Measurement uncertainty
By international agreement, this uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity value.
This particular single choice is usually called the measured value, which may be optimal in some well-defined sense (e.g., a mean, median, or mode).
Measurands on ratio or interval scales include the size of a cylindrical feature, the volume of a vessel, the potential difference between the terminals of a battery, or the mass concentration of lead in a flask of water.
Then, no matter how many times the person's mass were re-measured, the effect of this offset would be inherently present in the average of the values.
The "Guide to the Expression of Uncertainty in Measurement" (commonly known as the GUM) is the definitive document on this subject.
The GUM has been adopted by all major National Measurement Institutes (NMIs) and by international laboratory accreditation standards such as ISO/IEC 17025 General requirements for the competence of testing and calibration laboratories, which is required for international laboratory accreditation, and is employed in most modern national and international documentary standards on measurement methods and technology.
The American Society of Mechanical Engineers (ASME) has produced a suite of standards addressing various aspects of measurement uncertainty.
[6] The above discussion concerns the direct measurement of a quantity, which incidentally occurs rarely.
A simple measurement model (for example for a scale, where the mass is proportional to the extension of the spring) might be sufficient for everyday domestic use.
Alternatively, a more sophisticated model of a weighing, involving additional effects such as air buoyancy, is capable of delivering better results for industrial or scientific purposes.
In general there are often several different quantities, for example temperature, humidity and displacement, that contribute to the definition of the measurand, and that need to be measured.
Neither the alignment of the instrument nor the ambient temperature is specified exactly, but information concerning these effects is available, for example the lack of alignment is at most 0.001° and the ambient temperature at the time of measurement differs from that stipulated by at most 2 °C.
Some such data relate to quantities representing physical constants, each of which is known imperfectly.
There are often other relevant data given in reference books, calibration certificates, etc., regarded as estimates of further quantities.
These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available knowledge concerning
, obtained from certificates and reports, manufacturers' specifications, the analysis of measurement data, and so on.
The use of available knowledge to establish a probability distribution to characterize each quantity of interest applies to the
For the domestic bathroom scale, the fact that the person's mass is positive, and that it is the mass of a person, rather than that of a motor car, that is being measured, both constitute prior knowledge about the possible values of the measurand in this example.
is inferred from repeated measured values ("Type A evaluation of uncertainty"), or scientific judgement or other information concerning the possible values of the quantity ("Type B evaluation of uncertainty").
In Type A evaluations of measurement uncertainty, the assumption is often made that the distribution best describing an input quantity
In such a case, knowledge of the quantity can be characterized by a rectangular probability distribution[11] with limits
, but these terms combined in quadrature,[1] namely by an expression that is generally approximate for measurement models
contain dependencies, the above formula is augmented by terms containing covariances,[1] which may increase or decrease
The main stages of uncertainty evaluation constitute formulation and calculation, the latter consisting of propagation and summarizing.
The formulation stage constitutes The calculation stage consists of propagating the probability distributions for the input quantities through the measurement model to obtain the probability distribution for the output quantity
When the measurement model is multivariate, that is, it has any number of output quantities, the above concepts can be extended.
[13] The output quantities are now described by a joint probability distribution, the coverage interval becomes a coverage region, the law of propagation of uncertainty has a natural generalization, and a calculation procedure that implements a multivariate Monte Carlo method is available.
The most common view of measurement uncertainty uses random variables as mathematical models for uncertain quantities and simple probability distributions as sufficient for representing measurement uncertainties.
[citation needed] A more robust representation of measurement uncertainty in such cases can be fashioned from intervals.
Distributions of such measurement intervals can be summarized as probability boxes and Dempster–Shafer structures over the real numbers, which incorporate both aleatoric and epistemic uncertainties.
