Mean square of weighted deviations

The MSWD and probability of fit All EWLS algorithms calculate a statistical parameter from which the observed scatter of the data points about the regression line can be quantitatively compared with the average amount of scatter to be expected from the assigned analytical errors. Arguably the most convenient and intuitively accessible of these is the so-called ATS ITD parameter (Mean Square of Weighted Deviates McIntyre et al. 1966 Wendt and Carl 1991), defined as ... 

Mean Squares of Weighted Deviates — used as a measure of the goodness of fit of an isochron Neutron Activation Analysis... 

The selected values of virial coefficients for each compound were fitted to a smoothing function of temperature by the least-squares criterion, whenever there were three or more data values over an appreciable temperatme range. The squares of the deviation between calculated and observed values were weighted in proportion to the reciprocal of the square of the estimated uncertainties. The data were then scaled by subtracting the means of each term in the polynomial from the given values. This eliminated the constant term. The singular value decomposition technique [88-pre/fla] was used for the remaining... 

Wendt I, Carl C (1991) The statistical distribution of the mean squared weighted deviation. Chem Geol 86 275-285... 

The parameters calculated for the acids are given in Table 3 and those for the bases in Table 4. The weighted root-mean-square deviation between the experimental enthalpies and those calculated from the parameters in Tables 3 and 4 using Eq. (13) is about 0.016 corresponding to a deviation of about 0.2 kcal/mole for a heat of 8 kcal/mole. The excellent agreement between the experimental enthalpies of adduct formation and the calculated enthalpies for aU of the interactions are reported in the literature (40). 

However, if equation 5 is used (weights wj rather than wj2), then equation 6 does equal zero. When a zero sum of deviations is desirable, function 5 may be minimized, often without increasing the root-mean-square-error by an undue amount. 

If A/w and of a polymer sample are known, we have information about the standard deviation and the variance of the number distribution. There is no quantitative information about the breadth of the weight distribution of the same sample unless and are known. As mentioned earlier, it is often assumed that the weight and number distributions will change in a parallel fashion and in this sense the Mw/M ratio is called the breadth of the distribution although it actually reflects the ratio of the variance to the square of the mean of the number distribution of the polymer (Eq. 2-34). 

Each individual measurement of any physical quantity yields a value A. But, independently of any possible observation errors associated with imperfect experimental measurements, the outcomes of identical measurements in identically prepared microsystems are not necessarily the same. The results fluctuate around a central value. It is this collection or Spectrum of values that characterizes the observable A for the ensemble. The fraction of the total number of microsystems leading to a given A value yields the probability of another identical measurement producing that result. Two parameters can be defined the mean value (later to be called the expected value ) and the indeterminacy (also called uncertainty by some authors). The mean value A) is the weighted average of the different results considering the frequency of their occurrence. The indeterminacy AA is the standard deviation of the observable, which is defined as the square root of the dispersion. In turn, the dispersion of the results is the mean value of the squared deviations with respect to the mean (A). Thus,... 

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