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## Propagation of Errors

Calculus provides insight into propagation of errors.  Let $y =f (x_1,x_2,x_3,\ldots , x_n)$  be a derived function of $n$ independent variables with measurement uncertainty $\Delta x_j$.  The uncertainty could be average deviation or standard deviation. Two approaches to assessment of the uncertainty in $y$ ($\Delta y$) are:

$\Delta y =\sum_{j=1}^n |\frac{\partial f}{\partial x_j}|\Delta x_j$

$\Delta y =\sqrt{\sum_{j=1}^n (\frac{\partial f}{\partial x_j})^2{\Delta x_j}^2}$

This yields rules for sums, products, powers and other calculations. Some examples of the first approach follow:

• Sums: $y = x_1 +x_2 \rightarrow \Delta y=\Delta x + \Delta y$
•  Products:  $y =x_1 x_2 \rightarrow \Delta y = |x_2|\Delta x_1 +|x_1|\Delta x_2\Leftrightarrow \frac{\Delta y}{y} =\frac{\Delta x_1}{x_1} +\frac{\Delta x_2}{x_2}$
• Division: $y =\frac{x_1}{x_2} \rightarrow \frac{\Delta y}{y} =\frac{\Delta x_1} {x_1} + \frac{\Delta x_2}{x_2}$
• Power:  $y ={x_1}^m{x_2}^n \rightarrow \frac{\Delta y}{y} = |m|\frac{\Delta x_1}{x_1} +|n|\frac{\Delta x_2}{x_2}$
• Exponential:  $y = e^{x}\rightarrow \Delta y = y \Delta x$
• Logarithm: $y =ln x \rightarrow \Delta y = \frac{\Delta x}{x}$
The second approach can be applied easily.