  There exist at least a few different kind of averaging of the differences of
the energy-momentum and angular momentum in normal coordinates {\bf NC(P)}
which give tensorial quantities. The obtained averaged quantities are
equivalent mathematically because they differ only by constant scalar
dimensional factors. One of these averaging was used in our papers [1-8] giving
the {\it canonical superenergy and angular supermomentum tensors}.
  In this paper we present one other averaging of the energy-momentum and
angular momentum differences which gives tensorial quantities with proper
dimensions of the energy-momentum and angular momentum densities. But these
averaged energy-momentum and angular momentum tensors, closely related to the
canonical superenergy and angular supermomentum tensors, {\it depend on some
fundamental length L}.
  The averaged energy-momentum and angular momentum tensors of the
gravitational field obtained in the paper can be applied, like the canonical
superenergy and angular supermomentum tensors, to coordinate independent local
(and also global) analysis of this field.
