  We study the distribution of finite size pseudo-critical points in a
one-dimensional random quantum magnet with a quantum phase transition described
by an infinite randomness fixed point. Pseudo-critical points are defined in
three different ways: the position of the maximum of the average entanglement
entropy, the scaling behavior of the surface magnetization, and the energy of a
soft mode. All three lead to a log-normal distribution of the pseudo-critical
transverse fields, where the width scales as $L^{-1/\nu}$ with $\nu=2$ and the
shift of the average value scales as $L^{-1/\nu_{typ}}$ with $\nu_{typ}=1$,
which we related to the scaling of average and typical quantities in the
critical region.
