Error statistical

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

Ferrenberg A M, Landau D P and Swendsen R H 1995 Statistical errors in histogram reweighting Phys. Rev. E 51 5092-100... 

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

If all sources of systematic error can be eliminated, there will still remain statistical errors. These errors are often reported as stcindard deviations. What we would particularly like to estimate is the error in the average value, (A). The standard deviation of the average value is calculated as follows ... 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

Andraos, J. On the Propagation of Statistical Errors for a function of Several Variables, /. Chem. Educ. 1996, 73, 150-154. 

Estimations based on statistics can be made for total accuracy, precision, and reproducibility of results related to the sampling procedure being applied. Statistical error is expressed in terms of variance. Total samphng error is the sum of error variance from each step of the process. However, discussions herein will take into consideration only step (I)—mechanical extraction of samples. Mechanical-extracdion accuracy is dependent on design reflecding mechanical and statistical factors in carrying out efficient and practical collection of representative samples S from a bulk quantity B,... 

Measurement Error Uncertainty in the interpretation of unit performance results from statistical errors in the measurements, low levels of process understanding, and differences in unit and modeled performance (Frey, H.C., and E. Rubin, Evaluate Uncertainties in Advanced Process Technologies, Chemical Engineering Progress, May 1992, 63-70). It is difficult to determine which measurements will provide the most insight into unit performance. A necessary first step is the understanding of the measurement errors hkely to be encountered. 

Computer simulation is an experimental science to the extent that calculated dynamic properties are subject to systematic and statistical errors. Sources of systematic error consist of size dependence, poor equilibration, non-bond interaction cutoff, etc. These should, of course, be estimated and eliminated where possible. It is also essential to obtain an estimate of the statistical significance of the results. Simulation averages are taken over runs of finite length, and this is the main cause of statistical imprecision in the mean values so obtained. 

Statistical errors of dynamic properties could be expressed by breaking a simulation up into multiple blocks, taking the average from each block, and using those values for statistical analysis. In principle, a block analysis of dynamic properties could be carried out in much the same way as that applied to a static average. However, the block lengths would have to be substantial to make a reasonably accurate estimate of the errors. This approach is based on the assumption that each block is an independent sample. 

The statistical error can thus be reduced by averaging over a larger ensemble. How well the calculated average (from eq. (16.9)) resembles the true value, however, depends on whether the ensemble is representative. If a large number of points is collected from a small part of the phase space, the property may be calculated with a small statistical error, but a large systematic error (i.e. the value may be precise, but inaccurate). As it is difficult to establish that the phase space is adequately sampled, this can be a very misleading situation, i.e. the property appears to have been calculated accurately but may in fact be significantly in error. 

Computationally it is therefore difficult to achieve a reasonable statistical error for entropic quantities such A, S and G. 

At a later stage, the basic model was extended to comprise several organic substrates. An example of the data fitting is provided by Figure 8.11, which shows a very good description of the data. The parameter estimation statistics (errors of the parameters and correlations of the parameters) were on an acceptable level. The model gave a logical description of aU the experimentally recorded phenomena. 

With this database in hand, a simple question is asked [29] How different is a knowledge-based potential derived from this lattice database compared to the actual energy function used to construct the database If statistical errors are negligible and the knowledge-based method is perfect, the answer is expected to be They are exactly the same. ... 

For measurements by AS, the errors of the isotope ratio will be dominated by counting statistics for each isotope. For measurements by TIMS or ICP-MS, the counting-statistic errors set a firm lower limit on the isotopic measurement errors, but more often than not contribute only a part of the total variance of the isotope-ratio measurements. For these techniques, other sources of (non-systematic) error include ... 

When specifying atomic coordinates, interatomic distances etc., the corresponding standard deviations should also be given, which serve to express the precision of their experimental determination. The commonly used notation, such as d = 235.1(4) pm states a standard deviation of 4 units for the last digit, i.e. the standard deviation in this case amounts to 0.4 pm. Standard deviation is a term in statistics. When a standard deviation a is linked to some value, the probability of the true value being within the limits 0 of the stated value is 68.3 %. The probability of being within 2cj is 95.4 %, and within 3ct is 99.7 %. The standard deviation gives no reliable information about the trueness of a value, because it only takes into account statistical errors, and not systematic errors. 

The main problem in using this method is the statistical error of the Compton profiles which calculates to... 

After calculating the Fourier transform of the Compton profiles one observes that the amplitude of its oscillations becomes smaller than this statistical error when r is greater than 15 a.u. and therefore the B(r) function cannot be used for r > 15 a.u. On the other hand if one wants to get results for Cu with a similar statistical error compared to the results of the Li reconstruction the number of counts needed is given by... 

Table 7-1. Representative results from statistical analyses used to determine the values and statistical errors of 9 AGch(D)Oh/3 - for the pKa calculation of the zinc-bound water in CAIIa...

Number without any parentheses is the size of the block (in ps), number with parentheses is the number of blocks these are determined after the equilibration sections of the trajectories are removed. d In kcal/mol the number in parentheses is the statistical error evaluated based on block average. 

The computed PMF surface is shown in Figure 13-10 as a 20-level contour plot. The statistical errors are less than 0.5 kcal/mol in most of the regions except the... 

Errors Inherent to the Radiocarbon Dating Method. The decay of radiocarbon is radioactive, involving discrete nuclear disintegrations taking place at random dates derived from the measurement of radiocarbon levels are therefore subject to statistical errors intrinsic to the measurement, which cannot be ignored. It is because of these errors that radiocarbon dates are expressed as a time range, in the form... 

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