Statistical analyses were applied at the Hanford Site, USA, to assess groundwater contamination problems that included (1) determining local backgrounds to ascertain whether a facility is affecting the groundwater quality and (2) determining a 'pre-Hanford' groundwater background to allow formulation of background-based cleanup standards. The primary purpose of this paper is to extend the random effects models for (1) assessing the spatial, temporal, and analytical variability of groundwater background measurements; (2) demonstrating that the usual variance estimate s2, which ignores the variance components, is a biased estimator; (3) providing formulas for calculating the amount of bias; and (4) recommending monitoring strategies to reduce the uncertainty in estimating the average background concentrations. A case study is provided. Results indicate that (1) without considering spatial and temporal variability, there is a high probability of false positives, resulting in unnecessary remediation and/or monitoring expenses; (2) the most effective way to reduce the uncertainty in estimating the average background, and enhance the power of the statistical tests in general, is to increase the number of background wells; and (3) background for a specific constituent should be considered as a statistical distribution, not as a single value or threshold. The methods and the related analysis of variance tables discussed in this paper can be used as diagnostic tools in documenting the extent of inherent spatial and/or temporal variation and to help select an appropriate statistical method for testing purposes.