What sqrt(N) Buys, and Where It Stops
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.
See the relationship before changing it
The figure reads from left to right. The blue card is independent readings. The middle card applies this page's rule. The green card is white-noise improvement. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only independent readings, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 1000 samples.
- 2
Name the relationship. improvement = sqrt(sample count)
- 3
Substitute with units. sqrt(1,000) = 31.62 times
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change independent readings
Try Predict the direction of improvement = sqrt(sample count). Test another independent readings, then compare white-noise improvement.
Observe White noise follows the square-root law until correlated low-frequency drift sets the floor. Reset independent readings to 1000 and compare white-noise improvement.
Explain White noise follows the square-root law until correlated low-frequency drift sets the floor.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Use the fixed N = 10, 100, 1000, and 10000 rows to compare sqrt(N), then follow the displayed 0.5 Hz corner-frequency derivation.
The white-noise improvement climbs from 3.2× to 100×, but the illustrated 1/f column flattens near 6× instead of following sqrt(N).
Independent white-noise samples average down by sqrt(N); correlated low-frequency drift does not cancel, so averaging beyond the corner adds samples without proportional precision.
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.
Technical boundaries: The table uses an illustrative 1/f plateau and a single 0.5 Hz corner; it does not model the sensor's measured noise spectrum, sample correlation, filter transients, quantisation, or changing drift.