TheInfoList

In
metrology Metrology is the scientific study of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with oth ...

, measurement uncertainty is the expression of the
statistical dispersion In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin w ...
of the values attributed to a measured quantity. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by a statement of the associated uncertainty, such as the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

. By international agreement, this uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity value. It is a non-negative parameter. The measurement uncertainty is often taken as the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

of a state-of-knowledge probability distribution over the possible values that could be attributed to a measured quantity. Relative uncertainty is the measurement uncertainty relative to the magnitude of a particular single choice for the value for the measured quantity, when this choice is nonzero. This particular single choice is usually called the measured value, which may be optimal in some well-defined sense (e.g., a
mean There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
,
median In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

, or
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections that expresses an attitude of mind ** Imperative mood ** Subjunctive mo ...
). Thus, the relative measurement uncertainty is the measurement uncertainty divided by the absolute value of the measured value, when the measured value is not zero.

# Background

The purpose of measurement is to provide information about a
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ...
of interest – a measurand. For example, the measurand might be the size of a cylindrical feature, the
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

of a vessel, the
potential difference Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is def ...

between the terminals of a battery, or the
mass concentrationMass concentration or mascon may refer to: * Mass concentration (chemistry), the mass of a constituent divided by the volume of a mixture, this formula is generally learned in year 8 * Mass concentration (astronomy), a region of a planet or moon's cr ...
of lead in a flask of water. No measurement is exact. When a quantity is measured, the outcome depends on the measuring system, the measurement procedure, the skill of the operator, the environment, and other effects. Even if the quantity were to be measured several times, in the same way and in the same circumstances, a different measured value would in general be obtained each time, assuming the measuring system has sufficient resolution to distinguish between the values. The dispersion of the measured values would relate to how well the measurement is performed. Their
average In colloquial language, an average is a single number taken as representative of a non-empty list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divide ...

would provide an estimate of the true value of the quantity that generally would be more reliable than an individual measured value. The dispersion and the number of measured values would provide information relating to the average value as an estimate of the true value. However, this information would not generally be adequate. The measuring system may provide measured values that are not dispersed about the true value, but about some value offset from it. Take a domestic bathroom scale. Suppose it is not set to show zero when there is nobody on the scale, but to show some value offset from zero. 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. See Joint Committee for Guides in Metrology. Measurement uncertainty has important economic consequences for calibration and measurement activities. In calibration reports, the magnitude of the uncertainty is often taken as an indication of the quality of the laboratory, and smaller uncertainty values generally are of higher value and of higher cost. The
American Society of Mechanical Engineers#REDIRECT American Society of Mechanical Engineers {{rcat shell, {{R from other capitalisation ...
(ASME) has produced a suite of standards addressing various aspects of measurement uncertainty. For example, ASME standards are used to address the role of measurement uncertainty when accepting or rejecting products based on a measurement result and a product specification, provide a simplified approach (relative to the GUM) to the evaluation of dimensional measurement uncertainty, resolve disagreements over the magnitude of the measurement uncertainty statement, or provide guidance on the risks involved in any product acceptance/rejection decision.

# Indirect measurement

The above discussion concerns the direct measurement of a quantity, which incidentally occurs rarely. For example, the bathroom scale may convert a measured extension of a spring into an estimate of the measurand, the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
of the person on the scale. The particular relationship between extension and mass is determined by the
calibration In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a Standard (metrology), calibration standard of known accuracy. Such a standard could be another measuremen ...

of the scale. A measurement
model In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...
converts a quantity value into the corresponding value of the measurand. There are many types of measurement in practice and therefore many models. 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 Buoyancy (), or upthrust, is an upward exerted by a that opposes the of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bo ...

, is capable of delivering better results for industrial or scientific purposes. In general there are often several different quantities, for example
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

,
humidity Humidity is the concentration of water vapor, water vapour present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation (meteorology), precipitation, d ...

and
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
, that contribute to the definition of the measurand, and that need to be measured. Correction terms should be included in the measurement model when the conditions of measurement are not exactly as stipulated. These terms correspond to systematic errors. Given an estimate of a correction term, the relevant quantity should be corrected by this estimate. There will be an uncertainty associated with the estimate, even if the estimate is zero, as is often the case. Instances of systematic errors arise in height measurement, when the alignment of the measuring instrument is not perfectly vertical, and the ambient temperature is different from that prescribed. 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. As well as raw data representing measured values, there is another form of data that is frequently needed in a measurement model. Some such data relate to quantities representing
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant (mathematics), constant value in time. It is contrasted with a ...
s, each of which is known imperfectly. Examples are material constants such as
modulus of elasticity An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defi ...
and
specific heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quant ...
. There are often other relevant data given in reference books, calibration certificates, etc., regarded as estimates of further quantities. The items required by a measurement model to define a measurand are known as input quantities in a measurement model. The model is often referred to as a functional relationship. The output quantity in a measurement model is the measurand. Formally, the output quantity, denoted by $Y$, about which information is required, is often related to input quantities, denoted by $X_1,\ldots,X_N$, about which information is available, by a measurement model in the form of :$Y = f\left(X_1,\ldots,X_N\right),$ where $f$ is known as the measurement function. A general expression for a measurement model is :$h\left(Y,$ $X_1,\ldots,X_N\right) = 0.$ It is taken that a procedure exists for calculating $Y$ given $X_1,\ldots,X_N$, and that $Y$ is uniquely defined by this equation.

# Propagation of distributions

The true values of the input quantities $X_1,\ldots,X_N$ are unknown. In the GUM approach, $X_1,\ldots,X_N$ are characterized by
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
s and treated mathematically as
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s. These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available knowledge concerning $X_1,\ldots,X_N$. Sometimes, some or all of are interrelated and the relevant distributions, which are known as
joint A joint or articulation (or articular surface) is the connection made between bone A bone is a Stiffness, rigid tissue (anatomy), tissue that constitutes part of the skeleton in most vertebrate animals. Bones protect the various organs of th ...
, apply to these quantities taken together. Consider estimates $x_1,\ldots,x_N$, respectively, of the input quantities $X_1,\ldots,X_N$, obtained from certificates and reports, manufacturers' specifications, the analysis of measurement data, and so on. The probability distributions characterizing $X_1,\ldots,X_N$ are chosen such that the estimates $x_1,\ldots,x_N$, respectively, are the expectationsJCGM 101:2008. Evaluation of measurement data – Supplement 1 to the "Guide to the expression of uncertainty in measurement" – Propagation of distributions using a Monte Carlo method
Joint Committee for Guides in Metrology.
of $X_1,\ldots,X_N$. Moreover, for the $i$th input quantity, consider a so-called ''standard uncertainty'', given the symbol $u\left(x_i\right)$, defined as the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

of the input quantity $X_i$. This standard uncertainty is said to be associated with the (corresponding) estimate $x_i$. The use of available knowledge to establish a probability distribution to characterize each quantity of interest applies to the $X_i$ and also to $Y$. In the latter case, the characterizing probability distribution for $Y$ is determined by the measurement model together with the probability distributions for the $X_i$. The determination of the probability distribution for $Y$ from this information is known as the ''propagation of distributions''. The figure below depicts a measurement model $Y = X_1 + X_2$ in the case where $X_1$ and $X_2$ are each characterized by a (different) rectangular, or
uniform A uniform is a variety of clothing A kanga, worn throughout the African Great Lakes region Clothing (also known as clothes, apparel, and attire) are items worn on the body. Typically, clothing is made of fabrics or textiles, but over ti ...
, probability distribution. $Y$ has a symmetric trapezoidal probability distribution in this case. Once the input quantities $X_1,\ldots,X_N$ have been characterized by appropriate probability distributions, and the measurement model has been developed, the probability distribution for the measurand $Y$ is fully specified in terms of this information. In particular, the expectation of $Y$ is used as the estimate of $Y$, and the standard deviation of $Y$ as the standard uncertainty associated with this estimate. Often an interval containing $Y$ with a specified probability is required. Such an interval, a coverage interval, can be deduced from the probability distribution for $Y$. The specified probability is known as the coverage probability. For a given coverage probability, there is more than one coverage interval. The probabilistically symmetric coverage interval is an interval for which the probabilities (summing to one minus the coverage probability) of a value to the left and the right of the interval are equal. The shortest coverage interval is an interval for which the length is least over all coverage intervals having the same coverage probability. Prior knowledge about the true value of the output quantity $Y$ can also be considered. 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. Such additional information can be used to provide a probability distribution for $Y$ that can give a smaller standard deviation for $Y$ and hence a smaller standard uncertainty associated with the estimate of $Y$.

# Type A and Type B evaluation of uncertainty

Knowledge about an input quantity $X_i$ 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 $X$ given repeated measured values of it (obtained independently) is a
Gaussian distribution In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is : f ...

. $X$ then has expectation equal to the average measured value and standard deviation equal to the standard deviation of the average. When the uncertainty is evaluated from a small number of measured values (regarded as instances of a quantity characterized by a Gaussian distribution), the corresponding distribution can be taken as a ''t''-distribution.JCGM 104:2009. Evaluation of measurement data – An introduction to the "Guide to the expression of uncertainty in measurement" and related documents
Joint Committee for Guides in Metrology.
Other considerations apply when the measured values are not obtained independently. For a Type B evaluation of uncertainty, often the only available information is that $X$ lies in a specified interval math>a, b In such a case, knowledge of the quantity can be characterized by a rectangular probability distribution with limits $a$ and $b$. If different information were available, a probability distribution consistent with that information would be used.

# Sensitivity coefficients

Sensitivity coefficients $c_1,\ldots,c_N$ describe how the estimate $y$ of $Y$ would be influenced by small changes in the estimates $x_1,\ldots,x_N$ of the input quantities $X_1,\ldots,X_N$. For the measurement model $Y = f\left(X_1,\ldots,X_N\right)$, the sensitivity coefficient $c_i$ equals the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of first order of $f$ with respect to $X_i$ evaluated at $X_1 = x_1$, $X_2 = x_2$, etc. For a
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

measurement model :$Y = c_1 X_1 + \cdots + c_N X_N,$ with $X_1,\ldots,X_N$ independent, a change in $x_i$ equal to $u\left(x_i\right)$ would give a change $c_i u\left(x_i\right)$ in $y.$ This statement would generally be approximate for measurement models $Y = f\left(X_1,\ldots,X_N\right)$. The relative magnitudes of the terms $, c_i, u\left(x_i\right)$ are useful in assessing the respective contributions from the input quantities to the standard uncertainty $u\left(y\right)$ associated with $y$. The standard uncertainty $u\left(y\right)$ associated with the estimate $y$ of the output quantity $Y$ is not given by the sum of the $, c_i, u\left(x_i\right)$, but these terms combined in quadrature,JCGM 100:2008. Evaluation of measurement data – Guide to the expression of uncertainty in measurement
Joint Committee for Guides in Metrology.
namely by an expression that is generally approximate for measurement models $Y = f\left(X_1,\ldots,X_N\right)$: :$u^2\left(y\right) = c_1^2u^2\left(x_1\right) + \cdots + c_N^2u^2\left(x_N\right),$ which is known as the law of propagation of uncertainty. When the input quantities $X_i$ contain dependencies, the above formula is augmented by terms containing
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the less ...

s, which may increase or decrease $u\left(y\right)$.

# Uncertainty evaluation

The main stages of uncertainty evaluation constitute formulation and calculation, the latter consisting of propagation and summarizing. The formulation stage constitutes #defining the output quantity $Y$ (the measurand), #identifying the input quantities on which $Y$ depends, #developing a measurement model relating $Y$ to the input quantities, and #on the basis of available knowledge, assigning probability distributions — Gaussian, rectangular, etc. — to the input quantities (or a joint probability distribution to those input quantities that are not independent). 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 $Y$, and summarizing by using this distribution to obtain #the expectation of $Y$, taken as an estimate $y$ of $Y$, #the standard deviation of $Y$, taken as the standard uncertainty $u\left(y\right)$ associated with $y$, and #a coverage interval containing $Y$ with a specified coverage probability. The propagation stage of uncertainty evaluation is known as the propagation of distributions, various approaches for which are available, including #the GUM uncertainty framework, constituting the application of the law of propagation of uncertainty, and the characterization of the output quantity $Y$ by a Gaussian or a $t$-distribution, #analytic methods, in which mathematical analysis is used to derive an algebraic form for the probability distribution for $Y$, and #a
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
, in which an approximation to the distribution function for $Y$ is established numerically by making random draws from the probability distributions for the input quantities, and evaluating the model at the resulting values. For any particular uncertainty evaluation problem, approach 1), 2) or 3) (or some other approach) is used, 1) being generally approximate, 2) exact, and 3) providing a solution with a numerical accuracy that can be controlled.

## Models with any number of output quantities

When the measurement model is multivariate, that is, it has any number of output quantities, the above concepts can be extended. 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.

# Uncertainty as an interval

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. In some situations, however, a mathematical interval might be a better model of uncertainty than a probability distribution. This may include situations involving periodic measurements, binned data values, censoring, detection limits, or plus-minus ranges of measurements where no particular probability distribution seems justified or where one cannot assume that the errors among individual measurements are completely independent. A more
robustRobustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' can be ...
representation of measurement uncertainty in such cases can be fashioned from intervals.Manski, C.F. (2003); ''Partial Identification of Probability Distributions'', Springer Series in Statistics, Springer, New YorkFerson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007)
''Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty''
Sandia National Laboratories SAND 2007-0939
An interval 'a'',''b''is different from a rectangular or uniform probability distribution over the same range in that the latter suggests that the true value lies inside the right half of the range ''a'' + ''b'')/2, ''b''with probability one half, and within any subinterval of 'a'',''b''with probability equal to the width of the subinterval divided by ''b'' – ''a''. The interval makes no such claims, except simply that the measurement lies somewhere within the interval. Distributions of such measurement intervals can be summarized as
probability box 300px, alt=A continuous p-box depicted as a graph with abscissa labeled X and ordinate labeled Probability, A p-box (probability box). A probability box (or p-box) is a characterization of an uncertain number consisting of both uncertainty quanti ...
es and Dempster–Shafer structures over the real numbers, which incorporate both aleatoric and epistemic uncertainties.

# References

* Bich, W., Cox, M. G., and Harris, P. M. Evolution of the "Guide to the Expression of Uncertainty in Measurement". Metrologia, 43(4):S161–S166, 2006. * Cox, M. G., and Harris, P. M
SSfM Best Practice Guide No. 6, Uncertainty evaluation. Technical report DEM-ES-011
National Physical Laboratory, 2006. * Cox, M. G., and Harris, P.
. Software specifications for uncertainty evaluation. Technical report DEM-ES-010
National Physical Laboratory, 2006. * Grabe,
., Measurement Uncertainties in Science and Technology
Springer 2005. * Grabe, M.
Generalized Gaussian Error Calculus
Springer 2010. * * EA. Expression of the uncertainty of measurement in calibration. Technical Report EA-4/02, European Co-operation for Accreditation, 1999. * Elster, C., and Toman, B. Bayesian uncertainty analysis under prior ignorance of the measurand versus analysis using Supplement 1 to the ''Guide'': a comparison. Metrologia, 46:261–266, 2009. * * Lira., I. Evaluating the Uncertainty of Measurement. Fundamentals and Practical Guidance. Institute of Physics, Bristol, UK, 2002. * Majcen N., Taylor P. (Editors), Practical examples on traceability, measurement uncertainty and validation in chemistry, Vol 1, 2010; . * Possolo A and Iyer H K 2017 Concepts and tools for the evaluation of measurement uncertainty Rev. Sci. Instrum.,88 011301 (2017). *
UKAS The United Kingdom Accreditation Service (UKAS) is the sole national accreditation body Accreditation is a third-party attestation related to a conformity assessment body (such as certification body, inspection body or laboratory) conveying form ...

The expression of uncertainty in EMC testing. Technical Report LAB34
United Kingdom Accreditation Service, 2002. * UKA
M3003 The Expression of Uncertainty and Confidence in Measurement
(Edition 3, November 2012) UKAS
ASME PTC 19.1, Test Uncertainty
New York: The American Society of Mechanical Engineers; 2005 * * * *

NPLUnc

Estimate of temperature and its uncertainty in small systems, 2011.

Introduction to evaluating uncertainty of measurement

JCGM 200:2008. International Vocabulary of Metrology – Basic and general concepts and associated terms
3rd Edition. Joint Committee for Guides in Metrology.

ISO * ttp://www.bipm.org/utils/common/documents/jcgm/JCGM_106_2012_E.pdf JCGM 106:2012. Evaluation of measurement data – The role of measurement uncertainty in conformity assessment.Joint Committee for Guides in Metrology.
NIST. Uncertainty of measurement results.
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