Mean squares weighted deviates

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

Average age, Ma (+1ct) and Mean Square Weighted Deviation calculated using la errors. The majority of data reported from single samples are inconsistent with a single population. 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

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 ... 

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. 

Table II contains the optimal empirical resolutions calculated from the weighted least squares analysis of counted peak maxima. The empirical r from each set was used to calculate, from an unweighted east squares fit, a component number from each series in that set. The results for each set were averaged, and the mean and standard deviation for each set are reported In Table II.

The first step in validation is simply to verify that the remaining errors in the reproduction of the reference data are acceptably small. If the weight factors have been set, as suggested earlier, to the inverse of the acceptable error for each data type, the test is particularly simple. If the final penalty function is lower than the number of data points, the root mean square (rms) error will automatically fall within the acceptable range. The data should also be divided by type and retested, to make sure that the proper balance has been obtained. As before, outliers should be carefully scmtinized. Any errors in the reference data or deficiencies in the functional form are most easily detected at this stage. Plots of calculated vs. reference data can also give valuable information on trends in remaining deviations and possible systematic errors (20). 

The behavior of B(p) and C(p) for propane is shown in Figure 7. The number of PpT data used here for adjusting the equation of state is 843, with different least-squares weightings than in Refs. 3 and 5. Overall deviations, with equal weighting for all points, are 2.07 bar for the mean of absolute pressure deviations and 0.34% for the rms of relative density deviations. 

Number of degrees of freedom weighted mean square deviation between data and fit. 

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

---