  We present in detail a statistical approach for the
reference-frame-independent detection and characterization of multipartite
entanglement based on moments of randomly measured correlation functions. We
start by discussing how the corresponding moments can be evaluated with
designs, linking methods from group and entanglement theory. Then, we
illustrate the strengths of the presented framework with a focus on the
multipartite scenario. We discuss a condition for characterizing genuine
multipartite entanglement for three qubits, and we prove criteria that allow
for a discrimination of $W$-type entanglement for an arbitrary number of
qubits.
