Expression used in Out et al. (2022) to efficiently express the standard deviation of the calculated magnetic moment in spherical coordinates. When obtaining results from a MMT inversion, both magnetization, * M*, and the corresponding standard deviations, \bm{\sigma}, are given in a Cartesian reference frame (x, y, z). Per grain, these magnetization and standard deviations are bootstrapped to obtain a collection of magnetization vectors. Subsequently, a sphere with radius,

*u*, and its center at the original magnetization is constructed in such a way that 95% of the bootstrapped magnetization vectors are located within this sphere.

The radius of the encompassing sphere is combined with the mean magnetization of the grain to construct a uncertainty ratio:

uncertainty ratio = \frac{u}{\mathbf{M}}\times 100\%

This uncertainty ratio is volume independent. A high uncertainty ratio means that the grains is solved badly.