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Measurement uncertainty
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Measurement uncertainty
In metrology, the estimation of the dispersion of the values “attributable” to the measurand is defined as the uncertainty of a measure. Measurement uncertainty is the degree of uncertainty with which the value of a physical quantity or property is obtained through its direct or indirect measurement. The result of the measurement is therefore not a single value, but a set of values derived from the measurement (direct or indirect) of the physical size or property itself. It is associated with the measurement value as follows in the example (a measurement of a diameter):
\[D=47\pm 0.1\;\textrm{mm}\]
Uncertainty means the range of possible values within which the true value of the measurement lies. This definition changes the usage of some other commonly used terms. For example, the term accuracy is often used to mean the difference between a measured result and the actual or true value. Since the true value of a measurement is usually not known, the accuracy of a measurement is usually not known either. Because of these definitions, we modified how we report lab results. For example, when students report results of lab measurements, they do not calculate a percent error between their results and the actual value. Instead, they determine whether the accepted value falls within the range of uncertainty of their result.
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Components of uncertainty
In general, measurement uncertainty includes numerous sources of uncertainty, each of which is called a “component of uncertainty”. Some components arise from effects of a systematic nature (e.g., components associated with corrections, or values assigned to measurement samples), and among these is definitional uncertainty. To estimate the overall uncertainty, it may be necessary to examine each component of uncertainty and treat it separately to assess its contribution to the total uncertainty. Most of the time, however, it is possible to evaluate the simultaneous effect of several components, which allows us to simplify the uncertainty calculation. For a measurement result y we may have:
- Combined standard uncertainty, \(u_c(y)\), is the total uncertainty of the measurement result \(y\); it is a mean square deviation estimated as the positive square root of the total variance obtained by combining all components of uncertainty.
- Expanded uncertainty, \(U(y)\), obtained by multiplying the previous \(u_c(y)\), by a coverage factor \(k\): provides a range within which the value of the measurand is found with a higher confidence level; should be used in most cases of measurements in analytical chemistry. For a 95% confidence level, the coverage factor \(k = 2\).
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Uncertainty estimation procedure
The uncertainty estimation procedure, i.e. error estimation, from a practical point of view requires:
- Specification of the measurand, i.e., clear and unambiguous definition of what is being measured;
- Definition of the mathematical model, i.e. definition of the relationship that links the measurand to the quantities determined by the measurement procedure chosen;
- Identification of the sources of uncertainty; there are various techniques, from the compilation of a structured list, to the use of cause-effect diagrams; the effect of several sources can be evaluated cumulatively;
- Quantification of uncertainty components; it is generally sufficient to quantify only the most important sources. Category A uncertainties are estimated as standard deviations from experimental data distributions; category B uncertainties must be derived from existing data and must be expressed and treated as standard deviations;
- Combining uncertainty components: all uncertainty components must be converted into standard uncertainties;
- Compounding the standard uncertainty; the expanded uncertainty will be calculated from the latter by applying the coverage factor.