How do I calculate the experimental uncertainty in a function of two measured quantities

Question

I am performing an experiment where I'm measuring two variables, say $x$ and $y$, but I'm actually interested in a third variable which I calculate from those two, $$z=f(x,y).$$ In my experiment, of course, both $x$ and $y$ have experimental uncertainties, which are given by the resolution of my measurement apparatus among other considerations. I am also considering doing multiple runs of measurement to obtain good statistics on my measurement of $x$ and $y$, and therefore on $z$. I don't really know how the statistical spread will compare to my calculated (resolution-induced) uncertainty, though.

I would like to know what the final uncertainty for $z$ should be, and I am not very familiar with the error propagation procedures for this.

• What are the usual ways to combine the experimental uncertainties in measured quantities?
• When should I use the different approaches?
• How do I include statistical uncertainties when they are present?
• What happens if the statistical spread of a variable is comparable to the instrument's resolution, so that I can't neglect either contribution?
• What are good references where I can read further about this type of problem?

I would also appreciate answers to cite their sources - and particularly to use 'official' ones - where possible.

Answer 1

In my experimental courses, all uncertainties are calculated with the so called “sum in quadrature“: $$ \delta z = \sqrt {\Biggl(\dfrac{\partial f}{\partial x} \delta x\Biggr)^2+\Biggl(\dfrac{\partial f}{\partial y} \delta y\Biggr)^2+2\Biggl(\dfrac{\partial f}{\partial x}\cdot \dfrac{\partial f}{\partial y}\Biggr)\text{cov}(x,y)},$$ where the partial derivatives are calculated in the expected value.

The motivation of the formula is roughly as follows: for a linear function of two random variables $X,Y$, $$Z=aX+bY+c$$ the variance is exactly: $$\text{Var} (Z)=a^2\text {Var}(X)+b^2\text {Var} (Y)+2ab\text {cov}(X,Y).$$ For a general function $Z=f(X,Y)$, we reconduct to the linear case by taking it's Taylor expansion around $(E(X),E(Y))$. Turns out that $$E(Z)\approx f(E(X),E(Y))$$ (the calculation is not at all difficult, tell me if you need it for a more precise statement). In the same way: $$\text {Var} (Z)\approx a^2\text {Var}(X)+b^2 \text {Var} (Y)+2ab\text {cov}(X,Y),$$ where the “weights” $a^2$ and $b^2$ are the squares of the derivatives as I wrote in my first formula.

I suggest to do the calculations.

An elementary book, that I found useful, is Taylor's.