Top: median prestellar core mass versus background column density (black triangles). The error bars correspond to the inter-quartile range in observed masses for each bin of background column density. The solid, dashed and dotted blue lines are linear fits to the observed median, upper-quartile, and lower-quartile masses as function of background column density, respectively. The red curve shows how the mass at the 50% completeness level varies with background column density according to the model completeness curves of Appendix C (see Fig. C.4). The upper x-axis provides an approximate mass-per-unit-length scale, assuming the local backgroundaround each core is dominated by the contribution of a parent filament with a characteristic width W_fil = 0.1 pc (see end of Sect. 4.5). Bottom: same as upper panel after correcting the distribution of observed core masses for incompleteness effects before estimating the median core mass and inter-quartile range in each column density bin (see Sect. 4.2). Both the median prestellar core mass and the dispersion in core masses increase roughly linearly with background column density.