Cavalieri sections--and more recently Cavalieri slices, especially in combination with non-invasive scanning--are widely used to estimate volumes. Physical Cavalieri slices are also increasingly used to estimate neuron numbers via the optical fractionator. In either case, the prediction of the error variance is important to assess optimal sample sizes. The error variance consists of two components, one due to the variation among the true contents of sections or slices, and the other due to local or 'nugget' errors. The latter may arise for instance estimating section areas by point counting discrete particles in slices or disectors. In this paper, a fairly comprehensive set of prediction formulae is presented to separate both variance components.