Error estimated standard

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

The estimated standard deviation due to systematic errors between analysts is calculated from equation 14.18. 

Every measured quantity or component in the main equations, Eqs. (12.30) and (12.31), influence the accuracy of the final flow rate. Usually a brief description of the estimation of the confidence limits is included in each standard. The principles more or less follow those presented earlier in Treatment of Measurement Uncertainties. There are also more comprehensive error estimation procedures available.These usually include, beyond the estimation procedure itself, some basics and worked examples. 

The numbers in parentheses correspond to 1 estimated standard deviation (esd), except those for the Cpf Cn ratio, which correspond to 3 esd. The values of n and m shown are the smallest integers for which the ratio (n + m)/n lies within the experimental error limits of the measured axial ratio < Fe-<=R — t + e. The c parameters in the bottom line refer to the supercells corresponding to the underlined (n + ro)/n values ... 

Parameter Point Estimates, Standard Errors, and Coefficients of Variation ... 

Figure 5. Molecular dynamics simulation of the decay forward and backward in time of the fluctuation of the first energy moment of a Lennard-Jones fluid (the central curve is the average moment, the enveloping curves are estimated standard error, and the lines are best fits). The starting positions of the adiabatic trajectories are obtained from Monte Carlo sampling of the static probability distribution, Eq. (246). The density is 0.80, the temperature is Tq — 2, and the initial imposed thermal gradient is pj — 0.02. (From Ref. 2.)...

Figure 7.55 shows a plot of over 2000 2%DOPC/dodecane Pe measurements (10-6 cm/s units), each representing at least three intra-plate replicates, vs. the estimated standard deviations, o(Pe). Over 200 different drug-like compounds were measured. The %CV (coefficient of variation 100 x a(Pe)/Pe) is about 10% near Pe 10 x 10 6 cm/s, and slightly increases for higher values of permeability, but rapidly increases for Pe< 0.1 x 10 6 cm/s, as shown in Table 7.21. These statistics accurately reflect the errors that should be expected in general. For some molecules, such as caffeine and metoprolol, %CV has been typically about 3-6%. 

A variety of statistical parameters have been reported in the QSAR literature to reflect the quality of the model. These measures give indications about how well the model fits existing data, i.e., they measure the explained variance of the target parameter y in the biological data. Some of the most common measures of regression are root mean squares error (rmse), standard error of estimates (s), and coefficient of determination (R2). 

If a large number of readings of the same quantity are taken, then the mean (average) value is likely to be close to the true value if there is no systematic bias (i.e., no systematic errors). Clearly, if we repeat a particular measurement several times, the random error associated with each measurement will mean that the value is sometimes above and sometimes below the true result, in a random way. Thus, these errors will cancel out, and the average or mean value should be a better estimate of the true value than is any single result. However, we still need to know how good an estimate our mean value is of the true result. Statistical methods lead to the concept of standard error (or standard deviation) around the mean value. 

The estimated standard error for the adjusted difference between two groups is given by... 

Each stated uncertainty in this and other tables represents one estimated standard error, propagated to parameters from uncertainties of measurements of wave numbers the uncertainties of the latter measurements were provided by authors of papers [91,93] reporting those data, and the weight of each datum in the non-linear regression was taken as the reciprocal square of those uncertainties. As the reduced standard deviation of the fit was 0.92, so less than unity, the authors... 

Additional measurements were made with the 17-)im sizing screen to obtain more information on the variability of our measurement techniques. Eight lint samples from a single source of cotton were analyzed by the procedures outlined previously. The dust levels obtained in this test were 11.7, 12.1, 13.5, 11.8, 10.8, 11.2, 10.9, and 9.7 mg, respectively, per 20 g of lint. The mean and standard deviation of these measurements were 11.5 and 1.1, respectively. The estimated standard error of the mean was 0.42, and the interval from 10.5 to 12.5 represented a 95% confidence interval for the lot mean. 

Estimated standard error of slope of regression line 0.118... 

Measured by H NMR. Error estimates are based on the standard deviation of values measured in duplicate runs. 

The function BSE(/, n) therefore increases monotonically with n and asymptotes to the true standard error associated with error estimate has converged, which is not subject to the extremes of numerical uncertainty associated with the tail of a correlation function. Furthermore, the blockaveraging analysis directly includes all trajectory information (all frames). 

Methods of statistical meta-analysis may be useful for combining information across studies. There are 2 principal varieties of meta-analytic estimation (Normand 1995). In a hxed-effects analysis the observed variation among estimates is attributable to the statistical error associated with the individual estimates. An important step is to compute a weighted average of unbiased estimates, where the weight for an estimate is computed by means of its standard error estimate. In a random-effects analysis one allows for additional variation, beyond statistical error, making use of a htted random-effects model. 

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