Other pages in this series (7)
M4
Tail heaviness
Student-t degrees of freedom fitted to edition-to-edition z
Takeaway
- The distribution of moves between editions (z) has much heavier tails than the normal (bell curve): very large misses happen far more often than the normal predicts.
- The Student-t degrees of freedom are ν = 1.00, close to the Cauchy distribution (ν = 1); the normal has ν = ∞, and smaller means heavier tails.
- In practice, a move of more than three published standard uncertainties (|z| > 3) is not rare: 9.1% of pairs.
- Definition
- A zero-centered Student-t is fitted by maximum likelihood to the z (M2) of consecutive pairs whose value changed. The Student-t distribution sets the thickness of its tails (the two ends) through its degrees of freedom ν: the smaller ν, the thicker the tails and the more common very large moves (ν = 1 is the Cauchy distribution; ν → ∞ the normal).
- Formula
- z ~ s · t_ν, ν and s by maximum likelihood; 95% intervals from 2,000 cluster-bootstrap resamples of constants. Pairs whose printed value did not change (z = 0) are left out.
- Denominator
- 122 of the 132 consecutive pairs over 17 core constants: those whose value changed.
- How to read it
- ν = 1.00 (95% interval 0.75–1.54), scale s = 0.38 (0.20–0.71). A ν near 1 means tails close to the Cauchy distribution: very large moves far more common than under the normal. Pairs from 1998 on alone give ν = 0.92 (0.69–1.30). For reference, Bailey (2017) obtained ν = 1.81 ± 0.15 from pairs of individual measurements of seven constants.
- Caveats
- With the pairs whose printed value did not change (z = 0) kept, no maximum-likelihood estimate exists: the likelihood grows without bound as the scale s goes to zero (for ν below 0.08). Those 10 pairs are left out and counted in M2 instead. Bailey compares individual experimental results and this site compares editions of recommended values, so the two ν are not the same quantity: recommended values are weighted averages of results, some of which are dropped or have their uncertainties expanded (expansion factors). A scale below 1 reflects the data shared between editions (see M2). The sample is small and the intervals wide.
| Group | Pairs | Constants | ν | 95% interval | s | 95% interval |
|---|---|---|---|---|---|---|
| pairs with a change | 122 | 17 | 1.00 | 0.75–1.54 | 0.38 | 0.20–0.71 |
| pairs with a change, 1998–2022 | 87 | 17 | 0.92 | 0.69–1.30 | 0.29 | 0.15–0.54 |
| Bailey 2017 (constants, pairs of individual measurements) | 9,098 | 7 | 1.81 ± 0.15 | 0.89 ± 0.06 |