Physical Constants Metrics

How often did the error bars that scientists published miss the next edition's value? This site joins the nine editions of the CODATA recommended values of the fundamental physical constants (1973–2022) and the 1969 adjustment that preceded them, and counts how values and uncertainties changed from edition to edition.

Unchanging numbers, a changing record

We learn that the physical constants are numbers fixed by nature. The speed of light and the charge of a single electron are the same today as yesterday, the same in this laboratory as across the galaxy. The record in which people have written those numbers down, though, has never stood still. Every few years a new table appeared, and with each table the last digits shifted a little.

Take the inverse fine-structure constant (α⁻¹, a pure number that measures the strength of the electromagnetic force). The 1969 table gave 137.036 02 with an error bar of ±0.000 21. The 2022 table gives 137.035 999 177, with a bar of ±0.000 000 021. In half a century the bar shrank ten-thousand-fold, and the wide bar of 1969 already held today's value. This is measurement at its smoothest.

It did not always go so smoothly. The error bar on the gravitational constant G grew nearly twelvefold in the 1998 edition. When laboratories could not agree on their measurements, the committee that adjusts the constants widened the margin of what it did not know rather than claim to know more. The road to precision has its steps back.

And some numbers are no longer measured at all. In 1983 the meter was redefined as the distance light travels in vacuum in 1/299 792 458 of a second, and the speed of light became a definition rather than a measurement. In 2019 the Planck constant, the elementary charge, the Boltzmann constant and the Avogadro constant followed. That ended the era in which one platinum-iridium cylinder, kept in a vault at Sèvres outside Paris since 1889, simply was the kilogram. Now the constants define the units.

This site reads that record backwards. With each value, the scientists of each edition published a bar saying the true value is probably inside it. The site counts how often those bars missed the values of later editions, and whether that happened as often as a normal distribution predicts or more often. The atomic-weights chapter asks the same question of a different kind of number. Atomic weights are not constants of nature: isotope ratios differ from sample to sample, so the values wander, and since 2009 some elements have been given an interval instead of a single value.

Takeaway

Scientists publish each constant with a standard uncertainty (the half-width, ±1σ, of an error bar saying the true value is probably inside this range). Over half a century the relative standard uncertainty fell more than ten-thousand-fold.
Yet moves between consecutive editions exceeded twice the combined standard uncertainty (|z| > 2) about once in 8, against a normal expectation of once in 22; beyond three times (|z| > 3) about once in 11, against once in 370.
Hold the ±2σ ranges of earlier editions (generous error bars) against the 2022 values and about 3 in 10 miss. Recent editions, with far smaller uncertainties, miss no less often.
The cause (overconfidence, unknown systematic error, or the adjustment procedure) cannot be told from these data alone.

12.9%normal expectation 4.6%Moves between consecutive editions beyond twice the combined standard uncertainty (|z| > 2), about once in 8 (132 pairs, 17 core constants, 95% interval 9.4–16.3%) 9.1%normal expectation 0.27%Beyond three times (|z| > 3), about once in 11 (95% interval 5.1–13.3%) 29%normal expectation 4.6%Earlier ±2σ ranges (generous error bars) that miss the 2022 value, about 3 in 10 (134 combinations, 95% interval 19–40%) 5.4 yearsHalf-life of the relative standard uncertainty: years for it to halve (median over the 16 core constants whose uncertainty declined)

One dot per (edition, core constant); the line is the median per edition. Log scale. Median 4.4×10⁻⁶ in 1969 (10 constants), 2.9×10⁻¹⁰ in 2022 (15). The set of constants changes between editions (the proton charge radius enters in 2002, for example), so the median can rise from one edition to the next.

Uncertainties shrank by more than four orders of magnitude in half a century. Yet the moves between editions often exceeded what those uncertainties allow. This site publishes those rates by constant and by edition, next to the earlier studies that asked the same question (Henrion & Fischhoff 1986; Bailey 2017). Every number carries its sample size and a 95% interval. "Scientists were overconfident" is an interpretation that would require looking into the measurements themselves, so the metric pages do not use it; the Reading page takes it up with its qualifications.

Uncertainty — the width, expressed as a standard deviation, within which a measured value may differ from the true value; the length of the ±1σ error bars in the plots. Unlike the error, the actual difference from the true value, the uncertainty is an estimate of how large that difference may be. It is the standard term in metrology (ISO/IEC Guide 98-3, the Guide to the Expression of Uncertainty in Measurement). What this site measures is whether that estimated width covered the actual moves between editions.

Edition — one set of recommended values published by the CODATA Task Group on Fundamental Constants after a least-squares adjustment of all the measurements and theoretical calculations available at the time. An edition is named after the closing year of its data (the 2022 edition uses data to the end of 2022 and was published in 2024). After the 1973 and 1986 editions, a new edition has appeared every four years since 1998. The 1969 edition is the pre-CODATA adjustment by Taylor, Parker and Langenberg, included for comparison. With each new edition the values and uncertainties of the same constants change; this site counts those changes.

Metrics

M1 · Relative-uncertainty half-lifeYears for the relative uncertainty to halve M2 · Edition-to-edition exceedanceShare of moves between consecutive editions not covered by the two uncertainties M3 · Miss rate against the 2022 editionShare of earlier error bars that do not contain the 2022 value M4 · Tail heavinessStudent-t degrees of freedom fitted to edition-to-edition z M5 · Convergence plot by constantEach edition's value and error bar as a deviation from 2022 M6 · Exact-value transitionsConstants that became exact, and exact ones that became measured Recalculation · Henrion & FischhoffTheir 1986 indices recomputed on these data

Atomic-weights chapter 1.1

Atomic weights — the revision recordCIAAW 1902–2024, 701 revisions. The same question, asked of a record that defines uncertainty as an expanded range

Hubble-constant chapter 1.2

Hubble constant — published H0 estimates and their errorsHuchra’s list 1929–2010 (aggregates only) and Table A2 1976–2019. The same question, asked of individual measurements against two references (Planck 2018, Riess et al. 2022)