Musing on use and misuse of numerical data in measurement science F. Pavese IMEKO, Torino, Italy
[email protected] https://www.researchgate.net/profile/Franco_Pavese/
Quantity empirical numerical value The empirical numerical value of a quantity, is not a number that can be treated as a simple mathematical object. • The digits are, not only determined by the chosen reference (VIM), but also by the full process that allowed setting them, because the latter consists of an evaluation—i.e., inference— from measurement results. • The evaluation, not only is affected by uncertainty, but the uncertainty level can be chosen from many “levels of confidence” (or “degrees of believe”). • The latter establishes the degree of possibility that the available numerical value is reasonably acceptable as the actual numerical value of the quantity in question.
Empirical ⇒ Experimental values That type of evaluation is making use of experience. • • •
•
Experience mostly comes from “experimenting”, i.e. from interacting with the world where the properties of quantities extend, by using means called “instruments”. Instruments provide in various manners a relationship between some properties of these quantities and a “magnitude” to which numbers can be assigned as its quantitative “measure”, with sufficient trust. Contrarily to the mathematical field, the experimental field is further complicated by the fact that the indicated interaction cannot be fully dominated at will, and the instruments only approximate their expected behaviour. The consequence is the need to introduce the concept of uncertainty, associated to the data, i.e. to each (set of) experimental values, quantifying partial knowledge of several types: “ontologic”, “epistemic”, “definitional”, “systematic”, “random” …
Numerical values • Number of digits expressing numerical values: they can be used as obtained, but also being approximated (rounded, truncated), or fixed by consensus (stipulated exact). • Manipulation of the digits of a numerical value: in science, a few rules exists intended to ensure the correctness and integrity of their meaning. • Top rule: “significance” of the reported digits: the level of uncertainty associated to a numerical value determines which digits are unaffected by the estimated uncertainty. • This simple criterion is far from allowing univocal application, e.g., the estimates of uncertainty can be different for different purposes and cases—mostly subjective. • Consequently, there are different ways to treat the digits of the same numerical values expressing the magnitude of the same quantity.
Examples: h, k, e, NA In order to attract your maximum attention I will now use, in order to illustrate the above issues, the new “fundamental constants” proposed for use in the new SI definition: • Planck (h) • Boltzmann (kB) • Electrical charge (e) • Avogadro (NA)
CODATA original 2017 values (u, i.e. k = 1)
{h} ×1034
6.626 070 150 (69)
{e} ×1019
1.602 176 6341 (93)
{NA} ×10–23
6.022 140 758 (62)
{kB} ×1023
1.380 649 03 (51)
CODATA original 2017 values Showing “uncertainty-affected digits” (u, k = 1)
{h} ×1034
6.626 070 150 (69)
{e} ×1019
1.602 176 6341 (93)
{NA} ×10–23
6.022 140 758 (62)
{kB} ×1023
1.380 649 03 (51)
CODATA original 2017 values The uncertainty interval can affect a further digit
(u, k = 1)
{h} ×1034
6.626 070 1 50 (69)
{e} ×1019
1.602 176 6 3 41 (93)
{NA} ×10–23
6.022 140 7 58 (62)
{kB} ×1023
1.380 649 03 (51)
Stipulated values Stipulation of a numerical value means a formal consensus decision about the number of digits to be used to indicate it as exact. CODATA proposed
×1034
6.626 070 15 |—
{e} ×1019
1.602 176 634 |—
{NA} ×10–23
6.022 140 76 |—
{kB} ×1023
1.380 649 |—
{h}
How they come from ?
My notation
Stipulated values CODATA proposed
CODATA original 2017 (u, k = 1)
{h} ×1034
6.626 070 15
6.626 070 150 (69)
{e} ×1019
1.602 176 634
1.602 176 6 341 (93)
{NA} ×10–23
6.022 140 76
6.022 140 758 (62)
{kB} ×1023
1.380 649
1.380 649 03 (51) ⇐ by rounding the original
Stipulated values CODATA original 2017 “Uncertain digits” shown (u, k = 1)
{h} ×1034
6.626 070 15
6.626 070 150 (69)
{e} ×1019
1.602 176 634
1.602 176 6 341 (93)
{NA} ×10–23
6.022 140 76
6.022 140 758 (62)
{kB} ×1023
1.380 649
1.380 649 03 (51)
Notice kB … but …
Stipulated values (CODATA) • Why using the standard deviation? In science, more standard is the use of expanded uncertainty (k ≈ 2) … • … and beyond: more conservative when applied to more important cases—e.g., in particle physics or six-sigma systems, k = 6. • The SI should deserve a more conservative approach
Stipulated values with expanded uncertainty, k ≈ 2 “Uncertain digits” shown for CODATA 2017 data
{h} ×1034
6.626 070 15
6.626 070 150 (138)
{e} ×1019
1.602 176 634
1.602 176 6 341 (186)
{NA} ×10–23
6.022 140 76
6.022 140 758 (124)
{kB} ×1023
1.380 6 49
1.380 6 4903 (102)
Notice here kB …
How stipulation is set? • I.e., how an “exactly-known” value (should) would arise from the originally uncertain numerical value? • … also considering that the CODATA adjustment technique does also reduce the actual experimental uncertainties?
• What is the meaning of “exactly-known” numerical value? (and its notation)
EXACTLY-KNOWN NUMERICAL VALUES Relative precision computed with no rounding due when next digit < 5
{h} ×1034 {e} ×1019 {NA} ×10–23 {kB} ×1023
via TRUNCATION Full: CODATA stipulated Bold: truncated
6.626 070 |– 15
urel = 7.5
x10–8
with expanded uncertainty, k ≈ 2 for CODATA 2017 data
6.626 070 150 (138)
1.602 1766 |– 34 1.602 1766 341 (186) –8
urel = 3 x10
6.022 140 |– 76 urel = 8
x10–8
1.3806 |– 49
urel = 1.5
x10–4
6.022 140 758 (124) 1.3806 4903 (102)
Same to CODATA stipulated, having uncertain digits truncated
EXACTLY-KNOWN NUMERICAL VALUES via ROUNDING Full: CODATA 2017 stipulated Bold: with rounding
{h} ×1034 {e} ×1019
6.626 070 |– (| 15 ) urel = 2.3
x10–8
(1.2–18.8)
for CODATA 2017 data
6.626 070 150 (138)
1.602 176 |– ( 6 | 34 ) 1.602 176 6 341 (186) –7
urel = 2 x10
(15.5–53.7)
{NA} ×10–23 6.022 1 |––5( 40 | 76 ) urel = 1.3 x10
{kB} ×1023
with expanded uncertainty, k ≈ 2
(63.4–89.2)
1.380 |– ( 6 | 49 ) urel = 4
x10–4
(48.0–50.1)
Less precise except h
6.022 140 758 (124) 1.380 6 4903 (102)
General final considerations • The same numerical value affected by uncertainty could be expressed in different ways: – truncated, rounded —expressing only the exactly-known digits ?
– stipulated —with the digits affected by uncertainty included ? • The inclusion of uncertain digits would propagate the uncertainty of the experimental evidence that the stipulation aims at stopping: higher risk of subsequent detection of different ‘best’ values for those digits. o In other fields, e.g. in media, larger number of digits equals higher competence, even when not significant.
It induces the reader/user to believe in a higher credibility of their validity. Here not a valid argument. o In the specific case of constants stipulated for use in the SI definition, the exactness of the numerical value is the basic requirement of the SI regulatory context, since units need be used as defined, requiring their uniqueness, and the stability in time.
One more consideration on the constants I.
In the context of metrology, in addition to the requirement to stipulate the current “best value”—in a strict statistical sense, a second criterion needs to be satisfied: continuity of the magnitudes of the same units over the change of their definitions.
II.
Continuity coefficients computed from CODATA proposed stipulations:
[m(K)/(kg)rev]/1 = 1.000 000 001(10) [µ0/(Hm−1)rev]/(4π ×10−7) = 1.000 000 000 20(23) [M(12C)/(kgmol−1)rev]/0.012 = 1.000 000 000 37(45) [TTPW/(K)rev]/273.16 = 1.000 000 01(37)
[1.2 ×10−8] [2.3 ×10−10] [4.5 ×10−10] [5.7 ×10−7]
thus, CODATA stipulations look be needed to support the required continuity.
III.
Evidence is, instead, that the 2017 experimental results do not support the number of digits required to also ensure continuity within the present uncertainties, as required by CIPM.
One more consideration on the constants DILEMMA (so far unpublished and unresolved) ?? Accepting stipulated values exceeding present accuracy, but preserving continuity or ?? Requiring stipulated values to conform present accuracy, but inducing a unit magnitude small discontinuity (due to evidence III— actually, already not the only reason for volt, mole, …)
