Measuring the spatial balance of a sample: A new measure based on the Moran's I index

Measuring the degree of spatial spreading of a sample can be of great interest when sampling from a spatial population. The commonly used spatial balance index by Grafstr"om et al. (2012) is particularly effective in comparing the level of spatial spreading of different samples from the same population. However, its unbounded and uninterpretable scale of measurement does not allow to assess the level of spatial spreading in absolute terms and confines its use to only raw comparisons. In this paper, we introduce a new absolute measure of the spatial spreading of a sample using a normalized version of the Moran’s $I$ index. The properties and behaviour of the proposed measure are analysed through two simulation experiments, one based on artificial populations and the other on a population of real business units located in the province of Siena (Italy).