  We introduce the cut averaged entanglement entropy in disordered periodic
spin chains and prove it to be a concave function of subsystem size for
individual eigenstates. This allows us to identify the entanglement scaling as
a function of subsystem size for individual states in inhomogeneous systems.
Using this quantity, we probe the critical region between the many-body
localized (MBL) and ergodic phases in finite systems.
  In the middle of the spectrum, we show evidence for bimodality of the
entanglement distribution in the MBL critical region, finding both volume law
and area law eigenstates over disorder realizations as well as within
\emph{single disorder realizations}. The disorder averaged entanglement entropy
in this region then scales as a volume law with a coefficient below its thermal
value. We discover in the critical region, as we approach the thermodynamic
limit, that the cut averaged entanglement entropy density falls on a
one-parameter family of curves. Finally, we also show that without averaging
over cuts the slope of the entanglement entropy \vs subsystem size can be
negative at intermediate and strong disorder, caused by rare localized regions
in the system.
