Upper panels: differential column density distribution as a function of redshift. Both black and grey data points show the results from the high-order Lyman sample. The grey data points at log  N_H i < 12.75 mark the column densities that are affected by incompleteness. 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 black solid line gives the power law fit for log  N_H i = [12.75,14.0] at each redshift bin, whereas the dashed line is the fit to the z = [1.9,3.2] redshift range (see Fig. 10). The overdensity plotted on the top x-axis is calculated at the mean z for each redshift bin. Lower panels: residuals from the power law fit from the entire redshift range at log  N_H i = [12.75,14.0].