We report on spectropolarimetric observations of a near-IR line of Mn I located at 15262.702 Å whose intensity and polarization profiles are very sensitive to the presence of hyperfine structure. A theoretical investigation of the magnetic sensitivity of this line uncovers several interesting properties. The most important one is that the presence of strong Paschen-Back perturbations due to the hyperfine structure produces an intensity line profile whose shape changes according to the absolute value of the magnetic field strength. A line ratio technique is developed from the intrinsic variations of the line profile. This line ratio technique is applied to spectropolarimetric observations of the quiet solar photosphere in order to explore the probability distribution function of the magnetic field strength. Particular attention is given to the quietest area of the observed field of view, which was encircled by an enhanced network region. A detailed theoretical investigation shows that the inferred distribution yields information on the average magnetic field strength and on the spatial scale at which the magnetic field is organized. A first estimation gives ~250 G for the mean field strength and a tentative value of ~0.4" for the spatial scale at which the observed magnetic field is horizontally organized.