Effect of Region Size on Slope Distribution
Accessed with the Geomorphometry analysis, Algorithms tab, "Slope region size" option.
Because computed slopes vary with size of the region used, this lets you assess the impact. It will use regions various multiples times the grid size.
The locations of the 8 neighbors used to compute slope, with region sizes and 1 (3x3), 2 (5x5), and 3 (7x7).
Region 3  Region 3  Region 3  
Region 2  Region 2  Region 2  
Region 1  Region 1  Region 1  
Region 3  Region 2  Region 1  Point  Region 1  Region 2  Region 3 
Region 1  Region 1  Region 1  
Region 2  Region 2  Region 2  
Region 3  Region 3  Region 3 
The diagram represents a geographic grid away from the equator, where the eastwest spacing (dx) is less than the north south spacing (dy). In this case there is not a single grid spacing in meters, and the algorithm must use both dx and dy.
The most common slope algorithm uses 8 points surrounding the point to compute the slope. There are three east west lines shown in blue, and three north south lines shown in red. The elevation difference of the two end points provides a dz, and the average of the three provides dz. The dx and dy will be two times the grid spacing. There are two ways to look at the distance over which the slope is computed. The region size is twice the grid spacing, but could also be considered to be a region of "radius" equal to the grid spacing around the point. While this is partly a matter of semantics, if you want to compare the slopes computed from the DEM to those actually measured in the field the difference is crucial.
There are slope computations that use larger regions, but still only using 8 points, and the region size doubles or triples as shown in the diagrams with sizes of 2 and 3. This can be done by thinning (decimating) the grid. There are arguments that averaging is preferable to decimation, but decimation was ingrained in early DEM work through it use in DTED and SRTM. Software can retrieve the collect elevations without the need to actually decimate the DEM.
In general the slope will decrease as the region size increases, but some local geomorphic forms will see it increase.
There are slope algorithms that use the actual grid spacing (dx, dy, and the diagonal), such as only taking the slope particular directions, or computing the steepest of the eight. These methods actually use the elevation of the central point, and will produce steeper slopes. They are mostly only of historical use, and not widely used at present.
The example is for a 1" DEM in Alaska created from lidar.
Slope distribution histograms. 
Slope Region Size Moment Report

Slope distribution moments.
Note that the average slope decreases as the region size increases, and the significant change in the skewness and curtosis of the slope distribution. 
This will give the same results as thinning the DEM and doing a slope computation with the thinned DEM. However, this method will compute a slope for every point in the DEM (except those along the margins which do have have a sufficiently large neighborhood). This method may help to decrease the noise in very high resolution DEMs, like 1 m or 1/9" LIDAR.
This parameter can be set for the slope algorithm used in MICRODEM on the Analysis tab of the options form.
You can get a plot of slope by region size for a single location, Right click, Geomorphometry, Point slope by region size. Size plotted is the y direction size, which matters for a geographic DEM.
last revised 2/19/2021