The type of population encountered in the sampling of coal, ores and other products of natural phenomena is one in which the variance within a cluster of elements generally increases steadily as the size of the cluster increases. To enable estimation of variances of stratified random or systematic sampling for such populations one must model the serial correlation function or alternatively, the variogram function. The author has postulated a specific symmetry of the serial correlation function which leads to expressions with rich theoretical and practical implications. These expressions provide a spatial correlation function which explains power law relationships often observed in studies of dispersion variances. They also enable straightforward calculations of sampling variances. The importance of the role of finite population size in the equations leads one to examine previously held concepts of large' and small' sample populations._x000D_
The model based on the symmetry postulate is a fractal model with the serial correlation function of a hypothetically infinite population being of the same form as that developed by Mandelbrot for discrete fractional noise. Techniques used in fractal mathematics apply, including box counting and fitting of a dimension function._x000D_
The paper establishes a framework for the understanding of this fractal theory and demonstrates how to calculate sampling variances from the model variogram using simulated data fitting the model.