Confidence limits on the magnitude and direction of remanent magnetization measurements are derived asandwherep is probability; n is the number of unbiased independent measurements Xi, Yi, Zi, of each Cartesian component x, y, z, and is the calculated magnitude of the remanence. Comparable expressions are derived for the confidence limits on the difference between two vectors. An approximate test for randomness is described. The errors commonly amount to more than 2% in magnitude and 2° in direction, and are greater for inhomogeneously magnetized samples and for vector magnitudes close to the sensitivity limit of the magnetometer. The effect of inhomogeneity of remanence requires special attention for most magnetometers. Applications of the methods to elimination of imprecise data, and assessing whether data conform to particular directional and stability-spectra models, are described.