Propagation of errors

A final experimental result is rarely built from a single measured quantity -- it usually combines several quantities, each measured (possibly with a different instrument, and hence a different error). The error in the final result depends on (i) the individual errors in each measured quantity, and (ii) the mathematical operation used to combine them -- so different operations propagate error differently.

i) Sum, $Z=A+B$. If $A=A\pm\Delta A$ and $B=B\pm\Delta B$, the worst case is when both errors add in the same direction: $Z\pm\Delta Z=(A\pm\Delta A)+(B\pm\Delta B)=(A+B)\pm(\Delta A+\Delta B)$, so

Example: $R_1=(100\pm3)\,\Omega$ and $R_2=(150\pm2)\,\Omega$ in series: $R=R_1+R_2=(250\pm5)\,\Omega$.

ii) Difference, $Z=A-B$. Again the worst case adds the two absolute errors: $Z\pm\Delta Z=(A\pm\Delta A)-(B\pm\Delta B)=(A-B)\pm(\Delta A+\Delta B)$, so

Example: $t_1=(20\pm0.5)^\circ$C, $t_2=(50\pm0.5)^\circ$C: $t=t_2-t_1=(30\pm1)^\circ$C.

iii) Product, $Z=AB$. $Z\pm\Delta Z=(A\pm\Delta A)(B\pm\Delta B)=AB\pm A\Delta B\pm B\Delta A\pm\Delta A\,\Delta B$. Dividing by $Z=AB$ and dropping the doubly-small term $\dfrac{\Delta A}{A}\cdot\dfrac{\Delta B}{B}$, the maximum fractional error is

Example: length $A=(5.7\pm0.1)$ cm, breadth $b=(3.4\pm0.2)$ cm: area $=A\times b=19.4$ cm$^2$ (3 sig. figs.), $\dfrac{\Delta A_{\text{area}}}{A_{\text{area}}}=\dfrac{0.1}{5.7}+\dfrac{0.2}{3.4}=0.0175+0.0588=0.0763$, so $\Delta A_{\text{area}}=0.0763\times19.4\approx1.5$ cm$^2$; area $=(19.4\pm1.5)$ cm$^2$.

iv) Quotient, $Z=A/B$. Using $(1+x)^n\approx1+nx$ for small $x$, the same style of derivation gives the identical rule as for a product:

Example (Ohm's law): $V=(100\pm5)$ V, $I=(10\pm0.2)$ A: $R=V/I=10\,\Omega$; $\dfrac{\Delta R}{R}=\dfrac{5}{100}+\dfrac{0.2}{10}=0.05+0.02=0.07$, so $\Delta R=0.7\,\Omega$; $R=(10\pm0.7)\,\Omega$.

v) Power, $Z=A^n$. Since $A^n$ is really $A$ multiplied by itself $n$ times, applying the product rule $n$ times gives

The fractional error in the $n$-th power of a quantity is $n$ times the fractional error in the quantity itself. …