What sqrt(N) Buys, and Where It Stops
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
What sqrt(N) Buys, and Where It Stops
The chapter’s averaging table shows white noise dropping cleanly by 3.2× at 10 readings and 31.6× at 1000, but with 1/f noise added the same table stalls near 6× even at 10000 readings. It sets a practical stopping rule instead: for a sensor with a 0.5 Hz corner frequency, average no longer than about 1.0 second. This audit asks the question those two claims invite: does the white-noise column really track sqrt(N) as stated, and does the one-second stopping rule for a 0.5 Hz corner actually follow from the physics?
Companion to the chapter Sensor Fusion and Kalman Filtering — every number here comes from that chapter.
Ada: The white-noise column of the averaging table claims a clean square-root law, and the corner-frequency rule claims a one-second ceiling for a 0.5 Hz sensor. Both are checkable. Averaging N independent white-noise samples reduces the standard deviation by sqrt(N):
sqrt(10) = 3.162, which the table rounds to 3.2xsqrt(100) = 10.000gives 10xsqrt(1000) = 31.623gives 31.6xsqrt(10000) = 100.000gives 100x
Every white-noise entry checks out. The “with 1/f noise” column (3.0, 5, 6, 6) is an illustrative scenario, not a formula, and its whole point is that it stops tracking sqrt(N) and flattens near 6x. That is where the corner-frequency rule earns its place: stop averaging at about twice the corner. For a 0.5 Hz corner, 2 x 0.5 Hz = 1.0 Hz, so the useful window is 1 / 1.0 Hz = 1.0 second.
The design-meaningful reading is that sqrt(N) is a promise the physics only keeps above the 1/f corner: below it, every extra sample buys drift instead of precision, so the honest design pins the averaging window to the corner rather than chasing an unreachable 100x.
Every number above is taken from the chapter’s own material and re-derived step by step.