The algorithm used to determine the default search radius, also known as the bandwidth, is as follows:
Calculate the mean center of the input points. If a Population field other than None was selected, this, and all the folllowing calculations, will be weighted by the values in that field.
Calculate the distance from the (weighted) mean center for all points.
Calculate the (weighted) median of these distances, Dm.
Calculate the (weighted) Standard Distance, SD.
See the Standard Distance tool for more details on this.