Measurement of the anisotropy of magnetic susceptibility (AMS) in rock specimens is complicated when magnitude differences between two of the three principal susceptibilities which are conventionally used in defining the susceptibility ellipsoid approach the precision limits of the instrument used. To evaluate this problem, the precision is reported for 5 replicate measurements of low field AMS in 34 samples (including diamagnetic samples) with susceptibilities ranging between 5.73¿10-3 and 2.94¿10-5 in mass SI units using a torsion fiber magnetometer. (These samples are available upon request for interlaboratory calibration.) Precision appears to be controlled by the within-sample axial susceptibility ratios and not by the susceptibility magnitudes. It is observed for the instruments used in this study that when the axial ratios (&khgr;a/&khgr;c or &khgr;b/&khgr;c) are equal to or less than 1.002, then the replicate precision for those axes is poor. When the axial ratios are greater than 1.005, then the precision is excellent. Such replicate AMS measurements for any given instrument provide an empirical means to determine a practical method for rejecting AMS data that are not sufficiently anisotropic for reliable measurement.