Uncertainty propagation, addition, multiplication, power

For quantities measured independently (uncorrelated errors), contributions to uncertainty combine by quadratic sum, not by simple addition: arithmetic summation of uncertainties systematically overestimates the actual uncertainty on the result.

x, Δx

· measured value and its absolute uncertainty, expressed in the same unit.

Absolute uncertainty

· uncertainty of the result expressed in the same unit as the quantity, denoted Δf.

Relative uncertainty

· ratio Δf / |f|, expressed as a percentage. Dimensionless, it allows comparison of the precision of measurements of different nature.

Independent quantities

· measurements whose errors are uncorrelated with each other: this assumption justifies the quadratic sum rather than simple addition of uncertainties.

The quadratic sum assumes quantities measured independently: if two measurements share the same source of error (same poorly calibrated instrument, for example), their uncertainties are correlated and this method underestimates the actual uncertainty. A relative uncertainty is undefined for a null quantity (division by zero in Δx/x): the tool flags this rather than returning a misleading value.

For L = 12.0 cm ± 0.1 cm and l = 5.0 cm ± 0.2 cm, the area A = L × l equals 60.0 cm². The relative uncertainty on A equals √((0.1/12.0)² + (0.2/5.0)²) ≈ 4.2%, hence an absolute uncertainty of approximately 2.5 cm²: A ≈ 60.0 ± 2.5 cm².