Other pages in this series (7)
M1
Relative-uncertainty half-life
Years for the relative uncertainty to halve
Takeaway
- The relative standard uncertainty (the error-bar width relative to the value) halved roughly every 5.4 years: the median half-life over the 16 core constants whose uncertainty declined.
- The electron relative atomic mass has the shortest half-life (3.0 years) and the Newtonian constant of gravitation the longest (12.2 years).
- Definition
- For each constant, how fast the relative standard uncertainty (u/|x| — the error-bar width relative to the value) fell. log₂ of the relative standard uncertainty is regressed on edition year, giving the years for the uncertainty to halve (the half-life).
- Formula
- log₂(u/|x|) = a + b·year, half-life = −1/b (years). The 95% interval maps b ± t(0.975, n−2)·SE(b) through the same expression (a regression interval, not a bootstrap). A slope flatter than −0.001 per year (half-life above 1,000 years) is reported as no decline.
- Denominator
- Constants with a measured (not exact) value in at least four editions: 17 core constants, 28 including display constants.
- How to read it
- Over the 16 core constants whose uncertainty declined, the median half-life is 5.4 years: roughly every 5.4 years the relative standard uncertainty halved (5.7 years over all 17, counting the one with no decline). The electron relative atomic mass has the shortest half-life (3.0 years) and the Newtonian constant of gravitation the longest (12.2 years). The neutron magnetic moment kept almost the same uncertainty from 1986 on (slope −7×10⁻¹¹ per year, interval including zero) and is shown as “no decline.”
- Caveats
- Editions are 4 to 13 years apart. Constants that became exact along the way (fixed by definition rather than measured), such as the Planck and Boltzmann constants, are counted only up to that point. A straight line is a summary: constants whose uncertainty dropped in one step when a new method arrived fit it poorly (low R²).
| Name | Layer | Editions (n) | Span | Relative standard uncertainty (first) | Relative standard uncertainty (last) | Half-life (years) | 95% interval | R² |
|---|---|---|---|---|---|---|---|---|
| Faraday constant | Display constant | 8 | 1969–2014 | 5.597×10⁻⁶ | 6.115×10⁻⁹ | 4.92 | 4.17–6.00 | 0.969 |
| Avogadro constant | Display constant | 8 | 1969–2014 | 6.642×10⁻⁶ | 1.229×10⁻⁸ | 5.21 | 4.40–6.39 | 0.967 |
| molar gas constant | Display constant | 8 | 1969–2014 | 4.210×10⁻⁵ | 5.773×10⁻⁷ | 7.35 | 6.51–8.44 | 0.984 |
| inverse fine-structure constant | Display constant | 10 | 1969–2022 | 1.532×10⁻⁶ | 1.532×10⁻¹⁰ | 3.70 | 3.33–4.17 | 0.982 |
| elementary charge | Display constant | 8 | 1969–2014 | 4.369×10⁻⁶ | 6.117×10⁻⁹ | 5.02 | 4.25–6.15 | 0.968 |
| electron mass | Display constant | 10 | 1969–2022 | 5.928×10⁻⁶ | 3.074×10⁻¹⁰ | 3.96 | 3.04–5.69 | 0.878 |
| proton-electron mass ratio | Display constant | 10 | 1969–2022 | 5.991×10⁻⁶ | 1.743×10⁻¹¹ | 3.18 | 2.79–3.70 | 0.971 |
| Stefan–Boltzmann constant | Display constant | 8 | 1969–2014 | 1.693×10⁻⁴ | 2.293×10⁻⁶ | 7.34 | 6.49–8.44 | 0.983 |
| alpha particle relative atomic mass | Core constant | 7 | 1998–2022 | 2.499×10⁻¹⁰ | 1.549×10⁻¹¹ | 9.60 | 3.73–∞ | 0.349 |
| deuteron relative atomic mass | Core constant | 8 | 1986–2022 | 1.192×10⁻⁸ | 7.450×10⁻¹² | 3.70 | 2.80–5.48 | 0.905 |
| electron relative atomic mass | Core constant | 10 | 1969–2022 | 6.198×10⁻⁶ | 1.768×10⁻¹¹ | 3.03 | 2.66–3.52 | 0.972 |
| neutron relative atomic mass | Core constant | 10 | 1969–2022 | 9.914×10⁻⁸ | 3.966×10⁻¹⁰ | 6.32 | 4.79–9.28 | 0.867 |
| proton relative atomic mass | Core constant | 10 | 1969–2022 | 7.942×10⁻⁸ | 8.240×10⁻¹² | 4.50 | 3.61–5.99 | 0.915 |
| Newtonian constant of gravitation | Core constant | 10 | 1969–2022 | 4.645×10⁻⁴ | 2.247×10⁻⁵ | 12.18 | 7.12–42.21 | 0.568 |
| Rydberg constant | Core constant | 10 | 1969–2022 | 1.002×10⁻⁷ | 1.094×10⁻¹² | 3.07 | 2.48–4.03 | 0.921 |
| electron magnetic-moment anomaly | Core constant | 10 | 1969–2022 | 2.673×10⁻⁶ | 1.552×10⁻¹⁰ | 3.50 | 2.87–4.47 | 0.933 |
| muon magnetic-moment anomaly | Core constant | 9 | 1973–2022 | 2.658×10⁻⁴ | 3.517×10⁻⁷ | 5.71 | 3.85–11.04 | 0.774 |
| fine-structure constant | Core constant | 10 | 1969–2022 | 1.507×10⁻⁶ | 1.507×10⁻¹⁰ | 3.70 | 3.34–4.15 | 0.983 |
| Planck constant | Core constant | 8 | 1969–2014 | 7.546×10⁻⁶ | 1.222×10⁻⁸ | 5.11 | 4.33–6.23 | 0.968 |
| Boltzmann constant | Core constant | 8 | 1969–2014 | 4.273×10⁻⁵ | 5.722×10⁻⁷ | 7.30 | 6.44–8.41 | 0.983 |
| muon-electron mass ratio | Core constant | 9 | 1973–2022 | 2.273×10⁻⁶ | 2.225×10⁻⁸ | 7.94 | 5.40–15.03 | 0.782 |
| neutron magnetic moment to nuclear magneton ratio | Core constant | 8 | 1986–2022 | 2.352×10⁻⁷ | 2.352×10⁻⁷ | no decline | 0.006 | |
| proton magnetic moment to nuclear magneton ratio | Core constant | 10 | 1969–2022 | 6.087×10⁻⁶ | 2.936×10⁻¹⁰ | 4.48 | 3.51–6.18 | 0.897 |
| deuteron rms charge radius | Core constant | 6 | 2002–2022 | 1.309×10⁻³ | 1.269×10⁻⁴ | 6.27 | 3.49–31.09 | 0.752 |
| proton rms charge radius | Core constant | 6 | 2002–2022 | 7.771×10⁻³ | 7.612×10⁻⁴ | 6.39 | 3.52–34.55 | 0.744 |
| helion relative atomic mass | Sensitivity candidate | 7 | 1998–2022 | 2.852×10⁻¹⁰ | 2.454×10⁻¹¹ | 4.17 | 2.21–37.07 | 0.627 |
| triton relative atomic mass | Sensitivity candidate | 5 | 2006–2022 | 8.290×10⁻¹⁰ | 3.316×10⁻¹¹ | 2.93 | 1.45–∞ | 0.764 |
| deuteron magnetic moment to nuclear magneton ratio | Sensitivity candidate | 8 | 1986–2022 | 2.799×10⁻⁸ | 2.566×10⁻⁹ | 10.55 | 8.30–14.47 | 0.931 |