Indeterminate errors sources

Sources of Indeterminate Error Indeterminate errors can be traced to several sources, including the collection of samples, the manipulation of samples during the analysis, and the making of measurements. 

When collecting a sample, for instance, only a small portion of the available material is taken, increasing the likelihood that small-scale inhomogeneities in the sample will affect the repeatability of the analysis. Individual pennies, for example, are expected to show variation from several sources, including the manufacturing process, and the loss of small amounts of metal or the addition of dirt during circulation. These variations are sources of indeterminate error associated with the sampling process. 

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. 

Values of sx are a complex function of transmittance when indeterminate errors are dominated by the noise associated with photon transducers. Curve B in Figure 10.35 shows that the relative uncertainty in concentration is very large for low absorbances, but is less affected by higher absorbances. Although the relative uncertainty reaches a minimum when the absorbance is 0.96, there is little change in the relative uncertainty for absorbances between 0.5 and 2. This source of inde-... 

Finally, values of sx are directly proportional to transmittance for indeterminate errors due to fluctuations in source intensity and for uncertainty in positioning the sample cell within the spectrometer. The latter is of particular importance since the optical properties of any sample cell are not uniform. As a result, repositioning the sample cell may lead to a change in the intensity of transmitted radiation. As shown by curve C in Figure 10.35, the effect of this source of indeterminate error is only important at low absorbances. This source of indeterminate errors is usually the limiting factor for high-quality UV/Vis spectrophotometers when the absorbance is relatively small. 

Let s use a simple example to develop the rationale behind a one-way ANOVA calculation. The data in Table 14.7 show the results obtained by several analysts in determining the purity of a single pharmaceutical preparation of sulfanilamide. Each column in this table lists the results obtained by an individual analyst. For convenience, entries in the table are represented by the symbol where i identifies the analyst and j indicates the replicate number thus 3 5 is the fifth replicate for the third analyst (and is equal to 94.24%). The variability in the results shown in Table 14.7 arises from two sources indeterminate errors associated with the analytical procedure that are experienced equally by all analysts, and systematic or determinate errors introduced by the analysts. 

Errors in the analytical laboratory are basically of two types determinate errors and indeterminate errors. Determinate errors, also called systematic errors, are errors that were known to have occurred, or at least were determined later to have occurred, in the course of the lab work. They may arise from avoidable sources, such as contamination, wrongly calibrated instruments, reagent impurities, instrumental malfunctions, poor sampling techniques, errors in calculations, etc. Results from laboratory work in which avoidable determinate errors are known to have occurred must be rejected or, if the error was a calculation error, recalculated. 

Random, or indeterminate, errors exist in every measurement. They can never be totally eliminated and are often the major source of uncertainty in a determination. Random errors are caused by the many uncontrollable variables that are an inevitable part of every analysis. Most contributors to random error cannot be positively identified. Even if we can identify sources of uncertainty, it is usually impossible to measure them because most are so small that they cannot be detected individually. The accumulated effect of the individual uncertainties, however, causes replicate measurements to fluctuate randomly around the mean of the set. For example, the scatter of data in Figures 5-1 and 5-3 is a direct result of the accumulation of small random uncertainties. We have replotted the KJeldahl nitrogen data from Figure 5-3 as a three-dimensional plot in Figure 6-1 in order to better see the precision and accuracy of each analyst. Notice that the random error in the results of analysts 2 and 4 is much larger than that seen in the results of analysts 1 and 3. The results of analyst 3 show good precision, but poor accuracy. The results of analyst 1 show excellent precision and good accuracy. 

Figure 26-1 la is a plot of the relative standard deviation for experimentally determined concentrations as a function of absorbance. It was obtained with a spectrophotometer similar to the one shown in Figure 25-19. The striking similarity between this curve and curve A in Figure 26-10 indicates that the instrument studied is affected by an absolute indeterminate error in transmittance of about 0.003 and that this error is independent of transmittance. The source of this uncertainty is probably the limited resolution of the transmittance scale.

Infrared spectrophotometers also exhibit an indeterminate error that is independent of transmittance. The source of the error in these instruments lies in the... 

In the second type of quaniiiaiive mass spectrometry for molecular species, analyte concentrations are obtained directly from the heights of the mass spectral peaks. Tor simple mixtures, it is sometimes possible to find peaks at unique m/r values for each component. Under these circumstances, calibration curves of peak heights versus concentration can be prepared and used for analysis of unknowns. More accurate results can ordinarily be rcali/ed, however, by incorporating a lixed amount of an internal standard substance in both samples and calibration standards. The ratio of the peak intensity of the analyte species to that of the internal standard is then plotted as a function of analyle concentration. The internal standard tends to reduce uncertainties arising in sample preparation and introduction. These uncertainties are often a major source ol indeterminate error with ibe small samples needed for mass spectrometry. Internal standards are also used inOCVMS and f.C/MS. For these techniques, the ratio of peak areas serves as the analytical variable. 

We can evaluate the impact of indeterminate error due to instrumental noise on the information obtained from transmittance measurements. The following discussion apphes to UV/ VIS spectrometers operated in regions where the hght source intensity is low or the detector sensitivity is low and to IR spectrometers where noise in the thermal detector is signihcant. 

Analytical chemists make a distinction between error and uncertainty Error is the difference between a single measurement or result and its true value. In other words, error is a measure of bias. As discussed earlier, error can be divided into determinate and indeterminate sources. Although we can correct for determinate error, the indeterminate portion of the error remains. Statistical significance testing, which is discussed later in this chapter, provides a way to determine whether a bias resulting from determinate error might be present. 

The critical value for f(0.05,4), as found in Appendix IB, is 2.78. Since fexp is greater than f(0.05, 4), we must reject the null hypothesis and accept the alternative hypothesis. At the 95% confidence level the difference between X and p, is significant and cannot be explained by indeterminate sources of error. There is evidence, therefore, that the results are affected by a determinate source of error. 

The difference between precision and accuracy and a discussion of indeterminate and determinate sources of error is covered in the following paper. 

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

Errors of the second class, called indeterminate or random errors, are brought about by the effects of uncontrolled variables. Usually a relatively large number of experimental variables, each of which causes a small error, must be left uncontrolled. For example, if a correction for solubility loss of a precipitate is made to reduce the systematic error from this source, random errors due to fluctuations in temperature or volume of wash water, for instance, remain. Random errors are as likely to cause high as low results, and a small random error is more likely to occur than a large one. If observations were coarse enough, random errors would cease to exist. Every observation would give the same result, but the result would be less accurate than the average of a number of finer observations with random scatter. 

All spectrometric measurements are subject to indeterminate (random) error, which will affect the accuracy and precision of the concentrations determined using spectrometric methods. A very common source of random error in spectrometric analysis is instrumental noise . Noise can be due to instability in the light source of the instmment, instabihty in the detector, variation in placement of the sample in the hght path, and is often a combination of all these sources of noise and more. Because these errors are random, they cannot be eliminated. Errors in measurement of radiation intensity lead directly to errors in measurement of concentration when using cahbration curves and Beer s Law. 

Errors of measurement can be classified as determinate or indeterminate. The latter is random and can be treated statistically (Gaussian statistics) the former is not, and the source of the nonrandom error should be found and eliminated. 

Inspection of a set of replicate measurements or results may reveal that one or more is considerably higher or lower than the remainder and appears to be outside the range expected from the inherent effects of indeterminate (random) errors alone. Such values are termed outliers, or suspect values, because it is possible that they may have a bias due to a determinate error. On occasions, the source of error may already be known or it is discovered on investigation, and the outlier(s) can be rejected without recourse to a statistical test. Frequently, however, this is not the case, and a test of significance such as the Q-test should be applied to a suspect value to determine whether it should be rejected and therefore not included in any further computations and statistical assessments of the data. 

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