Phoebe guidesSee the relationship before changing it
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 samples averaged, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 10 samples.
- 2
Name the relationship. averaged noise = 4.41 mV / sqrt(sample count)
- 3
Substitute with units. 4.41 / sqrt(10) = 1.394 mV
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change independent samples averaged
Try Predict the direction of averaged noise = 4.41 mV / sqrt(sample count). Test another independent samples averaged, then compare rms input noise.
Observe Independent random noise falls with the square root of sample count. Reset independent samples averaged to 10 and compare rms input noise.
Explain Independent quantization noise partly cancels when readings are averaged, so the modeled input-noise magnitude falls with the inverse square root of sample count.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
1. Start with converter bins
An analog-to-digital converter (ADC) cannot return every possible voltage. A 12-bit result has 2¹² = 4,096 codes. Splitting 50,000 g evenly gives the ideal least-significant-bit step.
The lab's measured span uses 3,275 counts, so its actual step is 50,000/3,275 = 15.27 g/count.
2. Turn a step into RMS noise
When quantisation error wanders evenly between half a step below and half a step above, its root-mean-square (RMS) size is the step divided by √12.
3. Derive the averaging rule
Add independent errorsFor N uncorrelated readings, error variances add: variance(sum) = Nσ².
Divide to form the meanThe average divides the sum by N, so its variance divides by N²: Nσ²/N² = σ²/N.
Take the square rootRMS is the square root of variance, so e_rms(N) = e_rms(1)/√N.
4. Try the chapter's averaging window
TryMove the window from 1 to 64 readings and stop at the chapter's N = 10.
ObserveAt N = 10 the RMS error is 1.39 g, a √10 = 3.16-fold tightening.
ExplainEvery readout uses the same 1/√N, logarithm, and bit formulas derived above.
The averaging rule keeps the single-reading noise fixed and assumes independent random fluctuations between readings. It does not remove correlated drift or recover motion already folded into the samples by aliasing.
- Correlated drift and aliased motion do not average away by this rule
- Needs separate evidence
Use field evidence or a deeper model before release.
5. Work both chapter cases
Ideal: 12.2/√12 = 3.52 g RMS, then 3.52/√10 = 1.11 g. Measured: 15.27/√12 = 4.41 g RMS, then 4.41/√10 = 1.39 g. Both improve by √10 = 3.16 because the window is the same.
6. Name what the result buys
The average makes a slowly changing beehive mass more repeatable without changing hardware. It does not create a smaller physical ADC bin, and it cannot recover a fast signal that folded during sampling.
7. Check yourself
Why are there 4,096 codes?
What does N = 10 do to 4.41 g RMS?
Would this repair an aliased vibration?
The averaging rule keeps the single-reading noise fixed and assumes independent random fluctuations between readings. It does not remove correlated drift or recover motion already folded into the samples by aliasing.
- 50,000 g span
- Full-scale measured mass span
- 4,096 levels
- Ideal twelve-bit converter code count
- 3,275-count measured span
- Measured calibration span in converter counts
- N = 10
- Independent readings in averaging window
- stated
- Chapter input or worked result
The independent-noise model is appropriate for dither, not drift, interference, or aliased motion.