Best Choice for the Expected Error

  In the last blog we found two values for the estimate of σ, the expected value, μ₁, and the rms error, √μ₂ = √n, and we have to ask which is the best choice. The expected value is the mean that one would get for an estimate of the error but it has some uncertainty, sd, associated with it. If we want to combine these two uncertainties we have to add their squares and the result is the second moment, μ₂ = n.

If the sum involves a short string of numbers then we are most likely to random values for z near the peak of the distribution where
z_peak = √n-1.

The rms error, s, for the sum is then equal to σ√n-1. We can turn things around to get an estimate of σ for the probability distribution from that of s and we conclude that σ = s/√n-1.