Error, systematic

Let us recall the zero-mean hypothesis (9.2.4a). More generally, if the mean E(ej) = gjo 0 for some measured variable Xj then this e is called the bias of the random variable (measurement error). It is a theoretical definition relative to the assumed probability space. In practice, the presence of a bias is regarded as a systematic error of the measurement. It can be due to an imperfectly adjusted instmment, its inadequate placing, or also be a property of the measuring method itself see again Madron (1992). The present book is not concerned with problems of measurement proper. Let us only show how the presence of biases affects the reconciliation. 

If we know the vector Cq, it is natural to subtract this Cg from the measured vector X and reconcile the vector x -eg. This procedure (subtraction of systematic errors) is called compensation. The compensation is widely used in practice. A typical example is compensation of flowrates measured by orifices for the deviation of actual density from that supposed in the instrument design. 

The naive approach (see Section 9.1) leads to replacing the condition (9.1.9c) by 

The maximum likelihood principle is generally formulated again in the form (9.1.12). If the error vector is Gaussian, with (9.5.1) used in (E.2.5-9) where a = Cg we have, in lieu of (9.1.13) 

If we don t know Cg, we reconcile again according to (9.2.15) getting the estimate x, with adjustment v of x. Observe that by (9.3.1 and 3) we have again 

This is essentially the multi-dimensional definition of slope. It describes how changes in u depend on changes in x, y, and z. Note that we use du to examine systematic errors but (du) to examine random errors. 

Example. Calculate the systematic error in the volume of a cylinder resulting from a mis-calibrated ruler. A mis-calibrated ruler results in a systematic error by +0.2 cm in diameter and height are 20 cm, and 5 cm, respectively. The values of r and h in the cylinder must be changed by +0.2 cm. 

Example. Suppose that we have systematic errors for La concentrations in two samples with the following systematic errors as a result of miscalibrated standard. 

Note that the systematic error in x/ y is better then the precision of both X (0.3/15=2%) and y (0.3/10=3%). This is why geochemists like to use ratios instead of concentrations, as ratios can reduce systematic errors. 

For further reading of error analysis, the following books are useful Beers (1957), Baird (1988), Bevington and Robinson (1992), and Taylor (1997). 

Uncertainty in measurement can arise from many different sources because all of the activities that go into generating the result are subject to some variation. The first type of variation is called random error and is related to the amount of scatter within a data set. The second type of variation is related to differences between the mean value for the data set and the accepted value for the measurement this is referred to as systematic error. The total uncertainty of measurement for a particular measured quantity comprises the random element and the systematic element. 

Random errors are caused by uncontrollable variables within the measurement system and can be assessed through multiple measurements of the same sample and assessing the relative standard deviation. 

Systematic errors can usually be identified through careful investigation and therefore have an assignable cause and are constant for a particular data set determined under the same set of conditions. For example, a systematic error in detector wavelength will have a uniform impact on all measurements 

Exactly how the information is used in relation to uncertainty measurement is dependent on the laboratory environment and the level of accreditation or quality management programmes that exist within it. Individual laboratories will have quality systems and standard operating procedures that deal with the measurement and recording of uncertainty measurement. To deal with all of the possible outcomes and eventualities is beyond the scope of this book. Suffice to say that it is important to be aware of the requirement and reasoning behind the application of uncertainty measurement. 

Deviations from radial symmetry in the diffraction pattern u 

Neglect of (or wrong correction for) non-zero molecular beam 

Several review papers (see ref. 9 in Section 1) also give discussions of error sources and accuracy in electron-diffraction studies. Some earlier works dealing with these topics are given in ref. 95. 

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The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. 

At low pressures, it is often permissible to neglect nonidealities of the vapor phase. If these nonidealities are not negligible, they can have the effect of introducing a nonrandom trend into the plotted residuals similar to that introduced by systematic error. Experience here has shown that application of vapor-phase corrections for nonidealities gives a better representation of the data by the model, oven when these corrections... 

An apparent systematic error may be due to an erroneous value of one or both of the pure-component vapor pressures as discussed by several authors (Van Ness et al., 1973 Fabries and Renon, 1975 Abbott and Van Ness, 1977). In some cases, highly inaccurate estimates of binary parameters may occur. Fabries and Renon recommend that when no pure-component vapor-pressure data are given, or if the given values appear to be of doubtful validity, then the unknown vapor pressure should be included as one of the adjustable parameters. If, after making these corrections, the residuals again display a nonrandom pattern, then it is likely that there is systematic error present in the measurements. ... 

Fig. 5 systematic error of the simple formula (3) compared to the correct model according equation (2) depending on the ratio of film focus distance to pipe diameter. The wall thickness calculated according to (3) is smaller then (2) by the given error. 

Ferrenberg A M, Landau D P and Binder K 1991 Statistical and systematic errors in Monte-Carlo sampling J. Stat. Phys. 63 867-82... 

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

If all sources of systematic error can be eliminated, there will still remain statistical errors. These errors are often reported as stcindard deviations. What we would particularly like to estimate is the error in the average value, (A). The standard deviation of the average value is calculated as follows ... 

In practice, it is found that this simple implementation is not the most effective approach. There are two particular problems. First, when the two simulations are run in tandem then significant correlations can arise, which results in large systematic errors. There are a number of ways to avoid these correlations, such as moving the J-walker an extra nuniher... 

The discrepancy be I ween ealetilatiun and experiment in Exercise 5-.3 is within the tincertainty limits uf many Iherrnuehemical meastirernents and would probably not be noticed in a single experiment. A systematic error of this magnitude would certainly be detected by a careful examination of many experimental results such as those carried out by Allinger (1989). 

Even within a particular approximation, total energy values relative to the method s zero of energy are often very inaccurate. It is quite common to find that this inaccuracy is almost always the result of systematic error. As such, the most accurate values are often relative energies obtained by subtracting total energies from separate calculations. This is why the difference in energy between conformers and bond dissociation energies can be computed extremely accurately. 

There could also be systematic errors that are not indicated by this relationship. 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

Some density functional theory methods occasionally yield frequencies with a bit of erratic behavior, but with a smaller deviation from the experimental results than semiempirical methods give. Overall systematic error with the better DFT functionals is less than with HF. 

The electron alfinity (FA) and ionization potential (IP) can be computed as the difference between the total energies for the ground state of a molecule and for the ground state of the appropriate ion. The difference between two calculations such as this is often much more accurate than either of the calculations since systematic errors will cancel. Differences of energies from correlated quantum mechanical techniques give very accurate results, often more accurate than might be obtained by experimental methods. 

The next step is to obtain geometries for the molecules. Crystal structure geometries can be used however, it is better to use theoretically optimized geometries. By using the theoretical geometries, any systematic errors in the computation will cancel out. Furthermore, the method will predict as yet unsynthesized compounds using theoretical geometries. Some of the simpler methods require connectivity only. 

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. 

Example 7 A new method for the analysis of iron using pure FeO was replicated with five samples giving these results (in % Fe) 76.95, 77.02, 76.90, 77.20, and 77.50. Does a systematic error exist ... 

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

If the r-value falls short of the formal significance level, this is not to be interpreted as proving the absence of a systematic error. Perhaps the data were insufficient in precision or in number to establish the presence of a constant error. Especially when the calculated value for t is only slightly short of the tabulated value, some additional data may suffice to build up the evidence for a constant error (or the lack thereof). 

Special attention should be paid to one-sided deviation from the control limits, because systematic errors more often cause deviation in one direction than abnormally wide scatter. Two systematic errors of opposite sign would of course cause scatter, but it is unlikely that both would have entered at the same time. It is not necessary that the control chart be plotted in a time sequence. In any... 

Any systematic error that causes a measurement or result to always be too high or too small can be traced to an identifiable source. 

Gardone, M. J. Detection and Determination of Error in Analytical Methodology. Part 11. Gorrection for Gorrigible Systematic Error in the Gourse of Real Sample Analysis, /. Assoc. Off. Anal. Chem. 1983, 66, 1283-1294. 

When an analyst performs a single analysis on a sample, the difference between the experimentally determined value and the expected value is influenced by three sources of error random error, systematic errors inherent to the method, and systematic errors unique to the analyst. If enough replicate analyses are performed, a distribution of results can be plotted (Figure 14.16a). The width of this distribution is described by the standard deviation and can be used to determine the effect of random error on the analysis. The position of the distribution relative to the sample s true value, p, is determined both by systematic errors inherent to the method and those systematic errors unique to the analyst. For a single analyst there is no way to separate the total systematic error into its component parts. 

The goal of a collaborative test is to determine the expected magnitude of ah three sources of error when a method is placed into general practice. When several analysts each analyze the same sample one time, the variation in their collective results (Figure 14.16b) includes contributions from random errors and those systematic errors (biases) unique to the analysts. Without additional information, the standard deviation for the pooled data cannot be used to separate the precision of the analysis from the systematic errors of the analysts. The position of the distribution, however, can be used to detect the presence of a systematic error in the method. 

A validation method used to evaluate the sources of random and systematic errors affecting an analytical method. 

Partitioning of random error, systematic errors due to the analyst, and systematic error due to the method for (a) replicate analyses performed by a single analyst and (b) single determinations performed by several analysts. 

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