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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.
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