My preference would certainly be to use a statistical software, such as R. That said, it is possible to do this in ArcGIS.
First, create a random sample using the "Create Random Points" tool in the Data Management toolbox. If the topography is highly variable you will need a bigger sample than if it is homogeneous. Just make sure that the sample is large enough to capture the variation in the data.
Second, assign the raster values to your random points using the "Extract Multi Values to Points" tool in the Spatial Analyst toolbox. To ensure that the sample distribution is adequate one would examine the empirical distribution, but in ArcGIS the histogram will do. Compare the histogram of the DEM to that of your samples to make sure that you are not missing sample variation.
Finally, you will have a table with the raster values from each raster assigned to your random points. To derive the Root Mean Squared Error (RMSE) you would calculate
sqrt(mean((predicted - observed)^2)) which, when exported, should be quite straight forward in in a spreadsheet such as Excel. Since it is designed to operate on attribute tables, the ArcGIS field calculator will not return a single value.
If you want to examine the scale dependent correlation or covariance, I have a tool in the Geomorphometry & Gradient Metrics Toolbox that will do exactly this. Sometimes looking at these patterns across multiple scales can reveal unexpected findings in your error structure.