Calibration connects an indication with a reference quantity under specified conditions. Matching the reference points is necessary, but it does not establish that the chosen conversion remains adequate between them or beyond them.
A perfect fit can hide a wrong model
Consider an invented instrument with true response y = 2 + 3x + qx(10 − x)/10 for inputs between zero and ten. All quantities are dimensionless. The curvature parameter q lies between −1 and 1, so the response remains strictly increasing over this range.
At x = 0 the indication is always 2; at x = 10 it is always 32. A straight-line calibration through these two points therefore gives y = 2 + 3x regardless of q. It fits both calibration points perfectly. Using its inverse, x̂ = (y − 2)/3, nevertheless gives a bias of qx(10 − x)/30 between them.
With q = 0.8 and true x = 3, the indication is 12.68 and the straight-line estimate is 3.56. At the midpoint x = 5, the error reaches 2/3. More precise measurements of the two endpoints would not reveal this curvature: the missing information is between those endpoints.
Compare calibration with a reserved check
MODEL ADEQUACY · NO MEASUREMENT NOISE
True responseCalibration fit
Hollow circles mark calibration points; the black diamond marks the selected check.
The check at x = 3 is separate from the two calibration points. Their exact fit leaves a +0.560 input error here.
Inspect reference and check values
| Input | Indication | Role |
|---|---|---|
| 0.0 | 2.000 | Calibration |
| 10.0 | 32.000 | Calibration |
| 3.0 | 12.680 | Reserved check |
The three-point option recovers q from the midpoint and fits the true quadratic family. Its success is conditional on that family being adequate. If the selected check coincides with a calibration point, the demonstration labels it accordingly: reproducing a fitted point is not an independent validation.
Three points still do not prove the whole curve
Add a term ηx(x − 5)(x − 10) to the response. It vanishes at all three reference inputs but generally changes other readings. For sufficiently small η the response can remain monotone. Thus even an exact three-point fit cannot rule out every alternative response function.
A realistic calibration uses physical knowledge, a stated range, uncertainty estimates and checks selected to challenge plausible departures. One does not need infinitely many points; one needs evidence adequate for the intended accuracy and use. Extrapolation beyond the tested range demands additional justification.
Do not confuse correction with certainty
This demonstration removes random noise to isolate model inadequacy. Real references and readings have uncertainties, and instruments may drift with time or environment. Those contributions must accompany the conversion. A correction can remove an estimated bias while uncertainty about that correction remains.
Metrological traceability records how a result connects with a reference through calibrations, including their uncertainties. It does not guarantee that the instrument's model is adequate for every later task. See precision, accuracy and trustworthy timing for a worked uncertainty budget.
Some instruments also depend on how an input was reached. If increasing and decreasing inputs produce different indications at the same value, one static curve may be insufficient. A useful calibration protocol records the direction, waiting time and relevant history rather than forcing both branches into an apparently precise average.
Calibration in a fundamental theory
The same distinction applies to SMM/DTT. A physical scale fitted from one measured quantity is a calibration input. Another quantity becomes a prediction only if its additional model inputs were fixed independently. The expanded book's physical dictionary and dimensionless comparisons make that requirement explicit.
A useful next calculation would reserve a ratio, frequency range or preparation that was not used to choose the model. Its agreement would then test more than the ability to reproduce the calibration data.
Sources and further reading
- JCGM, metrological traceability and Developing and using measurement models.
- Recognizing circular derivations. The response functions above are original teaching constructions.