This paper presents a new method to diagnose the star-forming potential of a molecular cloud region from the probability density function of its column density (N-pdf). This method provides expressions for the column density and mass profiles of a symmetric filament having the same N-pdf as a filamentary region. The central concentration of this characteristic filament can distinguish regions and can quantify their fertility for star formation. Profiles are calculated for N-pdfs which are pure lognormal, pure power law, or a combination. In relation to models of singular polytropic cylinders, characteristic filaments can be unbound, bound, or collapsing depending on their central concentration. Such filamentary models of the dynamical state of N-pdf gas are more relevant to star-forming regions than are spherical collapse models. The star formation fertility of a bound or collapsing filament is quantified by its mean mass accretion rate when in radial free fall. For a given mass per length, the fertility increases with the filament mean column density and with its initial concentration. In selected regions the fertility of their characteristic filaments increases with the level of star formation.