Two examples of the PDF in our observational samples. Top left panel: Three-dimensional image of f_DMh for FRB 20181112A (see Sect. 4 and Table 1 for details). The x-axis is the DM_h parameter from 20 to 600 pc cm⁻³. The y-axis is the $A_{\tau }\stackrel{\sim }{F}G$ parameter from −3 to 1 ${({\mathrm{pc}}^2\mathrm{km})}^{-\frac{1}{3}}$ in log scale. For a given DM_h and $A_{\tau }\stackrel{\sim }{F}G$, the corresponding probability density in the log scale is presented by color. Bottom left panel: Integration of the PDF over the $A_{\tau }\stackrel{\sim }{F}G$ parameter from 0.001 to 10 ${({\mathrm{pc}}^2\mathrm{km})}^{-\frac{1}{3}}$, which provides us with the PDF of DM_h. The purple line shows the 50 percentile of the PDF. The dashed black lines correspond to the 84.2 and 15.8 percentiles of the PDF (±σ). The 50 percentile is DM_h = 85.6 pc cm⁻³, +σ = 78 pc cm⁻³, and −σ = 45.4 pc cm⁻³. Top right panel: Example of a 3D image of f_DMh by using FRB 20191001A (see Sect. 4 and Table 1 for details). Bottom left panel: Same as the bottom left panel, but using FRB 20191001A. The 50 percentile is DM_h = 143.3 pc cm⁻³, +σ = 89.3 pc cm⁻³, and −σ = 44.2 pc cm⁻³.