Random scatter

The second class, indeterminate or random errors, is brought about by the effects of uncontrolled variables. Truly random errors are as likely to cause high as low results, and a small random error is much more probable than a large one. By making the observation coarse enough, random errors would cease to exist. Every observation would give the same result, but the result would be less precise than the average of a number of finer observations with random scatter. 

Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magnitude and direction. Indeterminate errors need not affect the accuracy of an analysis. Since indeterminate errors are randomly scattered around a central value, positive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. 

Suppose that you need to add a reagent to a flask by several successive transfers using a class A 10-mL pipet. By calibrating the pipet (see Table 4.8), you know that it delivers a volume of 9.992 mL with a standard deviation of 0.006 mL. Since the pipet is calibrated, we can use the standard deviation as a measure of uncertainty. This uncertainty tells us that when we use the pipet to repetitively deliver 10 mL of solution, the volumes actually delivered are randomly scattered around the mean of 9.992 mL. 

Are the points randomly scattered, or are they clustered in certain regions of the plot If the latter, see if you can see any common structural motifs (see also Chapter 16, Problem 9). 

Purpose A technique to detect deviations from random scatter in the residuals (symmetrical about 0, frequent change of sign) Cumulative sum of residuals detects changes in trend or average. Here, an average is subtracted to yield residuals these residuals are then summed over points 1. .. k. .. N, with the sum being plotted at every point x k). Two uses are possible ... 

Test of Model Adequacy. The final step is to test the adequacy of the model. Figure 4 is a plot of the residual errors from the model vs. the observed values. The residuals are the differences between the observed and predicted values. Random scatter about a zero mean is desireable. 

All of the studies published so far have been aiming at the reconstruction of the total electron density in the crystal by redistribution of all electrons, under the constraints imposed by the MaxEnt requirement and the experimental data. After the acceptance of this paper, the authors became aware of valence-only MaxEnt reconstructions contained in the doctoral thesis of Garry Smith [58], The authors usually invoke the MaxEnt principle of Jaynes [23-26], although the underlying connection with the structural model, known under the name of random scatterer model, is seldom explicitly mentioned. 

When it is employed to specify an ensemble of random structures, in the sense mentioned above, the MaxEnt distribution of scatterers is the one which rules out the smallest number of structures, while at the same time reproducing the experimental observations for the structure factor amplitudes as expectation values over the ensemble. Thus, provided that the random scatterer model is adequate, deviations from the prior prejudice (see below) are enforced by the fit to the experimental data, while the MaxEnt principle ensures that no unwarranted detail is introduced. 

In this expression, G is the number of elements of the space group of the crystal, and / and n are the scattering power and number of the point random scatterers in... 

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 error-free likelihood gain, V,( /i Z2) gives the probability distribution for the structure factor amplitude as calculated from the random scatterer model (and from the model error estimates for any known substructure). To collect values of the likelihood gain from all values of R around Rohs, A, is weighted with P(R) ... 

As mentioned in Section 4.1.1, the number of random scatterers n has to be chosen in input. Five BUSTER runs used values of n in the series n = nvafcnce x N N = 60, 70, 80, 90, 100. The rms deviation from the reference map varied between 0.0317 and 0.0293 e A 3, the latter value pertaining to the run with N = 90 this value of n was then used in the calculation described below. 

If these M points are randomly scattered near Z(i), then the average error is near zero (positive and negative errors are equally likely), but the average square error is nonzero. An accurate Taylor expansion could be defined as one for which the root-mean-square error is less than some tolerance, Etoi,... 

In principle, all measurements are subject to random scattering. Additionally measurements can be affected by systematic deviations. Therefore, the uncertainty of each measurement and measured result has to be evaluated with regard to the aim of the analytical investigation. The uncertainty of a final analytical result is composed of the uncertainties of all the steps of the analytical process and is expressed either in the way of classical statistics by the addition of variances... 

Fig. 6.8. Typical plots of residual deviations random scattering (a), systematic deviations indicating nonlinearity (b), and trumpet-like form of heteroscedasticity (c)...

As a measuring science, analytical chemistry has to guarantee the quality of its results. Each kind of measurement is objectively affected by uncertainties which can be composed of random scattering and systematic deviations. Therefore, the measured results have to be characterized with regard to their quality, namely both the precision and accuracy and - if relevant - their information content (see Sect. 9.1). Also analytical procedures need characteristics that express their potential power regarding precision, accuracy, sensitivity, selectivity, specificity, robustness, and detection limit. 

Another proceeding is recommended in the case of time series or lateral line scans. When the values randomly scatter and - possibly - at the same time continuously slope up or down, then the mean between the two (or three) preceding and following should be taken see Fig. 8.2a (I). On the other hand, if an extreme value, e.g., a maximum has to be expected, then it should be interpolated after extrapolation, e.g., as illustrated in Figs. 8.2a (II) and 8.2b. Such a situation is indicated by successively increasing and then decreasing values (or vice versa), to be precise at least three each of them. 

Perhaps the most discouraging type of deviation from linearity is random scatter of the data points. Such results indicate that something is seriously wrong with the experiment. The method of analysis may be at fault or the reaction may not be following the expected stoichiometry. Side reactions may be interfering with the analytical procedures used to follow the progress of the reaction, or they may render the mathematical analysis employed invalid. When such plots are obtained, it is wise to reevaluate the entire experimental procedure and the method used to evaluate the data before carrying out additional experiments in the laboratory. 

Here fi2 = 1/Lv(t + r), L is the mean free path of radicals at thermal velocity v, and the initial spur radius r0 and the fictitious time T are related by r2 = Lvr. On random scattering, the probability per unit time of any two radicals colliding in volume dv will be ov/dv, where <7 is the collision cross section. The probability of finding these radicals in dv at the same time t is N(N - 1 )p2 dv2, giving the rate of reaction in that volume as crvN(n - 1 )p2 dv. Thus,... 

The average of a number of fine observations having random scatter is definitely more accurate, precise and, hence, more cogent than coarse data that appear to agree perfectly. 

FIGURE 4.8 Examples of residual plots from linear regression. In the upper left plot, the residuals are randomly scattered around 0 (eventually normally distributed) and fulfill a requirement of OLS. The upper right plot shows heteroscedasticity because the residuals increase with y (and thus they also depend on x). The lower plot indicates a nonlinear relationship between x and y. 

Plots of Residuals. Residuals can be plotted in many ways overall against a linear scale versus time that the observations were made versus fitted values versus any independent variable (3 ). In every case, an adequate fit provides a uniform, random scatter of points. The appearance of any stematic trend warns of error in the fitting method. Figures 4 and 5 shows a plot of area versus concentration and the associated plot of residuals. Also, the lower part of Figure 2 shows a plot of residuals (as a continuous line because of the large number of points) for the fit of the Gaussian shape to the front half of the experimental peak. In addition to these examples, plots of residuals have been used in SBC to examine shape changes in consecutive uv spectra from a diode array uv/vis spectrophotometer attached to an SBC euid the adequacy of linear calibration curve fits (1). 

To avoid penetration and filament formation via static and randomly scattered pinholes, one approach is to diminish the area of the junction until statistically the presence of a pinhole defect is near vanishing, as might be calculated by a Poisson distribution. An example of this strategy is the use of a nanopore junction of 50 nm diameter, though in this case the device fabrication yields were still reported to be quite low, down to a few percent [16]. 

Information concerning Sedan was obtained from the report by Lane (2). He reports equivalent fissions per gram for material from fallout trays between 5800 and 19,200 ft. from ground zero for a series of nuclides. The determinations show random scatter and do not indicate a trend with distance. Therefore, the values for different trays were averaged. A report by Nordyke and Williamson (3) provided the experimental determination of fallout-mass area density divided by gamma field readings in units of (kg./sq. meter)/(roentgens/hr.) over the same fallout area. These two sources provided the necessary input to calculate the fractionation indices for Sedan. 

---