Distribution of random errors

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

Precision is the closeness of agreement between independent test results obtained under stipulated conditions. Precision depends only on the distribution of random errors and does not relate to the true value. It is calculated by determining the standard deviation of the test results from repeat measurements. In numerical terms, a large number for the precision indicates that the results are scattered, i.e. the precision is poor. Quantitative measures of precision depend critically on the stipulated conditions. Repeatability and reproducibility are the two extreme conditions. 

B.2.1 Statistical treatment of finite samples 3B.2.2 Distribution of random errors 3B.2.3 Significant figures 3B.2.4 Comparison of results 3B.2.5 Method of least squares... 

In a situation whereby a large number of replicate readings, not less than 5 0, are observed of a titrimetric equivalence point (continuous variable), the results thus generated shall normally be distributed around the mean in a more or less symmetrical fashion. Thus, the mathematical model which not only fits into but also satisfies such a distribution of random errors is termed as the Normal or Gaussian distribution curve. It is a bell-shaped curve which is noted to be symmetrical about the mean as depicted in Figure 3.2. 

The features of an ideal chromatogram are the same as those obtained from a normal distribution of random errors (Gaussian distribution equation (1.2), cf. 21.3). In keeping with the classical notations, fi would correspond to the retention time of the eluting peak, a to the standard deviation of the peak (a2 represents the variance) and y represents the signal, as a function of time, from the detector located at the end of the column (see Fig. 1.3). 

Why have we gone to the trouble of classifying different types of error Because once we can identify the systemic errors we can correct for them, and a statistical treatment of the random error will allow us to estimate what the true result is and what uncertainty there may be about that result. Figure 1.5 brings together this discussion and shows the relationships between the true value of the measurand, the errors in a single measurement result, and the distribution of random errors. 

The distribution of random errors in these data is more easily understood if they are organized into equal-sized, adjacent data groups, or cells, as shown in Table a 1-2. The relative frequency of occurrence of results in each cell is then plotted as in Figure al-lA to give a bar graph called a histogram. 

What percentage of measurements should fall within +2(t of the true value for a data set with no determinate error, assuming a Gaussian distribution of random error ... 

The distribution of random errors should follow the Gaussian or normal curve if the number of measurements is large enough. The shape of Gaussian distribution was given in Chapter 3 (Fig. 3.4). It can be characterized by two variables—the central tendency and the symmetrical variation about tjie central tendency. Two measures of the central tendency are the mean, X, and the median. One of these values is usually taken as the correct value for an analysis, although statistically there is no correct value but rather the most probable value. The ability of an analyst to determine this most probable value is referred to as his accuracy. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Bell-shaped probability distribution curve for measurements and results showing the effect of random error. 

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. 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

The calculations discussed in the previous section fit the noise-free amplitudes exactly. When the structure factor amplitudes are noisy, it is necessary to deal with the random error in the observations we want the probability distribution of random scatterers that is the most probable a posteriori, in view of the available observations and of the associated experimental error variances. 

The normal distribution, as expressed by Eq. (7), can be employed in the analysis of random errors. If the error in a given measurement i is represented... 

These ten results represent a sample from a much larger population of data as, in theory, the analyst could have made measurements on many more samples taken from the tub of low-fat spread. Owing to the presence of random errors (see Section 6.3.3), there will always be differences between the results from replicate measurements. To get a clearer picture of how the results from replicate measurements are distributed, it is useful to plot the data. Figure 6.1 shows a frequency plot or histogram of the data. The horizontal axis is divided into bins , each representing a range of results, while the vertical axis shows the frequency with which results occur in each of the ranges (bins). 

Cases 4 and 5 deserve some special consideration. They were performed under the same conditions in terms of noise and initial parameter value, but in case 5 the covariances (weights) of the temperature measurements were increased with respect to those in the remaining measurements. For case 4 it was noticed that, although a normal random distribution of the errors was considered in generating the measurements, some systematic errors occur, especially in measurement numbers 6, 8,... 

On the other hand, random errors do not show any regular dependence on experimental conditions, since they are generated by many small and uncontrolled causes acting at the same time, and can be reduced but not completely eliminated. Thus, random errors are observed when the same measurement is repeatedly performed. In the simplest case, the universe of random errors is described by a continuous random variable e following a normal distribution with zero mean, i.e., for a univariate variable, the probability density function is given by... 

A straight-line model is the most used, but also the most misused, model in analytical chemistry. The analytical chemist should check five basic assumptions during method validation before deciding whether to use a straight-line regression model for calibration purposes. These five assumptions are described in detail by MacTaggart and Farwell [6] and basically are linearity, error-free independent variable, random and homogeneous error, uncorrelated errors, and normal distribution of the error. The evaluation of these assumptions and the remedial actions are discussed hereafter. 

Conclusions should always be tempered by the possible importance of untested or uncontrolled variables, and by the risks assumed in the testing protocol. The problem of outlying values and their effects needs to be considered also. Retrospective correlations of historical data are frequently employed with no consideration of weighting for unbalanced distributions, with the result that one often ends up with merely a mathematical description of random error. Finally, as noted above, there is the danger of comparisons at fixed points, or at fixed sets of... 

The next three subsections describe the background and principles of random error treatment, and they introduce two important quantities standard deviation a- and 95 percent confidence limits. The four subsections following these— Uncertainty in Mean Valne, Small Samples, Estimation of Limits of Error, and Presentation of Numerical Results—are essential for the kind of random error analysis most frequently required in the experiments given in this book. The Student t distribution is particularly important and useful. 

The normal probability function as expressed by Eq. (14) is useful in theoretical treatments of random errors. For example, the normal probability distribution function is used to establish the probability Hthat an error is less than a certain magnitude 8, or conversely to establish the limiting width of the range, —8 to 8, within which the integrated probability P, given by... 

In the analysis of the effect on the calculated quantity of random errors in measured quantities it is unfortunate that the only model susceptible to an exact statistical treatment is the linear one (II). Here we have attempted to characterize the frequency distribution of the error in the calculated vapor composition by the standard methods and have not included a co-variance term for each pair of dependent variables (12). Our approach has given a satisfactory result for the methanol-water-sodium chloride system but it has not been tested on other systems and perhaps of more importance, it has not been possible, so far, to confirm the essential correctness of the method by an independent procedure. Work is currently being undertaken on this project. 

The distribution function for the random walk gives us an immediate approach to the theory of random errors. In any physical experiment there may be a number of factors disturbing an observation, with each one contributing an error of magnitude, let us say, 8 (for the ith source) which may be positive or negative. The result of all of these individual errors is to produce a total error a = SS, such that our observed measurement, call it m, differs from what we presume is the true value m by the amount x... 

If we assume that each of the individual errors may be assigned an equal probability of being positive or negative and all act independently, then the question of finding the distribution of possible errors is given precisely by the answer to the random walk problem with varied step. That is, the chance of making an error x is given by... 

In the analyses of blood specimens from subjects participating in bioavailability studies, the FDA instructs laboratories to include quality control specimens (QC) at each of three known concentrations (low, mid, and high). The QC specimens are processed in duplicate with each batch of subject specimens. The acceptance criteria for the batch, based on the results of these QC specimens, is that at least four of the six values must fall within a specified range about their nominal concentrations. In addition, no more than one value at each of the three QC concentration levels can be outside its acceptance range. Combining binomial and normal distribution theory, we can estimate the number of batch runs we expect to reject because of random error. 

One important question regarding the distribution of measurements about their mean is the expected frequency of occurrence of an error as a function of the error magnitude. The most commonly utilized function, which describes well the relative frequency of occurrence of random errors in large sets of measurements, is given by Gauss formula (see also rel. (9) Section 5.2) ... 

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