Error distribution histogram

Experience gained in the ZAF analysis of major and minor constituents in multielement standards analyzed against pure element standards has produced detailed error distribution histograms for quantitative EPMA. The error distribution is a normal distribution centered about 0%, with a standard deviation of approximately 2% relative. Errors as high as 10% relative are rarely encountered. There are several important caveats that must be observed to achieve errors that can be expected to lie within this distribution ... 

Figure 10, (a,b,c) Horizontal Position Error (HPE) for Single Point Positioning(SPP) and tightly-coupled EKF with FOG and MEMS IMU, (d) error distribution histogram and (e) cumulative error histogram... 

An informative description of the prediction errors is a visual representation, for instance by a histogram or a probability density curve these plots, however, require a reasonable large number of predictions. For practical reasons, the error distribution can be characterized by a single number, for instance the standard deviation. 

The error model used in the minimization is based on the hypothesis that the residuals have zero mean and are normally distributed. The first is easily checked, the latter is only possible when sufficient data points are available and a distribution histogram can be constructed. An adequate model also follows the experimental data well, so if the residuals are plotted as a function of the dependent or independent variable(s) a random distribution around zero should be observed. Nonrandom trends in the residuals mean that systematic deviations exist and indicate that the model is not completely able to follow the course of the experimental data, as a good model should do. This residual trending can also be evaluated numerically be correlation calculations, but visual inspection is much more powerful. An example is given in Fig. 12 for the initial rate data of the metathesis of propene into ethene and 2-butenc [60], One expression was based on a dual-site Langmuir-Hinshelwood model, whereas the other... 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

What level of inaccuracy can be expected for a simulation with a certain sample size N1 This question can be transformed to another one what is the effective limit-perturbation Xf or xg in the inaccuracy model [(6.22) or (6.23)] To assess the error in a free energy calculation using the model, one may histogram / and g using the perturbations collected in the simulations, and plot x in the tail of the distribution. However, if Xf is taken too small the accuracy is overestimated, and the assessed reliability of the free energy is therefore not ideal. In the following, we discuss the most-likely analysis, which provides a more systematic way to estimate the accuracy of free energy calculations. 

The next three chapters deal with the most widely used classes of methods free energy perturbation (FEP) [3], methods based on probability distributions and histograms, and thermodynamic integration (TI) [1, 2], These chapters represent a mix of traditional material that has already been well covered, as well as the description of new techniques that have been developed only recendy. The common thread followed here is that different methods share the same underlying principles. Chapter 5 is dedicated to a relatively new class of methods, based on calculating free energies from nonequilibrium dynamics. In Chap. 6, we discuss an important topic that has not received, so far, sufficient attention - the analysis of errors in free energy calculations, especially those based on perturbative and nonequilibrium approaches. 

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). 

Sometimes a measurement involves a single piece of calibrated equipment with a known measurement uncertainty value o, and then confidence limits can be calculated just as with the coin tosses. Usually, however, we do not know o in advance it needs to be determined from the spread in the measurements themselves. For example, suppose we made 1000 measurements of some observable, such as the salt concentration C in a series of bottles labeled 100 mM NaCl. Further, let us assume that the deviations are all due to random errors in the preparation process. The distribution of all of the measurements (a histogram) would then look much like a Gaussian, centered around the ideal value. Figure 4.2 shows a realistic simulated data set. Note that with this many data points, the near-Gaussian nature of the distribution is apparent to the eye. 

Fig. 16.3 Histogram and normal distribution of calculated values for the dipole polarizability of liquid Ar, using Model 1. Average polarizability is (a) = 11.59 ao3 and statistical error, v=0.03 ao3. Also shown (below) are the individually calculated mean dipole polarizabilities of the different configurations used...

Fig. 10.15. Vector diagram (left frame) of feature misalignment in a composite stamp designed for an active matrix circuit for a display. The histogram in the right frame shows the distribution of alignment errors observed in this system. These micron-level...

Fig. 14. Experimental distribution of 7r+7r pairs produced in the thick (a) and thin (c) targets as a function of F (points with errors) and approximating distributions of free pairs on the same variable (dashed histogram) the ratio of the experimental to the approximating distribution for thick (b) and thin (d) targets. The deviation of the ratio from unity in the two first bins in (b) is due to extra pairs originating from ionization of. 4 f- in the thick target. The absence of extra pairs in the first two bins in (d) is caused by the low A2-k ionization probability in the thin target...

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