Note on the measurement of spatial imbalance

This note presents a new measure of spatial imbalance that uses the center of gravity (or the mean center) of a distribution and the geometric center (or the centroid) of a region to define a vector indicating the relative position of the distribution in the region. The spatial imbalance of a distribution in a region is defined as the distance and directional deviation of the center of gravity of the distribution from the geometric center of the region. The convex hull of a region is introduced is the reference for maximum spatial imbalance of the distribution in the region. The spatial imbalance is measured in terms of (1) the relative distance of the center of gravity between the geometric center and the convex hull; and (2) the direction from the geometric center to the center of gravity. This new measure is tested by quantifying the westward movement of the contiguous U.S. population between 1790 and 1950. It describes the well-known observation that the U.S. population was moving westward across the contiguous United States and toward, reduced spatial imbalance.