Does sqrt(n) Really Turn 74% Into 91%?

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Does sqrt(n) Really Turn 74% Into 91%?

A sleep-tracking wearable is only 74% accurate on any single night — a 26% error — yet the chapter claims a 7-night rolling average reaches about 91%, and a 30-night one about 95%. The lift comes from the standard-error rule, error = sigma / sqrt(n). This audit re-derives 26 / sqrt(7) = 9.8% to a 90.2% accuracy and asks whether that gain is genuine, or holds only when the nightly errors are independent and zero-mean rather than a hidden calibration bias.

Companion to the chapter Wearable IoT — every number here comes from that chapter.

Ada: This section makes a strong claim — that averaging seven nights lifts a 74% nightly accuracy to about 91%. The arithmetic is sound, but it rests on a condition the reader must not lose sight of. Let me carry the chapter’s own numbers through, then name the assumption that makes them true.

Start from the given nightly error of 26% and the standard-error rule, error = sigma / sqrt(n):

• sqrt(7) = 2.645751 (the chapter rounds this to 2.65)
• 7-night error: 26 / 2.645751 = 9.8271%, or using the chapter’s 2.65, 26 / 2.65 = 9.8113% — either way it rounds to the stated 9.8%
• 7-night accuracy: 100 - 9.8 = 90.2%

The chapter reports this as matching the empirical 91%, and the two agree to within 91 - 90.2 = 0.8 of a percentage point. The 30-night figure checks the same way: 26 / sqrt(30) = 26 / 5.477226 = 4.75%, giving 100 - 4.75 = 95.25% ~ 95%.

Now the condition. The 1/sqrt(n) law only shrinks error when the nightly errors are independent and zero-mean — genuine random noise that partly cancels when summed. If instead the band carries a per-user bias, say it systematically under-scores one sleeper’s deep sleep, that offset is common to every night, and averaging seven copies of the same offset returns it unchanged. Under that failure mode the 90.2% is unreachable no matter how long the window. So 90.2% is the best case, earned only when the 26% is truly random night-to-night, not a hidden calibration tilt.

There is also a design payoff hiding in the square root. Extending the window from 7 to 30 nights multiplies the data by 30 / 7 = 4.29 but cuts the error by only sqrt(30/7) = 2.07. Roughly four times the waiting for barely twice the precision — that punishing curve is exactly why the product settles on a 7-day rolling average: far enough up the 1/sqrt(n) slope to be statistically meaningful, not so far that the user loses patience for a diminishing gain.

Every number above is taken from the chapter’s own material and re-derived step by step.