Analytical error

Effective computer codes for the optimization of plants using process simulators require accurate values for first-order partial derivatives. In equation-based codes, getting analytical derivatives is straightforward, but may be complicated and subject to error. Analytic differentiation ameliorates error but yields results that may involve excessive computation time. Finite-difference substitutes for analytical derivatives are simple for the user to implement, but also can involve excessive computation time. 

The limits of quantification of an analytical procedure are the lowest and largest amounts of the targeted substance in the sample that can be quantitatively determined under the prescribed experimental conditions with well-established measurement error (analytical bias and precision). Consequently, the dynamic range of an analytical procedure is the range between the lower and the upper limits of quantification within which a measured result is expected to have acceptable levels of bias and precision. 

Zinc metabolic studies in which the difference between the total dietary intake and the total excretion in faeces and in urine, is measured over a period of several days. The difficulties of this procedure include problems of ensuring complete collections of excreta and measurement of losses in sweat, shed skin and hair. These losses although minor, may cause cumulative errors. Analytical problems caused by accidental zinc contamination are also a worry, although here the use of stable zinc isotopes and mass spectrometry or neutron activation is an advantage. 

Since all of the above information is available in the computer, we print it out on a daily basis on a Quality Control Report. This report is then reviewed by the Instrument Engineer (in charge of metering), the Plant Chemist (in charge of analyses) and the Technical Coordinator (in charge of software) to resolve whether discrepancies are a result of measurement errors, analytical error, or data input error (Figure 8). 

The analysis of solids is time consuming and is prone to many sources of error. Analytical chemists have been trying for many years to analyze solid samples directly, without having to dissolve than. For some types of samples, this can be done by AAS. Solids can be analyzed using a glow discharge (GD) atomizer, by inserting small pieces or particles of sample directly into the flame or furnace or by the use of laser ablation. 

The effects of pressure are especially sensitive at high temperatures. The analytical expression [4.71] given by the API is limited to reduced temperatures less than 0.8. Its average error is about 5%. 

To be more precise, this error occurs in the limit /c — oo with Ef = 0(1) and step-size k such that k /ii = const. 3> 1. This error does not occur if Ef = 0 for the analytic problem, i.e., in case there is no vibrational energy in the stiff spring which implies V,. = U. 

In analytical chemistry, a number of identical measurements are taken and then an error is estimated by computing the standard deviation. With computational experiments, repeating the same step should always give exactly the same result, with the exception of Monte Carlo techniques. An error is estimated by comparing a number of similar computations to the experimental answers or much more rigorous computations. 

Each observation in any branch of scientific investigation is inaccurate to some degree. Often the accurate value for the concentration of some particular constituent in the analyte cannot be determined. However, it is reasonable to assume the accurate value exists, and it is important to estimate the limits between which this value lies. It must be understood that the statistical approach is concerned with the appraisal of experimental design and data. Statistical techniques can neither detect nor evaluate constant errors (bias) the detection and elimination of inaccuracy are analytical problems. Nevertheless, statistical techniques can assist considerably in determining whether or not inaccuracies exist and in indicating when procedural modifications have reduced them. 

An analytical procedure is often tested on materials of known composition. These materials may be pure substances, standard samples, or materials analyzed by some other more accurate method. Repeated determinations on a known material furnish data for both an estimate of the precision and a test for the presence of a constant error in the results. The standard deviation is found from Equation 12 (with the known composition replacing /x). A calculated value for t (Eq. 14) in excess of the appropriate value in Table 2.27 is interpreted as evidence of the presence of a constant error at the indicated level of significance. 

Analytical chemists use a variety of glassware to measure volume, several examples of which are shown in Figure 2.4. The type of glassware used depends on how exact the volume needs to be. Beakers, dropping pipets, and graduated cylinders are used to measure volumes approximately, typically with errors of several percent. 

Analytical methods may be divided into three groups based on the magnitude of their relative errorsd When an experimental result is within 1% of the correct result, the analytical method is highly accurate. Methods resulting in relative errors between 1% and 5% are moderately accurate, but methods of low accuracy produce relative errors greater than 5%. 

When designing and evaluating an analytical method, we usually make three separate considerations of experimental error. First, before beginning an analysis, errors associated with each measurement are evaluated to ensure that their cumulative effect will not limit the utility of the analysis. Errors known or believed to affect the result can then be minimized. Second, during the analysis the measurement process is monitored, ensuring that it remains under control. Finally, at the end of the analysis the quality of the measurements and the result are evaluated and compared with the original design criteria. This chapter is an introduction to the sources and evaluation of errors in analytical measurements, the effect of measurement error on the result of an analysis, and the statistical analysis of data. 

An error due to limitations in the analytical method used to analyze a sample. 

Method Errors Determinate method errors are introduced when assumptions about the relationship between the signal and the analyte are invalid. In terms of the general relationships between the measured signal and the amount of analyte... 

Personal Errors Finally, analytical work is always subject to a variety of personal errors, which can include the ability to see a change in the color of an indicator used to signal the end point of a titration biases, such as consistently overestimating or underestimating the value on an instrument s readout scale failing to calibrate glassware and instrumentation and misinterpreting procedural directions. Personal errors can be minimized with proper care. 

Effect of Constant Positive Determinate Error on Analysis of Sample Containing 50% Analyte (%w/w)... 

A proportional determinate error, in which the error s magnitude depends on the amount of sample, is more difficult to detect since the result of an analysis is independent of the amount of sample. Table 4.6 outlines an example showing the effect of a positive proportional error of 1.0% on the analysis of a sample that is 50.0% w/w in analyte. In terms of equations 4.4 and 4.5, the reagent blank, Sreag, is an example of a constant determinate error, and the sensitivity, k, may be affected by proportional errors. 

Effect of a constant determinate error on the reported concentration of analyte. 

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. 

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 probability of a type 1 error is inversely related to the probability of a type 2 error. Minimizing a type 1 error by decreasing a, for example, increases the likelihood of a type 2 error. The value of a chosen for a particular significance test, therefore, represents a compromise between these two types of error. Most of the examples in this text use a 95% confidence level, or a = 0.05, since this is the most frequently used confidence level for the majority of analytical work. It is not unusual, however, for more stringent (e.g. a = 0.01) or for more lenient (e.g. a = 0.10) confidence levels to be used. 

---