20.1 are from the SDSS II DR7 at ⟨ z ⟩ = 3.02 by Noterdaeme et al. (2009) and from O’Meara et al. (2007) at ⟨ z ⟩ = 3.1, respectively. The solid line gives the power law fit to our data for log  N_H i = [12.75,14.0]. The overdensity plotted on the top x-axis is calculated at z = 2.55.