**1. ****Introduction**

In design stereology the volume of an object, or the total number of cells or particles inside it, is usually estimated by means of a Cavalieri design. To predict the error variance of the corresponding unbiased estimators is therefore an important problem. The methods available for this prediction may be classified as follows (the references correspond to the most recent error variance prediction formulae).

- (i)Systematic observations used as one subset in their natural, sequential order.
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*Parallel systematic planes*. The target is a volume. See Garcia-Fiñana & Cruz-Orive (2000, subsection 5.1). For an application including the point counting effect see Garcia-Fiñana*et al*. (2003, subsection 3.2). - •
*Systematic observations on the circle*. Here the target may be curve length, surface area or volume (see Cruz-Orive & Gual-Arnau, 2002, subsection 4.1). - •
*Parallel systematic slabs for volume*. See Gual-Arnau & Cruz-Orive (1998, subsections 5.4 and 5.5). For an application see McNulty*et al*. (2000). - •
*Parallel systematic slabs for number*. See Cruz-Orive (1999, eq. 3.7). The local within-slices error is handled either by assuming that the particle centroids follow approximately a Poisson model (same article, eq. 3.12) or by the double disector technique (same article, eq. 3.19).

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- (ii) Fractionator approach: the initial systematic sample is split into two systematic subsets, and the error variance is predicted from the corresponding two estimators (Cruz-Orive, 1990).

The purpose of this paper is to update the fractionator variance predictor proposed in Cruz-Orive (1990) in the light of more recent results.

In the latter paper the target quantity was the number *N* of particles in a fixed and bounded subset *X* ⊂ ℝ^{3}, for instance the number of neurons in a well-defined brain compartment. To estimate *N* the object *X* was first split into serial slices of a constant thickness. A systematic subset of slices was drawn with a known sampling fraction, and this subset was split into odd- and even-numbered slices (this design was originally proposed by Gundersen, 1986). After an arbitrary chain of further subsampling stages, the variance predictor was computed from the corresponding two particle number estimates.

The main shortcomings of the current approach may be summarized as follows.

- 1The basic probes should be slabs (because the target is particle number), but the model used, leading notably to assumption (7) in Cruz-Orive (1990), was based on plane sections. No account was therefore taken of the fact that the estimator variance should rapidly decrease as the gap between consecutive slices decreases. We introduce the slice thickness effect making use of the result (5.16) from Gual-Arnau & Cruz-Orive (1998).
- 2As mentioned above, the number of particles contained in the odd- and even-numbered slice subsets was estimated by further subsampling stages, leading to estimation or ‘local’ errors that were also not taken into account. This is achieved using the model described in Cruz-Orive (1999, eq. 3.4).

In addition, we generalize the design in Cruz-Orive (1990) in the following way.

- 3We develop a general variance predictor (Eq. 2.27) for the case in which the initial slice sample is split not necessarily into two, but into
*n*systematic subsets, (*n*= 2, 3, … ).

The mentioned general variance predictor is discussed in Section 4 for the particular case of *n* = 2 in two common situations, namely for ordinary Cavalieri disectors (Subsection 4.1) and for double disectors (Subsection 4.3). The latter have the advantage that the local errors can be estimated directly without resorting to particle arrangement models such as Poisson’s. The formulae are illustrated in each case with numerical examples.