Indeterminate sampling error

FIGURE 27-1 Relation between the indeterminate sampling error a, (in percentage) and the number of particles n for the fraction p ranging from 0.0001 to 0.999. Firom Harris and Kratochvil. )... 

A consideration of statistics is required in a discussion of sampling because of the randomness with which samples are acquired. Just as random, indeterminate errors associated with laboratory work (Chapter 1) are dealt with by statistics because there is no other way to deal with them, so also with sampling errors. We may take pains to see that a sample randomly acquired is representative, but the compositions of a series of such samples taken from the same system always vary to some unknown extent. 

An error in an experimental measurement is defined as a deviation of an observed value from the true value. There are two types of errors, determinate and indeterminate. Determinate errors are those that can be controlled by the experimenter and are associated with malfunctioning equipment, improperly designed experiments, and variations in experimental conditions. These are sometimes called human errors because they can be corrected or at least partially alleviated by careful design and performance of the experiment. Indeterminate errors are those that are random and cannot be controlled by the experimenter. Specific examples of indeterminate errors are variations in radioactive counting and small differences in the successive measurements of glucose in a serum sample. 

All spectrometric measurements are subject to indeterminate (random) error, which will affect the accuracy and precision of the concentrations determined using spectrometric methods. A very common source of random error in spectrometric analysis is instrumental noise . Noise can be due to instability in the light source of the instmment, instabihty in the detector, variation in placement of the sample in the hght path, and is often a combination of all these sources of noise and more. Because these errors are random, they cannot be eliminated. Errors in measurement of radiation intensity lead directly to errors in measurement of concentration when using cahbration curves and Beer s Law. 

Control charts The purpose of a control chart is to monitor data from an ongoing series of quantitative measurements so that the occurrence of determinate (systematic) errors (bias), or any changes in the indeterminate (random) errors affecting the precision of replicates can be detected and remedial action taken. The predominant use of control charts is for quality control (QC) in manufacturing industries where a product or intermediate is sampled and analyzed continually in a process stream or periodically from batches. They may also be used in analytical laboratories, such as those involved in clinical or environmental work, to monitor the condition of reagents, standards and instrument components, which may deteriorate over time. 

Sources of Indeterminate Error Indeterminate errors can be traced to several sources, including the collection of samples, the manipulation of samples during the analysis, and the making of measurements. 

When collecting a sample, for instance, only a small portion of the available material is taken, increasing the likelihood that small-scale inhomogeneities in the sample will affect the repeatability of the analysis. Individual pennies, for example, are expected to show variation from several sources, including the manufacturing process, and the loss of small amounts of metal or the addition of dirt during circulation. These variations are sources of indeterminate error associated with the sampling process. 

During the analysis numerous opportunities arise for random variations in the way individual samples are treated. In determining the mass of a penny, for example, each penny should be handled in the same manner. Cleaning some pennies but not cleaning others introduces an indeterminate error. 

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. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

Finally, values of sx are directly proportional to transmittance for indeterminate errors due to fluctuations in source intensity and for uncertainty in positioning the sample cell within the spectrometer. The latter is of particular importance since the optical properties of any sample cell are not uniform. As a result, repositioning the sample cell may lead to a change in the intensity of transmitted radiation. As shown by curve C in Figure 10.35, the effect of this source of indeterminate error is only important at low absorbances. This source of indeterminate errors is usually the limiting factor for high-quality UV/Vis spectrophotometers when the absorbance is relatively small. 

When using a spectrophotometer for which the precision of absorbance measurements is limited by the uncertainty of reading %T, the analysis of highly absorbing solutions can lead to an unacceptable level of indeterminate errors. Consider the analysis of a sample for which the molar absorptivity is... 

A second way to work with the data in Table 14.7 is to treat the results for each analyst separately. Because the repeatability for any analyst is influenced by indeterminate errors, the variance, s, of the data in each column provides an estimate of O rand- A better estimate is obtained by pooling the individual variances. The result, which is called the within-sample variance (s ), is calculated by summing the squares of the differences between the replicates for each sample and that sample s mean, and dividing by the degrees of freedom. 

Errors in the analytical laboratory are basically of two types determinate errors and indeterminate errors. Determinate errors, also called systematic errors, are errors that were known to have occurred, or at least were determined later to have occurred, in the course of the lab work. They may arise from avoidable sources, such as contamination, wrongly calibrated instruments, reagent impurities, instrumental malfunctions, poor sampling techniques, errors in calculations, etc. Results from laboratory work in which avoidable determinate errors are known to have occurred must be rejected or, if the error was a calculation error, recalculated. 

The procedure used to determine whether a given result is unacceptable involves running a series of identical tests on the same sample, using the same instrument or other piece of equipment, over and over. In such a scenario, the indeterminate errors manifest themselves in values that deviate, positively and negatively, from the mean (average) of all the values obtained. Given this brief background, let us proceed to define some terms related to elementary statistics. 

The act of obtaining samples from a bulk system is subject to errors that can be neither detected nor compensated due to the bulk system often being nonhomogeneous and the sample therefore possibly not being exactly representative. Such errors are indeterminate and must be dealt with by statistics. 

Define Quality Control, Quality Assurance, sample, analyte, validation study, accuracy, precision, bias, calibration, calibration curve, systematic error, determinate error, random error, indeterminate error, and outlier. 

Indeterminate errors are errors that cannot be eliminated and are inherent in the analytical technique. Determinate errors are errors whose cause and magnitude can be determined, and they include poor sampling technique, decomposition of the column, change in detector response, improper recorder performance, calculation errors, and operator prejudice or error. 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

In chemical analysis, as in many other sciences, statistical methodologies are unavoidable. The calibration curve constitutes an everyday application, just as an analytical result can only be ascertained if an estimation of the possible error has been considered. Once a measurement has been repeated, a statistical exploitation becomes possible. However, the laws of sampling and tests based upon hypotheses must be understood to avoid non-value conclusions, or to ensure the meaningful quality tests. Systematic errors (user-based, instrumental) or gross errors which lead to results beyond reasonable limits do not enter into this chapter. For the tests most frequently met in chemistry only indeterminate errors are considered here. 

In the second type of quaniiiaiive mass spectrometry for molecular species, analyte concentrations are obtained directly from the heights of the mass spectral peaks. Tor simple mixtures, it is sometimes possible to find peaks at unique m/r values for each component. Under these circumstances, calibration curves of peak heights versus concentration can be prepared and used for analysis of unknowns. More accurate results can ordinarily be rcali/ed, however, by incorporating a lixed amount of an internal standard substance in both samples and calibration standards. The ratio of the peak intensity of the analyte species to that of the internal standard is then plotted as a function of analyle concentration. The internal standard tends to reduce uncertainties arising in sample preparation and introduction. These uncertainties are often a major source ol indeterminate error with ibe small samples needed for mass spectrometry. Internal standards are also used inOCVMS and f.C/MS. For these techniques, the ratio of peak areas serves as the analytical variable. 

Problem 1J3. If the indeterminate error in weighing on a laboratory balance is 0.003 g, what size sample should you take to keep the relative error to 1.0% ... 

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