The main objective of this paper is to elucidate the characteristics of a new spatial association statistic, S, in comparison with Moran’s I and Geary’s c. A general statistic, S, is defined and two derivative statistics, S0 and S*, are subsequently proposed with a strong guidance for an exclusive use of the latter. S* is defined as a rate of the variance of one variable’s spatial moving average vector to the original variance suggesting that the presence of a strong positive spatial autocorrelation results in a smaller reduction in variance which leads to a higher S* with a culminating point of 1 in a theoretical sense. In order to examine the properties of S* , two methods are introduced; one is to derive the first four central moments and the other is to extract eigenvalues and eigenvectors. The former aims at determine the distributional characteristics of the statistics, and the latter seeks to the obtain their feasible ranges. Regular tessellations of triangles, squares, and hexagons with three different sample sizes (64, 256, 1,024) are generated and used for an investigation. The smallest administrative spatial units for the 7 big cities in South Korea are also utilized to examine the practical research implications. The major findings are twofold. First, S*, unlike other spatial association statistics, turns out to yield a constant feasible range of zero to one regardless of different spatial unit shapes, different contiguity types, and different spatial proximity matrices. This is the most important merit of the statistic convincing its usability. Second, the skewness and kurtosis of S* are considerably deviant form the norms with even a large sample size such that the normal approximation based on the first two moments may not be valid. This is the most important defect of the statistic precipitating the use of more advanced significance testing procedures.