The Propagation of Errors

In many chemical situations we deduce a value for a property of interest by placing experimentally measured values in the right-hand side of an appropriate formula. For example, if we use the ideal gas equation  

Note that we have taken the product of the partial derivatives, 5 and 55, with the derivative jg. This is perfectly legitimate because = jjg in the context of ihe original expression involving iwo independent variables. 

A Using the chain rule, with the substitution u — c + b2, we initially define the partial derivatives of u with respect to a and b, respectively  

Differentiating c with respect to the single variable, u, gives  

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Donato, H. Metz, G. A Direct Method for the Propagation of Error Using a Personal Gomputer Spreadsheet Program, ... 

If the errors in the variables are independent, then = 0, and the propagation of error equation can be written... 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

We apply the propagation of errors treatment to Eq. (2-93), where the quantities in parentheses are treated as known constants. The result is... 

A variation of the lUPAC method called the propagation of errors (PE) method has been discussed by Long and Winefordner. In the PE method, the LOD is defined as... 

A better alternative would be to use the propagation of errors definition, which takes into consideration values of both and si when calculating the MDL. This would involve generating at least five calibration curves in order to obtain an accurate measurement of si and Sm ... 

In derivatisation reactions with a KIE correction factors are first calculated this calculation introduces another step where errors propagate. The propagation of errors under these circumstances is calculated using Equation (14.5), where subscript s stands for the standard used in correction factor determination and sd stands for the derivatised standard. The magnitude of the errors associated with the correction factors themselves can be calculated using Equation (14.4), along with the precisions for each determination (Docherty et al. 2001). 

Opeea, T.L, Olah, M., Ostopovici, L., Rad, R., and Meacec, M. On the propagation of errors in the QSAR literature. In EuroQSAR 2002 - Designing Drugs and Crop Protectants Processes, Problems and Sdutions, Foed, M., Livingstone, D., Deaeden, J., and... 

The variance for the weight fraction w(i) can be obtained from equation 1H and the variance for M(i) can be calculated directly from equation 18. Note that the propagation of error analysis can be readily extended to other averages and and it can also be used to account for the errors associated with the calibration of columns and detectors. 

Using the rules of the propagation of errors [33,34] a measure of the robustness of the partition coefficient (C, ) and the robustness of the selectivity (C ) can be obtained. Below, a derivation of robustness of the partition coefficient P, of a compound i and the selectivity Uij for two compounds i and j with respect to variation in extraction liquid composition is given. The general form of a (Special Cubic) mixture model for three-component mixtures is given by ... 

Accuracy and precision depend on the propagation of error starting from the error in weighing, volumetrically preparing the sample, and delivering the titrant to the sample. 

Insisting that the propagation of error formula be used wherever appropriate in laboratory reports, we find it easier to introduce the total differential once the topic comes up during the thermodynamics portion of the course. The similarity between the propagation of error formula and the total differential provides a more intuitive model for our students. Because our order of topics delays thermodynamics to later in the semester, we have time to emphasize the more concrete example used to determine the uncertainty in a measurement. 

By minimizing the error between b ) and the b simultaneously, which are respectively the intercept and the slope of the graph, one can find the best fit for the calculated data points. In (18.3), w, is the weight of the point on the line determined from the error bars in each isothermal-isobaric simulation. For the propagation of error, and in particular, by using the uncertainty in sums and differences and the uncertainty in products and quotients rules, the intersection of the lines, x, can be shown to be ... 

The propagation of error has been given for the subtraction of two numbers. In this section, the propagation of error for other mathematical operations is given. 

Computations, whether by hand calculator or by computer, should be done with awareness of certain principles concerning the effects of arithmetic operations. These follow directly from the principles underlying the propagation of errors, discussed in Part B of this chapter. 

This part of the chapter is concerned with the evaluation of nncertainties in data and in calculated results. The concepts of random errors/precision and systematic errors/accuracy are discussed. Statistical theory for assessing random errors in finite data sets is summarized. Perhaps the most important topic is the propagation of errors, which shows how the error in an overall calculated result can be obtained from known or estimated errors in the input data. Examples are given throughout the text, the headings of key sections are marked by an asterisk, and a convenient summary is given at the end. 

One final problem that must be addressed in the interpretation of sensor-array data is the reliability of the final result. Each sensor response contains a certain degree of error, and the propagation of error for a sensor array is not trivial. This fact has important ramifications in terms of identification of an analyte in the presence of interferences, as well as in the selection of coatings for inclusion in the array. The efficacy of the array depends on the uniqueness of coating responses as colinearity increases, error in the final result is amplified and the detection limit is adversely affected. These concerns have been addressed and the effects on the analytical result from the sensor array have been described quantitatively [260,272]. 

In order to carry out the weighting of y values, we need crj, the variance of y. Applying the propagation of errors treatment (Eq. 2-66) to the function y = In we have... 

The estimation of the error of a computed result R from the errors of the component terms or factors A, B, and C depends on whether the errors are determinate or random. The propagation of errors in computations is summarized in Table 26-2. The absolute determinate error e or the variance V = s for a random error is transmitted in addition or subtraction. (Note that the variance is additive for both a sum and a difference.) On the other hand, the relative determinate error ejx or square of the relative standard deviation (sJxY is additive in multiplication. The general case R = f A,. ) is valid only if A, B,C,... are independently variable it is... 

To further check the consistency of the method, we compare the copolymer s diffusion coefficient in THF, calculated from equation 3, with an independent measurement. Using our experimentally determined values of My and [ry], equation 3 predicts a D value of 9.85 X 10 cm /s this is in good agreement with the value of 9.79 X 10 cm s measured independently using NMR by workers at Exxon. (A similar comparison in toluene cannot be made because an independent value of D in toluene is not available.) A treatment of the propagation of errors is summarized in Table II. Here, uncertainties in My and Xa propagated by estimated uncertainties in the independent variables are listed. The largest uncertainty comes from the Dt values, but accurate retention data are also critical. For example, a 2% uncertainty in retention parameter X translates to a 9-10% uncertainty in My and a 5-8% uncertainty inXA. 

A treatment of the propagation of errors in the toluene-cyclohexane system is summarized in Table IV and Figure 5. In this solvent pair, the... 

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