Bias, RMSE, and the Variance Split

Bias, RMSE, and the Variance Split

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

foundations
math-foundations
calculation-audit
sensor-applications
Ada ADA · CALCULATION AUDIT

Bias, RMSE, and the Variance Split

A sensor lab logs five paired errors against a reference — +0.4, +0.6, +0.5, +0.3, +0.7 — for an RMSE of 0.52, and claims almost all of it is a 0.50 offset that a single calibration could cut to 0.14. That rests on the identity MSE = bias² + variance. This audit carries every digit through the bias, the MSE, and the variance split, and asks whether the numbers really justify removing 0.50 of the 0.52 with one zero-offset correction.

Companion to the chapter Lab: Sensor Application Workflow — every number here comes from that chapter.

Try

A sensor lab logs five paired errors against a reference — +0.4, +0.6, +0.5, +0.3, +0.7 — for an RMSE of 0.52 , and claims almost all of it is a 0.50 offset that a single calibration could cut to 0.14 . Calculate this case.

Observe

This audit carries every digit through the bias, the MSE, and the variance split, and asks whether the numbers really justify removing 0.50 of the 0.52 with one zero-offset correction. Check shows this.

Explain

The audit shows the number that matters is the split, not the headline: because the variance (0.02) is tiny beside the squared bias (0.25), a single zero-offset calibration is justified in removing 0.50 of the 0.52 -- and the honest lab reports "0.14 residual after offset," a claim a bare "accurate enough" cannot make. Check confirms it.

See the relationship before changing it

The figure reads from left to right. The blue input is offset removed. The middle card names the page’s rule. The green output is residual rmse. The arrow matters: change the input, apply the rule once, then read the result with its unit.

Offset Removed changes residual rmse A three-part teaching diagram connects offset removed, the rule RMSE = sqrt((0.50 - removed offset)^2 + 0.02), and residual rmse. INPUT Offset removed APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrow. Removing bias cannot remove the separate random variance.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 0.5 units.

  2. 2

    Name the relationship. RMSE = sqrt((0.50 - removed offset)^2 + 0.02)

  3. 3

    Substitute with units. sqrt((0.50 - 0.50)^2 + 0.02) = 0.141 units

  4. 4

    Read the result. Keep the unit beside the value, then use the result only inside the technical boundary below.

Predict, then change offset removed

Try Predict how residual rmse responds when offset removed moves. Calculate offset removed; compare residual rmse with that prediction.

0.5 units
Chapter baseline
Residual RMSE

Observe Return to 0.5 units. Recheck residual rmse with offset removed at its chapter value.

Explain Removing bias cannot remove the separate random variance.

Check yourself

What should you do before trusting a moved-slider result?
Answer: Predict its direction, apply the displayed relationship, keep the units, and compare the reset value with the chapter’s worked baseline.
What does this small model leave out?
Answer: Only offset removed moves here. The residual rmse calculation excludes field effects listed below.

Honesty note. The residual rmse result holds offset removed as the only changing input. It is not a field prediction.

Technical boundaries

Not represented by “Bias, RMSE, and the Variance Split” are time correlation, calibration drift, non-Gaussian errors, missing samples, or changing ground truth outside the supplied residual set; “Bias, RMSE, and the Variance Split” therefore reports only its named fixtures.

Ada: This chapter’s worked comparison makes a strong claim – that an RMSE of 0.52 is almost all a 0.50 offset, so calibration would cut the error to 0.14. That rests on the identity MSE = bias^2 + variance, so let me carry every digit and check it. The five paired errors (measured minus reference) are +0.4, +0.6, +0.5, +0.3, +0.7.

  • Bias: (0.4 + 0.6 + 0.5 + 0.3 + 0.7) / 5 = 2.5 / 5 = 0.50.
  • Squared errors: 0.16, 0.36, 0.25, 0.09, 0.49, summing to 1.35.
  • MSE: 1.35 / 5 = 0.27, so RMSE = sqrt(0.27) = 0.519615..., which rounds to 0.52.
  • Variance of the errors: MSE - bias^2 = 0.27 - 0.25 = 0.02.
  • Residual after removing the offset: sqrt(0.02) = 0.141421..., rounding to 0.14.

The identity closes exactly: bias^2 + variance = 0.25 + 0.02 = 0.27 = MSE, so the split loses nothing.

The audit shows the number that matters is the split, not the headline: because the variance (0.02) is tiny beside the squared bias (0.25), a single zero-offset calibration is justified in removing 0.50 of the 0.52 – and the honest lab reports “0.14 residual after offset,” a claim a bare “accurate enough” cannot make.

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