  We consider the macroscopic system of free lattice fermions in one dimension
assuming that the one-body Hamiltonian of the system is the one dimensional
discrete Schr\"odinger operator with independent identically distributed random
potential. We show analytically and numerically that the variance of the
entanglement entropy of the segment $[-M,M]$ of the system is bounded away from
zero as $M\rightarrow \infty $. This manifests the absence of the selfaveraging
property of the entanglement entropy in our model, meaning that in the
one-dimensional case the complete description of the entanglement entropy is
provided by its whole probability distribution. This also may be contrasted the
case of dimension two or more, where the variance of the entanglement entropy
per unit surface area vanishes as $% M\rightarrow \infty $ \cite{El-Co:17},
thereby guaranteeing the representativity of its mean for large $M$ in the
multidimensional case.
