Combination of errors

When an experiment involves several measured quantities that must be combined (added, multiplied, divided, or raised to a power) to compute a final result, it is essential to know how the individual errors of each measured quantity combine to determine the error in that final result. For example, when the resistance of a conductor is found using Ohm's law from separately measured potential difference and current, the errors in those two individual measurements combine to produce an overall error in the derived resistance.

1. Errors in sum and in difference. Suppose two quantities $A$ and $B$ have measured values $A \pm \Delta A$ and $B \pm \Delta B$. For their sum, $Z = A + B$: $Z \pm \Delta Z = (A \pm \Delta A) + (B \pm \Delta B) = (A+B) \pm \Delta A \pm \Delta B$, so $\pm\Delta Z = \pm\Delta A \pm \Delta B$. For their difference, $Z = A - B$: similarly $Z \pm \Delta Z = (A-B) \pm \Delta A \pm \Delta B$. In both cases there are four possible sign combinations for $\Delta Z$ (namely $+\Delta A - \Delta B$, $+\Delta A + \Delta B$, $-\Delta A - \Delta B$, $-\Delta A + \Delta B$), and the maximum possible value of $|\Delta Z|$ in every case is $\Delta Z = \Delta A + \Delta B$. So: whenever two quantities are added or subtracted, the maximum absolute error in the result is the sum of the absolute errors in the two individual quantities — errors never cancel when we want the worst-case bound.
2. Errors in product and in division. Suppose $Z = AB$, with measured values $A \pm \Delta A$ and $B \pm \Delta B$. Then $Z \pm \Delta Z = (A \pm \Delta A)(B \pm \Delta B) = AB \pm A\Delta B \pm B\Delta A \pm \Delta A \Delta B$. Dividing the left side by $Z$ and the (equal) right side by $AB$, and neglecting the tiny product term $\Delta A\, \Delta B / AB$ (since both $\Delta A/A$ and $\Delta B/B$ are small), gives the maximum relative error in $Z$ as

   $\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B} \qquad \text{--- (1.7)}$

   Exactly the same formula applies to division of two quantities. So: whenever two quantities are multiplied or divided, the maximum relative error in the result is the sum of the relative errors in each of the individual quantities.
3. Errors due to a power (index) of a measured quantity. If $Z = A^3 = A \cdot A \cdot A$, applying the product rule above three times over gives $\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta A}{A} + \dfrac{\Delta A}{A} = 3\dfrac{\Delta A}{A}$ — the relative error is tripled. This generalises: for $Z = A^n$,

   $\dfrac{\Delta Z}{Z} = n\dfrac{\Delta A}{A} \qquad \text{--- (1.8)}$

   and more generally still, for a quantity built from several measured factors each raised to its own power, $Z = \dfrac{A^p B^q}{C^r}$, $\dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B} + r\dfrac{\Delta C}{C} \qquad \text{--- (1.9)}$ …