# Where does the error propagation formula comes from?

As an engineering student I have come several times across the formula $$\sigma_{f(\vec{x})}=\sqrt{\sum_{i} \big (\dfrac{\partial f}{\partial x_{i}}\sigma_{x_{i}}\big )^{2}}$$ for the propagation of errors, but I never understood where does it come from. Could you provide some reference for me to understand it?

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## 1 Answer

If you have just one error, $\dfrac{\partial f}{\partial x_{i}}\sigma_{x_{i}}$ gives the error in $f$. It essentially comes from the Taylor series, linearizing the value of $f$ around the correct value. If you have a number of errors, we think of them as uncorrelated, so it is a random walk. The root sum square is the expected distance of a random walk.

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Thank you! It is a very intuitive way of seing it. Do you know if there is any generalization to this formula without assuming uncorrelation, i.e., assuming you have ways to compute the correlation (a covariance or whatever) between variables? – Francisco Feb 16 '13 at 1:26
@Francisco: if you know all the errors will drive the value of $f$ in the same direction, you should add them linearly. This will give a larger (sometimes much larger) $\sigma_f$ It is the worst case, and in some applications it is insisted on. – Ross Millikan Feb 16 '13 at 1:47