Uncertainty calculation propagation

A standard solution of Mn + was prepared by dissolving 0.250 g of Mn in 10 ml of concentrated HNO3 (measured with a graduated cylinder). The resulting solution was quantitatively transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10-mL aliquot of the solution was pipeted into a 500-mL volumetric flask and diluted to volume, (a) Express the concentration of Mn in parts per million, and estimate uncertainty by a propagation of uncertainty calculation, (b) Would the uncertainty in the solution s concentration be improved... 

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

Dust composition is bom Jessberger et al. (198S). gas composition is from Delsemme (1988). Bulk abundances were calculated assuming that the dust/gas ratio of typical comets is within 0.5-1.3 (Delscmmc 1988). Uncertainties were propagated accordingly. 

It is important to note that Table 111-2 and Table lV-2 include only those species for which the primary selected data are reaction data. The formation data derived from them and listed in Table Ill-l are obtained using auxiliary data, and their uncertainties are propagated aecordingly. In order to maintain the uncertainties originally assigned to the selected data in this review, the user is advised to make direct use of the reaction data presented in Table I1I-2 and Table lV-2, rather than taking the derived values in Table III-l and Table IV-1 to calculate the reaction data with Eq.(II.55). The... 

Section VII.3.6.1. (Uncertainties calculated by error propagation would be much too large.)... 

In fact, the error term in eqn (2.1), a,-, produces uncertainty in the estimators bg and bj. In order for these estimates to be useful, it is necessary to know how important their uncertainties are. As in any uncertainty (error propagation) calculation, the variance of the data has to be propagated to the estimates. Unfortunately, the LS method does not provide unbiased estimators for the variance of the y-values (cr unless there is no lack of fit between the data and the line. ... 

Propagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are sr, sa, sb) and sq. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 

Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing. As shown in the following example, the propagation of uncertainty is easily calculated by treating each operation separately using equations 4.6 and 4.7 as needed. 

Given the complexity of determining a result s uncertainty when several measurements are involved, it is worth examining some of the reasons why such calculations are useful. A propagation of uncertainty allows us to estimate an ex-... 

A propagation of uncertainty also helps in deciding how to improve the uncertainty in an analysis. In Example 4.7, for instance, we calculated the concentration of an analyte, obtaining a value of 126 ppm with an absolute uncertainty of 2 ppm and a relative uncertainty of 1.6%. How might we improve the analysis so that the absolute uncertainty is only 1 ppm (a relative uncertainty of 0.8%) Looking back on the calculation, we find that the relative uncertainty is determined by the relative uncertainty in the measured signal (corrected for the reagent blank)... 

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. 

I. 000 X 10- 1.000 X 10-k 1.000 X 10-k and 1.000 X 10- M from a 0.1000 M stock solution. Calculate the uncertainty for each solution using a propagation of uncertainty, and compare to the uncertainty if each solution was prepared by a single dilution of the stock solution. Tolerances for different types of volumetric glassware and digital pipets are found in Tables 4.2 and 4.4. Assume that the uncertainty in the molarity of the stock solution is 0.0002. 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

These equations identify the dominant source and loss processes for HO and H02 when NMHC reactions are unimportant. Imprecisions inherent in the laboratory measured rate coefficients used in atmospheric mechanisms (for instance, the rate constants in Equation E6) can, themselves, add considerable uncertainty to computed concentrations of atmospheric constituents. A Monte-Carlo technique was used to propagate rate coefficient uncertainties to calculated concentrations (179,180). For hydroxyl radical, uncertainties in published rate constants propagate to modelled [HO ] uncertainties that range from 25% under low-latitude marine conditions to 72% under urban mid-latitude conditions. A large part of this uncertainty is due to the uncertainty (la=40%) in the photolysis rate of 0(3) to form O D, /j. 

Cabaniss, S., Propagation of Uncertainty in Aqueous Equilibrium Calculations Non-Gaussian Output Distributions, Anal. Chem. 69, 1997, 3658-3664. 

In this method, each assessment factor is considered uncertain and characterized as a random variable with a lognormal distribution with a GM and a GSD. Propagation of the uncertainty can then be evaluated using Monte Carlo simulation (a repeated random sampling from the distribution of values for each of the parameters in a calculation to derive a distribution of estimates in the population), yielding a distribution of the overall assessment factor. This method requires characterization of the distribution of each assessment factor and of possible correlations between them. As a first approach, it can be assumed that all factors are independent, which in fact is not correct. 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

In the ordinary weighted least squares method, the most probable values of source contributions are achieved by minimizing the weighted sum of squares of the difference between the measured values of the ambient concentration and those calculated from Equation 1 weighted by the analytical uncertainty of those ambient measurements. This solution provides the added benefit of being able to propagate the measured uncertainty of the ambient concentrations through the calculations to come up with a confidence interval around the calculated source contributions. 

This solution provides two benefits. First, it propagates a confidence interval around the calculated source contributions which reflects the cumulative uncertainty of the input observables. The second benefit provided by this "effective variance" weighting is to give those chemical properties with larger uncertainties, or chemical properties which are not as unique to a source type, less weight in the fitting procedure than those properties having more precise measurements or a truly unique source character. 

Since the uncertainties in and > 238 independent and random, then the uncertainty in the age of the zircon can be calculated using error propagation for independent random errors as... 

Chapter 3 gave rules for propagation of uncertainty in calculations. For example, if we were dividing a mass by a volume to find density, the uncertainty in density is derived from the uncertainties in mass and volume. The most common estimates of uncertainty are the standard deviation and the confidence interval. 

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