When averaging becomes worse: an exact test of measurement drift

Incoprea Research · 30 October 2026 · Mathematical/model test

Finding: Averaging can have a best stopping point when an instrument wanders during acquisition. In a declared random-walk model, the root-mean-square error falls to 0.10769 units at 173 samples, then rises to 0.57748 units at 10,000 samples. The same calculation without drift continues improving. This supports a conditional mechanism in Uncertainty Has a Floor, rather than a universal lower bound on every measurement.

This is an exact mathematical result with a synthetic simulation check, not a measurement of an actual instrument. The Incoprea Research hypothesis collection remains the source of the question. No novelty or external peer review is claimed.

What was tested

The target is a fixed baseline at the beginning of acquisition. At sample i, the reading is y_i = mu + d_i + e_i. The observation noise e_i is independent Gaussian noise with standard deviation sigma. Instrument drift d_i is a cumulative sum of independent Gaussian steps with standard deviation q. Drift and observation noise are independent. The estimate is the ordinary mean of the first n readings.

The time-zero target matters. A long average of a moving instrument is not necessarily a better estimate of its initial offset. This test does not assess an estimator that tracks the current state, performs recalibration, or knows and subtracts the realized drift.

Before calculating outcomes, the plan specified sigma = 1 unit, q = 0, 0.003, 0.01 and 0.03 units per square-root sample, a 1–10,000-sample grid, and a 5,000-path simulation with fixed seed 2026103012. The random-walk step itself has standard deviation q units at the fixed sample interval. The square-root-sample notation describes accumulation with elapsed sample count.

Derivation and result

Every drift step affects several later readings. In the mean, their weights are n/n, (n−1)/n, down to 1/n. Summing their independent variances gives the exact mean-squared error:

MSE(n) = sigma²/n + q²(n+1)(2n+1)/(6n).

For positive q, the continuous optimum is n* = sqrt(3 sigma²/q² + 1/2). The best integer is obtained by checking its two neighboring integers. An exhaustive search of the declared grid agrees. The error initially falls because the independent observation term shrinks; eventually the accumulated drift term grows approximately as q²n/3.

At q = 0.01, the optimum is 173 samples and its exact RMSE is 0.107689878. At 10,000 samples, exact RMSE is approximately 0.57748: about 5.36 times larger. Lowering q to 0.003 shifts the minimum to 577 samples; raising it to 0.03 shifts it to 58. These are model outputs in arbitrary measurement units, not recommended acquisition lengths for a real device.

The no-drift negative control is decisive. With q = 0, MSE is simply sigma²/n and decreases throughout the grid, reaching RMSE 0.01 at 10,000 samples. An optimum is therefore conditional on the noise process and the target, not guaranteed by averaging itself.

Numerical check

Five thousand independently simulated drift trajectories were accumulated directly. At each selected length, an independently generated Gaussian mean represents the white-noise contribution, whose exact standard deviation is sigma/sqrt(n). This avoids storing an unnecessary second full noise trajectory while preserving the marginal distribution being checked.

For n = 1, 10, 100, 173, 1,000 and 10,000, simulated mean-squared errors differ from the exact formula by 0.87%, 1.57%, 1.47%, 2.53%, −0.88% and 1.57%, respectively. Every point meets the prespecified 5% agreement tolerance. The formula, not the Monte Carlo minimum, determines the reported optimum.

Interpretation and limits

NIST's frequency-stability handbook describes methods for analyzing noise and stability; its time and frequency guide explains why averaging can stop helping. Those are established precedents, not evidence that this particular synthetic process matches a given instrument.

Our RMSE is also not Allan deviation. It quantifies a specified estimator against a specified initial baseline. Drift spectra, sampling intervals, correlations, outliers and correction methods can change both curve and optimum. The original hypothesis's “drift time” needs an operational definition before empirical testing. The useful finding is narrower and exact: more samples can worsen an uncorrected baseline estimate under cumulative drift, and the balance can be computed rather than guessed.

The accompanying runnable code, frozen plan and results provide the complete assumptions and fixed random seed. A real-instrument study would next estimate its noise components and test prediction on held-out acquisition periods.