You can use either approach inside a script. If you use ParaView’s DescriptiveStatistics filter, your python script should be run with pvpython or pvbatch (for distributed-memory parallelism). If you use the VTK filter directly, you will run your script with vtkpython. (If you have your PYTHONPATH set properly, you can use the plain python interpreter whose python libraries ParaView/VTK link against).
To understand how the statistics filters work in general, this blog post does a good job of explaining the process. As far as interpreting the vtkTable data that the filters output, here is a description. In your example, the mean and standard deviation will be in the table output by the “Derive” phase of the filter. The second link describes how to find the right entries in the table.