The assumption that the time-averaged geomagnetic field closely approximates that of a geocentric axial dipole (GAD) is valid for at least the last 5 million years and most paleomagnetic studies make this implicit assumption. Inclination anomalies observed in several recent studies have called the essential GAD nature of the ancient geomagnetic field into question, calling on large (up to 20%) contributions of the axial octupolar term to the geocentric axial dipole in the spherical harmonic expansion to explain shallow inclinations for even the Miocene. In this paper, we develop a simplified statistical model for paleosecular variation (PSV) of the geomagnetic field that can be used to predict paleomagnetic observables. The model predicts that virtual geomagnetic pole (VGP) distributions are circularly symmetric, implying that the associated directions are not, particularly at lower latitudes. Elongation of directions is North-South and varies smoothly as a function of latitude (and inclination). We use the model to characterize distributions expected from PSV to distinguish between directional anomalies resulting from sedimentary inclination error and from non-zero non-dipole terms, in particular a persistent axial octupole term. We develop methodologies to correct the shallow bias resulting from sedimentary inclination error. Application to a study of Oligo-Miocene redbeds in central Asia confirms that the reported discrepancies from a GAD field in this region are most probably due to sedimentary inclination error rather than a non- GAD field geometry or undetected crustal shortening. Although non-GAD fields can be imagined that explain the data equally well, the principle of least astonishment requires us to consider plausible mechanisms such as sedimentary inclination error as the cause of persistent shallow bias before resorting to the very "expensive" option of throwing out the GAD hypothesis.