Equation 3 · How Scientific Instruments and Metrology Actually Work

What does this equation mean?

y = f(x_1, x_2, \ldots, x_N)

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Inputs and operationsf(x_1, x_2, ldots, x_N)

Result or conditiony

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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y

Symbol y

the if a measurement result.

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f

Symbol f

f is an input to the expression that computes the quantity on the left.

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x_1

Symbol x_1

$x_1$ is an input to the expression that computes the quantity on the left.

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x_2

Symbol x_2

$x_2$ is an input to the expression that computes the quantity on the left.

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x_N

Symbol x_N

$x_N$ is an input to the expression that computes the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

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:

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Sources cited in the article section

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