Measurement Uncertainty
Measurement Uncertainty
Measurement Uncertainty
In metrology, measurement uncertainty is a non-negative parameter characterizing the dispersion of the values attributed to a measured quantity. The uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and a measured value is only complete if it is accompanied by a statement of the associated uncertainty. Fractional uncertainty is the measurement uncertainty divided by the measured value. Codex has guidelines on Measurement Uncertainty, CAC/GL 54-2004.
Contents
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1 Background 2 Random uncertainties and systematic errors 3 GUM approach 4 Measurement model 5 Propagation of distributions 6 Type A and Type B evaluation of uncertainty 7 Sensitivity coefficients 8 Stages of uncertainty evaluation 9 Joint Committee for Guides in Metrology 10 Alternative Perspective 11 See also 12 Further reading 13 References
[edit]Background
The purpose of measurement is to provide information about a quantity of interest - a measurand. For example, the measurand might be the volume of a vessel, the potential differencebetween the terminals of a battery, or the mass concentration 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.
[1]
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 that 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 made. Their average 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.
and
systematic
There are two types of measurement faults, systematic error and random uncertainty. A systematic error (an estimate of which is known as a measurement bias) is associated with the fact that a measured value contains an offset. In general, a systematic error, regarded as a quantity, is a component of error that remains constant or depends in a specific manner on some other quantity. A random uncertainty is associated with the fact that when a measurement is repeated it will generally provide a measured value that is different from the previous value. It is random in that the next measured value cannot be predicted exactly from previous such values. (If a prediction were possible, allowance for the effect could be made.) In general, there can be a number of contributions to each type of error.
GUM approach
The Guide to the Expression of Uncertainty in Measurement (GUM)[2] is a document published by the JCGM that establishes general rules for evaluating and expressing uncertainty in measurement.[3] The GUM provides a way to express the perceived quality of the result of a measurement. Rather than express the result by providing a best estimate of the measurand along with information about systematic and random error values (in the form of an "error analysis"), the GUM approach is to express the result of a measurement as a best estimate of the measurand along with an associated measurement uncertainty. One of the basic premises of the GUM approach is that it is possible to characterize the quality of a measurement by accounting for both systematic and random errors on a comparable footing, and a method is provided for doing that. This method refines the information previously provided in
an "error analysis", and puts it on a probabilistic basis through the concept of measurement uncertainty. Another basic premise of the GUM approach is that it is not possible to state how well the true value of the measurand is known, but only how well it is believed to be known. Measurement uncertainty can therefore be described as a measure of how well one believes one knows the true value of the measurand. This uncertainty reflects the incomplete knowledge of the measurand. The notion of "belief" is an important one, since it moves metrology into a realm where results of measurement need to be considered and quantified in terms of probabilities that express degrees of belief.
Measurement model
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 of the person on the scale. The particular relationship between extension and mass is determined by the calibration of the scale. A
measurement model 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, 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. 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 representingphysical constants, each of which is known imperfectly. Examples are material constants such as modulus
of elasticity and specific heat. 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 input quantities, denoted by
that
Y is
Propagation of distributions
The true values of the input quantities approach,
characterized
mathematically as random variables. These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available of knowledge concerning
Sometimes,
some
or
all
X1, ... , XN are interrelated and the relevant distributions, which are known as joint, x1, ... , xN,
and respectively, of the input quantities
obtained from certificates and reports, manufacturers' specifications, the analysis of measurement characterizing data, so on. The probability
X1, ... , XN are chosen such that the estimates x1, ... , xN, respectively, are the expectations of X1, ... , XN. Moreover, for the ith input quantity, consider a socalled standard uncertainty, given the symbol u(xi), defined as the standard deviation of the input quantity Xi. This standard uncertainty is said to be associated 2 with the (corresponding) estimate xi. The estimate xi is best in the sense that u (xi) is smaller than the expected squared difference of Xi from any other value.
[4] [4]
The use of available knowledge to establish a probability distribution to characterize each quantity of interest applies to the probability distribution for
X i.
Y = X1 + X2 in
the case
probability distribution.
probability distributions, and the measurement model has been developed, the probability distribution for the measurand particular, the expectation of of fully specified in terms of this information. In
Y with a specified probability is required. Such an interval, a coverage interval, can be deduced from the probability distribution for Y. The specified
Often an interval containing 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
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 can give a smaller standard deviation for associated with the estimate of .[5][6][7]
Y that
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
X given repeated measured values of it (obtained independently) is a Gaussian distribution. X then has expectation equal to the
distribution best describing an input quantity 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 -distribution.[8] 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 in a specified interval [
X lies
a, b].
a and b.
If different
information were available, a probability distribution consistent with that information would be used.[9]
Sensitivity coefficients
Main article: Sensitivity analysis
c1, ... , cN describe how the estimate y of Y would be influenced by small changes in the estimates x1,..., xN of the input quantities X1, ... , XN. For the measurement function Y = f(X1, ... , XN), the sensitivity coefficient ci equals the partial derivative of first order of f with respect to Xi evaluated at X1 = x1, X2 = x2, etc. For a linearmeasurement function
Sensitivity coefficients ,
X1, ... , XN independent, a change in xi equal to u(xi) would give a change ciu(xi) in y. This statement would generally be approximate for measurement functions Y = f(X1, ... ,XN). The relative magnitudes of the terms | ci | u(xi) are useful in assessing the respective contributions from the input quantities to the standard uncertainty u(y) associated with y.
with
u(y) associated with the estimate y of the output quantity Y is not given by the sum of the | ci | u(xi), but these terms combined in
The standard uncertainty quadrature,[2] namely by (an expression that is generally approximate for measurement functions
which is known as the law of propagation of uncertainty. When the input quantities augmented decrease by terms
Xi contain
u(y).
1. 2. 3.
4.
Y (the measurand), identifying the input quantities on which Y depends, developing a measurement model relating Y to
defining the output quantity
the input
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 to obtain
Y, taken as an estimate y of Y, 2. the standard deviation of Y, taken as the uncertainty u(y) associated with y, and 3. a coverage interval containing Y with a specified
1.
the expectation of probability.
standard
coverage
The propagation stage of uncertainty evaluation is known as the propagation of distributions, various approaches for which are available, including
1.
the law of propagation of uncertainty, and the characterization of the output quantity
2.
3.
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.
Electrotechnical Commission (IEC), the International Federation of Clinical Chemistry and Laboratory Medicine(IFCC), the International Laboratory for
Accreditation
Cooperation (ILAC),
Applied Applied
Physics (IUPAP), and the International Organization of Legal Metrology (OIML). JCGM has two Working Groups. Working Group 1, "Expression of uncertainty in measurement", has the task to promote the use of the GUM and to prepare Supplements and other documents for its broad application. Working Group 2, "Working Group on International vocabulary of basic and general terms in metrology (VIM)", has the task to revise and promote the use of the VIM. For further information on the activity of the JCGM, see www.bipm.org. Revision by Working Group 1 of the GUM itself has started in parallel with work on preparing documents in a series of JCGM documents under the generic heading Evaluation of measurement data. The parts in the series are
Alternative Perspective
Most of this article represents the most common view of measurement uncertainty, which assumes that random variables are proper mathematical models for uncertain quantities and simple probability distributions are sufficient for representing all forms of measurement uncertainties. In some situations, however, a mathematical interval rather than a probability distribution might be a better model of uncertainty. 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 measuresments are completely independent.
A more robust representation of measurement uncertainty in such cases can be fashioned from intervals.[10][11] 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 ba. 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 boxes and Dempster-Shafer structures over the real numbers, which incorporate both aleatoric and epistemic uncertainties.