# Regarding the propagation of uncertainty

Not sure if I should post this here or in the statistic section.

I have a question regarding the propagation of uncertainty. In this book , a dilution ratio is expressed as a function of the sample flow $Q_s$, total flow, $Q_t$ and the dilution flow $Q_d$ via (eq-6-82). It is then stated that the uncertainty associated with the dilution ratio is $DR = \frac{Q_t}{Q_s}$ and the propagated uncertainty

$U(DR) = \sqrt{U(Q_s)^2 + U(Q_t)^2}$

In general, propagated uncertainty is $\Delta f(x_1,x_2,\ldots,x_n) = \sqrt{ \left(\frac{\partial f}{\partial x_1} \Delta x_1\right)^2 + \left(\frac{\partial f}{\partial x_2} \Delta x_2\right)^2 + \ldots + \left(\frac{\partial f}{\partial x_n} \Delta x_n\right)^2 } = \sqrt{ \sum_{i=1}^n \left(\frac{\partial f}{\partial x_i} \Delta x_i \right)^2 }$

so I'm not really sure how they come to this conclusion. I would have thought the uncertainty should be

$U(DR) = \sqrt{ (\frac{U(Q_t)}{Q_s})^2 + (\frac{Q_t U(Q_s)}{Q_s^2})^2 }$

Also, regarding eq 6-84 and 6-85, the above general equation for uncertainty propagation, the variables should be independent, but $Q_s, Q_d$ and $Q_t$ are not independent so are equations 6-84 and 6-85 really valid?

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They're probably defining $U$ to be the relative uncertainty, $U(x)=\Delta(x)/x$. With a little manipulation, it should turn out that the given equation for $U(DR)$ is equivalent to your equation for $\Delta(DR)$.