The problem that we encountered fitting the distribution for the standard deviation is a systematic error due to the fact that the Theory of Errors does not work out exactly because the estimates of the moments are off. Bessel's correction is needed for the standard deviation and we saw that a second correction was needed for the standard deviation's own distribution function. One can plot the residuals for the original plot, the corrected plot and plot with the standard deviation calculated using its known mean.
The original plot shows evidence of a shift of the standard deviation distribution to left while the correction has greater standard deviations towards the sides and lower standard deviations near the peak as already mentioned. The plot using the known mean of the standard deviation shows a non-uniform distribution with greater fluctuations at the center of the plot. The probability distribution is responsible for reducing the counts on the sides.
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