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While doing an experiment I've came upon the need to calculate the error of my value that was calculated using observed values with known observational error. While I know how to calculate the error for expressions with two observed values:

$$ \frac{\mathrm{d}F}{F}=\sqrt{\left(\frac{\mathrm{d}x}{x}\right)^2+\left(\frac{\mathrm{d}y}{y}\right)^2} \quad \text{for } F=x y \text{ or } F=\frac{x}{y} $$

Is there a formula for more complex expressions, like $F=xyz$, of $F=\frac{xy}{z}$ and such, or am I forced to calculate the value of $x y$ and its error and then use it with $z$ and its error?

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Assuming independence of these errors, it would be $\frac{\mathrm{d}F}{F}=\sqrt{\left(\frac{\mathrm{d}x}{x}\right)^2+\left(\frac{ \mathrm{d}y}{y}\right)^2 + \left(\frac{\mathrm{d}z}{z}\right)^2}$ – Sasha Jan 30 '12 at 18:42
Write it as the answer so I could accept it. – Ilya Melamed Jan 30 '12 at 19:10
up vote 1 down vote accepted

Let $w = F(x_1, x_2, \ldots, x_n)$ represent the relationship between actual measured quantities $x_k$ and the quantity of interest $w$. Assuming $F$ is a analytic function, we get $$ \Delta(F) \approx \sum_{k=1}^n \frac{\partial F}{\partial x_k} \Delta x_k $$ Is we further assume that sources of error $\Delta x_k$ for each measured quantity $x_k$ are normally distributed with zero mean and independent, we get $$ \mathbb{E}\left( (\Delta F)^2 \right) \approx \sum_{k=1}^n \sum_{\ell=1}^n \frac{\partial F}{\partial x_k} \frac{\partial F}{\partial x_\ell} \mathbb{E}\left( \Delta x_k \Delta x_\ell\right) $$ Due to independence, for $k \not= \ell$, $\mathbb{E}(\Delta x_k \Delta x_\ell) = \mathbb{E}(\Delta x_k) \mathbb{E}(\Delta x_\ell) = 0$, thus $$ \mathbb{E}((\Delta F)^2) = \sum_{k=1}^n \left( \frac{\partial F}{\partial x_k} \right)^2 \mathbb{E}((\Delta x_k)^2) $$

In the case of $F$ that is a simple product of powers $F=x_1^{p_1} \cdots x_n^{p_n}$, $$ \frac{\partial F}{\partial x_k} = p_k \frac{F}{x_k} $$

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