Rules for random aggregation

In this paper we derive the effect of aggregation on common statistics when the geographic areas used in the analysis are equivalent to randomly formed groups of individuals. Simple rules of aggregation are provided for use when analysis of such groups is performed. The expectations of common statistics such as means, variances, regression and correlation coefficients, are not affected by aggregation. However, the variation of these statistics is affected, mainly as a result of changes in the number of groups. This variation is related solely to random fluctuations associated with the generation of variate values. Weighting by the group population sizes is shown to be important in the calculation of statistics. Generally, unweighted statistics have larger variation than the corresponding weighted version, and the variation depends not only on the number of groups but also on the distribution of group population sizes. Methods for conducting statistical analysis of aggregate data in this situation are described and statistical inferences based on unweighted statistics are shown to be invalid.