In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a quantity measured on an interval or ratio scale.
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. 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 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, median, or mode). 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 of interest – a measurand. 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.
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.Bell, S. Measurement Good Practice Guide No. 11. A Beginner's Guide to Uncertainty of Measurement. Tech. rep., National Physical Laboratory, 1999. 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. If measured on a ratio or interval scale, 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.
The Guide to the Expression of Uncertainty in Measurement "Guide to the Expression of Uncertainty in Measurement" (commonly known as the GUM) is the . 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 (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,ASME B89.7.3.1, Guidelines for Decision Rules in Determining Conformance to Specifications to provide a simplified approach (relative to the GUM) to the evaluation of dimensional measurement uncertainty,ASME B89.7.3.2, Guidelines for the Evaluation of Dimensional Measurement Uncertainty to resolve disagreements over the magnitude of the measurement uncertainty statement,ASME B89.7.3.3, Guidelines for Assessing the Reliability of Dimensional Measurement Uncertainty Statements and to provide guidance on the risks involved in any product acceptance/rejection decision.ASME B89.7.4, Measurement Uncertainty and Conformance Testing: Risk Analysis
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 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 representing physical 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 , about which information is required, is often related to input quantities, denoted by , about which information is available, by a measurement model in the form of
where is known as the measurement function. A general expression for a measurement model is
It is taken that a procedure exists for calculating given , and that is uniquely defined by this equation.
Propagation of distributions
The true values of the input quantities are unknown. In the GUM approach, are characterized by probability distributions and treated mathematically as random variables. These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available knowledge concerning . Sometimes, some or all of are interrelated and the relevant distributions, which are known as joint, apply to these quantities taken together.
Consider estimates , respectively, of the input quantities , obtained from certificates and reports, manufacturers' specifications, the analysis of measurement data, and so on. The probability distributions characterizing are chosen such that the estimates , 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 . Moreover, for the th input quantity, consider a so-called standard uncertainty, given the symbol , defined as the standard deviation of the input quantity . This standard uncertainty is said to be associated with the (corresponding) estimate .
The use of available knowledge to establish a probability distribution to characterize each quantity of interest applies to the and also to . In the latter case, the characterizing probability distribution for is determined by the measurement model together with the probability distributions for the . The determination of the probability distribution for from this information is known as the propagation of distributions.
The figure below depicts a measurement model in the case where and are each characterized by a (different) rectangular, or uniform, probability distribution. has a symmetric trapezoidal probability distribution in this case.
Once the input quantities have been characterized by appropriate probability distributions, and the measurement model has been developed, the probability distribution for the measurand is fully specified in terms of this information. In particular, the expectation of is used as the estimate of , and the standard deviation of as the standard uncertainty associated with this estimate.
Often an interval containing with a specified probability is required. Such an interval, a coverage interval, can be deduced from the probability distribution for . 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 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 that can give a smaller standard deviation for and hence a smaller standard uncertainty associated with the estimate of .Bernardo, J., and Smith, A. "Bayesian Theory". John Wiley & Sons, New York, USA, 2000. 3.20EURACHEM/CITAC. "Quantifying uncertainty in analytical measurement". Tech. Rep. Guide CG4, EU-RACHEM/CITEC, EURACHEM/CITAC Guide], 2000. Second edition.
Type A and Type B evaluation of uncertainty