Propagation of errors method

The estimated excess activity A and associated uncertainty for each slice of core are listed in Table II. The error, associated with each count is estimated using the propagation-of-errors method (Bevington, 1969). Counting statistics account for most of the imprecision, although uncertainties associated with counter background and efficiency also contribute. 

The magnitude of stochastic errors in the simulation averages can be quantified using standard analyses [15] and their influence on the precision of the coexistence line gauged via standard propagation-of-error method [58]. 

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

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... 

Fish, W. and Morel, F.M.M. (1985) Propagation of error in fulvic acid titration data a comparison of three analytical methods. Can.J. Chem., 63, 1185-1193. 

ACp°) changes for both the processes are summarized in Tables X, XI, and XII (on the molal scale), and in Tables XIII, XIV, and XV (on the mole-fraction basis), together with the respective estimates of the errors. The standard deviations of these quantities were estimated by the application of the method of propagation of errors as used by Please (26). 

Calculation of the Rate Constant from Concentrations Treatment of Instrument Response Data 34 Methods When the Final Value is Unknown 36 Propagation of Errors 40... 

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. 

In these equations P is the headspace pressure in atmospheres, a is the gas-to-seawater ratio (Vg/V ), and is the total volume of the flask (here 1 L). By this procedure the initial concentration of a dissolved gas (C f) in seawater can be determined. Normally single extractions are done, but subsequent extractions can be used to check the amount of gas remaining in the seawater. The uncertainties associated with the headspace method can be evaluated by referring to Equation 2. In addition to the headspace gas concentration, Cg, the uncertainty in C f is dependent on the uncertainties in H and the ratio a. By using a propagation of error procedure on Equation 2, the uncertainties associated with the vacuum extraction flask can be estimated for different gases (2). The results show that for a gas that is not very soluble (H > 3), the total uncertainty is 5 %. For a more soluble gas H < 0.5) whose Henry s constant is not accurately known in seawater, the uncertainty is 30 %. 

The internal standard technique is based on the covariance term in the error propagation of this method 89) and avoids the error caused by applying small volumes to the plate. But there is another important aspect of error propagation in the comparison of internal and external standard methods. If the error in measurement is smaller than the error in spotting, the internal standard method is more precise than the external standard method 90>. However, if the error in measurement becomes dominant, the external standard leads to better results. This is the reason why in early quantitative TLC, where peaks had been triangulated and evaluated by the squared... 

The estimation of the systematic errors sj (which, of course, have to relate to X and be expressed in the same unit) can only be made by a person who is familiar with the experimental method. The uncertainty a has to correspond to the 95% confidence level preferred in this review. It should be noted that for all the corrections and recalculations made (e.g., temperature or ionic strength corrections) the rules of the propagation of errors have to be followed, as outlined in Section C.6.2. 

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters... 

We also discussed graphical and numerical data reduction procedures. The most important numerical data reduction procedure is the least squares, or regression, method, which finds the best member of a family of functions to represent a set of data. We discussed the propagation of errors through this procedure and presented a version of the procedure in which different data points are given different weights, or importances, in the procedure. 

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