# Theory of random measurement errors: concepts, uses, and misuses

## TL;DR

This paper explains the theory of random measurement errors, highlighting how it differs from statistics and addressing common misunderstandings that lead to incorrect usage.

## Contribution

The paper clarifies conceptual differences between random measurement error theory and statistics, emphasizing proper interpretation of mean and uncertainty.

## Key findings

- The mean in random measurement error theory represents the best estimate of the true value, differing from its role in statistics.
- Uncertainty of the mean, not standard deviation, is the key parameter for determining the probability of true value intervals.
- Misunderstandings of these concepts lead to frequent misuses in experimental analysis and presentation.

## Abstract

We present an overview of the theory of random measurement errors, focusing on the underlying concepts rather than on a strict mathematical formulation. Although the related literature is extensive, one can frequently encounter partly or completely wrong usages of the theory. In many cases, these misuses stem from incomplete understanding of the basic principles. Our presentation is based on a discussion on similarities and differences between this theory and statistics, as they are used differently in analysing the results of an experiment. In statistics, the central parameters are the mean and standard deviation, which are related to a given statistical distribution. In the theory of random measurement errors, the mean has a different meaning, representing the best estimate of the true value of a measured quantity. The second parameter of importance is not standard deviation but the uncertainty of the mean, which sets the probability that the true value lies in a given interval around the mean. These conceptual differences are seldom pointed out, which sometimes results in doubtful or wrong analyses and presentations of measurement results. We illustrate our theoretical considerations with examples of proper and improper use of the theory.