Bias, RMSE, and the Variance Split

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

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.

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.