The Moran Statistic

The Moran statistic, or spatial autocorrelation coefficient, is given by

where

 the number of points or spatial units

 the variable of interest

 the mean of x, and

 the spatial weight describing the adjacency or distance between the i-th and j-th point.

The expected value of Moran’s  that would be obtained when there is no spatial autocorrelation is .

The significance of  can be judged by calculating the variance of  and then comparing the following statistic to the standard normal distribution

                                           .

Under the assumption that the observed pattern of points is just one out of many possible patterns composed of  points, the variance of I is given by

where

, and

.

In practice, values of  or  (p-values < 0.05) indicate significant spatial autocorrelation.