Comparison

We have seen four different ways to take an average - two simple averages (the second skipping zero values) and two weighted averages (using straight Poisson and Poisson-Mighell [6] $\chi ^2$ formulations). We know that the simple average (not skipping zeros) is the best possible result. However, there are inconveniences with it. If for instance we need to scale our data before averaging, then the simple average is no more usable (it will give the correct average but a bad estimate of the s.d.) . In any case, the passage to normal statistics (using Mighell's correction) needs to be done before or later. Therefore a comparison is due in order to ascertain how wrong can it be using the different methods.

We have to give a measure of what is negligible first. The relative error is a measure of the smallest relative variation of an estimate $x$ that is not negligible:

$$\epsilon_x = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{x}}}}}}}$$

We shall then consider negligible (w.r.t. $x$ ) terms whose relative magnitude is $O(\epsilon_x^2)$ . As the s.d. of $x$ is $\propto\epsilon_x$ , we may not discard terms $O(\epsilon_x^2)$ on the s.d.; there instead we may neglect terms $O(\epsilon_x^3)$ .