Random errors sources

The observed frequencies with 3, 5, 7 and 14 different and equally probable random error sources are shown in Fig. 3,6c-f. With many different sources of experimental error, it is seen that the frequency of the experimental response data can be approximately described by the bell-shaped curve in Fig. 3.6g. 

If appropriate proficiency scheme results are not available this detailed examination of each step in the analytical process, the so-called bottom-up approach to uncertainty estimates, must be used. In doing so, it is necessary to use a number of equations that allow the calculation of the overall random error arising in a process containing two or more steps. This is technically referred to as the propagation of errors. The basic problem in such cases is that, if the two or more experimental steps have random error sources that are independent of each other, such errors will partly, but not wholly, cancel each other out. In the following equations the final result of the analysis is called X, and the individual analytical steps contributing to X are A, B, etc. The random errors in X, A, B, etc. (normally standard deviations) are given the symbols SX (which we wish to determine), M, 5B, etc. Then,... 

The biological test result represents a lower degree of precision reflecting one or more random error sources not associated with the other chemicals studied. 

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. 

The goal of a collaborative test is to determine the expected magnitude of ah three sources of error when a method is placed into general practice. When several analysts each analyze the same sample one time, the variation in their collective results (Figure 14.16b) includes contributions from random errors and those systematic errors (biases) unique to the analysts. Without additional information, the standard deviation for the pooled data cannot be used to separate the precision of the analysis from the systematic errors of the analysts. The position of the distribution, however, can be used to detect the presence of a systematic error in the method. 

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... 

The flowsheet shown in the introduction and that used in connection with a simulation (Section 1.4) provide insights into the pervasiveness of errors at the source, random errors are experienced as an inherent feature of every measurement process. The standard deviation is commonly substituted for a more detailed description of the error distribution (see also Section 1.2), as this suffices in most cases. Systematic errors due to interference or faulty interpretation cannot be detected by statistical methods alone control experiments are necessary. One or more such primary results must usually be inserted into a more or less complex system of equations to obtain the final result (for examples, see Refs. 23, 91-94, 104, 105, 142. The question that imposes itself at this point is how reliable is the final result Two different mechanisms of action must be discussed ... 

Gross errors are generated by human mistakes or by instrumental or computational error sources. Depending on whether they are short- or longterm effects, they may have systematic or random character. Frequently, it is easy to perceive and to correct for them. They will not play any role in the following discussion. 

Repeated measurements of the same measurand on a series of identical measuring samples result in random variations (random errors), even under carefully controlled constant experimental conditions. These should include the same operator, same apparatus, same laboratory, and short interval of the time between measurements. Conditions such as these are called repeatability conditions (Prichard et al. [2001]). The random variations are caused by measurement-related technical facts (e.g., noise of radiation and voltage sources), sample properties (e.g., inhomogeneities), as well as chemical or physical procedure-specific effects. 

If you answered (b), perhaps you were thinking of the spread of values obtained from replicate measurements. While these do indeed form a range, one such range will relate to only one source of uncertainty and there may be several sources of uncertainty affecting a particular measurement. The precision of a measurement is an indication of the random error associated with it. This takes no account of any systematic errors that may be connected with the measurement. It is important to realize that uncertainty covers the effects of both random error and systematic error and, moreover, takes into account multiple sources of these effects where they are known to exist and are considered significant. 

Measurements can contain any of several types of errors (1) small random errors, (2) systematic biases and drift, or (3) gross errors. Small random errors are zero-mean and are often assumed to be normally distributed (Gaussian). Systematic biases occur when measurement devices provide consistently erroneous values, either high or low. In this case, the expected value of e is not zero. Bias may arise from sources such as incorrect calibration of the measurement device, sensor degradation, damage to the electronics, and so on. The third type of measurement... 

How to best describe this broadening we expect to occur One way is by analogy to random error in measurements. We know or assume there is a truly correct answer to any measurement of quantity present and attempt to determine that number. In real measurements there are real, if random, sources of error. It is convenient to talk about standard deviation of the measurement but actually the error in measurement accumulates as the sum of the square of each error process or variance producing mechanism or total variance = sum of the individual variances. If we ignore effects outside the actual separation process (e.g. injection/spot size, connecting tubing, detector volume), this sum can be factored into three main influences ... 

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. 

The relationship takes into account that the day-to-day samples determined are subject to error from several sources random error, instrument error, observer error, preparation error, etc. This view is the basis of the process of fitting data to a model, which results in confidence intervals based on the intrinsic lack of fit and the random variation in the data. 

Random error is the divergence, due to chance alone, of an observation on a sample from the true population value, leading to lack of precision in the measurement of an association. There are three major sources of random error individual/biological variation, sampling error, and measurement error. Random error can be minimized but can never be completely eliminated since only a sample of the population can be studied, individual variation always occurs, and no measurement is perfectly accurate. 

Validation - Simulated data sets created from known, source contributors and perturbed by random error should be presented to models, and their source contribution predictions should be compared to the known contributions. Several models should be applied to the same data set and their results compared. 

The experiments are performed according to the chosen design and a response or a number of responses are measured. The sequence in which the experiments are performed can influence the estimation of the effect of a factor [36]. The reason for this lies in the fact that the measurements can be influenced by different sources of error. Each measurement is influenced by uncontrolled factors that cause random error. Measurements can also be influenced by systematic errors or by systematic errors caused by drift (linear drift due to time-dependent factors). The occurrence of systematic errors or of drift will affect the estimation of the effects of the factors fi om the design [36]. 

There is, however, one obvious difference between a mathematical model and a physical model (or the real system itself). The response of the former to the same set of conditions is always identical. In physical experiments, where results are measured rather than calculated, there are inevitably random errors which may be appreciable. As already pointed out, mathematical models are usually to some extent imperfect in other words, they do contain systematic errors. The important point is that these imperfections are always reproduced in the same way, even though their ultimate source may have been random errors in data on which the model was based. This point has been stressed because it is important to recognize that only partial use of methods from statistical treatments of design of experiments is involved in what follows. The use of these methods here is only for the purpose of studying the geometry of response with respect to the controllable variables. No consideration of probability or of error enters into the discussion. 

What is the uncertainty in the molecular mass of 02 On the inside cover of this book, we find that the atomic mass of oxygen is 15.9994 0.000 3g/mol. The uncertainty is not mainly from random error in measuring the atomic mass. The uncertainty is predominantly from isotopic variation in samples of oxygen from different sources. That is, oxygen from one source could have a mean atomic mass of 15.999 1 and oxygen from another source could have an atomic mass of 15.999 7. The atomic mass of oxygen in a particular lot of reagent has a systematic uncertainty. It could be relatively constant at 15.999 7 or 15.999 1, or any value in between, with only a small random variation around the mean value. 

Calibration of FAGE1 from a static reactor (a Teflon film bag that collapses as sample is withdrawn) has been reported (78). In static decay, HO reacts with a tracer T that has a loss that can be measured by an independent technique T necessarily has no sinks other than HO reaction (see Table I) and no sources within the reactor. From equation 17, the instantaneous HO concentration is calculated from the instantaneous slope of a plot of ln[T] versus time. The presence of other reagents may be necessary to ensure sufficient HO however, the mechanisms by which HO is generated and lost are of no concern, because the loss of the tracer by reaction with whatever HO is present is what is observed. Turbulent transport must keep the reactor s contents well mixed so that the analytically measured HO concentration is representative of the volume-averaged HO concentration reflected by the tracer consumption. If the HO concentration is constant, the random error in [HO] calculated from the tracer decay slope can be obtained from the slope uncertainty of a least squares fit. Systematic error would arise from uncertainties in the rate constant for the T + HO reaction, but several tracers may be employed concurrently. In general, HO may be nonconstant in the reactor, so its concentration variation must be separated from noise associated with the [T] measurement, which must therefore be determined separately. 

Although the computed and experimentally estimated [84] (extrapolated to zero frequency) hyperpolarizability for Ne are in perfect agreement, the results for Ar do not agree within their respective uncertainties. Actually, this is due to an experimental uncertainty estimate that is almost certainly too optimistic. The uncertainty quoted for the experimental value is derived from the experimental statistics, that is, it is a measure of random error in the measurement. It contains essentially no contribution from any possible source of systematic error. In fact, Shelton believes that a more realistic uncertainty would be 20 or perhaps 30 [85]. If that were the case there would be no disagreement between theory and experiment. This is an excellent illustration of the dangers of relying on a given experimental estimate or uncertainty. It is always necessary to ascertain exactly what the experimentalist means by his/her error bars. 

Glazov and Vigdorovich (1969) classified systematic and random errors according to their source inaccuracy inherent in loading and in the optical system. 

Dither. Starting from Robert s pioneering paper [Roberts, 1976], the use of dither in audio was seriously analyzed by Vanderkooy and Lipshitz [Vanderkooy and Lipshitz, 1984], The basic idea is simple to whiten the quantization error, a random error signal is introduced. While the introduction of noise will make the signal noisier , it will also decorrelate the quantization error from the input signal (but not totally). Vanderkooy and Lipshitz also propose the use of triangular dither derived from the sum of two uniform random sources [Vanderkooy and Lipshitz, 1989],... 

A typical example of subtractive dither use in an A/D converter can be found in a Teac patent [Nakahashi and Ono, 1990], Notable features include the use of limiting and overload detectors as well as the ability to control the dither amplitude. A clever example of non-subtractive dither [Frindle, 1995][Frindle, 1992] separates the input signal into two paths. One of the paths is now inverted and then both paths are added to a random noise source. After the DAC, the two analog signals are subtracted the result is the sum of the conversion errors and twice the input signal. 

In order to estimate the uncertainty in the value obtained for the rate parameter K, one should also consider random errors. There are two sources of random error in this case the least-squares fitting which gives an error (dK)ls due to random noise in the experimental spectrum [equation (164)] and the measurement of w0 the line-width of the standard employed. The total random error in K is approximated by ... 

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). 

The outcome of the different exercises should be discussed among all participants in technical meetings, in particular to identify random and/or systematic errors in the procedures. Whereas random errors can be detected and minimised by intralaboratory measures, systematic errors can only be identified and eliminated by comparing results with other laboratories/techniques. When all steps have been successfully evaluated, i.e. all possible sources of systematic errors have been removed and the random errors have been minimised, the methods can be considered as valid. This does not imply that the technique(s) can directly be used routinely and further work is likely to be needed to test the robustness and ruggedness of the method before being used by technicians for daily routine measurements . 

As mentioned earlier, the complete analytical process involves sampling, sample preservation, sample preparation, and finally, analysis. The purpose of quality assurance (QA) and quality control (QC) is to monitor, measure, and keep the systematic and random errors under control. QA/QC measures are necessary during sampling, sample preparation, and analysis. It has been stated that sample preparation is usually the major source of variability in a measurement process. Consequently, the QA/QC during this step is of utmost importance. The discussion here centers on QC during sample preparation. 

Some of the most important factors affecting the precision and the accuracy of 14 MeV activation are reviewed below. The random errors as listed in the section on precision are generally reduced in importance by running replicate analyses on each of several aliquants of the sample and averaging all the results. Consideration given to the reduction of these sources of error will, of course, result in a reduction of the number... 

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