What Averaging Buys, and What It Cannot

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

ADA · CALCULATION AUDIT

What Averaging Buys, and What It Cannot

Five raw temperature readings of 24.8°C, 25.2°C, 24.9°C, 25.1°C, and 25.0°C carry noise of 0.14°C, and the chapter’s 5-sample moving average brings that down to 0.063°C — roughly a 2.24x improvement. This audit asks the question that improvement invites: does averaging away that random noise also fix a systematic offset in the readings, or only make the wrong answer look steadier?

Companion to the chapter Sensor Data Processing — every number here comes from that chapter.

Ada: The section above takes five readings and reports noise of 0.14 C falling to 0.063 C after averaging – a 2.24x gain. Let me rebuild that from the raw numbers, then name the one error averaging can never touch.

• Five readings 24.8, 25.2, 24.9, 25.1, 25.0 have mean 125.0 / 5 = 25.00 C.
• Squared deviations: 0.04 + 0.04 + 0.01 + 0.01 + 0 = 0.10; variance 0.10 / 5 = 0.02; so sigma = sqrt(0.02) = 0.141421 C (the chapter’s 0.14).
• Averaging N independent samples divides that by sqrt(N): 0.141421 / sqrt(5) = 0.063246 C, an improvement of sqrt(5) = 2.2361x. Both match.

The sqrt(N) law is a law of diminishing returns: halving the noise again needs N to quadruple, because sqrt(4N) = 2 sqrt(N). But the sharper point is what stays fixed – averaging attacks only the random spread. If all five readings shared a +2 C offset, the mean would still land 2 C high; sqrt(N) shrinks the jitter around the wrong answer without moving it. That is the dividing line the chapter draws next: filtering earns precision, and only calibration can earn accuracy.

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