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Equation 4 · Part 2 · How Scientific Instruments and Metrology Actually Work

Symbol y

uc2(y)=∑i=1N(∂f∂xi)2u2(xi)  +  2∑i=1N−1∑j=i+1N∂f∂xi∂f∂xj u(xi,xj)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)
yy

What this part means

the if a measurement result.

Its job in the formula

y is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Where the article explains it

If a measurement result y is a function of N input quantities, y = f(x1x_1, x2x_2, …\ldots, xNx_N) , the combined standard uncertainty is obtained from a first-order Taylor expansion of that function: uc2(y)=∑i=1N(∂f∂xi)2u2(xi)  +  2∑i=1N−1∑j=i+1N∂f∂xi∂f∂xj u(xi,xj)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 passage around this formula

…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(x1x_1, x2x_2, …\ldots, xNx_N) , the combined standard uncertainty is obtained from a first-order Taylor expansion of that function: uc2(y)=∑i=1N(∂f∂xi)2u2(xi)  +  2∑i=1N−1∑j=i+1N∂f∂xi∂f∂xj u(xi,xj)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…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.