On the probability distribution of the experimental results

Abstract

The analysis of Tables of particle properties shows that the probability distribution of the results of physical measurements is far from the conventional Gaussian $\rho(\xi)=exp(-\xi^2/2) $, but is more likely to follow the simple exponential law $\rho(\xi)=exp(-\xi)$ ($\xi$ is the deviation of the measured from the true value in units of the presented standard error). A gap between the expected and actual probabilities grows with $\xi$ very rapidly, amounting to $ 10^7 $ at $\xi\approx 6 $, and is significant even at $\xi=2 $. A more detailed study reveals the two-component structure of the distribution: the $exp(-\xi)$ law is closely fulfilled up to $\xi=3$, but then, at $\xi$ larger than that, the decrease is retarded drastically. This behaviour can be associated with the existence of two various types of systematic errors, the detected and undetected ones. Within some model, both types of errors are seen to affect the form of the distribution, one at moderate $\xi$ and the other at large $\xi$. The first type (detected) errors are shown in some natural-looking assumptions to yield the distribution not quite equal but close to the simple exponential.