Uncertainty combined standard

Pauwels (1999) argues that the certified values of CRMs should be presented in the form of an expanded combined uncertainty according to the ISO Guide on the expression of uncertainty in measurement, so that coverage factor should always be clearly mentioned in order to allow an easy recalculation of the combined standard uncertainty. This is needed for uncertainty propagation when the CRM is used for calibration and the ISO Guide should be revised accordingly. The use of the expanded uncertainty has been pohcy in certification by NIST since 1993 (Taylor and Kuyatt 1994). 

Figure 1.1 Results from an international intercomparison, IMEP 9, wl expanded uncertainty and ue the combined standard uncertainty [1] (see Chapter 6, Section 6.3). Reproduced by permission of EC-JRC-IRMM (Philip Taylor) from IMEP 9, Trace elements in water III, Cd, certified range 81.0-85.4nmoll-1 .

If the model contains a mixture of operations, e.g. y = (a — b)/(c + d), it should be broken down into expressions which consist solely of operations covered by the cases shown in equations (6.12)-(6.15). For example, the expression above should be broken down into the elements (a — b) and (c + d) and the uncertainty for each element calculated by using equation (6.12). The results from these calculations can then be combined by using equation (6.13) to give the combined standard uncertainty. 

Table 6.5 Expressions for combining standard uncertainties (c.f. equation (6.11))...

One consequence of the equations for combining standard uncertainties is that the combined standard uncertainty will be dominated by the largest uncertainty components. This is illustrated in Figure 6.14. 

The combined standard uncertainty associated with T is obtained from ... 

Which of the following equations is the correct one to use to combine standard uncertainties when the measurement model involves only multiplication and/or division ... 

Consider the previous example of calculating the concentration of a standard solution. The combined standard uncertainty of 2.69 mg l-1 would be multiplied by a coverage factor of 2 to give an expanded uncertainty of 5.38 mg l-1. We can now report the result as follows concentration of solution = (1004 5) mg 1 1, where the reported uncertainty is an expanded uncertainty calculated using a coverage factor of 2, which gives a level of confidence of approximately 95%. Note that the coverage factor is applied only to the final combined uncertainty. 

Combined Standard Uncertainty is the standard uncertainty that is obtained by combining (root of the sum of squares) individual standard measurement uncertainties associated with the input quantities in a measurement model. 

Expanded Uncertainty is the combined standard uncertainty multiphed with the coverage factor k. Often k is chosen to be 2 or sometimes 3. With k=2 about 95%, and with k=3 about 99% of all likely values are encompassed. 

In most cases a confidence level of 68% is not enough to take reliable decisions. To increase this confidence the combined standard uncertainty is multiplied with a factor to get an expanded uncertainty. If we choose a factor of 2 we get a level of confidence of approx. 95%. This is the most widely used expanded uncertainty in analytical chemistry. 

U is obtained by muitipiying Uc(y), the combined standard uncertainty, with a coverage factork. 

The general relationship between the combined standard uncertainty Uc(y) of a value y and the uncertainty arising from the independent parameters x. X2.. ..Xn on which it depends is according to the law of propagation of errors... 

To get an expanded uncertainty with a higher level of confidence we multiply the combined standard uncertainty with a coverage factor k. If we choose k=2, we get a confidence level of approximately 95%. 

Once the uncertainty components have been identified and quantified as standard uncertainties, the remainder of the procedure to estimate uncertainty is a somewhat complicated but mostly straightforward. Most software products on the market will perform this task. Otherwise, some spreadsheet manipulation or mathematics must be done to reach the uncertainty. The combined standard uncertainty of a result is obtained by mathematical manipulation of the standard uncertainties as part of the uncertainty budget. These standard uncertainties may also be combinations of other uncertainties, and so on, as the branches and sub-branches of the cause-and-effect diagram are worked through. A combined standard uncertainty of a quantity is written uc(y). 

The result [uc(Ptest) = 0.0224] may be compared with that calculated from equation 6.31, which is calculated in spreadsheet 6.3, cell C9, from the relative combined standard uncertainty in cell D9. 

There is a problem when some components of the combined standard uncertainty are assessed from measurements or estimates with finite degrees of freedom. A type A estimate from a standard deviation of n repeated measurements has n - 1 degrees of freedom. Usually Type estimates will be based on data that have essentially infinite degrees of freedom, but if the standard uncertainty is open to doubt, the effective degrees of freedom can be determined from... 

When assessing measurement uncertainty as part of a method validation, enough experiments are done to have degrees of freedom that do not adversely affect the coverage factor, and usually is taken as 2. As long as subsequent field measurements followed the validated method, a measurement uncertainty can be then quoted with = 2. For the most part, therefore, the expanded uncertainty should be calculated from the combined standard uncertainty by... 

The final combined standard uncertainty, whether obtained using algebra or a spreadsheet or other software, is the answer, and can be quoted as such. I recommend using the wording Result x units [with a] standard uncertainty [of] uc units [where standard uncertainty is as defined in the International Vocabulary of Basic and General Terms in Metrology, 3rd edition, 2007, ISO, Geneva, and corresponds to one standard deviation]. ... 

For a measurement result to be metrologically traceable, the measurement uncertainty at each level of the calibration hierarchy must be known. Therefore, a calibration standard must have a known uncertainty concerning the quantity value. For a CRM this is included in the certificate. The uncertainty is usually in the form of a confidence interval (expanded uncertainty see chapter 6), which is a range about the certified value that contains the value of the measurand witha particular degree of confidence (usually 95%). There should be sufficient information to convert this confidence interval to a standard uncertainty. Usually the coverage factor ( see chapter 6) is 2, corresponding to infinite degrees of freedom in the calculation of measurement uncertainty, and so the confidence interval can be divided by 2 to obtain uc, the combined standard uncertainty. Suppose this CRM is used to calibrate... 

Measurement anomalies due to nonlinearities of the gas analysers have been determined and assigned a value of 0.1% (at 68% confidence level). These values are added to the combined standard uncertainties after correcting for probability distribution as square root sum of squares. 

Overall uncertainty can be estimated by identifying all factors which contribute to the uncertainty. Their contributions are estimated as standard deviations, either from repeated observations (for random components), or from other sources of information (for systematic components). The combined standard uncertainty is calculated by combining the variances of the uncertainty components, and is expressed as a standard deviation. The combined standard uncertainty is multiplied by a coverage factor of 2 to give a 95% level of confidence (approximately). 

Precision %RSD< 15 % at>3 levels n>5 at each level %RSD<20 % at LLOQ %RSD<20% n=4-7 CV<2.2 % Uncertainty evaluation Coefficient of variation (CV) 1 % Uncertainty of volumetric error 0.3 % Uncertainty of reference standard 0.1 % Uncertainty of weighing 0.5 % Uncertainty of other systematic errors Combined standard uncertainty Coverage factor Expanded uncertainty Relative expanded uncertainty (%) Intraassay CV=2-9 % Interassay CV=4-12%... 

The analyst normally requires the total uncertainty of the result. This is termed the combined standard uncertainty and is made up of all the individual standard uncertainty components. 

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