Errors treatment

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

In the following example, the measured chemical quantities are fixed by four continuous factors age, treatment A, treatment B and analytical error. Treatment B has been applied to only one sample of the class. Age and treatment A produce variations in composition that cannot be interpreted as deviations from the model they are the inner factors. On the other hand, treatment B cannot be identified as an inner factor because of the lack of information, and its effects fall, with the analytical error, in the outer space. Besides, if the effects of treatment B are noticeably greater than those of the analytical error, the B-treated object can be identified as an outlier, as it really is. 

An error analysis dealing with the uncertainty in the final result due to random errors in the measurements will normally be part of the Results section. The type of error analysis undertaken will depend a great deal on the nature of the experiment see Chapters IIB and XXI for more details. The analysis given in the sample report is typical of a straightforward propagation-of-errors treatment. If a long and complex propagation-of-errors... 

The above-mentioned considerations are not a substitute for a propagation-of-errors treatment to determine the uncertainty in the final result, especially when the calculations have been carried out on a computer and the result is given with an artificially large number of digits. 

The next three subsections describe the background and principles of random error treatment, and they introduce two important quantities standard deviation a- and 95 percent confidence limits. The four subsections following these— Uncertainty in Mean Valne, Small Samples, Estimation of Limits of Error, and Presentation of Numerical Results—are essential for the kind of random error analysis most frequently required in the experiments given in this book. The Student t distribution is particularly important and useful. 

Many propagation-of-error treatments are not so simple as those illustrated by Eqs. (43) to (47). The expression for F is often complicated enough or the functional form sufficiently awkward to justify a breakdown of the procedure into steps ... 

Given below are three numerical examples of a propagation of errors treatment. 

Suppose, however, that we are slightly unsure about the effect of the neglect of the v V term on the limit of error. Let us disregard the extremely small terms involving a and (B — Bg) and otherwise carry out a full-fledged limit-of-error treatment. The partial derivatives and their values are... 

None of the experiments described in this book will involve mathematical treatments for limit-of-error estimation more complicated than this example. With careful judgment, most limit-of-error treatments can be made much simpler. 

A propagation-of-error treatment of Eq. (13) yields for the estimated standard deviation in parameter a, the expression... 

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 resistance in the medical community to switching from the trial-and-error treatment approach to the gene-based approach is still prominently evident. Physicians in clinical practice, trainee physicians, and medical undergraduates have not been adequately educated in the field the concept of pharmacogenetics has not been incorporated in the curriculum of medical courses worldwide. They are, thus,... 

Neuschaefer-Rube U., Hozapfel W., Wirth F. Surface measurement applying focusing reflection ellipsometry configurations and error treatment. Measurement 2003 33 163-171... 

Two generally accepted models for the vapor phase were discussed in Chapter 3 and one particular model for the liquid phase (UNIQUAC) was discussed in Chapter 4. Unfortunately, these, and all other presently available models, are only approximate when used to calculate equilibrium properties of dense fluid mixtures. Therefore, any such model must contain a number of adjustable parameters, which can only be obtained from experimental measurements. The predictions of the model may be sensitive to the values selected for model parameters, and the data available may contain significant measurement errors. Thus, it is of major importance that serious consideration be given to the proper treatment of experimental measurements for mixtures to obtain the most appropriate values for parameters in models such as UNIQUAC. 

Extended defects range from well characterized dislocations to grain boundaries, interfaces, stacking faults, etch pits, D-defects, misfit dislocations (common in epitaxial growth), blisters induced by H or He implantation etc. Microscopic studies of such defects are very difficult, and crystal growers use years of experience and trial-and-error teclmiques to avoid or control them. Some extended defects can change in unpredictable ways upon heat treatments. Others become gettering centres for transition metals, a phenomenon which can be desirable or not, but is always difficult to control. Extended defects are sometimes cleverly used. For example, the smart-cut process relies on the controlled implantation of H followed by heat treatments to create blisters. This allows a thin layer of clean material to be lifted from a bulk wafer [261. 

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

The precision of a result is its reproducibility the accuracy is its nearness to the truth. A systematic error causes a loss of accuracy, and it may or may not impair the precision depending upon whether the error is constant or variable. Random errors cause a lowering of reproducibility, but by making sufficient observations it is possible to overcome the scatter within limits so that the accuracy may not necessarily be affected. Statistical treatment can properly be applied only to random errors. 

Evaluating Indeterminate Error Although it is impossible to eliminate indeterminate error, its effect can be minimized if the sources and relative magnitudes of the indeterminate error are known. Indeterminate errors may be estimated by an appropriate measure of spread. Typically, a standard deviation is used, although in some cases estimated values are used. The contribution from analytical instruments and equipment are easily measured or estimated. Indeterminate errors introduced by the analyst, such as inconsistencies in the treatment of individual samples, are more difficult to estimate. 

Irvin, J. A. Quickenden, T. L. Linear Least Squares Treatment When There Are Errors in Both x and y, J. Chem. Educ. 1983, 60, 711-712. 

The error is approximately equal to the FAC at pH 7.2—7.8. However, it is significant only at high FAC, eg, during superchlorination or shock treatment. 

Randomization means that the sequence of preparing experimental units, assigning treatments, miming tests, taking measurements, and so forth, is randomly deterrnined, based, for example, on numbers selected from a random number table. The total effect of the uncontrolled variables is thus lumped together into experimental error as unaccounted variabiUty. The more influential the effect of such uncontrolled variables, the larger the resulting experimental error, and the more imprecise the evaluations of the effects of the primary variables. Sometimes, when the uncontrolled variables can be measured, their effect can be removed from experimental error statistically. 

Erythrocyte Entrapment of Enzymes. Erythrocytes have been used as carriers for therapeutic enzymes in the treatment of inborn errors (249). Exogenous enzymes encapsulated in erythrocytes may be useful both for dehvery of a given enzyme to the site of its intended function and for the degradation of pathologically elevated, diffusible substances in the plasma. In the use of this approach, it is important to determine that the enzyme is completely internalized without adsorption to the erythrocyte membrane. Since exposed protein on the erythrocyte surface may ehcit an immune response following repeated sensitization with enzyme loaded erythrocytes, an immunologic assessment of each potential system in animal models is required prior to human trials (250). 

More detailea descriptions of small-scale sedimentation and filtration tests are presented in other parts of this section. Interpretation of the results and their conversion into preliminary estimates of such quantities as thickener size, centrifuge capacity, filter area, sludge density, cake diyness, and wash requirements also are discussed. Both the tests and the data treatment must be in experienced hands if error is to be avoided. 

Method used to clean specimens after exposure and the extent of any error expected by this treatment... 

If the comparison shows that the measurement is inconsistent with the comparison information, the measurement is considered suspecl. If a measurement can be compared to more than one set of information and found to be inconsistent with all, it is likely that the measurement is in error. The measurement should then be excluded from the measurement set. In this section, validation is extended to include comparison of the measurements to the constraints and initial adjustment in the measurements. Validation functions as an initial screening procedure before the more comphcated procedures begin. Oftentimes, vahdation is the only measurement treatment required prior to interpretation. 

Other cases, neglecting heat effects would cause serious errors. In such cases the mathematical treatment requires the simultaneous solution of the diffusion and heat conductivity equations for the catalyst pores. 

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