  Entanglement measures quantify nonclassical correlations present in a quantum
system, but can be extremely difficult to calculate, even more so, when
information on its state is limited. Here, we consider broad families of
entanglement criteria that are based on variances of arbitrary operators and
analytically derive the lower bounds these criteria provide for two relevant
entanglement measures: the best separable approximation (BSA) and the
generalized robustness (GR). This yields a practical method for quantifying
entanglement in realistic experimental situations, in particular, when only few
measurements of simple observables are available. As a concrete application of
this method, we quantify bipartite and multipartite entanglement in
spin-squeezed Bose-Einstein condensates of $\sim 500$ atoms, by lower bounding
the BSA and the GR only from measurements of first and second moments of the
collective spin operator.
