Not Normal: the uncertainties of scientific measurements

TL;DR

This study analyzes a large dataset of scientific measurements across various fields to understand uncertainties, revealing heavy-tailed distributions and common outliers, which impact the interpretation and reproducibility of scientific results.

Contribution

It provides a comprehensive empirical analysis of measurement uncertainties across disciplines, highlighting the heavy-tailed nature of errors and implications for scientific significance criteria.

Findings

Outliers are common, with 5σ disagreements up to five orders of magnitude more frequent than expected.

Uncertainty differences follow heavy-tailed Student-t distributions, often near Cauchy.

Measurement errors from mistakes follow power-law distributions, indicating systematic failure patterns.

Abstract

Judging the significance and reproducibility of quantitative research requires a good understanding of relevant uncertainties, but it is often unclear how well these have been evaluated and what they imply. Reported scientific uncertainties were studied by analysing 41000 measurements of 3200 quantities from medicine, nuclear and particle physics, and interlaboratory comparisons ranging from chemistry to toxicology. Outliers are common, with 5{\sigma} disagreements up to five orders of magnitude more frequent than naively expected. Uncertainty-normalized differences between multiple measurements of the same quantity are consistent with heavy-tailed Student-t distributions that are often almost Cauchy, far from a Gaussian Normal bell curve. Medical research uncertainties are generally as well evaluated as those in physics, but physics uncertainty improves more rapidly, making feasible simple significance criteria such as the 5{\sigma} discovery convention in particle physics. Contributions to measurement uncertainty from mistakes and unknown problems are not completely unpredictable. Such errors appear to have power-law distributions consistent with how designed complex systems fail, and how unknown systematic errors are constrained by researchers. This better understanding may help improve analysis and meta-analysis of data, and help scientists and the public have more realistic expectations of what scientific results imply.