Error, analytical propagation

Of all the requirements that have to be fulfilled by a manufacturer, starting with responsibilities and reporting relationships, warehousing practices, service contract policies, airhandUng equipment, etc., only a few of those will be touched upon here that directly relate to the analytical laboratory. Key phrases are underlined or are in italics Acceptance Criteria, Accuracy, Baseline, Calibration, Concentration range. Control samples. Data Clean-Up, Deviation, Error propagation. Error recovery. Interference, Linearity, Noise, Numerical artifact. Precision, Recovery, Reliability, Repeatability, Reproducibility, Ruggedness, Selectivity, Specifications, System Suitability, Validation. 

The simplest method for propagation of the analytical errors into the °Th/U age equation (Eqn. 1) involves linear expansion (Albarede 1995, ch. 4.3) of the effect on the calculated age of very small perturbations of the measured ratios (in effect, a Taylor expansion using only the first term). As long as the effect of the errors of the measured ratios on the age is not a large fraction of the age itself, this method will yield acceptably accurate age-errors with minimal effort. 

Ignore the problem completely and simply quote the errors propagated from the analytical errors of the data points. 

It is not always possible to tell strictly the difference between random and systematic deviations, especially as the latter are defined by random errors. The total deviation of an analytical measurement, frequently called the total analytical error , is, according to the law of error propagation, composed of deviations resulting from the measurement as well as from other steps of the analytical process (see Chap. 2). These uncertainties include both random and systematic deviations, as a rule. 

Traditionally, analytical chemists and physicists have treated uncertainties of measurements in slightly different ways. Whereas chemists have oriented towards classical error theory and used their statistics (Kaiser [ 1936] Kaiser and Specker [1956]), physicists commonly use empirical uncertainties (from knowledge and experience) which are consequently added according to the law of error propagation. Both ways are combined in the modern uncertainty concept. Uncertainty of measurement is defined as Parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand (ISO 3534-1 [1993] EURACHEM [1995]). 

A realistic uncertainty interval has to be estimated, namely by considering the statistical deviations as well as the non-statistical uncertainties appearing in all steps of the analytical process. All the significant deviations have to be summarized by means of the law of error propagation see Sect. 4.2. 

Equation (4.20) was proposed by Hoskuldsson [65] many years ago and has been adopted by the American Society for Testing and Materials (ASTM) [59]. It generalises the univariate expression to the multivariate context and concisely describes the error propagated from three uncertainty sources to the standard error of the predicted concentration calibration concentration errors, errors in calibration instrumental signals and errors in test sample signals. Equations (4.19) and (4.20) assume that calibrations standards are representative of the test or future samples. However, if the test or future (real) sample presents uncalibrated components or spectral artefacts, the residuals will be abnormally large. In this case, the sample should be classified as an outlier and the analyte concentration cannot be predicted by the current model. This constitutes the basis of the excellent outlier detection capabilities of first-order multivariate methodologies. 

Appendixes Tables of solubility products, acid dissociation constants (updated to 2001 values), redox potentials, and formation constants appear at the back of the book. You will also find discussions of logarithms and exponents, equations of a straight line, propagation of error, balancing redox equations, normality, and analytical standards. 

Analyze analytical errors and their effect on the results, e. g. a sensitivity analysis with the Gaussian error propagation method. 

Standard deviations in unit-cell parameters may be calculated analytically by error propagation. In these programs, however, the Jacobian of the transformation from Sj,. .., s6 to unit-cell parameters and volume is evaluated numerically and used to transform the variance-covariance matrix of Si,. .., s6 into the variances of the cell parameters and volume from which standard deviations are calculated. If suitable standard deviations are not obtained for certain of the unit cell parameters, it is easy to program the computer to measure additional reflections which strongly correlate with the desired parameters, and repeat the final calculations with this additional data. 

Lorber A. Error propagation and figures of merit for quantification by solving matrix equations. Analytical Chemistry 1986, 58, 1167-1172. 

According to the law of error propagation (see Eq. 4-21), the total variance of the analytical result can be expressed as ... 

The description of an object in the sense of environmental investigation may be the determination of the gross composition of an environmental compartment, for example the mean state of a polluted area or particular location. If this is the purpose, the number of individual samples required and the required mass or size of these increments have to be determined. The relationship between the variance of sampling and that of analysis must be known and both have to be optimized. The origin of the variance of the samples can be investigated by the study of variance contribution of the different steps of the analytical process by means of the law of error propagation (Eq. 4-21) according to Section 4.3.4. 

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. 

For more complex models or for input distributions for which exact analytical methods are not applicable, approximate methods might be appropriate. Many approximation methods are based on Taylor series expansion solutions, in which the series is truncated depending on the desired amount of solution accuracy and whether one wishes to consider covariance among the input distributions (Hahn Shapiro, 1967). These methods often go by names such as generation of system moments , statistical error propagation , delta method and first-order methods , as discussed by Cullen Frey (1999). 

All of the considerations discussed lead naturally to the question of what price the analyst pays for this less-than-ideal spike/sample ratio. In most cases, error in the measurement of Rm makes the largest contribution to analytical uncertainty the isotopic compositions of sample and spike are usually well known in comparison to Rm. The matter of error propagation in isotope dilution analyses has been extensively treated by Adriaens et al., [13], and Patterson et al. used Monte Carlo simulation to study the problem [14]. Using propagation of error laws, Heumann derived the following relationship with which to calculate tfopt, the optimum spike-to-sample ratio (neglecting cost and availability) [8] ... 

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