Systematic and Random Errors

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. 

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

These measurements with their inherent errors are the bases for numerous fault detection, control, and operating and design decisions. The random and systematic errors corrupt the decisions, amplifying their uncertainty and, in some cases, resulting in substantially wrong decisions. 

Single-Module Analysis Consider the single-module unit shown in Fig. 30-10. If the measurements were complete, they would consist of compositions, flows, temperatures, and pressures. These would contain significant random and systematic errors. Consequently, as collected, they do not close the constraints of the unit being studied. The measurements are only estimates of the actual plant operation. If the actual operation were known, the analyst could prepare a scatter diagram comparing the measurements to the actual values, which is a useful analysis tool Figure 30-19 is an example. 

The methodical elaboration is included for estimation of random and systematic errors by using of single factor dispersion analysis. For this aim the set of reference samples is used. X-ray analyses of reference samples are performed with followed calculation of mass parts of components and comparison of results with real chemical compositions. Metrological characteristics of x-ray fluorescence silicate analysis are established both for a-correction method and simplified fundamental parameter method. It is established, that systematic error of simplified FPM is less than a-correction method, if the correction of zero approximation for simplified FPM is used by preliminary established correlation between theoretical and experimental set data. 

The means and habit of making highly precise measurements, with careful attention to the identification of sources of random and systematic error, were well established by the period I am discussing. According to a recent historical essay by... 

Gardner [6] has reported a detailed statistical study involving ten laboratories of the determination of cadmium in coastal and estuarine waters by atomic absorption spectrometry. The maximum tolerable error was defined as 0.1 ptg/1 or 20% of sample concentration, whichever is the larger. Many laboratories participating in this work did not achieve the required accuracy for the determination of cadmium in coastal and estuarine water. Failure to meet targets is attributable to both random and systematic errors. 

Write down what you think the difference is between random and systematic errors. 

Figure 6.11 Illustration of the differences between random and systematic errors, using the example of delivering liquid from a 25 ml Class A pipette.

If you identified most of the differences between random and systematic errors correctly, you obviously have a good understanding of the nature of error in chemical measurement. If you had difficulty with this do not worry, but now is the time to get these ideas clear in your mind. 

As we have seen in previous sections, the result of a measurement is not complete unless an estimate of the uncertainty associated with the result is available. In any measurement procedure, there will be a number of aspects of the procedure that will contribute to the uncertainty. Uncertainty arises due to the presence of both random and systematic errors. To obtain an estimate of the uncertainty in a result, we need to identify the possible sources of uncertainty, obtain an estimate of their magnitude and combine them to obtain a single value which encompasses the effect of all the significant sources of error. This section introduces a systematic approach to evaluating uncertainty. 

Because measurements always contain some type of error, it is necessary to correct their values to know objectively the operating state of the process. Two types of errors can be identified in plant data random and systematic errors. Random errors are small errors due to the normal fluctuation of the process or to the random variation inherent in instrument operation. Systematic errors are larger errors due to incorrect calibration or malfunction of the instruments, process leaks, etc. The systematic errors, also called gross errors, occur occasionally, that is, their number is small when compared to the total number of instruments in a chemical plant. 

By plotting the sum T and difference D in time ordered sequence the variation of random and systematic errors can be monitored between analytical runs. 

The primary purpose of any quality control scheme is to identify ("flag") significant performance changes. The two-sample quality control scheme described above effectively identifies performance changes and permits separation of random and systematic error contributions. It also permits rapid evaluation of a specific analytical result relative to previous data. Graphical representation of these data provide effective anomaly detection. The quality control scheme presented here uses two slightly different plot formats to depict performance behavior. 

Error (of measurement) is the sum of random and systematic errors of one measurement. Since a true value cannot be determined, in practice a reference quantity value is used. Each individual result of a measurement will have its own associated error. 

It must always be remembered that optimisation is not an exact science and, therefore, it is sometimes difficult to define confidence limits in the final optimised values for the coefficients used in the thermodynamic models. The final outcome is at least dependent on the number of experimental measurements, their accuracy and the ability to differentiate between random and systematic errors. Concepts of quality can, therefore, be difficult to define. It is the author s experience that it is quite possible to have at least two versions of an optimised diagram with quite different underlying thermodynamic properties. This may be because only experimental enthalpy data were available and different entropy fiinctions were chosen for the different phases. Also one of the versions may have rejected certain experimental measurements which the other version accepted. This emphasises the fact that judgement plays a vital role in the optimisation process and the use of optimising codes as black boxes is dangerous. 

Obtairiing kinetic data is very tedious, and it requires great care to avoid both random and systematic errors. For this reason, it is very common to assemble computer-based data acquisition systems, frequently with simple personal computers equipped with data acquisition analog-to-digital capabilities and graphics. These computers can be programmed patienfly to acquire the necessary data, make frequent calibrations, vary parameters such as temperature and concentration, analyze data statistics, and print out parameters. 

The analyst conducting a method validation must assess any systematic effects that need to be corrected for or included in the measurement uncertainty of the results. The interplay between random and systematic error is complex and something of a moveable feast. The unique contribution of each analyst to the result of a chemical measurement is systematic, but in a large... 

All equipment used should be checked regularly and monitored carefully to reduce random and systematic errors. The checking and maintenance should be documented and reviewed regularly. 

The accuracy of an analytical method is estimated as the percentage difference (bias) between the mean values generated by the method and the true or known concentrations. Accuracy is usually synonymous with systematic errors. Systematic errors cause all the results in a series of replicates to deviate from the true value of the measured quantity in a particular sense (i.e., all the results are too high or all are too low) (20). Accuracy has also been used in recent years to refer to any error causing a single measurement to deviate from the true value (i.e., to encompass elements of random and systematic errors) (21). 

Establishment of the error analysis and deviation of the experimentally measured values resulting from random and systematic errors in the present investigation has been made previously by Jaques and Furter (28). The fluctuations in the barometric pressure are indicated in Tables I-XVI for each system, and... 

A detailed reassessment of several multi-parameter correlations describing the solvent effects on the rates of four solvolytic SN reactions has shown that great caution should be exercised when using these relationships.85 The conclusions based on the multi-parameter correlations are not reliable because (i) both random and systematic errors have been underestimated, (ii) mechanisms may change when different substrates are involved, (iii) data extrapolated from different temperatures, and (iv) only small numbers of samples have been used in establishing these relationships. 

Garcia Alonso, J. I., Determination of fission products and actinides by inductively coupled plasma mass spectrometry using isotope dilution analysis A study of random and systematic errors, Anal. Chim. Acta, 312, 57-78, 1995. 

The procedure outlined above has many possible sources of both random and systematic error. The measurements of volume and of mass will not be perfectly accurate if the equipment has been correctly calibrated and the laboratory technique is good, these errors are random (equally likely to be positive or negative). The masses of the two nuclei are not perfectly known, but these errors can be assumed to be random as well (the error bars are the results of many careful measurements). The silver chloride and water will both have impurities, which will tend to make systematic errors. Some impurities (e.g., chloride ions in the water) would tend to make the measured solubility product smaller than the true value. Some impurities (e.g., sodium chloride in the silver chloride) would tend to make the measured solubility product larger than the true value. Finally, even without impurities, there is one (probably small) systematic error... 

Before a new analytical method or sample preparation technique is to be implemented, it must be validated. The various figures of merit need to be determined during the validation process. Random and systematic errors are measured in terms of precision and bias. The detection limit is established for each analyte. The accuracy and precision are determined at the concentration range where the method is to be used. The linear dynamic range is established and the calibration sensitivity is measured. In general, method validation provides a comprehensive picture of the merits of a new method and provides a basis for comparison with existing methods. 

Analytical measurements should be made with properly tested and documented procedures. These procedures should utilise controls and calibration steps to minimise random and systematic errors. There are basically two types of controls (a) those used to determine whether or not an analytical procedure is in statistical control, and (b) those used to determine whether or not an analyte of interest is present in a studied population but not in a similar control population. The purpose of calibration is to minimise bias in the measurement process. Calibration or standardisation critically depends upon the quality of the chemicals in the standard solutions and the care exercised in their preparation. Another important factor is the stability of these standards once they are prepared. Calibration check standards should be freshly prepared frequently, depending on their stability (Keith, 1991). No data should be reported beyond the range of calibration of the methodology. Appropriate quality control samples and experiments must be included to verify that interferences are not present with the analytes of interest, or, if they are, that they be removed or accommodated. 

Data values are subject to both random and systematic errors. The effect of random errors can be reduced by increasing the number N of observations, but this is an inefficient process since, as shown in Chapter II, the error in the average of N values varies as 1 /. Thus one must quadraple the number of measurements in order to cut the random error in half A much better approach is to improve the design and sensitivity of the apparatus. Even then, you must guard against systematic errors and personal bias in recording the data. 

If there are no deviations from the Lambert-Beer law, the accuracy of the results is limited only by measuring errors. An analysis of the random and systematic errors that occur during the determination of the IR absorption coefficient of liquids and their influence on the precision and accuracy of the results has been provided by Staat et al. 

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