Statistical error with large number of particles in weak measurements

Question

Consider a measurement process. If $\Delta \pi$ and $\Delta x_n$ is the uncertainty in momentum and position of the measuring device.

Aharonov, Albert, et al. ask us to consider the opposite limit: an imprecise measurement where $\Delta \pi \gg \Delta x_n$. Here we are "measuring" the difference in shifts of the Gaussian curve which are much smaller than the width of the original Gaussian. They note that though a single measurement cannot yield any information due to the large uncertainty in the measuring device, measurements on a large ensemble of $N$ particles similarly prepared can reduce the statistical error by $N^{-1/2}$. A sufficiently large ensemble will produce an arbitrarily accurate determination of the average $\langle X \rangle$ for the ensemble. THey proceed to analyze the expected results of such measurements of PPS ensembles.

How can one prove the highlighted statement? The passage is taken from the paper "The Curious Quantum Mechanics of Pre- and Post-Selected Ensembles" by Wayne Hu (Found. Phys. 20, 447 (1990); eprint).

Answer 1

This is a standard result in statistics: the standard error of the mean produced by a sample of $N$ copies of a given system will scale with $1/\sqrt{N}$. The details of the proof depend on exactly what hypotheses are supplied, but every statistics textbook includes at least one such proof.