We present a new, uniform analysis of the H i transmitted flux (F) and H i column density ($N_{\mathrm{H\,{\small I}}}$) distribution in the low-density IGM as a function of redshift z for 0 < z < 3.6 using 55 HST/COS FUV (Δz = 7.2 at z < 0.5), five HST/STIS + COS NUV (Δz = 1.3 at z ∼ 1) and 24 VLT/UVES, and Keck/HIRES (Δz = 11.6 at 1.7 < z < 3.6) AGN spectra. We performed a consistent, uniform Voigt profile analysis to combine spectra taken with different instruments, to reduce systematics and to remove metal-line contamination. We confirm previously known conclusions on firmer quantitative grounds in particular by improving the measurements at z ∼ 1. Two flux statistics at 0 < F < 1, the mean H i flux and the flux probability distribution function (PDF), show that considerable evolution occurs from z = 3.6 to z = 1.5, after which it slows down to become effectively stable for z < 0.5. However, there are large sightline variations. For the H i column density distribution function (CDDF, f ∝ $N_{\rm H\,{\small I}}^{-\beta }$) at $\log (N_{\mathrm{H\,{\small I}}}/1\, {\mathrm{cm}^{-2}})$ ∈ [13.5, 16.0], β increases as z decreases from β = 1.60 at z ∼ 3.4 to β = 1.82 at z ∼ 0.1. The CDDF shape at lower redshifts can be reproduced by a small amount of clockwise rotation of a higher-z CDDF with a slightly larger CDDF normalization. The absorption line number per z (dn/dz) shows a similar evolutionary break at z ∼ 1.5 as seen in the flux statistics. High-$N_{\mathrm{H\,{\small I}}}$ absorbers evolve more rapidly than low-$N_{\mathrm{H\,{\small I}}}$ absorbers to decrease in number or cross-section with time. The individual dn/dz shows a large scatter at a given z. The scatter increases towards lower z, possibly caused by a stronger clustering at lower z.