1.3: Error Estimates for Sums, Products, and Powers

When analyzing experiments, you often need to take sums, products, and powers of measured data. For instance, you might measure the voltage $V$ and current $I$ across a circuit, and use them to find the resistance $R = V/I$. Errors in $V$ and $I$ will turn into errors in $R$. This is called error propagation.

The basic rules of error propagation are easy to summarize (if you want the detailed derivations, refer to Ref. ). Consider two independent quantities $X \pm \Delta X$ and $Y \pm \Delta Y$. We call $\pm\Delta X$ and $\pm \Delta Y$ the standard errors for $X$ and $Y$, regardless of whether they originate from measurement error or sampling error. Now, suppose we derive $Z \pm \Delta Z$ by addition or subtraction of $X$ and $Y$: $$\left\{\begin{array}{l}Z = X + Y \;\;\mathrm{or}\\ Z = X - Y \end{array} \right. \;\;\; \leftrightarrow \;\;\; \Delta Z = \sqrt{(\Delta X)^2 + (\Delta Y)^2}.$$ More generally, for any two (error-free) constants $\alpha$ and $\beta$, $$\;\;\qquad Z = \alpha X + \beta Y \qquad \leftrightarrow \;\;\; \Delta Z = \sqrt{\alpha^2(\Delta X)^2 + \beta^2 (\Delta Y)^2}.$$ For products and fractions, $$\quad\qquad\;\left\{\begin{array}{l}Z = \alpha XY \;\;\mathrm{or}\\ Z = \alpha X/Y \end{array} \right. \quad \leftrightarrow \;\;\; \frac{\Delta Z}{|Z|} = \sqrt{\left(\frac{\Delta X}{X}\right)^2 + \left(\frac{\Delta Y}{Y}\right)^2}.$$ For powers, exponentials and logarithms, $$\begin{align} Z &= X^n \quad\qquad\; \leftrightarrow \;\;\; \frac{\Delta Z}{|Z|} = |n| \,\frac{\Delta X}{|X|} \\ Z &= \ln(X) \qquad\, \leftrightarrow \;\;\; \Delta Z \,= \frac{\Delta X}{|X|} \\ Z &= \log_{10}(X) \;\;\; \leftrightarrow \;\;\; \Delta Z \, = \frac{1}{\ln(10)} \, \frac{\Delta X}{|X|} \\ Z &= \exp(\alpha X) \;\;\; \leftrightarrow \;\;\; \frac{\Delta Z}{|Z|} = |\alpha|\,\Delta X. \end{align}$$

More complicated formulas can usually be handled by combining these rules. For example, suppose you seek an object’s kinetic energy $T = \frac{1}{2}mv^2$, using data $m \pm \Delta m$ and $v \pm \Delta v$. First, find the error in the quantity $v^2$ using the rule for powers: $$\frac{\Delta (v^2)}{|v^2|} = 2 \frac{\Delta v}{|v|}.$$ Then, plug this into the rule for products: $$\begin{align} \begin{aligned} \Delta T &= |T|\, \sqrt{\left(\frac{\Delta m}{m}\right)^2 + \left(\frac{\Delta (v^2)}{v^2}\right)^2} \\ &= \frac{1}{2} m v^2\, \sqrt{\left(\frac{\Delta m}{m}\right)^2 + 4 \left(\frac{\Delta v}{v}\right)^2}. \end{aligned} \end{align}$$