Uncertainty combined

It is easy to see that combining uncertainties in this way overestimates the total uncertainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that both volumes are either greater than 9.992 mL or less than 9.992 mL. At the other extreme, we might assume that the two deliveries will always be on opposite sides of the pipet s mean volume. In this case we subtract the uncertainties for the two deliveries,... 

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). 

From the formulae presented it is clearly evident that the uncertainty of the CRM may become a strong component in the user s combined uncertainty, but it is not the only component. 

On most occasions CRMs are used as Quality Control materials, rather than as calibrations . As outlined above, this common application adds significantly to the user s uncertainty budget, since at a minimum it is necessary to consider at least two independent measurement events (Um). so increasing the combined uncertainty of the results. Again this process rapidly increases the combined uncertainty with increasing complexity of the analytical system and so the usefulness of a control analysis may be downgraded when a correct uncertainty budget is formulated. 

Users may conclude that the analytical procedure gives correct results if the difference between the analyst s experimental mean(s) (Xe) and the certified value(s) (xj is less than the combined uncertainty (is standard deviation) of the experimental and certified means (Equation. 7-1), with Se and Sc representing the estimates of the respective standard deviations. 

U(x) Extended combined uncertainty (limit) of an analytical value x ... 

In the case that the parameters pi are independent from each other, the combined uncertainty is given by... 

From the combined uncertainty the extended uncertainty is calculated. The extended combined uncertainty U(y) represents an interval that con-... 

In many cases the general expression can be reduced to relatively simple expressions for combining uncertainties. The general expression, in equation (6.11), should be used for cases not covered by the equations shown in... 

Equation for calculating result, y Combined uncertainty Equation number... 

Figure 6.14 Illustration of the combination of standard uncertainties u(a) is much greater than 11(b) and so the combined uncertainty is approximately equal to u(a).

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. 

A second use for uncertainty values lies in their potential for helping us to improve our experimental procedures. In calculating the uncertainty for a measurement, we will have assembled a list of standard uncertainties for the variables of the measurement model. If we wish to improve the quality of our measurement, we must look first at the component of the measurement system contributing the largest uncertainty. If this is the dominant contribution to the combined uncertainty, then any attempt to improve other aspects of the measurement process will be a waste of time. By attempting to reduce the size of the dominant uncertainty first, we will produce the greatest return for our effort. 

Uncertainty factors 2 for interspecies 3 for intraspecies combined uncertainty factor of 6... 

Descriptions of the experimental scattering and microscopy conditions have been published elsewhere and are referenced in each section. Throughout this report certain conventions will be used when describing uncertainties in measurements. Plots of small angle scattering data have been calculated from circular averaging of two-dimensional files. The uncertainties are calculated as the estimated standard deviation of the mean. The total combined uncertainty is not specified in each case since comparisons are made with data obtained under... 

We deal with two types of numbers in chemistry—exact and measured. Exact values are just that—exact, by definition. There is no uncertainty associated with them. There are exactly 12 items in a dozen and 144 in a gross. Measured values, like the ones you deal with in the lab, have uncertainty associated with them because of the limitations of our measuring instruments. When those measured values are used in calculations, the answer must reflect that combined uncertainty by the number of significant figures that are reported in the final answer. The more significant figures reported, the greater the certainty in the answer. 

Figure 13.5a compares calculated and experimental VPIE s for H20/D20. Experiment and calculation overlap within the combined uncertainties. The thin line through the small circles in Fig. 13.5a represents VdWl choosing Aa/a = 0. It departs radically from experiment, but the corresponding calculated liquid molar density IE s are insensitive to the choice of Aa/a. Therefore the corresponding states treatment is most useful for analysis of molar density IEs because it avoids the necessity of introducing a fourth isotope sensitive parameter, Aco/co (equivalently Aa/a). 

Values for the yields are affected by combined uncertainties in the basic physics of nuclear reactions and convection effects, during both pre-supernova evolution and the explosion itself. Nuclear contributions from stars depend primarily on the following ... 

The last step is the combination of all sources and the review of the largest contribution. Perhaps it will be possible to get a better estimation of these components that influence the magnitude of the combined uncertainty most. Finally the expanded uncertainty is calculated to adjust it to the requested level of confidence. 

The combined uncertainty we get from the square root of the smn of squares of all rmcerlainty contributions. 

The combined uncertainty is multiplied with the coverage factor... 

The mathematics are even simpler when contributions to the uncertainty of a single quantity are combined. Here the sensitivity coefficient is 1, and the individual uncertainties are just squared and summed. For example, for the combination of the standard uncertainties of the effects on the volume delivered by a pipette discussed above, which are repeatability, calibration uncertainty, and the effect of temperature, the square of the combined uncertainty is simply the sum of the squares of each effect ... 

For more than one contributing uncertainty component, the u(y) calculated one x at a time can be squared and summed to give the square of the combined uncertainty. 

The simple equations used to combine uncertainty rely on the independence of the values of the components. Consider a titration volume calculated from the difference between initial and final readings of a burette and the three uncertainty components identified for volume in table 6.2. Repeatability should be genuinely random, and so the combined uncertainty of a difference measurement with repeatability u(r) is... 

There is still some debate about what is and what is not correlated. The assumption there is no correlation leads at worst to an overestimate of the uncertainty. Also, because the repeatability, which is always uncorrelated (or should be), is often the greatest component of the combined uncertainty, the contributions of correlated effects are small. 

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