A correlation between the anisotropy of the critical current density J c in ex situ monofilamentary MgB2/Fe tapes and the degree of texturing of the MgB2 filaments has been found. After removing the Fe sheath, X-ray diffraction measurements were performed at various depths of MgB2 filaments. A texture gradient was detected, the degree of texturing being highest for the MgB2 grains close to the Fe interface, the grains at the centre being almost untextured. The J c anisotropy Gamma = J c(par)/J c(perp), was found to be highest for the larger initial particle grain size (average 4 mum), a strong decrease being observed for the smaller ones (1 mum). The texturing of MgB2 grains (and thus the anisotropy) is caused by the contact with Fe sheath surface during the cold deformation of the tapes, while in hot rolled tapes, a random orientation of the MgB2 grains was observed in the whole filament. From our data, it follows that almost no J c anisotropy would be expected in ex situ Fe/MgB2 tapes with submicron particle sizes and a sharp distribution. An anisotropic behavior of the irreversibility field B irr was also found by inductive measurements, the highest value of B irr being observed for the smallest particle diameters. The pinning force was found to scale only for measurements performed in perpendicular field. Using a simple model which takes into account the observed texture gradient, i.e. the filament volume consisting of a textured and an untextured part, scaling was also obtained in the case of fields parallel to the tape surface.