Uncertainty Has a Floor

Metrology · notebook volume 12 · Simulation n = 300,000

The claim

The uncertainty of a measurement that is averaged over time reaches a minimum set by the ratio of averaging time to drift time, rather than decreasing indefinitely with more averaging.

What was tested

Allan deviation should show an interior minimum in averaging time; past it, longer averages should increase error.

Falsified by: Uncertainty decreasing monotonically with averaging would remove the drift term.

Method

300,000 samples at 1 s spacing. Frequency noise is white FM (1e-11 at 1 s) plus a random walk (2e-15 per sqrt(s)) plus a linear fractional drift (4e-16 per s). Overlapping Allan deviation at averaging factors from 1 to about 25,000 s. A white-noise-only record is the control.

Result

Figure generated by code/tests/measurement-floor-drift.py from the recorded run.

Optimum averaging time (s)	17
Allan deviation at the floor	3.41e-12
Worse at ten times the optimum	32.3
Slope before / after	-0.39
Slope after the floor	0.49

supported

Allan deviation falls as tau^-0.39 to a minimum of 3.41e-12 at tau = 17 s, then rises as tau^0.49. Past the minimum, a tenfold longer average is 32.3 times worse. The white-only control falls as tau^-0.50 with no minimum.

Limits

This is the standard Allan-variance result; the test proves the posted claim is internally correct, not that any particular instrument is drift-limited.
The floor's location depends on the noise mix. A clock with no random-walk or drift term has no floor, and the claim would not apply.
Real time scales use ensembles and steering to push the floor below a single clock's.

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Working result, 30 October 2026. Not peer reviewed. Generated by code/build_site.py from results/measurement-floor-drift.json. All tests