Error, analytical random

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. 

X and Y for each method. When the difference between X and Y is calculated (as d) the systematic error drops out so that the difference (d) between X and Y contains no systematic errors, only random errors. We then estimate the precision by using the difference quantities. The difference between the true analyte concentrations of X and Y represents the true analyte difference between X and Y without the systematic error, but... 

If error is random and follows probabilistic (normally distributed) variance phenomena, we must be able to make additional measurements to reduce the measurement noise or variability. This is certainly true in the real world to some extent. Most of us having some basic statistical training will recall the concept of calculating the number of measurements required to establish a mean value (or analytical result) with a prescribed accuracy. For this calculation one would designate the allowable error (e), and a probability (or risk) that a measured value (m) would be different by an amount (d). 

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. 

Accuracy (absence of systematic errors) and uncertainty (coefficient of variation or confidence interval) as caused by random errors and random variations in the procedure are the basic parameters to be considered when discussing analytical results. As stressed in the introduction, accuracy is of primary importance however, if the uncertainty in a result is too high, it cannot be used for any conclusion concerning, e.g. the quality of the environment or of food. An unacceptably high uncertainty renders the result useless. When evaluating the performance of an analytical technique, all basic principles of calibration, of elimination of sources of contamination and losses, and of correction for interferences should be followed (Prichard, 1995). 

In all measurements there are errors that may be determinate or indeterminate. Indeterminate errors are random errors that cannot be eliminated and are inherent in the analytical technique. When indeterminate errors are minimized, high precision is possible. Determinate errors are errors whose cause and magnitude can be determined. If the determi-... 

Random analytical method errors (as random fluctuations in a chemical laboratory procedure),... 

Expression and Interpretation of Results. Archaeological interpretation of a radiocarbon age may depend critically on the error associated with that age. Errors are commonly expressed as a variance range attached to the central number (e.g., 2250 80 years). The 80 years in this example may correspond to the random error for a single analytical step. Both decay and direct-atom counting are statistical in nature, and lead to errors that vary as the square root of the number of counts. The error may also be expressed as the overall random experimental error (the sum of individual errors.) Overall random error can be determined only by analyzing replicate samples. 

The problem of matrix mismatch is always attendant when one analyses an unknown sample with the same matrix using a fixed, previously determined, calibration function. Not uncommonly, an analytical procedure is developed to cover a range of sample matrices in such a way that an overall calibration function can be used. An error due to matrix mismatch is therefore inevitable if not necessary significant. Commonly regarded as systematic for a sample with a particular matrix, the error becomes random when a population of samples to which the procedure applies is considered this in fact constitutes an inherent part of the total variability associated with the analytical procedure. 

As analytical methods usually have to be applicable over a wide range of concentrations, a new method is often compared with a standard method by analysis of samples in which the analyte concentration may vary over several powers of 10. In this case it is inappropriate to use the paired t-test since its validity rests on the assumption that any errors, either random or systematic, are independent of concentration. Over wide ranges of concentration this assumption may no longer be true. An alternative method in such cases is linear regression (see Section 5.9) but this approach also presents difficulties. 

Error sources in HPLC are so manifold that they cannot all be discussed completely. Nevertheless. one can distinguish between random errors and systematic errors. Whereas random errors are relatively easy to recognize and avoid, systematic errors often lead to false quantitative interpretation of chromatograms. Two simple ways of recognizing a systematic error in a newly developed (or newly applied) analytical method are either to test this new method with certified materials or—if an old (i.e., validated) method already exists—to compare the result from the old and the new methods. If the new method shows good reproducibility, a correction factor can be calculated easily by... 

The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 

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

The design of a collaborative test must provide the additional information needed to separate the effect of random error from that due to systematic errors introduced by the analysts. One simple approach, which is accepted by the Association of Official Analytical Chemists, is to have each analyst analyze two samples, X and Y, that are similar in both matrix and concentration of analyte. The results obtained by each analyst are plotted as a single point on a two-sample chart, using the result for one sample as the x-coordinate and the value for the other sample as the -coordinate. ... 

The "feedback loop in the analytical approach is maintained by a quality assurance program (Figure 15.1), whose objective is to control systematic and random sources of error.The underlying assumption of a quality assurance program is that results obtained when an analytical system is in statistical control are free of bias and are characterized by well-defined confidence intervals. When used properly, a quality assurance program identifies the practices necessary to bring a system into statistical control, allows us to determine if the system remains in statistical control, and suggests a course of corrective action when the system has fallen out of statistical control. 

In the previous section we described several internal methods of quality assessment that provide quantitative estimates of the systematic and random errors present in an analytical system. Now we turn our attention to how this numerical information is incorporated into the written directives of a complete quality assurance program. Two approaches to developing quality assurance programs have been described a prescriptive approach, in which an exact method of quality assessment is prescribed and a performance-based approach, in which any form of quality assessment is acceptable, provided that an acceptable level of statistical control can be demonstrated. 

The principal tool for performance-based quality assessment is the control chart. In a control chart the results from the analysis of quality assessment samples are plotted in the order in which they are collected, providing a continuous record of the statistical state of the analytical system. Quality assessment data collected over time can be summarized by a mean value and a standard deviation. The fundamental assumption behind the use of a control chart is that quality assessment data will show only random variations around the mean value when the analytical system is in statistical control. When an analytical system moves out of statistical control, the quality assessment data is influenced by additional sources of error, increasing the standard deviation or changing the mean value. 

In this case results don t depend on random errors. The shape and size of analytical signal can vary smoothly. Physicochemical simulation is difficult because of irreproducibility many experimental factors. 

Data collected by modern analytical instalments are usually presented by the multidimensional arrays. To perform the detection/identification of the supposed component or to verify the authenticity of a product, it is necessary to estimate the similarity of the analyte to the reference. The similarity is commonly estimated with the use of the distance between the multidimensional arrays corresponding to the compared objects. To exclude within the limits of the possible the influence of the random errors and the nonreproductivity of the experimental conditions and to make the comparison of samples more robust, it is possible to handle the arrays with the use of the fuzzy set theory apparatus. 

The molecular absoi ption spectra, registered at a lower temperature (e.g. 700 °C for iodide or chloride of potassium or sodium), enable one to find the absorbance ratio for any pair of wavelengths in the measurement range. These ratios can be used as a correction factor for analytical signal in atomic absoi ption analysis (at atomization temperatures above 2000 °C). The proposed method was tested by determination of beforehand known silicon and iron content in potassium chloride and sodium iodide respectively. The results ai e subject to random error only. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

X-ray emission spectrography, in common with other analytical methods, is subject to errors of different kinds. Lacking better information, we shall usually assume these errors to be independent and random. (Drift caused by changes in the electronic system is definitely not random.) Before we consider errors in general, we shall examine one that is not only important and unavoidable, but that also sets x-ray... 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

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