To be complete, one should realize that there are at least two types of statistical groupings when working with raster data...those that work with overlapping neighborhoods and those that work with adjacent neighborhoods
...focal vs block statistics. Both have their place. Details of both are documented in their respective help topics
How block statistics works the neighborhood 'jumps' by its size and shape to the right (then down)
...How focal statistics works the neighborhood 'moves' the neighborhood shape one cell to the right (then down)
Then don't forget there is Line Statistics and Point Statistics ... so the choice really depends upon the neighborhood type, shape and configuration and how you want to implement the movement.
