Maximum uncertainty

We see that the uncertainty in a derived quantity is fixed by the uncertainties in the measurements that must be combined. For an addition or a subtraction, the maximum uncertainty is simply the sum of the uncertainties in the components 0.2 + 0.2 = 0.4. 

Suppose the weight of the can is 61 1 grams and the weight of the can plus water is 406 1 grams. Calculate the weight of the water and the maximum uncertainty in the weight caused by the uncertainties in each of the two weighings. 

We need convenient rules for estimating the maximum uncertainty in derived quantities. This is rather easy for a sum or a difference. In either case, merely add up the uncertainties in the components. Fortunately there is an equally sim-... 

Assuming an analytical error of 2 % which is based on the analysis of the results of a number of proficiency tests and collaborative trials (Daas and Miller 1998), a maximum uncertainty of 0.24% would imply a 0.1% probability of rejecting a good result. Given that the uncertainty of a content to be assigned is below a predetermined value, the results of the collaborative trial are acceptable otherwise it is recommended to repeat the trial in whole or in part. 

In a typical experiment, benzene vapor was admitted into the evacuated balance chamber by means of stopcock D. After achieving the desired pressure, stopcock D was closed. The maximum uncertainty in p/p during the course of any one experiment was 0.008 units. Equilibrium was judged to be established when there was less than a 1 percent change in weight over a 12 hour period. Total sorption values were reproducible to within 1%. 

It is also noted that there is overlap in the individual UFs and that the application of five UFs of ten for the chronic reference value (yielding a total UF of 100,000) is inappropriate. In fact, in cases where maximum uncertainty exists in all five areas, it is unlikely that the database is sufficient to derive a reference value. Uncertainty in four areas may also indicate that the database is insufficient to derive a reference value. In the case of the RfC, the maximum UF would be 3,000, whereas the maximum would be 10,000 for the RfD. This is because the derivation of RfCs and RfDs has evolved somewhat differently. The RfC methodology (US-EPA 1994) recommends dividing the interspecies UF in half, one-half (10° ) each for toxicokinetic and toxicodynamic considerations, and it includes a Dosimetric Adjustment Factor (D AF, represents a multiplicative factor used to adjust an observed exposure concentration in a particular laboratory species to an exposure concentration for humans that would be associated with the same delivered dose) to account for toxicokinetic differences in calculating the Human Equivalent Concentration (HEC), thus reducing the interspecies UF to 3 for toxicodynamic issues. RfDs, however, do not incorporate a DAF for deriving a Human Equivalent Dose (HED), and the interspecies UF of 10 is typically applied, see also Section 5.3.4. It is recommended to limit the total UF applied for any particular chemical to no more than 3000, for both RfDs and RfCs, and avoiding the derivation of a reference value that involves application of the full 10-fold UF in four or more areas of extrapolation. 

The decision that does not take into account the uncertainty, does not create any guard bands , i.e. seller and purchaser are sharing the risk. But in order to know what risk they are sharing, they should know about a maximum uncertainty. 

The maximum-entropy (maxEnt) approach involves the use of a measure of the uncertainty in a distribution (Shannon-Weaver entropy). The idea is to choose the distribution type that has maximum uncertainty subject to specification of some features of the distribution such as the range or a few moments or percentiles. Warren-Hicks and Moore (1998) list maxEnt solutions for a number of situations. In particular when only a min and max is available the maxEnt solution is the uniform distribution. The solution when the information available is the mean and variance, and the min and max are infinite, is the normal distribution. 

If Dx is the maximum uncertainty in position of the particle during the measurement of p, then the minimum uncertainty in the value of p will be... 

Fig. 11. Plot of rate constant vs. exothermicity for the reaction P + Q->P+ +Q and for P++Q - P + Q, where P = porphyrin and Q = quinone. Band T denote data obtained in butyronitrile or in toluene, respectively. The maximum uncertainty in any given rate constant is 20%...

Since the estimated half-life has a maximum uncertainty factor of 3, the hydrolysis half-life at 20 °C and pH 6.8 may be as high as 900 days. The hydrolysis rate is expected to increase with increasing temperature and pH (Navy 1984b). Hydrolysis products include picrate ion, N-methylpicramide, methylnitramine, nitrite ion, and nitrate ion. In the dark and under buffered, alkaline conditions (pH 9), methylnitramine formation dominated (66%) picrate ion (28%), nitrite (4.1%), nitrate (3.1%), and N-methylpicramide (4.1%) were also formed. Under laboratory light and more acidic conditions (pH 4-6), N-methylpicramide (41%) and nitrate (35.2%) were the major products picrate ion (3.9%), nitrite (9.4%), and methylnitramine (0.01%) were also formed (Navy 1984b). In sea water at 25 °C and pH 8.1, 88% of initial tetryl hydrolyzed in 101 days, yielding picric acid as a hydrolysis product. This hydrolysis rate corresponds to a first-order half-life of 33 days (Hoffsommer and Rosen 1973 HSDB 1994). 

At this time it is not possible to make an accurate estimate of the absolute uncertainty in the situ values of the apparent solubility constants for calcite and aragonite. An approximate maximum uncertainty can be determined for at 5000 m water depth and 2°C by using two different data sets. The maximum value for is obtained by using value and temperature coefficient calculated by Berner (26) and the effect of pressure of Ingle (32). The value for obtained by this method is 19.8 X 10" mole kg . The minimum value is obtained by using the v ue and temperature coefficient of Ingle. (25),... 

The worst-case method is useful for estimating the maximum uncertainty expected when the results of several measurements are combined to obtain a result. We assume the maximum uncertainty in each measurement and then calculate the minimum and maximum possible results. These extreme values describe the range and thus the maximum error limit associated with a particular determination. 

This amounts to a range of maximum uncertainty of approximately 7 kcal mol". The products of... 

The maximum uncertainty factor used with the minimum confidence database is generally 10000. 

The surface area of the inner Baltic Sea, bordered at the bottlenecks of the Danish Straits, amounts toA = 382 486 km if computed using RANGS shorelines on the WGS-84 ellipsoid (Hagen and Feistel, 2001), Equation 20.10. The absolute maximum uncertainty of the sea surface is about 10%, while the likely uncertainty lies in the range between 0.1% and 1 %. The length of the border line enclosing this surface area computed at the RANGS resolution of about 100 m is about L = 70 000 km. 

Alveoli are the tiny sacs of air in the lungs (see Problem 5.122) whose average diameter is 5.0 X 10 m. Consider an oxygen molecule (5.3 X 10 kg) trapped within a sac. Calculate the uncertainty in the velocity of the oxygen molecule. (Hint The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)... 

For any individual measurement, a maximum uncertainty in the weight would be 0.5 percent based upon experience. 

The chemical recovery of Cul for this system is determined gravimetrically. Each of the weight terms has an associated uncertainty which are listed below. These uncertainties are based upon the precision allowed (e.g. in a carrier calibration), the maximum uncertainty in measuring the stable iodide concentration and an estimate of the effect of variation of the Cul precipitation solution pH causing the co-precipitation of CuCl. 

Maximum uncertainty 1 0.5 ppm typical uncertainty t 0.2 ppn. hreference 15 reference 16 < reference 17... 

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