Propagation of Errors

When $R = f(x, y)$, uncertainty propagates:

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

Common cases

• Sum/difference ($R = x \pm y$):

$$\delta R = \sqrt{(\delta x)^2 + (\delta y)^2}$$

• Product or quotient ($R = xy$ or $x/y$):

$$\frac{\delta R}{R} = \sqrt{(\delta x/x)^2 + (\delta y/y)^2}$$

• Power ($R = x^n$): $\delta R / R = |n|\cdot\delta x / x$

Example — $v = d/t$ with $d = 2.50 \pm 0.02$ m, $t = 1.20 \pm 0.05$ s:

$$v = 2.08 \pm 0.09\;\text{m/s}$$

Key insight: The quantity with the largest relative uncertainty dominates. Improve that measurement first.

Common pitfall: Independent uncertainties combine in quadrature ($\sqrt{a^{2}+b^{2}}$), not by addition — and powers multiply relative errors by the exponent. Adding everything linearly overstates the error; ignoring exponents understates it.