  We study the out-of-equilibrium probability distribution function of the
local order parameter in the transverse field Ising quantum chain. Starting
from a fully polarised state, the relaxation of the ferromagnetic order is
analysed: we obtain a full analytical description of the late-time stationary
distribution by means of a remarkable relation to the partition function of a
3-states classical model. Accordingly, depending on the phase whereto the
post-quench Hamiltonian belongs, the probability distribution may locally
retain memories of the initial long-range order. When quenching deep in the
broken-symmetry phase, we show that the stationary order-parameter statistics
is indeed related to that of the ground state. We highlight this connection by
inspecting the ground-state equilibrium properties, where we propose an
effective description based on the block-diagonal approximation of the
$n$-point spin correlation functions.
