Observed mean squares

The mean square torque is another test of the pair potentials used. The calculated mean square torques are very potential dependent they range from 7 x 10"31 to 36 x 10"28 (dyne-cm)2. The experimental values51 of the mean square torque in solid CO at 68°K and in liquid CO at 77.5°K are 19 x 10 28 and 21 x 10 28 (dyne-cm)2, respectively. Therefore, the Stockmayer potential clearly does not represent the noncentral forces in liquid CO, i.e., this potential is much too weak. On the other hand, the noncentral part of the modified Stockmayer potential is too strong. However, as pointed out previously, this problem can easily be solved by using a smaller quadrupole moment. The mean square torques from the other two potentials agree quite favorably with the experimental values. We conclude from the above that the quadrupole-quadrupole interaction can easily account for observed mean square torques in liquid CO. 

Figure 33.3 Different results of motion picture with different video frame speeds for the identical trajectory of random walk. In contrast to the case of normal diffusion, observed mean square displacements (MSD) significantly depend on the frame speed in the case of anomalous diffusion.

Method of moments estimates (also known as ANOVA estimates) can be calculated directly from the raw data as long as the design is balanced. The reader is referred to Searle et al. (11) for a thorough but rather technical presentation of variance components analysis. The equations that follow show the ANOVA estimates for the validation example. First, a two-factor with interaction ANOVA table is computed (Table 7). Then the observed mean squares are equated to the expected mean squares and solved for the variance components (Table 8 and the equations that follow). 

The expected mean squares are equated to the observed mean squares (those from the data) and solved for the variance components. Thus, =MSE is the variance estimate... 

Test rule For Levene s test we conduct an analysis of variance (ANOVA) of the absolute deviations from each sample average. Details on the ANOVA procedure are given in another chapter of this handbook. If the observed mean square ratio exceeds the appropriate critical value of the F statistic, we reject the hypothesis that all variances are equal. 

The standard deviation takes the difference between each observation and the observation mean, squares it to remove plus and minus values, then divides by the number of observations, and finally takes the square root of the result. Both means and standard deviations can be calculated with spreadsheet programs. 

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

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... 

Several ways may be used to characterize the spread or dispersion in the originai data. The range is the difference between the iargest vaiue and the smaiiest vaiue in a set of observations. However, aimost aiways the most efficient quantity for characterizing variabiiity is the standard deviation (aiso caiied the root mean square). 

Figiire 8-38 ihust rates the typical spread of values of the controhed variable that might be expected to occur under steady-state operating conditions. The mean and root mean square (BMS) deviation are identified in Fig. 8-38 and can be computed from a series of n observations Cl, C9,. . . c as fohows ... 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

Variance The mean square of deviations, or errors, of a set of observations the sum of square deviations, or errors, of individual observations with respect to their arithmetic mean divided by the number of observations less one (degree of freedom) the square of the standard deviation, or standard error. 

Another characteristic of a polymer surface is the surface structure and topography. With amorphous polymers it is possible to prepare very smooth and flat surfaces (see Sect. 2.4). One example is the PMIM-picture shown in Fig. 7a where the root-mean-square roughness is better than 0.8 ran. Similar values are obtained from XR-measurements of polymer surfaces [44, 61, 62], Those values compare quite well with observed roughnesses of low molecular weight materials. Thus for instance, the roughness of a water surface is determined by XR to 0.32 nm... 

For the data the squared correlation coefficient was 0.93 with a root mean square error of 2.2. The graph of predicted versus actual observed MS(1 +4) along with the summary of fit statistics and parameter estimates is shown in Figure 16.7. 

Root mean square error Mean of response Observations (or sum wt)... 

Some alternative method had to be devised to quantify the TCDD measurements. The problem was solved with the observation, illustrated in Figure 9, that the response to TCDD is linear over a wide concentration range as long as the size and nature of the sample matrix remain the same. Thus, it is possible to divide a sample into two equal portions, run one, then add an appropriate known amount of TCDD to the other, run it, and by simply noting the increase in area caused by the added TCDD to calculate the amount of TCDD present in the first portion. Figure 9 illustrates the reproducibility of the system. Each point was obtained from four or five independent analyses with an error (root mean square) of 5-10%, as indicated by the error flags, which is acceptable for the present purposes. 

The relative change of the mean-square nuclear radius in going from the excited to the ground state, A r )/ r ), is positive for u. An increase in observed isomer shifts S therefore reflects an increase of the s-electron density at the Ru nucleus caused by either an increase in the number of s-valence electrons or a decrease in the number of shielding electrons, preferentially of d-character. 

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 ... 

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

---