[Paper Review] On the probability distribution of the experimental results

A.P. Bukhvostov|ArXiv.org|May 22, 1997

Diffusion and Search Dynamics1 references3 citations

TL;DR

This paper analyzes the probability distribution of experimental results in high-energy physics and finds that measured deviations from true values follow an exponential decay (ρ(ξ) = exp(−ξ)) rather than the assumed Gaussian distribution. The discrepancy with Gaussian expectations grows rapidly—reaching 10⁷ at ξ ≈ 6—and is attributed to two types of systematic errors: detected (moderate ξ) and undetected (large ξ), which together shape the observed distribution's two-component structure.

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 $ρ(ξ)=exp(-ξ^2/2) $, but is more likely to follow the simple exponential law $ρ(ξ)=exp(-ξ)$ ($ξ$ 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 $ξ$ very rapidly, amounting to $ 10^7 $ at $ξ\approx 6 $, and is significant even at $ξ=2 $. A more detailed study reveals the two-component structure of the distribution: the $exp(-ξ)$ law is closely fulfilled up to $ξ=3$, but then, at $ξ$ 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 $ξ$ and the other at large $ξ$. 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.

Motivation & Objective

• To investigate the actual probability distribution of experimental measurement results in high-energy physics.
• To compare the observed distribution against the conventional Gaussian assumption used in data analysis.
• To identify the origin of the discrepancy between theoretical expectations and empirical data in particle physics measurements.
• To model how detected and undetected systematic errors influence the shape of the distribution at different deviation levels (ξ).

Proposed method

• Analyzes published particle physics data from the Particle Data Group's tables to extract the frequency of measurement deviations (ξ) in units of standard error.
• Compares the empirical distribution of ξ to the standard Gaussian ρ(ξ) = exp(−ξ²/2) and the simple exponential ρ(ξ) = exp(−ξ).
• Identifies a two-component structure in the distribution: exponential decay up to ξ ≈ 3, followed by a significant slowdown in decay for ξ > 3.
• Proposes a phenomenological model where two distinct types of systematic errors—detected and undetected—affect the distribution at moderate and large ξ, respectively.
• Uses natural assumptions to show that detected errors yield a distribution close to exponential, while undetected errors dominate the tail behavior.
• Employs statistical analysis and graphical comparison (via 3 figures and .eps files) to validate the two-component model against empirical data.

Experimental results

Research questions

• RQ1 How does the actual distribution of experimental deviations in high-energy physics compare to the assumed Gaussian distribution?
• RQ2 Why does the discrepancy between expected and observed probabilities grow so rapidly with increasing ξ?
• RQ3 What physical or methodological factors could explain the observed two-component structure in the distribution of measurement deviations?
• RQ4 To what extent do detected and undetected systematic errors shape the form of the probability distribution at different deviation levels?
• RQ5 Can a simple exponential model (ρ(ξ) = exp(−ξ)) adequately describe the distribution of measurement results, and if so, under what conditions?

Key findings

• The empirical probability distribution of measurement deviations in particle physics is better described by an exponential law ρ(ξ) = exp(−ξ) than by the conventional Gaussian ρ(ξ) = exp(−ξ²/2).
• The discrepancy between Gaussian and empirical probabilities reaches a factor of 10⁷ at ξ ≈ 6, indicating a severe underestimation of tail events under the Gaussian assumption.
• The distribution exhibits a two-component structure: exponential decay holds up to ξ ≈ 3, after which the decay rate slows significantly.
• The observed behavior is attributed to two types of systematic errors: detected errors (affecting moderate ξ) and undetected errors (dominating at large ξ).
• Under natural assumptions, detected systematic errors produce a distribution close to the simple exponential, validating its use in the moderate-ξ regime.
• The model explains the observed distribution as a combined effect of both error types, with the exponential form emerging naturally from the interplay of detectable and undetectable systematic effects.