Systematic deviation

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. 

Systematic deviations, in addition to random deviations, may be produced 

Several of the types of systematic deviations mentioned can frequently occur in combination with each other  

In analytical practice, they are best recognized by the determination of xtest as a function of the true value xtrue, and thus, by analysis of certified reference materials (CRMs). If such standards are not available the use of an independent analytical method or a balancing study may provide information on systematic errors (Doerffel et al. [1994] Kaiser [1971]). In simple cases, it may be possible, to estimate the parameters a, / , and y, in Eq. (4.5) by eliminating the unknown true value through appropriate variation of the weight of the test portions or standard additions to the test sample. But in the framework of quality assurance, the use of reference materials is indispensable for validation of analytical methods. 

Commonly the absence of systematic errors is tested by recovery studies. The estimation of recovery by analyzing only one standard, as frequently be done, can give misleading results as can be seen from Fig. 4.2. In case of combined and nonlinear systematic deviations there can occur points of accidental congruence that can be hit. Therefore, if at all possible, there should be estimated the entire recovery function. However, at least two different standards should be analyzed. 

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It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

The problem with plant data becomes more significant when sampling, instrument, and cahbration errors are accounted for. These errors result in a systematic deviation in the measurements from the actual values. Descriptively, the total error (mean square error) in the measurements is... 

An inspection of the residuals should indicate no systematic deviation of the calculated curve from the data points. 

The fit with H= 1.53 is quite good. The results for the fits with n = 1 andn = 2 show systematic deviations between the data and the fitted model. The reaction order is approximately 1.5, and this value could be used instead of n= 1.53 with nearly the same goodness of fit, a = 0.00654 versus 0.00646. This result should motivate a search for a mechanism that predicts an order of 1.5. Absent such a mechanism, the best-fit value of 1.53 may as well be retained. 

Pseudo-first order kinetics was assumed to interpret the experimental data. Fig. 5.4-36 shows that the fit of the experimental data using first order-kinetics is acceptable. However, a systematic deviation is observed in the curve obtained at 110 C. This indicates inadequacy of the first-order kinetics, which is inappropriate from the view of theory. On the other hand, the kinetic equation seems to describe the... 

Parity diagrams the quantity calculated y, fc vs. the quantity observed yexp or plots of residual deviations (>> ,/, - y, xp) vs. predicted values should show uniform bands the scatter of points should be uniform any systematic deviations disqualify the model, which should then be rejected. The data points on plots of linearized equations should scatter uniformly. 

It should be kept in mind that an objective function which does not require any phase equilibrium calculations during each minimization step is the basis for a robust and efficient estimation method. The development of implicit objective functions is based on the phase equilibrium criteria (Englezos et al. 1990a). Finally, it should be noted that one important underlying assumption in applying ML estimation is that the model is capable of representing the data without any systematic deviation. Cubic equations of state compute equilibrium properties of fluid mixtures with a variable degree of success and hence the ML method should be used with caution. 

When applying these methods to the study of molybdenum complexes Voityuk and Rosch [25, 41] found that the use of the AMI core-repulsion function (Eq. 5-6) led to some systematic deviations for some Mo—X bond lengths. To address these problems, important changes to the core-repulsion function were made by the introduction of bond-specific parameters (omo-x and 8 io-x, Eq. 5-9) [22, 25], The idea of using bond-specific core repulsion parameters is not new, since the AMI parameterisation of boron used bond-specific Gaussian functions to improve the final results [42],... 

Bias (systematic deviation) of an individual test value x (or y) from the (conventional) true value xtrui, (or ytrue)... 

In principle, all measurements are subject to random scattering. Additionally measurements can be affected by systematic deviations. Therefore, the uncertainty of each measurement and measured result has to be evaluated with regard to the aim of the analytical investigation. The uncertainty of a final analytical result is composed of the uncertainties of all the steps of the analytical process and is expressed either in the way of classical statistics by the addition of variances... 

Systematic deviations (errors) displace the individual results of measurement one-sided to higher or lower values, thus leading to incorrect results. In contrast to random deviations, it is possible to avoid or eliminate systematic errors if their causes become known. The existence and magnitude of systematic deviations are characterized by the bias. The bias of a measured result is defined as a consistent difference between the measured value ytest and the true value ytrue ... 

The relation between systematic and random deviations as well as the character of outliers is shown in Fig. 4.1. The scattering of the measured values is manifested by the range of random deviations (confidence interval or uncertainty interval, respectively). Measurement errors outside this range are described as outliers. Systematic deviations are characterized by the relation of the true value p and the mean y of the measurements, and, in general, can only be recognized if they are situated beyond the range of random variables on one side. 

Measurement caused by concurrent reactions or incomplete reaction processes in the case of chemical principles, and by instrumental deviations and wrong adjustment in the case of physical methods. A frequently encountered reason for the occurrence of systematic deviations is erroneous calibration due to unsuitable calibration standards, matrix effects, or insufficient methodical or theoretical foundation. 

Even data evaluation, often thought free of errors to a large extent, can generate systematic deviations by reason of incorrect or incomplete algorithms. 

Consequences of various types of systematic deviations between xtest and xtruey and therefore, xstandard and xsampie are discussed in Danzer [1995]. 

Fig. 6.8. Typical plots of residual deviations random scattering (a), systematic deviations indicating nonlinearity (b), and trumpet-like form of heteroscedasticity (c)...

Accuracy. In general, the accuracy of analytical results is assured by recovery studies (Wegscheider [1996] Danzer [1995] Burns et al. [2002]). According to the recovery function in the general three-dimensional calibration model (see Fig. 6.3), common studies on systematic deviations (Fig. 4.3), and Eqs. (4.2) and (4.3) the following recovery formulae... 

Systematic deviations are also detected if the corresponding confidence intervals of the validation coefficients do not include 0 or 1, respectively, namely... 

Linearity. Whether the chosen linear model is adequate can be seen from the residuals ey over the x values. In Fig. 6.8a the deviations scatter randomly around the zero fine indicating that the model is suitable. On the other hand, in Fig. 6.8b it can be seen that the errors show systematic deviations and even in the given case where the deviations alternate in the real way, it is indicated that the linear model is inadequate and a nonlinear model must be chosen. The hypothesis of linearity can be tested ... 

As a measuring science, analytical chemistry has to guarantee the quality of its results. Each kind of measurement is objectively affected by uncertainties which can be composed of random scattering and systematic deviations. Therefore, the measured results have to be characterized with regard to their quality, namely both the precision and accuracy and - if relevant - their information content (see Sect. 9.1). Also analytical procedures need characteristics that express their potential power regarding precision, accuracy, sensitivity, selectivity, specificity, robustness, and detection limit. 

Accuracy is defined in the name of BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML as the closeness of agreement between the result of a measurement and the true value of the measurand (ISO 3534-1 [1993] Fleming et al. [1996b]). Accuracy characterizes the absence of a relevant bias of a measured value. As can be seen from Fig. 4.1, accuracy is a measure that combines the effects of both random and systematic deviations. Therefore, a bias can only be detected if it exceeds the range of random error. 

The advantage of the measures as suggested here is that they point into the same direction as it is done by the verbal definitions. High precision and a high degree of accuracy, respectively, are characterized by high numerical values of the measures which approximate to 1 in the ideal case (absence of random and systematic deviations, respectively) and approximate to 0 if the deviations approach to 100%. In the worst cases, the numerical values prec(x) and acc(x) become negative which indicates that the relative random or systematic error exceeds 100%. 

Classical Evaluation of Porod s Law. The practical evaluation of Porod s law in the most common case of an isotropic sample measured with point-focus is demonstrated in Fig. 8.11. By variation of the fluctuation background a long and linear Porod region can be received. Nevertheless, the line still shows a negative slope (Fig. 8.11). Ruland s theory of the systematic deviations from Porod s law [132] explains this finding. 

In general, several models should be tested. Systematic deviations between model and data indicate, how the model should be varied in order to improve the fit. If anisotropic materials have been studied, the CDF frequently exhibits the ingredients of a good model. 

Fig. 10). With the completion of the structure transition, the current should drop to zero, which is indeed the case except for peak B, where a slight leak current is seen (ascribed to the side reaction Cu++ I c > Cu+). According to the theory by Bewick, Fleischmann and Thirsk (BFT) the transients can be used to distinguish between instantaneous and progressive nucleation [45], A corresponding analysis revealed that the falling part of the transients agrees well with the model for instantaneous nucleation, while the rising part shows a systematic deviation. This was explained by the existence of surface defects on a real electrode in contrast to the ideal case of a defect-free surface assumed in the theoretical model. By including an adsorption term in the BFT theory to account for Cu deposition at defects, the experimentally obtained transients could indeed be reproduced very well [44], We shall return to the important role of surface defects in metal deposition later (sec. 3.2).

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