A display can show many digits while repeating a biased result. A trustworthy measurement explains what was measured, how the instrument was calibrated and how uncertainty was assessed. Precision is valuable, but it answers only part of that question.
Separate the questions
The International Vocabulary of Metrology distinguishes precision, accuracy, measurement error and measurement uncertainty. Precision concerns agreement among repeated measurements under stated conditions. Accuracy concerns closeness to the quantity value being sought. Uncertainty characterizes the dispersion of values reasonably attributed to the measurand using the available information.
The vocabulary does not treat accuracy as a quantity with a numerical value. It also distinguishes an error from uncertainty about a result. These distinctions help prevent a small spread of readings from being mistaken for complete knowledge of the measured quantity.
An invented calibration
Suppose a timing instrument measures a reference interval assigned a value of 5.000 seconds. Four invented readings are 5.010, 5.012, 5.008 and 5.010 seconds. Their mean is 5.010 seconds, ten milliseconds above the reference value.
The sample standard deviation is approximately 1.633 milliseconds. If the fluctuations are independent with a stable distribution, the estimated standard uncertainty of the mean from this repeatability component is 1.633/√4, or approximately 0.816 milliseconds.
| Quantity | Result | Meaning |
|---|---|---|
| Mean reading | 5.010 s | Average indication |
| Estimated offset from reference | +10.0 ms | Calibration difference |
| Sample standard deviation | 1.633 ms | Observed repeatability spread |
| Repeatability uncertainty of mean | 0.816 ms | Conditional on the stated statistical model |
The compact cluster around 5.010 seconds is compatible with a systematic offset. Taking more readings can improve the estimate of that cluster's centre without moving it toward 5.000 seconds.
Include the reference
Now suppose the reference interval itself has a standard uncertainty of 2.0 milliseconds, independent of the repeatability component. In the simplified model where these are the only two contributions, the standard uncertainty of the estimated offset is
The root-sum-of-squares calculation uses independent components with unit sensitivity coefficients in this difference. Correlations or a different measurement equation change the calculation. NIST Technical Note 1297 describes the general propagation approach, including covariances where appropriate.
If 400 independent readings had the same individual spread, their repeatability contribution to the mean would be approximately 0.0816 milliseconds. The reference contribution would still be 2.0 milliseconds. Improving one term can leave another as the practical limit.
A correction has a domain
The calibration suggests subtracting ten milliseconds from indications near the tested interval under the tested conditions. It does not establish that the same correction works at every duration, temperature or operating mode. An offset, a scale error and a nonlinear response require different calibration models.
For example, an instrument that reads 0.2% high gives the same ten-millisecond error at five seconds but a twenty-millisecond error at ten seconds. A second calibration point can distinguish that scale error from a constant offset. This is an experimentally useful difference between two explanations of the same first result.
Resolution is another property
Adding decimal places changes a display. It does not automatically improve the reference, reduce drift or remove a transmission asymmetry. The network example shows an ambiguity that survives exact timestamps; the sampling example shows different signals with identical exact readings.
Those examples distinguish limited numerical resolution from missing information about the measurement process. A sensible improvement targets the actual limiting contribution rather than the most visible number on the screen.
Make a new effect testable
A proposed SMM/DTT timing effect needs a prediction that differs from the ordinary model under declared conditions. Its estimated size should be compared with a justified uncertainty budget and with plausible alternative effects. Repeating a precise result is useful only if the apparatus could distinguish the proposed difference.
The experimental pathway is a natural place to develop that comparison. State the measured output, calibration data, independently fixed parameters, proposed additional term and reserved test conditions. This turns a broad idea about finer time into a bounded investigation.
Sources and further reading
- JCGM, International Vocabulary of Metrology, measurement concepts, especially entries 2.13–2.16 and 2.26.
- NIST, Technical Note 1297: Combined Standard Uncertainty.
- Matching claims with evidence; what a proof proves about a model.