Evaluating standard uncertainty

Evaluating standard uncertainty

Modelling the measurement

1.1 In most cases, a measured is not measured Y directly, but is determined from N other quantities  through a functional relationship ƒ :


NOTE 1  
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.

1.3 The set of input quantities may be categorized as: 
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.

1.4 An estimate of the measured Y, denoted by y,is obtained from Equation (1) using input estimates for the values of the N quantities . Thus the output estimate y, which is the result of the measurement, is given by 

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 Yk of  Y, each determination having the same uncertainty and each being based on a complete set of observed values of the N input quantities Xi obtained at the same time. This way of averaging, rather than

              , where

is the arithmetic mean of the individual observations Xik, 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 Xi.

1.5 The estimated standard deviation associated with the output estimate or measurement result y, termed combined standard uncertainty and denoted by uc(y), is determined from the estimated standard deviation associated with each input estimate Xi, termed standard uncertainty and denoted by u(Xi).

1.6 Each input estimate Xi and its associated standard uncertainty  u(Xi) are obtained from a distribution of possible values of the input quantity Xi. This probability distribution may be frequency based, that is, based on a series of observations  Xik of Xi, or it may be an a priori distribution. Type A evaluations of standard uncertainty components are founded on frequency distributions while Type B evaluations are founded on a priori distributions. It must be recognized that in both cases the distributions are models that are used to represent the state of our knowledge.