systems engineering//error budget//root sum square

Root sum square is the rule for combining independent, zero-mean error sources by adding their variances and taking the square root of the sum, and it is used in every error budget, tolerance stack-up and measurement uncertainty statement to say how large the total error of a chain will typically be. Three independent sources of 3, 4 and 12 cm combine to \(\sqrt{9+16+144}=13\) cm, not 19: the large one almost alone decides the total.

Root sum square is the rule for combining independent, zero-mean error sources by adding their variances and taking the square root of the sum, and it is used in every error budget, tolerance stack-up and measurement uncertainty statement to say how large the total error of a chain will typically be. Three independent sources of 3, 4 and 12 cm combine to 9+16+144=13\sqrt{9+16+144}=139+16+144​=13 cm, not 19: the large one almost alone decides the total.

The rule follows from a property of variance: for independent errors the variance of a sum is the sum of the variances, so standard deviations add like the sides of a right triangle.

σtotal=∑iσi2againstemax⁡=∑i∣ei∣\sigma_{\text{total}}=\sqrt{\sum_i \sigma_i^2}\qquad\text{against}\qquad e_{\max}=\sum_i |e_i|σtotal​=i∑​σi2​​againstemax​=i∑​∣ei​∣

The left expression is the realistic total for random, independent errors; the right one, the worst-case linear sum, is what happens when every error points the same way at once, and it is the correct combination for biases and for errors that share a cause.

Because squares are added, the dominant source rules. Halving a source that carries 5 % of the variance changes the total by about 2 %; halving the one that carries 60 % cuts it by about a quarter. So the first thing an error budget does after combining is rank the sources by their share of the variance, and the first fix goes to the top of that list.

Independence is the assumption that fails silently. Two errors with the same cause (one temperature drifting two sensors, one vibration shaking two mounts, one clock timing two measurements) are correlated, add linearly, and make a root-sum-square budget optimistic. On a moving drone, uncompensated latency and clock offset both grow with speed along the direction of motion, so they belong in the linear sum together (error budget). The same trap undermines redundancy built from identical parts (common-mode failure).

Zero mean matters too. A bias shifts every sample the same way and survives averaging, so the usual practice is biases in a linear sum, random terms by root sum square, and the two combined at the end, without pretending everything is Gaussian.

With correlations known, the general form is the covariance propagation JΣJ⊤J\Sigma J^\topJΣJ⊤, of which root sum square is the diagonal case (uncertainty propagation); with strong nonlinearity or awkward distributions, a Monte Carlo run replaces both.

The total is a standard deviation, and a requirement is usually a percentile. For a roughly Gaussian total, 95 % of samples fall within about 1.96 σ1.96,\sigma1.96σ, which is how a 0.5 m requirement at 95 % becomes a budget of about 0.25 m.

The rule is the arithmetic of every error budget, and the same sum appears in mechanical tolerance stack-ups and in the uncertainty statements of measurement standards.