$$u_c^2(y) = \sum_{i=1}^{N} \left(\frac{\partial f}{\partial x_i}\right)^2 u^2(x_i) \;+\; 2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N} \frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_j}\, u(x_i, x_j)$$

Equation 4 · Scientific Methods

How Scientific Instruments and Metrology Actually Work

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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”).…

Meanings in this article

• $y$: the if a measurement result.

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