Bessel's Correction

  Student published a paper on The Probable Error of a Mean in 1908 in the journal Biometrika. In it he discusses the definition of mean and standard deviation and the distribution of their errors. It is based on Airy's Theory of Errors which contains a discussion of the error for the sum of two random numbers. So what has been presented in these blogs makes use of Airy's approach to the problem. The use of n-1 instead of n in the formula for the standard deviation is known as Bessel's correction.

  It is necessary to make two assumptions in order to derive formulas for the mean, μ, and standard deviation, σ. Assuming that a set of random numberss has a mean value and a purely random component imposes a constraint on possible values for them. If we take the average data set, x_k, it will be equal to μ plus σ times the average of the z-values. We have two unknowns and the average of the z-values which is subject to some error. The same is true for the mean square of the x_k. Both assumptions yield and equation with an unknown sum involving z-values, Eqns (2) and (3) below.

For a normal distribution of errors the expected values of the sum of the z-values and their squares are approximately 0 and n. Making these substitutions we get the usual formulas for the mean and standard deviations. Near the peak the sum of the squares is approximately n-1 instead of n and we get Bessel's correction for the standard deviation.

  Using random numbers to check of these two formulas we see that the average for Bessel's correction is closer to chosen value for the standard deviation but its variation is slightly larger.