Random Errors in Chemical Analysis

Chapter 3 Using Spreadsheets in Analytical Chemistry 54 Chapter 4 Calculations Used in Analytical Chemistry 71 Chapter 5 Errors in Chemical Analyses 90 Chapter 6 Random Errors in Chemical Analysis 105 Chapter 7 Statistical Data Treatment and Evaluation 142 Chapter 8 Sampling, Standardization, and Calibration 175... 

A final point is the value of earlier (old) validation data for actual measurements. In a study about the source of error in trace analysis, Horwitz et al. showed that systematic errors are rare and the majority of errors are random. In other words, the performance of a laboratory will vary with time, because time is related to other instruments, staff, chemicals, etc., and these are the main sources of performance variation. Subsequently, actual performance verification data must be generated to establish method performance for all analytes and matrices for which results will be reported. 

As shown, various calibration methods can be applied in chemical analysis. The choice of method depends on the kind of analytical problems and sources of random errors expected in the course of analysis. Nevertheless, it is hard to say that any of the discussed methods is especially adapted to trace analysis. However, because of its specificity, trace analysis does require special attention in the choice of calibration method, as well as special care in realization of the selected method at every step of the calibration procedure. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

One common characteristic of many advanced scientific techniques, as indicated in Table 2, is that they are applied at the measurement frontier, where the net signal (S) is comparable to the residual background or blank (B) effect. The problem is compounded because (a) one or a few measurements are generally relied upon to estimate the blank—especially when samples are costly or difficult to obtain, and (b) the uncertainty associated with the observed blank is assumed normal and random and calculated either from counting statistics or replication with just a few degrees of freedom. (The disastrous consequences which may follow such naive faith in the stability of the blank are nowhere better illustrated than in trace chemical analysis, where S B is often the rule [10].) For radioactivity (or mass spectrometric) counting techniques it can be shown that the smallest detectable non-Poisson random error component is approximately 6, where ... 

Equation (8.5) for independent random errors has been frequently used in the uncertainty analysis of many physical and chemical experiments. 

In the text the word sample is used in two ways. The sample referred to in Part 1 is the portion of material submitted for chemical analysis, e.g. a 10 mL sample of olive oil removed from a one litre bottle of oil for the purpose of ehemical analysis. When statistieal procedures are applied to the results of analytical measurements (Part 6) the term sample refers to the segment of all the possible results, i.e. the population that is being used to calculate the statistic. Ten measurements are used to calculate the sample standard deviation (see random errors, Section 6.2) the ten results are only a portion of the infinite number of possible measurements. 

The accuracy of a measurement refers to how close it is to the true value. An inaccurate result occurs as a result of some flaw (systematic error) in the measurement the presence of an interfering substance, incorrect calibration of an instrument, operator error, and so on. The goal of chemical analysis is to eliminate systematic error, but random errors can only be minimized. In practice, an experiment is almost always done in order to find an unknown value (the true value is not known—someone is trying to obtain that value by doing the experiment). In this case the precision of several replicate determinations is used to assess the accuracy of the result. The results of the replicate experiments are expressed as an average (which we assume is close to the true value) with an error limit that gives some indication of how close the average value may be to the true value. The error limit represents the uncertainty of the experimental result. 

Even if all systematic error could be eliminated, the exact value of a chemical or physical quantity still would not be obtained through repeated measurements, due to the presence of random error (Barford, i985). Random error refers to random differences between the measured value and the exact value the magnitude of the random error is a reflection of the precision of the measuring device used in the analysis. Often, random errors are assumed to follow a Gaussian, or normal, distribution, and the precision of a measuring device is characterized by the sample standard deviation of the distribution of repeated measurements made by the device. [By contrast, systematic errors are not subject to any probability distribution law (Velikanov, 1965).] A brief review of the normal distribution is provided below to provide background for a discussion of the quantification of random error. 

Another methodological problem concerns the question whether the dispersion of the random error components is constant or changes with the analyte concentration as considered previously in the difference plot sections. For most clinical chemical compounds, the analytical SDs vary with the measured concentration, and this relationship may also apply to the random-bias components. In cases with a considerable range (i.e., a decade or more), this phenomenon should also be taken into account when applying a regression analysis. Figure 14-20 schematically shows how the dispersions may increase proportionally with concentration. 

Measurements invariably involve errors and uncertainties. Only a few of these are due to mistakes on the part of the experimenter. More commonly, errors are caused by faulty calibrations or standardizations or random variations and uncertainties in results. Frequent calibrations, standardizations, and analyses of known samples can sometimes be used to lessen all but the random errors and uncertainties. In the limit, however, measurement errors are an inherent part of the quantized world in which we live. Because of this, it is impossible to peiform a chemical analysis that is totally free of errors or uncertainties. We can only hope to minimize errors and estimate their size with acceptable accuracy. In this and the next two chapters, we explore the nature of experimental errors and their effects on the results of chemical analyses. 

Another important characteristic is that of precision. This becomes evident only when repeat measurements are made, because precision refers to the amount of agreement between repeated measurements (the standard deviation around the mean estimate). Precision is subject to both random and systematic errors. In industrial quality control and chemical analysis, Shewhart Control Charts provide a means of assessing the precision of repeat measurements but these approaches are rarely used in ecotoxicity testing. The effect is that we generally understand little about either the accuracy or the precision of most bioassays. 

In order to provide a means for the precise recalculation of nitrogen chemical shifts reported since 1972, it is necessary to have accurate values of the differences in the screening constants between neat CH3N02 and the large number of reference compounds which have so far been used. Table VII shows the results of precise, 4N measurements (61) which have been carried out in concentric spherical sample and reference containers in order to eliminate bulk susceptibility effects on the shifts. Since the technique adopted (61, 63) involves the accumulation of a large number of individually calibrated spectra with the subsequent use of a full-lineshape analysis by the differential saturation method, (63) the resulting random errors comprise those from minor temperature variations, phase drifts, frequency instability, sweep nonlinearity, etc. so that systematic errors should be insignificant as compared with random errors. 

The underlying assumption in statistical analysis is that the experimental error is not merely repeated in each measurement, otherwise there would be no gain in multiple observations. For example, when the pure chemical we use as a standard is contaminated (say, with water of crystallization), so that its purity is less than 100%, no amount of chemical calibration with that standard will show the existence of such a bias, even though all conclusions drawn from the measurements will contain consequent, determinate or systematic errors. Systematic errors act uni-directionally, so that their effects do not average out no matter how many repeat measurements are made. Statistics does not deal with systematic errors, but only with their counterparts, indeterminate or random errors. This important limitation of what statistics does, and what it does not, is often overlooked, but should be kept in mind. Unfortunately, the sum-total of all systematic errors is often larger than that of the random ones, in which case statistical error estimates can be very misleading if misinterpreted in terms of the presumed reliability of the answer. The insurance companies know it well, and use exclusion clauses for, say, preexisting illnesses, for war, or for unspecified acts of God , all of which act uni-directionally to increase the covered risk. 

Lastly, the quality of a statistical correlation alone cannot be taken as an indication of correctness of the assumptions. For example, a model with slightly larger, but random error is more likely correct than its rival with smaller, but systematic error, and primitive statistics programs do not take this into account. As Connors puts it, "the human eye, in combination with chemical knowledge, is a more subtle qualitative judge of data than is regression analysis" [57]. 

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