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

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?

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

• For an in-depth look at the combination of experimental and statistical uncertainties, see How to combine measurement error with statistic error. – Emilio Pisanty Jan 13 '14 at 18:09
• I had a good look and it seems we don't really have a good, canonical question to point people to when they ask how to combine experimental uncertainties. I'm therefore proposing we take this as a place for that. Feel free to improve the question if you have good ideas. – Emilio Pisanty Jan 13 '14 at 18:11
• Hi Emilio, I suggest to split this in different questions, since a comprensive answer should be very long, I fear. I've tried to address to what seems to me the main question “What is the common procedure...“, considering the case where the two variables can be correlated. – pppqqq Jan 13 '14 at 19:09
• Any version of the "error analysis" books by Bevington. – Carl Witthoft Jan 13 '14 at 19:46

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.
• Correct. Could be retrieved by a Taylor serie (around $\langle X\rangle, \langle Y\rangle$) of $V(f(X, Y)) = \langle f^2(X,Y)\rangle - \langle f(X,Y)\rangle^2$, at second order in $X - \langle X\rangle, Y - \langle Y\rangle$. – Trimok Jan 13 '14 at 19:52
• How do you take into account that for each time you vary the pair $(x,y)$ the value $z$ will change. Wouldn't that increase dispersion? – jinawee Dec 12 '14 at 18:20
• You mean if you have a set of $(x_i,y_i)$ couples? Here I assumed that the experimenter has previously made two estimation of $x_{\text {best}}$ and $y_{\text {best}}$, with uncertainties $\sigma _x, \sigma _y$ and has to estimate $z$ from these two. If the two uncertainties are little (for example if $(\partial f / \partial x)\cdot \sigma _x + (\partial f / \partial y )\sigma _y << f$ at that point $(x,y)$) it is reasonable to make a Taylor expansion. – pppqqq Dec 12 '14 at 21:46