Mean, uncertainty

A detailed discussion and a table can be found in Ref. 26. First of all, we note that the mean uncertainty for the experimental values in the G2-1 set is itself 0.6 kcal/mol. MAD values for W1 and W2 theory stand at 0.6 and 0.5 kcal/mol, respectively, suggesting that these theoretical methods have a reliability comparable to the experimental data themselves. 

Standard deviation from the mean (uncertainty in individual analysis <2%). 

Uncertainty on minimum requirements induces uncertainty on the specification of function events and on top events. Uncertainty on top events means uncertainty on the structure of the respective fault tree model. That means, the uncertainty on the results from... 

Table 7-5 shows results for two of seven compounds sent to many labs to compare their performance in combustion analysis. For each compound, the first row gives the theoretical wt% for each element and the second row shows the measured wt%. Accuracy is excellent Mean wt% C, H, N, and S are usually within 0.1 wt% of theoretical values. The 95% confidence intervals for uncertainty for C for the first compound is 0.63 wt% and the uncertainty for the second compound is 0.33 wt%. The mean uncertainty for C listed in the bottom row of the table for all seven compounds in the study was 0.47 wt%. Mean 95% confidence intervals for H, N, and S are 0.24, 0.31, and 0.76 wt%, respectively. Chemists consider a result within 0.3 wt% of theoretical to be good evidence that the compound has the expected formula. This criterion can be difficult to meet for C and S with a single analysis because the 95% confidence intervals are larger than 0.3. 

Diffraction methods have also been used in aqueous salt solutions for establishing the distances di.o between the centers of ions and those of the oxygen atoms of adjacent water molecules. They have a mean uncertainty of 0.002 nm, and if the radius of a water molecule, rw = 0.138 nm, is deducted from the di.o values, the results correspond quite well with the rj set for the bare ions, as noted by Marcus [3]. 

Readers may notice the absence of certain terms in common use. The exclusion of some such terms is a deliberate choice. For example, instead of photopeak we prefer full-energy peak we have avoided the statisticians use of error to mean uncertainty and reserve that word to indicate bias or error in the sense of mistake . Branching ratio we avoid altogether. This is often used ambiguously and without definition. In other texts, it may mean the relative proportions of different decay modes, or the proportions of different beta-particle transitions, or the ratio of de-excitation routes from a nuclear-energy level. Furthermore, it sometimes appears as a synonym for gamma-ray emission probabihty , where it is not always clear whether or not internal conversion has been taken into account. 

From Table 2.26b the area under the normal curve from — 1.5cr to -I- 1.5cr is 0.866, meaning that 86.6% of the measurements will fall within the range 30.00 0.45 and 13.4% will lie outside this range. Half of these measurements, 6.7%, will be less than 29.55 and a similar percentage will exceed 30.45. In actuality the uncertainty in z is about 1 in 15 therefore, the value of z could lie between 1.4 and 1.6 the corresponding areas under the curve could lie between 84% and 89%. 

The degree of uncertainty of 10 per cent or more, inseparable from estimates of specific surface from adsorption isotherms, even those of nitrogen, may seem disappointing. In fact, however, attainment of this level of accuracy is a notable achievement in a field where, prior to the development of the BET method, even the order of magnitude of the specific surface of highly disperse solids was in doubt. The adsorption method still provides the only means of determining the specific surface of a mass of non-... 

Suppose that you need to add a reagent to a flask by several successive transfers using a class A 10-mL pipet. By calibrating the pipet (see Table 4.8), you know that it delivers a volume of 9.992 mL with a standard deviation of 0.006 mL. Since the pipet is calibrated, we can use the standard deviation as a measure of uncertainty. This uncertainty tells us that when we use the pipet to repetitively deliver 10 mL of solution, the volumes actually delivered are randomly scattered around the mean of 9.992 mL. 

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,... 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/VTj. 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

Evaluation of reactivity ratios from the copolymer composition equation requires only composition data—that is, analytical chemistry-and has been the method most widely used to evaluate rj and t2. As noted in the last section, this method assumes terminal control and seeks the best fit of the data to that model. It offers no means for testing the model and, as we shall see, is subject to enough uncertainty to make even self-consistency difficult to achieve. 

Effect of Uncertainties in Thermal Design Parameters. The parameters that are used ia the basic siting calculations of a heat exchanger iaclude heat-transfer coefficients tube dimensions, eg, tube diameter and wall thickness and physical properties, eg, thermal conductivity, density, viscosity, and specific heat. Nominal or mean values of these parameters are used ia the basic siting calculations. In reaUty, there are uncertainties ia these nominal values. For example, heat-transfer correlations from which one computes convective heat-transfer coefficients have data spreads around the mean values. Because heat-transfer tubes caimot be produced ia precise dimensions, tube wall thickness varies over a range of the mean value. In addition, the thermal conductivity of tube wall material cannot be measured exactiy, a dding to the uncertainty ia the design and performance calculations. 

If a heat exchanger is sized usiag the mean values of the design parameters, then the probabiUty, or the confidence level, of the exchanger to meet its design thermal duty is only 50%. Therefore, in order to increase the confidence level of the design, a proper uncertainty analysis must be performed for all principal design parameters. 

The values of CJs are experimentally determined for all uncertain parameters. The larger the value of O, the larger the data spread, and the greater the level of uncertainty. This effect of data spread must be incorporated into the design of a heat exchanger. For example, consider the convective heat-transfer coefficient, where the probabiUty of the tme value of h falling below the mean value h is of concern. Or consider the effect of tube wall thickness, /, where a value of /greater than the mean value /is of concern. 

The pumped-discharge case is generally more difficult to solve because of the uncertainty in deahng with negative numerical results. As a final answer, a negative value could indicate that the pump has completely emptied the tank however, as an intermediate value, it could mean that it is not a true solution. A simple check is to try a different initial estimate and see if the intermediate negative results disappear. 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

Consider a Nyquist contour for the nominal open-loop system Gm(iLu)C(iuj) with the model uncertainty given by equation (9.119). Let fa( ) be the bound of additive uncertainty and therefore be the radius of a disk superimposed upon the nominal Nyquist contour. This means that G(iuj) lies within a family of plants 7r(C(ja ) e tt) described by the disk, defined mathematically as... 

The fraction 0.1% is chosen to be so low that individuals living near a nuclear plant should have no special concern because of the closeness. Uncertainties in the analysis of risk are not caused by the "quantitative methodology" but are highlighted by it. Uncertainty reduction will be achieved by methodological improvements mean values should be calculated. As a guideline for rcinilatory implementation, the following is recommended ... 

Also, presented are the level-1 uncertainty analysis, results. The MLO mean core damage frequency from internal events is about an order of magnitude lower than that of full power operation. The mean core damage frequency due... 

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