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1.3: Error Estimates for Sums, Products, and Powers

  • Page ID
    34676
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    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: \[\begin{align} \begin{aligned} \Delta T &= |T|\, \sqrt{\left(\frac{\Delta m}{m}\right)^2 + \left(\frac{\Delta (v^2)}{v^2}\right)^2} \\ &= \frac{1}{2} m v^2\, \sqrt{\left(\frac{\Delta m}{m}\right)^2 + 4 \left(\frac{\Delta v}{v}\right)^2}. \end{aligned} \end{align}\]


    This page titled 1.3: Error Estimates for Sums, Products, and Powers is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.