Modern methods of geomagnetic field analysis can be applied to paleomagnetic data provided account is taken of the relatively sparse and inaccurate nature of paleomagnetic results. Paleomagnetism tends to produce data sets dominated by inclination measurements, which do not provide a unique field model. As an example, two magnetic fields are found with closely similar inclinations but completely different declinations and strengths. Directional measurements are difficult to deal with because they are nonlinearly related to the field model. The equations are simplified by linearizing the field about an axial dipole; the approximation is adequate for paleomagnetic data and can be used to show that the VGP scatter function is not an appropriate estimate of the non-axial-dipole field even when averaged over longitude. Four assertions made in the paleomagnetic literature are shown to be correct within this linearization but are not true in general. In particular, equatorial symmetry of the VGP scatter functions for the separate symmetric (''quadrupole'') field holds within the approximation. The virtual geomagnetic pole (VGP) scatter functions for the modern geomagnetic field has strong north-south asymmetry, but historical field models show it to be a relatively recent feature. The VGP scatter function for the symmetric part of the field, Sq, is dominated by the equatorial dipole and is approximately equal to the angular distance between the geomagnetic and geographic poles; that for the antisymmetric part of the field, Sd, is dominated by geomagnetic coefficients of degree 2 and order 1. Surprisingly, the axisymmetric quadrupole and octupole terms contribute very little, highlighting the dangers of modeling the paleomagnetic field without regard to departures from axial symmetry. ¿ American Geophysical Union 1995