207:. 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.
159:(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, to provide a simplified approach (relative to the GUM) to the evaluation of dimensional measurement uncertainty, to resolve disagreements over the magnitude of the measurement uncertainty statement, and to provide guidance on the risks involved in any product acceptance/rejection decision.
1609:. 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.
1379:
154:
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
67:
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.
3542:
representation of measurement uncertainty in such cases can be fashioned from intervals. An interval 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 with probability one
127:
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
3495:
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
1776:
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
131:
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
1378:
183:
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,
115:
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
1632:
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
3518:
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
225:
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.
3486:
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.
3041:
167:
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
139:(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
1728:
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").
3200:
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
3535:, 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.
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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
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360:
2905:
2614:
2132:
469:
3717:
JCGM 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
210:
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
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3551:. 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
2844:
2670:
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Da Silva, R.B.; Bulska, E.; Godlewska-Zylkiewicz, B.; Hedrich, M.; Majcen, N.; Magnusson, B.; Marincic, S.; Papadakis, I.; Patriarca, M.; Vassileva, E.; Taylor, P. (2012).
2530:
1295:
3920:
Ellison S. L. R., Williams A. (Eds). Eurachem/CITAC guide: Quantifying
Uncertainty in Analytical Measurement, Third edition, (2012) ISBN 978-0-948926-30-3. Available from
3127:
The main stages of uncertainty evaluation constitute formulation and calculation, the latter consisting of propagation and summarizing. The formulation stage constitutes
2290:
2250:
674:. These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available knowledge concerning
2484:
1067:
80:). 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.
56:
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
3854:
JCGM 102: Evaluation of
Measurement Data – Supplement 2 to the "Guide to the Expression of Uncertainty in Measurement" – Extension to Any Number of Output Quantities
3074:
2448:
2210:
2159:
1726:
1434:
1407:
1349:
1322:
1222:
1155:
1125:
1098:
867:, obtained from certificates and reports, manufacturers' specifications, the analysis of measurement data, and so on. The probability distributions characterizing
3339:
3107:
2748:
2699:
415:
1835:
60:. By international agreement, this uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity value. It is a non-negative parameter.
2553:
3482:
is established numerically by making random draws from the probability distributions for the input quantities, and evaluating the model at the resulting values.
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3359:
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have been characterized by appropriate probability distributions, and the measurement model has been developed, the probability distribution for the measurand
116:
different measured value would in general be obtained each time, assuming the measuring system has sufficient resolution to distinguish between the values.
203:
Correction terms should be included in the measurement model when the conditions of measurement are not exactly as stipulated. These terms correspond to
144:
3392:
the GUM uncertainty framework, constituting the application of the law of propagation of uncertainty, and the characterization of the output quantity
188:, is capable of delivering better results for industrial or scientific purposes. In general there are often several different quantities, for example
3910:
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.
3796:
JCGM 104:2009. Evaluation of measurement data – An introduction to the "Guide to the expression of uncertainty in measurement" and related documents
3388:
The propagation stage of uncertainty evaluation is known as the propagation of distributions, various approaches for which are available, including
2913:
4013:
222:. There are often other relevant data given in reference books, calibration certificates, etc., regarded as estimates of further quantities.
3661:
3934:
EA. Expression of the uncertainty of measurement in calibration. Technical Report EA-4/02, European Co-operation for
Accreditation, 1999.
1732:
In Type A evaluations of measurement uncertainty, the assumption is often made that the distribution best describing an input quantity
17:
3952:
Majcen N., Taylor P. (Editors), Practical examples on traceability, measurement uncertainty and validation in chemistry, Vol 1, 2010;
3523:
might be a better model of uncertainty than a probability distribution. This may include situations involving periodic measurements,
1589:
with a specified probability is required. Such an interval, a coverage interval, can be deduced from the probability distribution for
3914:
3949:
Lira., I. Evaluating the
Uncertainty of Measurement. Fundamentals and Practical Guidance. Institute of Physics, Bristol, UK, 2002.
3963:
Possolo A and Iyer H K 2017 Concepts and tools for the evaluation of measurement uncertainty Rev. Sci. Instrum.,88 011301 (2017).
148:
1130:
The use of available knowledge to establish a probability distribution to characterize each quantity of interest applies to the
119:
The dispersion of the measured values would relate to how well the measurement is performed. If measured on a ratio or interval
132:
times the person's mass were re-measured, the effect of this offset would be inherently present in the average of the values.
3997:
3957:
147:, and is employed in most modern national and international documentary standards on measurement methods and technology. See
68:
This particular single choice is usually called the measured value, which may be optimal in some well-defined sense (e.g., a
3435:
analytic methods, in which mathematical analysis is used to derive an algebraic form for the probability distribution for
4034:
4035:
ISO 3534-1:2006. Statistics – Vocabulary and symbols – Part 1: General statistical terms and terms used in probability.
2846:, but these terms combined in quadrature, namely by an expression that is generally approximate for measurement models
2302:
1881:. If different information were available, a probability distribution consistent with that information would be used.
3583:
1838:
1352:
3496:
natural generalization, and a calculation procedure that implements a multivariate Monte Carlo method is available.
769:
301:
109:
2849:
2558:
2076:
4041:
JCGM 106:2012. Evaluation of measurement data – The role of measurement uncertainty in conformity assessment.
3122:
419:
2672:
are useful in assessing the respective contributions from the input quantities to the standard uncertainty
3691:
ASME B89.7.3.3, Guidelines for
Assessing the Reliability of Dimensional Measurement Uncertainty Statements
2380:
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of the person on the scale. The particular relationship between extension and mass is determined by the
3618:
3593:
3560:
3556:
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half, and within any subinterval of with probability equal to the width of the subinterval divided by
569:
4040:
4029:
JCGM 200:2008. International
Vocabulary of Metrology – Basic and general concepts and associated terms
3852:
3795:
3716:
3648:
3649:
JCGM 100:2008. Evaluation of measurement data – Guide to the expression of uncertainty in measurement
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1254:
667:
2255:
2215:
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Bernardo, J., and Smith, A. "Bayesian Theory". John Wiley & Sons, New York, USA, 2000. 3.20
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ISO/IEC 17025 General requirements for the competence of testing and calibration laboratories
120:
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is determined by the measurement model together with the probability distributions for the
1133:
1103:
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136:
105:
3928:
3673:
ASME B89.7.3.1, Guidelines for
Decision Rules in Determining Conformance to Specifications
3315:
3083:
2724:
2675:
391:
8:
4018:
3940:"Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty"
3783:. Tech. Rep. Guide CG4, EU-RACHEM/CITEC, EURACHEM/CITAC Guide], 2000. Second edition.
3780:
3738:
Elster, Clemens (2007). "Calculation of uncertainty in the presence of prior knowledge".
3662:
Measurement Good
Practice Guide No. 11. A Beginner's Guide to Uncertainty of Measurement.
3578:
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1814:
1753:
3821:
3751:
2535:
1100:. This standard uncertainty is said to be associated with the (corresponding) estimate
249:, about which information is required, is often related to input quantities, denoted by
3894:
Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty
3833:
3763:
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3465:
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3438:
3415:
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3180:
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1964:
1944:
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1784:. Other considerations apply when the measured values are not obtained independently.
385:
is known as the measurement function. A general expression for a measurement model is
4046:
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3953:
3837:
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3759:
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3509:
1787:
For a Type B evaluation of uncertainty, often the only available information is that
211:
204:
77:
31:
3829:
3682:
ASME B89.7.3.2, Guidelines for the Evaluation of Dimensional Measurement Uncertainty
200:, that contribute to the definition of the measurand, and that need to be measured.
3915:
SSfM Best Practice Guide No. 6, Uncertainty evaluation. Technical report DEM-ES-011
3825:
3755:
3532:
1506:
is fully specified in terms of this information. In particular, the expectation of
3552:
2293:
671:
215:
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295:, about which information is available, by a measurement model in the form of
214:, each of which is known imperfectly. Examples are material constants such as
93:
4055:
3938:
Ferson, S.; Kreinovich, V.; Hajagos, J.; Oberkampf, W.; Ginzburg, L. (2007).
3891:
Ferson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007);
3808:
Weise, K.; Woger, W. (1993). "A Bayesian theory of measurement uncertainty".
3598:
3036:{\displaystyle u^{2}(y)=c_{1}^{2}u^{2}(x_{1})+\cdots +c_{N}^{2}u^{2}(x_{N}),}
1633:
additional information can be used to provide a probability distribution for
3700:
ASME B89.7.4, Measurement Uncertainty and Conformance Testing: Risk Analysis
3524:
3987:
3613:
3608:
3076:
contain dependencies, the above formula is augmented by terms containing
1673:
and hence a smaller standard uncertainty associated with the estimate of
189:
180:
converts a quantity value into the corresponding value of the measurand.
173:
49:
of the values attributed to a quantity measured on an interval or ratio
3850:
3077:
1837:]. In such a case, knowledge of the quantity can be characterized by a
1177:. In the latter case, the characterizing probability distribution for
3975:
2555:
This statement would generally be approximate for measurement models
140:
38:
3939:
3892:
768:
are interrelated and the relevant distributions, which are known as
4019:
Estimate of temperature and its uncertainty in small systems, 2011.
3781:
EURACHEM/CITAC. "Quantifying uncertainty in analytical measurement"
1752:
given repeated measured values of it (obtained independently) is a
1375:
has a symmetric trapezoidal probability distribution in this case.
193:
185:
89:
3968:
M3003 The Expression of Uncertainty and Confidence in Measurement
124:
3937:
101:
73:
3990:
Analytical measurement: measurement uncertainty and statistics
3490:
1696:
3462:, in which an approximation to the distribution function for
88:
The purpose of measurement is to provide information about a
1612:
Prior knowledge about the true value of the output quantity
3921:
1566:
as the standard uncertainty associated with this estimate.
1382:
An additive measurement function with two input quantities
169:
156:
69:
3046:
which is known as the law of propagation of uncertainty.
1224:. The determination of the probability distribution for
3977:
Propagation of Uncertainties in Experimental Measurement
1351:
are each characterized by a (different) rectangular, or
4031:, 3rd Edition. Joint Committee for Guides in Metrology.
3225:, and summarizing by using this distribution to obtain
137:"Guide to the Expression of Uncertainty in Measurement"
1981:
would be influenced by small changes in the estimates
1436:
characterized by rectangular probability distributions
4024:
Introduction to evaluating uncertainty of measurement
3468:
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1987:
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255:
235:
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It is taken that a procedure exists for calculating
3882:, Springer Series in Statistics, Springer, New York
3880:
Partial Identification of Probability Distributions
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63:The measurement uncertainty is often taken as the
4053:
3885:
3851:Joint Committee for Guides in Metrology (2011).
2367:{\displaystyle Y=c_{1}X_{1}+\cdots +c_{N}X_{N},}
3664:Tech. rep., National Physical Laboratory, 1999.
1653:that can give a smaller standard deviation for
563:
100:include the size of a cylindrical feature, the
3559:over the real numbers, which incorporate both
3897:, Sandia National Laboratories SAND 2007-0939
3499:
1251:The figure below depicts a measurement model
772:, apply to these quantities taken together.
3872:
3791:
3789:
3491:Models with any number of output quantities
1884:
1697:Type A and Type B evaluation of uncertainty
108:between the terminals of a battery, or the
3807:
3798:. Joint Committee for Guides in Metrology.
3719:. Joint Committee for Guides in Metrology.
3712:
3710:
3708:
3706:
3651:, Joint Committee for Guides in Metrology.
3154:identifying the input quantities on which
229:Formally, the output quantity, denoted by
27:Factor of lower probability in measurement
4047:NIST. Uncertainty of measurement results.
3112:
355:{\displaystyle Y=f(X_{1},\ldots ,X_{N}),}
184:involving additional effects such as air
4043:Joint Committee for Guides in Metrology.
3844:
3786:
3177:developing a measurement model relating
2900:{\displaystyle Y=f(X_{1},\ldots ,X_{N})}
2609:{\displaystyle Y=f(X_{1},\ldots ,X_{N})}
2127:{\displaystyle Y=f(X_{1},\ldots ,X_{N})}
1029:th input quantity, consider a so-called
821:, respectively, of the input quantities
574:The true values of the input quantities
157:American Society of Mechanical Engineers
3703:
2616:. The relative magnitudes of the terms
464:{\displaystyle X_{1},\ldots ,X_{N})=0.}
162:
149:Joint Committee for Guides in Metrology
14:
4054:
3737:
3384:with a specified coverage probability.
1244:from this information is known as the
560:is uniquely defined by this equation.
145:international laboratory accreditation
3917:, National Physical Laboratory, 2006.
3644:
3642:
3640:
3561:aleatoric and epistemic uncertainties
3312:, taken as the standard uncertainty
1839:rectangular probability distribution
3929:Generalized Gaussian Error Calculus
2416:{\displaystyle X_{1},\ldots ,X_{N}}
2066:{\displaystyle X_{1},\ldots ,X_{N}}
2020:{\displaystyle x_{1},\ldots ,x_{N}}
1934:{\displaystyle c_{1},\ldots ,c_{N}}
1479:{\displaystyle X_{1},\ldots ,X_{N}}
1002:{\displaystyle X_{1},\ldots ,X_{N}}
952:{\displaystyle x_{1},\ldots ,x_{N}}
913:are chosen such that the estimates
906:{\displaystyle X_{1},\ldots ,X_{N}}
860:{\displaystyle X_{1},\ldots ,X_{N}}
814:{\displaystyle x_{1},\ldots ,x_{N}}
760:{\displaystyle X_{1},\ldots ,X_{N}}
713:{\displaystyle X_{1},\ldots ,X_{N}}
659:{\displaystyle X_{1},\ldots ,X_{N}}
620:are unknown. In the GUM approach,
613:{\displaystyle X_{1},\ldots ,X_{N}}
533:{\displaystyle X_{1},\ldots ,X_{N}}
288:{\displaystyle X_{1},\ldots ,X_{N}}
24:
3904:
3810:Measurement Science and Technology
3637:
1701:Knowledge about an input quantity
1377:
96:. Measurands on ratio or interval
25:
4073:
4007:
3584:Experimental uncertainty analysis
3080:, which may increase or decrease
128:would not generally be adequate.
1546:, and the standard deviation of
3970:(Edition 3, November 2012) UKAS
3801:
3774:
3364:a coverage interval containing
2839:{\displaystyle |c_{i}|u(x_{i})}
2790:is not given by the sum of the
2665:{\displaystyle |c_{i}|u(x_{i})}
1777:distribution can be taken as a
3913:Cox, M. G., and Harris, P. M.
3731:
3722:
3694:
3685:
3676:
3667:
3654:
3328:
3322:
3096:
3090:
3027:
3014:
2977:
2964:
2933:
2927:
2894:
2862:
2833:
2820:
2813:
2798:
2737:
2731:
2688:
2682:
2659:
2646:
2639:
2624:
2603:
2571:
2519:
2506:
2473:
2460:
2134:, the sensitivity coefficient
2121:
2089:
1355:, probability distribution.
1056:
1043:
670:and treated mathematically as
452:
398:
346:
314:
13:
1:
3630:
3131:defining the output quantity
3123:Quality of analytical results
2750:associated with the estimate
2525:{\displaystyle c_{i}u(x_{i})}
1569:Often an interval containing
1290:{\displaystyle Y=X_{1}+X_{2}}
720:. Sometimes, some or all of
112:of lead in a flask of water.
83:
3197:to the input quantities, and
2073:. For the measurement model
1246:propagation of distributions
564:Propagation of distributions
176:of the scale. A measurement
7:
3566:
2721:. The standard uncertainty
2285:{\displaystyle X_{2}=x_{2}}
2245:{\displaystyle X_{1}=x_{1}}
1526:is used as the estimate of
10:
4078:
3760:10.1088/0026-1394/44/2/002
3619:Uncertainty quantification
3594:Propagation of uncertainty
3557:Dempster–Shafer structures
3503:
3500:Uncertainty as an interval
3292:the standard deviation of
3116:
3049:When the input quantities
1941:describe how the estimate
1888:
1440:Once the input quantities
570:Propagation of uncertainty
567:
29:
18:Uncertainty of measurement
3830:10.1088/0957-0233/4/1/001
2423:independent, a change in
1895:Sensitivity coefficients
668:probability distributions
45:is the expression of the
3860:(Technical report). JCGM
2479:{\displaystyle u(x_{i})}
2027:of the input quantities
1885:Sensitivity coefficients
1062:{\displaystyle u(x_{i})}
959:, respectively, are the
143:, which is required for
30:Not to be confused with
3249:, taken as an estimate
2770:of the output quantity
43:measurement uncertainty
3589:History of measurement
3574:Accuracy and precision
3476:
3449:
3426:
3406:
3378:
3355:
3335:
3306:
3283:
3263:
3243:
3219:
3191:
3168:
3145:
3113:Uncertainty evaluation
3103:
3070:
3037:
2901:
2840:
2784:
2764:
2744:
2715:
2695:
2666:
2610:
2549:
2526:
2480:
2444:
2417:
2368:
2286:
2246:
2206:
2179:
2155:
2128:
2067:
2021:
1975:
1955:
1935:
1875:
1855:
1831:
1801:
1770:
1746:
1722:
1687:
1667:
1647:
1626:
1603:
1583:
1560:
1540:
1520:
1500:
1480:
1437:
1430:
1403:
1369:
1345:
1318:
1291:
1238:
1218:
1191:
1171:
1151:
1121:
1094:
1073:of the input quantity
1063:
1023:
1003:
953:
907:
861:
815:
761:
714:
660:
614:
554:
534:
488:
465:
411:
379:
356:
289:
243:
47:statistical dispersion
3878:Manski, C.F. (2003);
3624:Random-fuzzy variable
3477:
3450:
3427:
3407:
3379:
3356:
3336:
3307:
3284:
3264:
3244:
3220:
3192:
3169:
3146:
3104:
3071:
3069:{\displaystyle X_{i}}
3038:
2902:
2841:
2785:
2765:
2745:
2716:
2696:
2667:
2611:
2550:
2527:
2481:
2445:
2443:{\displaystyle x_{i}}
2418:
2369:
2287:
2247:
2207:
2205:{\displaystyle X_{i}}
2180:
2156:
2154:{\displaystyle c_{i}}
2129:
2068:
2022:
1976:
1956:
1936:
1876:
1856:
1832:
1802:
1771:
1754:Gaussian distribution
1747:
1723:
1721:{\displaystyle X_{i}}
1688:
1668:
1648:
1627:
1604:
1584:
1561:
1541:
1521:
1501:
1481:
1431:
1429:{\displaystyle X_{2}}
1404:
1402:{\displaystyle X_{1}}
1381:
1370:
1346:
1344:{\displaystyle X_{2}}
1319:
1317:{\displaystyle X_{1}}
1292:
1239:
1219:
1217:{\displaystyle X_{i}}
1192:
1172:
1152:
1150:{\displaystyle X_{i}}
1122:
1120:{\displaystyle x_{i}}
1095:
1093:{\displaystyle X_{i}}
1064:
1024:
1009:. Moreover, for the
1004:
954:
908:
862:
816:
762:
715:
666:are characterized by
661:
615:
555:
535:
489:
466:
412:
380:
357:
290:
244:
216:modulus of elasticity
92:of interest – a
3466:
3439:
3416:
3396:
3368:
3345:
3334:{\displaystyle u(y)}
3316:
3296:
3273:
3253:
3233:
3209:
3181:
3158:
3135:
3119:Uncertainty analysis
3102:{\displaystyle u(y)}
3084:
3053:
2914:
2850:
2794:
2774:
2754:
2743:{\displaystyle u(y)}
2725:
2705:
2694:{\displaystyle u(y)}
2676:
2620:
2559:
2536:
2490:
2486:would give a change
2454:
2427:
2381:
2303:
2256:
2216:
2189:
2169:
2138:
2077:
2031:
1985:
1965:
1945:
1899:
1891:Sensitivity analysis
1865:
1845:
1815:
1807:lies in a specified
1791:
1760:
1736:
1705:
1677:
1657:
1637:
1616:
1593:
1573:
1550:
1530:
1510:
1490:
1444:
1413:
1386:
1359:
1328:
1301:
1255:
1228:
1201:
1181:
1161:
1134:
1104:
1077:
1037:
1031:standard uncertainty
1013:
967:
917:
871:
825:
779:
725:
678:
624:
578:
544:
498:
478:
420:
410:{\displaystyle h(Y,}
392:
369:
302:
253:
233:
163:Indirect measurement
106:potential difference
3974:Rouaud, M. (2013),
3822:1993MeScT...4....1W
3752:2007Metro..44..111E
3579:Confidence interval
3506:Confidence interval
3412:by a Gaussian or a
3229:the expectation of
3003:
2953:
1830:{\displaystyle a,b}
1033:, given the symbol
775:Consider estimates
3604:Set identification
3472:
3460:Monte Carlo method
3445:
3422:
3402:
3374:
3351:
3331:
3302:
3279:
3259:
3239:
3215:
3187:
3164:
3141:
3099:
3066:
3033:
2989:
2939:
2897:
2836:
2780:
2760:
2740:
2711:
2691:
2662:
2606:
2548:{\displaystyle y.}
2545:
2522:
2476:
2440:
2413:
2364:
2296:measurement model
2282:
2242:
2202:
2175:
2165:of first order of
2163:partial derivative
2151:
2124:
2063:
2017:
1971:
1951:
1931:
1871:
1851:
1827:
1797:
1766:
1742:
1718:
1683:
1663:
1643:
1622:
1599:
1579:
1556:
1536:
1516:
1496:
1476:
1438:
1426:
1399:
1365:
1341:
1314:
1297:in the case where
1287:
1234:
1214:
1187:
1167:
1147:
1117:
1090:
1071:standard deviation
1059:
1019:
999:
949:
903:
857:
811:
757:
710:
656:
610:
550:
530:
484:
461:
407:
375:
352:
285:
239:
212:physical constants
110:mass concentration
65:standard deviation
58:standard deviation
3999:978-92-79-23070-7
3958:978-92-79-12021-3
3553:probability boxes
3514:Fiducial interval
3510:Credible interval
3475:{\displaystyle Y}
3448:{\displaystyle Y}
3425:{\displaystyle t}
3405:{\displaystyle Y}
3377:{\displaystyle Y}
3354:{\displaystyle y}
3305:{\displaystyle Y}
3282:{\displaystyle Y}
3262:{\displaystyle y}
3242:{\displaystyle Y}
3218:{\displaystyle Y}
3190:{\displaystyle Y}
3167:{\displaystyle Y}
3144:{\displaystyle Y}
2783:{\displaystyle Y}
2763:{\displaystyle y}
2714:{\displaystyle y}
2178:{\displaystyle f}
1974:{\displaystyle Y}
1954:{\displaystyle y}
1874:{\displaystyle b}
1854:{\displaystyle a}
1800:{\displaystyle X}
1769:{\displaystyle X}
1745:{\displaystyle X}
1686:{\displaystyle Y}
1666:{\displaystyle Y}
1646:{\displaystyle Y}
1625:{\displaystyle Y}
1602:{\displaystyle Y}
1582:{\displaystyle Y}
1559:{\displaystyle Y}
1539:{\displaystyle Y}
1519:{\displaystyle Y}
1499:{\displaystyle Y}
1368:{\displaystyle Y}
1237:{\displaystyle Y}
1190:{\displaystyle Y}
1170:{\displaystyle Y}
1069:, defined as the
1022:{\displaystyle i}
553:{\displaystyle Y}
487:{\displaystyle Y}
378:{\displaystyle f}
242:{\displaystyle Y}
205:systematic errors
104:of a vessel, the
32:Measurement error
16:(Redirected from
4069:
4003:
3984:
3983:(short ed.)
3982:
3946:
3944:
3931:, Springer 2010.
3922:www.eurachem.org
3898:
3889:
3883:
3876:
3870:
3869:
3867:
3865:
3859:
3848:
3842:
3841:
3805:
3799:
3793:
3784:
3778:
3772:
3771:
3735:
3729:
3726:
3720:
3714:
3701:
3698:
3692:
3689:
3683:
3680:
3674:
3671:
3665:
3658:
3652:
3646:
3533:detection limits
3481:
3479:
3478:
3473:
3454:
3452:
3451:
3446:
3431:
3429:
3428:
3423:
3411:
3409:
3408:
3403:
3383:
3381:
3380:
3375:
3360:
3358:
3357:
3352:
3341:associated with
3340:
3338:
3337:
3332:
3311:
3309:
3308:
3303:
3288:
3286:
3285:
3280:
3268:
3266:
3265:
3260:
3248:
3246:
3245:
3240:
3224:
3222:
3221:
3216:
3196:
3194:
3193:
3188:
3173:
3171:
3170:
3165:
3151:(the measurand),
3150:
3148:
3147:
3142:
3108:
3106:
3105:
3100:
3075:
3073:
3072:
3067:
3065:
3064:
3042:
3040:
3039:
3034:
3026:
3025:
3013:
3012:
3002:
2997:
2976:
2975:
2963:
2962:
2952:
2947:
2926:
2925:
2906:
2904:
2903:
2898:
2893:
2892:
2874:
2873:
2845:
2843:
2842:
2837:
2832:
2831:
2816:
2811:
2810:
2801:
2789:
2787:
2786:
2781:
2769:
2767:
2766:
2761:
2749:
2747:
2746:
2741:
2720:
2718:
2717:
2712:
2701:associated with
2700:
2698:
2697:
2692:
2671:
2669:
2668:
2663:
2658:
2657:
2642:
2637:
2636:
2627:
2615:
2613:
2612:
2607:
2602:
2601:
2583:
2582:
2554:
2552:
2551:
2546:
2531:
2529:
2528:
2523:
2518:
2517:
2502:
2501:
2485:
2483:
2482:
2477:
2472:
2471:
2449:
2447:
2446:
2441:
2439:
2438:
2422:
2420:
2419:
2414:
2412:
2411:
2393:
2392:
2373:
2371:
2370:
2365:
2360:
2359:
2350:
2349:
2331:
2330:
2321:
2320:
2291:
2289:
2288:
2283:
2281:
2280:
2268:
2267:
2251:
2249:
2248:
2243:
2241:
2240:
2228:
2227:
2211:
2209:
2208:
2203:
2201:
2200:
2185:with respect to
2184:
2182:
2181:
2176:
2160:
2158:
2157:
2152:
2150:
2149:
2133:
2131:
2130:
2125:
2120:
2119:
2101:
2100:
2072:
2070:
2069:
2064:
2062:
2061:
2043:
2042:
2026:
2024:
2023:
2018:
2016:
2015:
1997:
1996:
1980:
1978:
1977:
1972:
1960:
1958:
1957:
1952:
1940:
1938:
1937:
1932:
1930:
1929:
1911:
1910:
1880:
1878:
1877:
1872:
1860:
1858:
1857:
1852:
1836:
1834:
1833:
1828:
1806:
1804:
1803:
1798:
1775:
1773:
1772:
1767:
1751:
1749:
1748:
1743:
1727:
1725:
1724:
1719:
1717:
1716:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1652:
1650:
1649:
1644:
1631:
1629:
1628:
1623:
1608:
1606:
1605:
1600:
1588:
1586:
1585:
1580:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1505:
1503:
1502:
1497:
1485:
1483:
1482:
1477:
1475:
1474:
1456:
1455:
1435:
1433:
1432:
1427:
1425:
1424:
1408:
1406:
1405:
1400:
1398:
1397:
1374:
1372:
1371:
1366:
1350:
1348:
1347:
1342:
1340:
1339:
1323:
1321:
1320:
1315:
1313:
1312:
1296:
1294:
1293:
1288:
1286:
1285:
1273:
1272:
1243:
1241:
1240:
1235:
1223:
1221:
1220:
1215:
1213:
1212:
1196:
1194:
1193:
1188:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1146:
1145:
1126:
1124:
1123:
1118:
1116:
1115:
1099:
1097:
1096:
1091:
1089:
1088:
1068:
1066:
1065:
1060:
1055:
1054:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
998:
997:
979:
978:
958:
956:
955:
950:
948:
947:
929:
928:
912:
910:
909:
904:
902:
901:
883:
882:
866:
864:
863:
858:
856:
855:
837:
836:
820:
818:
817:
812:
810:
809:
791:
790:
767:
766:
764:
763:
758:
756:
755:
737:
736:
719:
717:
716:
711:
709:
708:
690:
689:
672:random variables
665:
663:
662:
657:
655:
654:
636:
635:
619:
617:
616:
611:
609:
608:
590:
589:
559:
557:
556:
551:
539:
537:
536:
531:
529:
528:
510:
509:
493:
491:
490:
485:
470:
468:
467:
462:
451:
450:
432:
431:
416:
414:
413:
408:
384:
382:
381:
376:
361:
359:
358:
353:
345:
344:
326:
325:
294:
292:
291:
286:
284:
283:
265:
264:
248:
246:
245:
240:
21:
4077:
4076:
4072:
4071:
4070:
4068:
4067:
4066:
4052:
4051:
4010:
4000:
3980:
3973:
3942:
3907:
3905:Further reading
3902:
3901:
3890:
3886:
3877:
3873:
3863:
3861:
3857:
3849:
3845:
3806:
3802:
3794:
3787:
3779:
3775:
3736:
3732:
3727:
3723:
3715:
3704:
3699:
3695:
3690:
3686:
3681:
3677:
3672:
3668:
3659:
3655:
3647:
3638:
3633:
3628:
3569:
3516:
3502:
3493:
3467:
3464:
3463:
3440:
3437:
3436:
3417:
3414:
3413:
3397:
3394:
3393:
3369:
3366:
3365:
3346:
3343:
3342:
3317:
3314:
3313:
3297:
3294:
3293:
3274:
3271:
3270:
3254:
3251:
3250:
3234:
3231:
3230:
3210:
3207:
3206:
3182:
3179:
3178:
3159:
3156:
3155:
3136:
3133:
3132:
3125:
3115:
3085:
3082:
3081:
3060:
3056:
3054:
3051:
3050:
3021:
3017:
3008:
3004:
2998:
2993:
2971:
2967:
2958:
2954:
2948:
2943:
2921:
2917:
2915:
2912:
2911:
2888:
2884:
2869:
2865:
2851:
2848:
2847:
2827:
2823:
2812:
2806:
2802:
2797:
2795:
2792:
2791:
2775:
2772:
2771:
2755:
2752:
2751:
2726:
2723:
2722:
2706:
2703:
2702:
2677:
2674:
2673:
2653:
2649:
2638:
2632:
2628:
2623:
2621:
2618:
2617:
2597:
2593:
2578:
2574:
2560:
2557:
2556:
2537:
2534:
2533:
2513:
2509:
2497:
2493:
2491:
2488:
2487:
2467:
2463:
2455:
2452:
2451:
2434:
2430:
2428:
2425:
2424:
2407:
2403:
2388:
2384:
2382:
2379:
2378:
2355:
2351:
2345:
2341:
2326:
2322:
2316:
2312:
2304:
2301:
2300:
2276:
2272:
2263:
2259:
2257:
2254:
2253:
2236:
2232:
2223:
2219:
2217:
2214:
2213:
2196:
2192:
2190:
2187:
2186:
2170:
2167:
2166:
2145:
2141:
2139:
2136:
2135:
2115:
2111:
2096:
2092:
2078:
2075:
2074:
2057:
2053:
2038:
2034:
2032:
2029:
2028:
2011:
2007:
1992:
1988:
1986:
1983:
1982:
1966:
1963:
1962:
1946:
1943:
1942:
1925:
1921:
1906:
1902:
1900:
1897:
1896:
1893:
1887:
1866:
1863:
1862:
1846:
1843:
1842:
1816:
1813:
1812:
1792:
1789:
1788:
1761:
1758:
1757:
1737:
1734:
1733:
1712:
1708:
1706:
1703:
1702:
1699:
1678:
1675:
1674:
1658:
1655:
1654:
1638:
1635:
1634:
1617:
1614:
1613:
1594:
1591:
1590:
1574:
1571:
1570:
1551:
1548:
1547:
1531:
1528:
1527:
1511:
1508:
1507:
1491:
1488:
1487:
1470:
1466:
1451:
1447:
1445:
1442:
1441:
1420:
1416:
1414:
1411:
1410:
1393:
1389:
1387:
1384:
1383:
1360:
1357:
1356:
1335:
1331:
1329:
1326:
1325:
1308:
1304:
1302:
1299:
1298:
1281:
1277:
1268:
1264:
1256:
1253:
1252:
1229:
1226:
1225:
1208:
1204:
1202:
1199:
1198:
1182:
1179:
1178:
1162:
1159:
1158:
1141:
1137:
1135:
1132:
1131:
1111:
1107:
1105:
1102:
1101:
1084:
1080:
1078:
1075:
1074:
1050:
1046:
1038:
1035:
1034:
1014:
1011:
1010:
993:
989:
974:
970:
968:
965:
964:
943:
939:
924:
920:
918:
915:
914:
897:
893:
878:
874:
872:
869:
868:
851:
847:
832:
828:
826:
823:
822:
805:
801:
786:
782:
780:
777:
776:
751:
747:
732:
728:
726:
723:
722:
721:
704:
700:
685:
681:
679:
676:
675:
650:
646:
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4008:External links
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3599:Repeatability
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2212:evaluated at
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1783:
1782:-distribution
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1072:
1051:
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1016:
994:
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936:
933:
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921:
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829:
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792:
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771:
752:
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741:
738:
733:
729:
705:
701:
697:
694:
691:
686:
682:
673:
669:
651:
647:
643:
640:
637:
632:
628:
605:
601:
597:
594:
591:
586:
582:
571:
561:
547:
525:
521:
517:
514:
511:
506:
502:
481:
458:
455:
447:
443:
439:
436:
433:
428:
424:
404:
401:
395:
388:
387:
386:
372:
349:
341:
337:
333:
330:
327:
322:
318:
311:
308:
305:
298:
297:
296:
280:
276:
272:
269:
266:
261:
257:
236:
227:
223:
221:
220:specific heat
217:
213:
208:
206:
201:
199:
195:
191:
187:
181:
179:
175:
171:
160:
158:
152:
150:
146:
142:
138:
133:
129:
126:
122:
117:
113:
111:
107:
103:
99:
95:
91:
81:
79:
75:
71:
66:
61:
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54:
52:
48:
44:
40:
33:
19:
3989:
3976:
3893:
3887:
3879:
3874:
3862:. Retrieved
3853:
3846:
3813:
3809:
3803:
3776:
3743:
3739:
3733:
3724:
3696:
3687:
3678:
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3656:
3548:
3544:
3537:
3517:
3494:
3485:
3387:
3204:
3126:
3048:
3045:
2376:
1894:
1841:with limits
1786:
1779:
1731:
1700:
1611:
1568:
1439:
1250:
1245:
1157:and also to
1129:
1030:
961:expectations
774:
573:
473:
364:
228:
224:
209:
202:
198:displacement
182:
166:
153:
134:
130:
118:
114:
87:
62:
55:
42:
36:
4062:Measurement
3927:Grabe, M.
3864:13 February
3816:(1): 1–11.
3614:Uncertainty
3609:Test method
3078:covariances
2161:equals the
540:, and that
190:temperature
174:calibration
3740:Metrologia
3631:References
3504:See also:
3117:See also:
568:See also:
155:cost. The
84:Background
3838:250751314
3768:123445853
3660:Bell, S.
3529:censoring
2984:⋯
2879:…
2588:…
2450:equal to
2398:…
2336:⋯
2106:…
2048:…
2002:…
1916:…
1461:…
984:…
934:…
888:…
842:…
796:…
742:…
695:…
641:…
595:…
515:…
437:…
331:…
270:…
94:measurand
39:metrology
4056:Category
3567:See also
3521:interval
3174:depends,
1809:interval
194:humidity
186:buoyancy
123:, their
90:quantity
3818:Bibcode
3748:Bibcode
3538:A more
1353:uniform
125:average
4014:NPLUnc
3996:
3956:
3836:
3766:
3540:robust
3525:binned
3512:, and
2294:linear
494:given
365:where
102:volume
98:scales
74:median
3981:(PDF)
3966:UKAS
3943:(PDF)
3858:(PDF)
3834:S2CID
3764:S2CID
3455:, and
3361:, and
2377:with
770:joint
178:model
121:scale
76:, or
51:scale
3994:ISBN
3954:ISBN
3866:2013
3555:and
3121:and
1861:and
1409:and
1324:and
218:and
196:and
170:mass
135:The
78:mode
70:mean
4037:ISO
3826:doi
3756:doi
3269:of
2532:in
1961:of
963:of
53:.
37:In
4058::
3992:.
3832:.
3824:.
3812:.
3788:^
3762:.
3754:.
3744:44
3742:.
3705:^
3639:^
3563:.
3531:,
3508:,
3458:a
3109:.
2907::
2252:,
1756:.
1693:.
1248:.
1127:.
459:0.
192:,
151:.
72:,
41:,
4002:.
3960:.
3945:.
3924:.
3868:.
3840:.
3828::
3820::
3814:4
3770:.
3758::
3750::
3549:a
3545:b
3470:Y
3443:Y
3420:t
3400:Y
3372:Y
3349:y
3329:)
3326:y
3323:(
3320:u
3300:Y
3289:,
3277:Y
3257:y
3237:Y
3213:Y
3185:Y
3162:Y
3139:Y
3097:)
3094:y
3091:(
3088:u
3062:i
3058:X
3031:,
3028:)
3023:N
3019:x
3015:(
3010:2
3006:u
3000:2
2995:N
2991:c
2987:+
2981:+
2978:)
2973:1
2969:x
2965:(
2960:2
2956:u
2950:2
2945:1
2941:c
2937:=
2934:)
2931:y
2928:(
2923:2
2919:u
2895:)
2890:N
2886:X
2882:,
2876:,
2871:1
2867:X
2863:(
2860:f
2857:=
2854:Y
2834:)
2829:i
2825:x
2821:(
2818:u
2814:|
2808:i
2804:c
2799:|
2778:Y
2758:y
2738:)
2735:y
2732:(
2729:u
2709:y
2689:)
2686:y
2683:(
2680:u
2660:)
2655:i
2651:x
2647:(
2644:u
2640:|
2634:i
2630:c
2625:|
2604:)
2599:N
2595:X
2591:,
2585:,
2580:1
2576:X
2572:(
2569:f
2566:=
2563:Y
2543:.
2540:y
2520:)
2515:i
2511:x
2507:(
2504:u
2499:i
2495:c
2474:)
2469:i
2465:x
2461:(
2458:u
2436:i
2432:x
2409:N
2405:X
2401:,
2395:,
2390:1
2386:X
2362:,
2357:N
2353:X
2347:N
2343:c
2339:+
2333:+
2328:1
2324:X
2318:1
2314:c
2310:=
2307:Y
2278:2
2274:x
2270:=
2265:2
2261:X
2238:1
2234:x
2230:=
2225:1
2221:X
2198:i
2194:X
2173:f
2147:i
2143:c
2122:)
2117:N
2113:X
2109:,
2103:,
2098:1
2094:X
2090:(
2087:f
2084:=
2081:Y
2059:N
2055:X
2051:,
2045:,
2040:1
2036:X
2013:N
2009:x
2005:,
1999:,
1994:1
1990:x
1969:Y
1949:y
1927:N
1923:c
1919:,
1913:,
1908:1
1904:c
1869:b
1849:a
1825:b
1822:,
1819:a
1811:[
1795:X
1780:t
1764:X
1740:X
1714:i
1710:X
1681:Y
1661:Y
1641:Y
1620:Y
1597:Y
1577:Y
1554:Y
1534:Y
1514:Y
1494:Y
1472:N
1468:X
1464:,
1458:,
1453:1
1449:X
1422:2
1418:X
1395:1
1391:X
1363:Y
1337:2
1333:X
1310:1
1306:X
1283:2
1279:X
1275:+
1270:1
1266:X
1262:=
1259:Y
1232:Y
1210:i
1206:X
1185:Y
1165:Y
1143:i
1139:X
1113:i
1109:x
1086:i
1082:X
1057:)
1052:i
1048:x
1044:(
1041:u
1017:i
995:N
991:X
987:,
981:,
976:1
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945:N
941:x
937:,
931:,
926:1
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899:N
895:X
891:,
885:,
880:1
876:X
853:N
849:X
845:,
839:,
834:1
830:X
807:N
803:x
799:,
793:,
788:1
784:x
753:N
749:X
745:,
739:,
734:1
730:X
706:N
702:X
698:,
692:,
687:1
683:X
652:N
648:X
644:,
638:,
633:1
629:X
606:N
602:X
598:,
592:,
587:1
583:X
548:Y
526:N
522:X
518:,
512:,
507:1
503:X
482:Y
456:=
453:)
448:N
444:X
440:,
434:,
429:1
425:X
405:,
402:Y
399:(
396:h
373:f
350:,
347:)
342:N
338:X
334:,
328:,
323:1
319:X
315:(
312:f
309:=
306:Y
281:N
277:X
273:,
267:,
262:1
258:X
237:Y
34:.
20:)
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