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# Evaluating standard uncertainty

**Evaluating standard uncertainty**

For economy of notation, in thisGuidethe same symbol is used for the physical quantity (the measured) and for the random variable (see 2.1) that represents the possible outcome of an observation of that quantity. When it is stated that Xi has a particular probability distribution, the symbol is used in the latter sense; it is assumed that the physical quantity itself can be characterized by an essentially unique value.

In a series of observations, thekth observed value of Xi is denoted by Xik; hence if R denotes the resistance of a resistor, thekth observed value of the resistance is denoted by Rk .

The estimate of Xi (strictly speaking, of its expectation) is denoted by Xi.

If a potential difference V is applied to the terminals of a temperature-dependent resistor that has a resistance R0 at the defined temperature t0 and a linear temperature coefficient of resistance α , the power P (the measured) dissipated by the resistor at the temperature t depends on V , R0 , α , and t according to

Other methods of measuring P would be modelled by different mathematical expressions.

**1.2 **The *input
quantities * upon which the *output quantity y** *depends may themselves
be viewed as measured and may themselves depend on other quantities, including
corrections and correction factors for systematic effects, thereby leading to a
complicated functional relationship ƒ that may never be written down explicitly. Further,ƒ may be determined experimentally or exist only as an algorithm that must
be evaluated numerically. The functionƒ as it appears in this *Guide *is to be interpreted in this broader context, in particular as that
function which contains every quantity, including all corrections and
correction factors, that can contribute a significant component of uncertainty
to the measurement result.

Thus, if data indicate that ƒ does not model the measurement to the degree imposed by the required accuracy of the measurement result, additional input quantities must be included in ƒ to eliminate the inadequacy. This may require introducing an input quantity to reflect incomplete knowledge of a phenomenon that affects the measured. In the example of 1.1, additional input quantities might be needed to account for a known nonuniform temperature distribution across the resistor, a possible nonlinear temperature coefficient of resistance, or a possible dependence of resistance on barometric pressure.

Nonetheless, Equation (1) may be as elementary as

This expression models, for example, the comparison of two determinations of the same quantity X.

quantities whose values and uncertainties are directly determined in the current measurement. These values and uncertainties may be obtained from, for example, a single observation, repeated observations, or judgement based on experience, and may involve the determination of corrections to instrument readings and corrections for influence quantities, such as ambient temperature, barometric pressure, and humidity;

quantities whose values and uncertainties are brought into the measurement from external sources, such as quantities associated with calibrated measurement standards, certified reference materials, and reference data obtained from handbooks.

In some cases, the estimate y may be obtained from

That is,y is taken as the arithmetic mean or average (see 2.1) of n independent determinations Y_{k} of Y, each determination having the same uncertainty and each being based on a complete set of observed values of the N input quantities X_{i} obtained at the same time. This way of averaging, rather than

, where

is the arithmetic mean of the individual observations *X _{ik}*, may be preferable when ƒ is a nonlinear function of the input quantities xxx, but the two approaches are identical if ƒ is a linear function of the