Relative variance

As a result of the linear regression, the values of fo (r) are now known for a sequence of discrete r. From Eq. (8.27) it is clear that fo (r) itself represents a weighted relative variance of the lattice distortions. If it is found to be almost constant, its square root directly describes the total amount of relative lattice distortion ( in percent ). 

Repeat the design procedure of Example 3.10.2 assuming constant relative variances. 

Equation 28-4 told us that when n particles are drawn from a mixture of two kinds of particles (such as liver tissue particles and droplets of water), the sampling standard deviation will be cr = s/npq, where p and q are the fraction of each kind of particle present. The relative standard deviation is crJn = /npq/n = y/pqtn. The relative variance. ([Pg.648]

More dramatic evidence is shown in Figure 2 where the relative variance of the diffusion coefficient distribution is plotted against the percent rejected cycles. The unfiltered case starts out very broad and approaches the correct value after more than 1J0 of the cycles are rejected. However, no plateau is evident, and without the once-filtered results to guide the interpretation, it would not be clear where the correct answer occurs. The once-filtered case shows a dramatic decrease in the relative variance after approximately 3 percent of the cycles are rejected. 

Figure 2. Relative variance versus percent rejected cycles due to dust. Nominal 8 nm latex in unfiltered, local tap water (solid circles) and once-filtered (0.22 micron) water (open circles).

The first row shows the results using the exact baseline an average diameter of 272nm an infinitely narrow distribution with a relative variance in the diffusion coefficient distribution, POLY, of zero and a typical pre-exponential factor of 0.5. These were the values used to generate the data. 

For experiment durations of a few minutes with strong scatterers a random error of 0.1 in the baseline is typical, and the second row shows the effects of this. The average diameter is hardly changed, and the relative variance is 0.02. This is a fairly typical value found for narrow standards. However, an error of as little as 0.3 in the baseline results in a relative variance of 0.056. This is equivalent to a relative standard deviation of 0.24, or 24. An error of 1 in the baseline results in an error of only 1.9 in the average diameter, but yields a distribution with a relative standard deviation of 37.4. Thus, very small errors in the baseline result in large errors in the calculated width of the distribution. 

Note that the relative variance of the timing probability distribution is considerably larger in the equilibrium case than in the physiological case. However, even in the physiological case, the system behavior is far from that of a perfect timer. In this near ideal case, r.v. 1 /2, which is the minimal value obtained by Equation (5.37) when 2,i 2,2. 

Combining these results we have the expression for the relative variance ... 

Thus the relative variance decreases in proportion to the square of the number of molecules in the system. For nA = n°B = 100 and Ka = 0.5, Equation (10.21) yields I = 86.8 and r.v. = 8.1 x 10-4. Thus, even for this relatively small number of molecules, the relative variance is small for most practical purposes. 

After calculating the value of the random variable F, we establish the reproducibility variances and carry out the test according to the procedure given in Table 5.6. Exceptionally, in cases when we do not have any experiment carried out in parallel, and when the statistical data have not been divided into two parts, we use the relative variance for the mean value (s ) instead of the reproducibility variance. This relative variance can be computed with the statistical data used for the identification of the coefficients using the relation (5.64) ... 

Table 5.9 Computed values of the residual and relative variances.

For statistical samples of small volume, an increase in the order of the polynomial regression of variables can produce a serious increase in the residual variance. We can reduce the number of the coefficients from the model but then we must introduce a transcendental regression relationship for the variables of the process. From the general theory of statistical process modelling (relations (5.1)-(5.9)) we can claim that the use of these types of relationships between dependent and independent process variables is possible. However, when using these relationships between the variables of the process, it is important to obtain an excellent ensemble of statistical data (i.e. with small residual and relative variances). 

Let s compare these results to a well-known statistical measure, the variance of the (sample) mean and the relative variance of the (sample) mean, defined as... 

The statistical variance of the critical content was higher in the second case, similar to the situation with CH. However, the statistical relative variance of the mean was lower, similar to the situation with the Var(FE). 

In other words, when the lot is large relative to the sample and the particle masses are about the same, then the statistical relative variance of the mean and Gy s variance of the FE will be very close numerically. 

This is the variation of all samples that are j time units apart. Division by makes it a relative variance. An example is given in Figure D.l with code for the Excel macro given in Table D.l. 

Third, the foregoing regression analysis is carried out under the assumption that the absolute variance of y is independent of x. This may not be true. An important situation in analytical chemistry is one in which the relative variance or coefficient of variation is independent of x. Thus, an instrument reading may be theoretically proportional to concentration with the same relative precision over a range of concentrations. With appropriate changes, this case can be handled. ... 

The difference plots in Figure 4.11 and Figure 4.12 point to the presence of two broad peaks near 35 and 41° 20. The overall improvement after these peaks were included in the fit is shown in Figure 4.13. We note that absolute differences between the observed and calculated profiles in the vicinities of strong reflections are usually greater when compared to those in the background and weak peaks regions. However, relative variances (AYi/Y,) do not differ substantially. 

An approximate value for the standard deviation in c, can then be obtained by assuming that the uncertainties in and V, are negligible with respect to those in in and b. Then, the relative variance of the result is assumed to be the sum... 

Consider now the relative variance rr., / (s ). Clearly, it does not decrease if wo make the system larger in the x- or j/-directions but will only decrease as we approach the bulk limit by letting approach infinity. However, tliis inevitably changes the physical nature of the confined fluid. 

Here, the relative uncertainties are additive, and the most probable error is represented by the square root of the sum of the relative varimices. That is, the relative variance of the answer is the sum of the individual relative variances. 

Applying the principles of propagation of error (absolute variances in numerator additive, relative variances in the division step additive), we calculate that x = 0.2 5 O.OI4 ppm. 

TABLE III The variances, covariances, and correlation coefficients of the values of a selected group of constants based on the 2006 CODATA adjustment. The numbers in bold above the main diagonal are 10 times the numerical values of the relative covariances the numbers in bold on the main diagonal are 10 times the numerical values of the relative variances and the numbers in italics below the main diagonal are the correlation coefficients. ... 

Proceeding in this same way yields the relationships shown in Table al-6 for other types of arithmetic operations. Note that in several calculations relative variances such as (.s,/x) and Sp p) are combined rather than absolute standard deviations. 

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