Experimental error, random systematic

We also discuss the analysis of the accuracy of experimental data. In the case that we can directly measure some desired quantity, we need to estimate the accuracy of the measurement. If data reduction must be carried out, we must study the propagation of errors in measurements through the data reduction process. The two principal types of experimental errors, random errors and systematic errors, are discussed separately. Random errors are subject to statistical analysis, and we discuss this analysis. 

The two types of experimental errors are systematic errors and random errors. 

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. 

When standardizing a solution of NaOH against potassium hydrogen phthalate (KHP), a variety of systematic and random errors are possible. Identify, with justification, whether the following are systematic or random sources of error, or if they have no effect. If the error is systematic, then indicate whether the experimentally determined molarity for NaOH will be too high or too low. The standardization reaction is... 

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. 

Experimental errors come from two different sources, termed systematic and random errors. However, it is sometimes difficult to distinguish between them, and many experiments have a combination of both types of error. 

Systematic Error and Random Error. An analytical result can be affected by a combination of two different kinds of experimental error systematic error and random error. Systematic errors are associated with the accuracy of an analytical method and are the difference of the mean value from the true value. The measure of this difference is the bias. The mean is only an estimate of the true value because only a limited number of experiments can be carried out. The estimate becomes better when more experiments have been carried out. In conclusion, since all the measurements are estimates, the true value can be approached only with replicate measurements. 

Every measurement has some uncertainty, which is called experimental error. Conclusions can be expressed with a high or a low degree of confidence, but never with complete certainty. Experimental error is classified as either systematic or random. 

Based on previous testing of the research subject, the design of the full factorial experiment 23 with one replication to determine experimental error has been chosen. To eliminate the influence of systematic error in doing the experiment, the sequence of doing design point-trials, in accord with theory of design of experiments, has been completely random. The outcomes are given in Table 2.107. 

In general, results from investigations based on measurements may be falsified by three principal types of errors gross, systematic, and random errors. In most cases gross errors are easily detected and avoidable. Systematic errors (so-called determinate errors) affect the accuracy and therefore the proximity of an empirical (experimental) result to the true result, which difference is called bias. Random errors (so-called indeterminate errors) influence the precision of analytical results. Sometimes precision is used synonymously with reproducibility and repeatability. Note that these are different measures of precision, which, in turn, is not related to the true value. 

All experimental measurements are affected by errors. In general, experimental errors are made out of systematic errors and random errors. Systematic errors show a dependence on the operating conditions and may be caused, e.g., by calibration errors of sensors. Since these errors are absent in a well-performed experimental campaign and can be corrected by an improved experimental practice, they are not considered any more in this context. 

This comparison never shows a perfect correspondence between models and experiments because of modeling and measurement errors. In fact, even if the presence of systematic experimental errors can be excluded, systematic errors generated by the inadequacy of the model must be added to random experimental errors for each measured variable (m = 1,..., Am) and each experimental time (j = 1,..., Ad), the errors generated by the model are defined as... 

If the model perfectly describes the experiments, the sample of residual errors does not contain systematic errors thus, it must be compatible with the statistical distribution of the random experimental errors. All the systematic discrepancies eventually observed are attributed to the mathematical model, thus allowing a comparison between alternative models, since systematic errors can be decreased if a better model becomes available. 

The theoretical results quoted here and below make use of the most recent calculations of the reduced mass and recoil corrections [30,31,32] and values of the fundamental constants (see [24,25]). We stress once more that systematic and random uncertainties due to the calibration procedure completely dominate the quoted experimental errors detailed examination of the data suggests that the total contribution from other sources is less than 50 kHz, despite the use of cell excitation and the need to extrapolate to... 

If the original model is sufficiently perfect, the linearization of the problem adequate, the measurements unbiased (no systematic error), and the covariance matrix of the observations, 0y, a true representation of the experimental errors and their correlations, then c2 (Eq. 21c) should be near unity [34], If 0y is indeed an honest assessment of the experimental errors, but a2 is nonetheless (much) larger than unity, model deficiencies are the most frequent source of this discrepancy. Relevant variables probably exist that have not been included in the model, and the experimental precision is hence better than can be utilized by the available model. Model errors have then been treated as if they were experimental random errors, and the results must be interpreted with great caution. In this often unavoidable case, it would clearly be meaningless to make a difference between a measurement with a small experimental error (below the useful limit of precision) and another measurement with an even smaller error (see ref. [41 ). A deliberate modification of the variance-covariance matrix y towards larger and more equal variances might then be indicated, which results in a more equally weighted and less correlated matrix. 

In the absence of systematic errors, the observed response, y, will be an unbiassed estimate of the true response, and the error term can be analyzed by statistical methods. In carefully executed synthesis experiments it is reasonable to assume that random errors occur independently of each other and that the observed variation of these random events are normally and independently distributed. A variation of the experimental conditions is considered to be significant if it produces a variation of the observed response outside the noise level given by the experimental error. Significant variations in this respect can then be analyzed by comparison to the error variation through known statistical distributions based on the normal distribution, e.g. the t distribution, the F distribution, and the %2 distribution. 

The objective of any review of experimental values is to evaluate the accuracy and precision of the results. The description of a procedure for the selection of the evaluated values (EvV) of electron affinities is one of the objectives of this book. The most recent precise values are taken as the EvV. However, this is not always valid. It is better to obtain estimates of the bias and random errors in the values and to compare their accuracy and precision. The reported values of a property are collected and examined in terms of the random errors. If the values agree within the error, the weighted average value is the most appropriate value. If the values do not agree within the random errors, then systematic errors must be investigated. In order to evaluate bias errors, at least two different procedures for measuring the same quantity must be available. 

If the variables do not have any influence whatsoever on the response, the response surface is completely flat and the true value of all BpS and Byzs is zero. The estimated effects in such cases would be nothing but different average summations of the experimental error. If we have randomized the order of execution of the experiments in the design, and have done all what we can do to avoid systematic errors, the set of estimated model parameters, [ft, b2,...b, would be a... 

In Chapter 3 it was discussed how the presence of a random error can be handled by statistical tools. The precautions which must be taken by the experimenter not to violate the assumption of independencies of the experimental error is randomization, which allows certain time-dependent systematic errors to be broken down and turned into random errors. There are, however, sources of error which can be suspected to produce systematic deviations which cannot be counteracted by randomization. In such cases, forseeable sources of systematic variation can be brought under control by dividing the whole set of experiments into smaller blocks which can be run under more homogeneous conditions. By a proper arrangement of these blocks, the systematic variation can be isolated through comparison of the between-block variation. Some examples where splitting the series of experiments into blocks is appropriate are ... 

In order to achieve satisfactory accuracy, it is essential to avoid errors. However, distinction should be made between experimental errors and mistakes. Mistakes are the type of errors ( blunders ) which necessitate the repetition of the test, whereas experimental errors are inherent to the test and may result from random error, from bias or from both. Random errors may be due to slight fluctuations in measuring enzyme activity, variations in temperature, ionic composition of the sample, etc., and can be minimized by the use of standards. Bias is a systematic error (storage effects, improper... 

Some of the more recent publications use statistical comparisons in which all errors between data and the rate equation are expressed by the magnitude of say, one standard deviation. The best ht is then identihed as the equation that gives a deviation value nearest to unity or similar criterion. Apparently, this is widely regarded as acceptable practice, but it may not establish whether the deviations arise from random experimental error or as a consequence of systematic variations, perhaps occurring within a limited a range and capable of some other explanation. 

The consideration of systematic experimental errors involves semantic problems as well as questions of proper statistical treatment. Although random errors are well-defined in the mathematical sense, the same cannot be said for systematic errors. 

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. 

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. 

All experimental uncertainty is due to either random errors or systematic errors. 

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