Propagation of Uncertainty vs Dividing Uncertainties I have a quick question!
When I’m calculating the uncertainty of a formula, like $v = d/t$ . . . What method do I use? Sticking with my example ($v = d/t$).
Do I convert the uncertainties of distance and time into their relative Uncertainties and add them up?
Or do I use one of the propagation of uncertainty formulas (see image below)?
I did both methods, but I received different results, albeit only a fraction or so off. I just want to be kosher and precise in my calculations. 
Thank you for your help!


 A: Consider a variable $U$ that depends on parameters $X_j$. We want the uncertainty (error) in $U$ expressed as $\Delta U$ as a function of the errors (uncertainties) in $X_j$ expressed as $\Delta X_j$.
The simplest approach is to sum relative uncertainties.
$$ \frac{\Delta U}{U} = \sum \frac{\Delta X_j}{X_j} $$
This is a rough, worse than first order approximation that has no basis in first principles. This approach should generally be avoided without at least an appreciation of how it arises in the next method.
The next level of analysis is to account for the functional form of $U = f(X_1, X_2 \ldots X_N)$. This adds a term that might be called the sensitivity factor $S_j$ as
$$ S_j \equiv \left(\frac{\partial U}{\partial X_j}\right)_{X_k \neq X_j} $$
Using this term, we write an improved first order approximation as:
$$ \Delta U = \sum |S_j| \Delta X_j $$
The equivalence of this equation to the expression for a total derivative $dU = \sum S_j dX$ is not without coincidence. The absolute function is required so that all errors add regardless of the sign of the partial derivative $S_j$. Finally, for a simple multiplication or division expression such as $U = XY$ or $U = X/Y$, we can prove that this expression becomes the same as the sum of relative uncertainties (the first expression).
The above expression is not uncommon and indeed is not incorrect to use as a first order approximation. It will provide an upper bounds to the uncertainty $\Delta U$. It is however not the fundamentally correct expression to use to propagate uncertainties in a rigorous manner.
The fundamentally correct approach to propagate uncertainties in a rigorous manner is to use the equation for the linear propagation of uncertainties. 
$$ \Delta^2 U = \sum S^2_j \Delta^2 X_j $$
This first principles expression basically states that we are adding the variances of the measurement distributions weighted by their contributing sensitivities. This is only exact when the uncertainties $\Delta X_j$ are uncorrelated. When the uncertainties in the terms are correlated, a more sophisticated approach is required.
The expression is called "linear" propagation because it sums the weighted variances. The variance of a random distribution of measurements is the square of the standard deviation of the distribution. 
The basic first order approach and the linear propagation approach will not give the same result, although with rounding they may appear the same. The first order approximation as a summation of relative uncertainties gives an upper bounds.
A thorough discussion of both approaches is given in the book Introduction to Error Analysis from Taylor. Examples are provided for different case studies. See also this link on Wikipedia.
