What does this equation mean?

The GUM’s central move is to treat every measured or estimated quantity a result depends on as a random variable with its own estimated standard uncertainty, and then to propagate those uncertainties through the mathematical relationship that produces the final result. If a measurement result y is a function of N input quantities, y = f($x_1$, $x_2$, $\ldots$, $x_N$) , the combined standard uncertainty is obtained from a first-order Taylor expansion of that function: $u_c^2(y) = \sum_{i=1}^{N} \left(\frac{\partial f}{\partial x_i}\right)^2 u^2(x_i) \;+\; 2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N} \frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_j}\, u(x_i, x_j)$. The first sum is the familiar root-sum-of-squares combination of independent contributions, each weighted by how sensitive the final result is to that input (the partial derivative, or “sensitivity coefficient”). The second sum is easy to forget and often the source of a badly wrong uncertainty budget: it accounts for covariance between input quantities that are not independent — for example, two lengths measured with the same miscalibrated ruler share a correlated error, and ignoring that correlation can make a combined uncertainty either too small or too large depending on the sign of the correlation [ 2 ] .