Uncertainty Equations Confusion

I am confused about the uncertainty equations calculations. 


If I have the equation X1 \* Y1 = X2 \* Y2 and I want to find the uncertainty of X1 , I convert the equation to X1 = (X2 \* Y2)/(Y1) . Now, I would know how to calculate the uncertainty if it was a straight division or multiplication uncertainty but this equation is both. How do I go about calculating the uncertainty if the equation contains both multiplication and division? Do I just do the multiplication uncertainty first and then the division uncertainty?


Thanks any advice is appreciated.




You have three variables, for simplicity let's call them, *x, y, z* each with a (standard deviation) uncertainty $\sigma\_x, \sigma\_y, \sigma\_z$. The formula you want to find the error on is then $f=xy/z$ which will have a standard deviation $\sigma\_f$ The method that always works is the the formula that gives the *square* of the uncertainty
$$ \sigma\_f ^2 = \left( \frac{\partial f}{\partial x}\right)^2\_{y,z}.\sigma\_x^2+\left( \frac{\partial f}{\partial y}\right)^2\_{x,z}.\sigma\_y^2+\left( \frac{\partial f}{\partial z}\right)^2\_{x,y}.\sigma\_z^2$$


which looks horrid and intimidating but is very easy to use as it only involves taking the derivative of *f* with *x, y* and *z* in turn, while keeping the other variables constant.


In your example $\partial f/\partial x = y/z$, $\partial f/\partial y =x/z$ and $\partial f/\partial z =-xy/z^2$. Putting the parts together gives
$$\sigma\_f^2= \frac{y^2}{z^2}\sigma\_x^2 + \frac{x^2}{z^2}\sigma\_y^2 +\frac{x^2y^2}{z^4}\sigma\_z^2 $$


You can simplify this a bit but you should have all the values to plug into this. Don't forget to take the square root at the end.


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$\%\text{RSD}\_{X1} = \sqrt{\%\text{RSD}^2\_{X2} + \%\text{RSD}^2\_{Y2} + \%\text{RSD}^2\_{Y1}}$


For multiplication and division the [error propagates](https://en.wikipedia.org/wiki/Propagation_of_uncertainty) as the % relative standard deviation. 


This ignores the covariances which means that there is no interaction between the variables. 


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