# Combining uncertainties - multiple measurements

I am trying to understand how to combine uncertainties when they are dependent and independent from each other.

Using this formula :

$$\delta z = \sqrt {\Biggl(\dfrac{\partial f}{\partial x} \delta x\Biggr)^2+\Biggl(\dfrac{\partial f}{\partial y} \delta y\Biggr)^2+2\Biggl(\dfrac{\partial f}{\partial x}\cdot \dfrac{\partial f}{\partial y}\Biggr)\text{cov}(x,y)}$$

Intuitively, if the covariance between the two is zero, the last term will disappear and the equation just becomes the square root of sums for combining uncertainty.

My question is does $\delta z$ then need to be divided by 2 to get the final uncertainty value.(i.e. divide by $N$)

• No. The square root takes care of that. The book(s) by Bevington is an excellent reference source. Commented Feb 10, 2014 at 14:53
• @CarlWitthoft: Do you think you might be able to expand your comment into an answer? Commented Feb 11, 2014 at 1:33
• @KyleKanos I'll take a whack at it, but it's pretty dang trivial if you just write out the derivative expansion. Commented Feb 11, 2014 at 1:56
• @CarlWitthoft: I'd say your comment plus a link to the Bevington book would suffice by my standards. I just hate unanswered questions when a decent one is in the comments. Commented Feb 11, 2014 at 1:57
• @Kyle -- how's this answer? Commented Feb 11, 2014 at 2:12

ok, here goes... Direct from "Data Reduction and Error Analysis for the Physical Sciences," Bevington, McGraw-Hill, Chapter Four, when $z = f(x,y)$ then since $\sigma_z^2 = \Sigma(x_j-<x>)^2$ and $\Delta z = \Delta x *\frac{\delta z}{\delta x} + \Delta y * \frac{\delta z}{\delta y}$ , some equation hacking leads to
$\sigma_z^2 = \sigma_x^2 * (\frac{\delta z}{\delta x})^2 + \sigma_y^2 * (\frac{\delta z}{\delta y} )^2$+ covariance term