Propagation of Errors

General Error Propagation

When a calculated result $R = f(x, y)$ depends on measured values, uncertainty propagates. The combined absolute uncertainty $\delta R$ is found using partial derivatives:

$$\delta R = \sqrt{\left(\frac{\partial f}{\partial x}\delta x\right)^2 + \left(\frac{\partial f}{\partial y}\delta y\right)^2}$$

Common algebraic cases simplify this general rule into handy formulas for sums, products, and powers.

Worked Example & Pitfalls

Example: Calculate velocity $v = d/t$ where $d = 2.50 \pm 0.02$ m and $t = 1.20 \pm 0.05$ s. Applying the quotient rule yields $v = 2.08 \pm 0.09$ m/s.

Key insight: The quantity with the largest relative uncertainty dominates the final error. Always improve that measurement first.

Common pitfall: Independent uncertainties combine in quadrature ($\sqrt{a^2 + b^2}$), not by simple addition. Adding errors linearly severely overstates uncertainty, while ignoring exponents understates it.