Limiting variance values

The experimental variance, S, obtained from these samples is of little value unless it can be related to limiting variance values. If the assumptions are made that the mixtu re is a two-component system, and that the component particles differ only in colour, then these limiting values can be predicted with the help of the binomial theorem. The binomial theorem is applicable for the particular cases where the value of y, is either 0 or I. This condition is applicable in two limiting cases. 

Using the two limiting values of variance of equations (2.1) and (2.2) and the experimental value of variance calculated from the samples a variety of Indices of Mixing can be calculated. 

The index will have a zero value for a completely segregated mixture and increase to unity for a fully randomized mixture. A criticism of the Lacey index is that it is insensitive to mixture quality. Even a very bad mixture will have a variance value much closer to than to So. and as a result practical values of the Lacey index are restricted to the range of 0.75 to 1.0. 

Taylor and Wall , omitted the value of Si from their index which is deflned as 

As the mixture approaches the random state the index will approach unity. Non-random mixtures will have a mixing index greater than unity. 

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As welt as providing essential limiting variance values, equations (2.1) and (2.2) indicate how the mixture quality will be affected by the scale of scrutiny, or sample size. The variance, and hence the quality, of a completely s regated mixture is independent of the scale of scrutiny. If a fully randomized mixture is attainable then the quality of that mixture is inversely proportional to the number of particles in the sample. In this case a reduction in the particle size of the mixture will increase the number of particles in each Hxed weight sample and produce a predictable improvement in the attainable mixture quality. For a randomized mixture, equation (2.2) quantifles the effect of changes in the scale of scrutiny on mixture quality. Between these extremes of mixture types there is no infallible guide as to the relationship between scale of scrutiny and mixture quality. 

The equations (2.1) and (2.2) defining the limiting variance values S and were based on the assumption that a two-component, equi-sized particulate system was being analysed. Real systems are unlikely to be so obliging ... 

Equations (2.2), (2.5) and (2.6) for random variance presuppose that all the individual particles within the mixture can move independently of their neighbours. It will be seen in Chapter 5 that for cohesive powder systems the structure of the powder mixture can prevent independent movement of individual particles and that these limiting variance values have to be modified. 

If it can be assumed that the mixture under consideration is ideal, in that the component particles differ only in colour, then the limiting variance values for the mixture can be estimated using the values of y and 5. From equation (2.1),... 

On average we would obtain an experimental variance value outside the range 0.00260 to 0.00957 only one time in twenty. These limiting variance values could be used to compare the mixture quality within a mixer at different mixing times or to compare the performance of two competing mixers. An F test would give a more refined statistical comparison of two mixture qualities as described by variance values. 

Figure 5.1 gives a two-dimensional representation of a binary powder mixture in varying states of mixedness. As was discussed in Chapter 2, it is possible to statistically define three limiting variance values for a non-structured powder mixture ... 

These vertical constant temperature lines (Fig. 2.1) are the same for a mixture as for a single component. Outside the phase envelope, the temperature lines tend to hold a constant enthalpy value or have very limited variance with variance of pressure. Compare Figs. 2.1 and 2.2. In particular, notice that the constant temperature line approaches verticality outside the phase envelope. This vertical temperature line indicates that outside the phase envelope there is no significant enthalpy variance with pressure variance. Again, outside the phase envelope, temperature variance is most effective to enthalpy. Observe how limited enthalpy is between 1.0 and 100.0 atm pressure. 

Other Systemic Effects. Workers exposed for less than 5 years to TWA concentrations of 4.8-8 ppm had significantly elevated plasma sodium and chloride ions and decreased erythrocyte potassium and calcium (Pines 1982). However, the large variance in the electrolyte measurements among workers, the concomitant exposure to other chemicals, the fluctuating exposure concentrations, and the lack of a dose response for blood electrolyte alterations limit the value of this study. 

The use of NOECs and LOECs has been questioned as they have some major limitations. Their values depend very much on the concentrations that have been selected in the test set-up. Large concentration steps will result in a large difference between NOEC and LOEC. Moreover, poor test design (e.g. low number of replicates) will result in increased variance of effects and the acceptance of the null hypothesis. As a consequence, the toxicity of a chemical may be underestimated. Statistical measurement endpoints obtained from ecotoxicity testing may be used to derive predicted no effect concentration (PNEC) levels that are employed in ecological risk assessment for chemicals. 

For the ideal system the measured proportions of the components will be constant no matter which particle characteristic is measured. The fraction of particles measured by weight will be exactly the same as that measured by volume, number of particles, surface area or any other characteristic of the mixture. Fora real system the values of the limiting variance will be dependent on the characteristic measured. An additional problem for multi-sized particulate systems is that the assumption made in defining Sj that each sample will contain the same number of particles is no longer valid and an estimate of the mean number of particles within the sample must be made. 

The denominator in equation (2.5) is the estimate of the mean number of particles in a sample and is directly comparable with the denominator value A of equation (2.2). In order to estimate the limiting variance by equation (2.5) the size analysis of the components is required along with a knowledge of particle shape and specific gravity. The use of equation (2.5) is illustrated in section 2.4. 

The PSR model fittings, also presented in Table 2, separate the curves in Fig. 2 into the ay and a terms. From the estimated values it is evident that the a acknowledged to be related widi the resistance to plastic deformation and the load-independent hardness (Hpsr), show a limited variance of 0.88OPa across the three samples, an improvement of only 3.83%. The ay, on the other hand, exhibits a substantial increase of 111.91% through U-SiC to H-SiC which implies that the ay holds greater significance to the total performance of the SiC ceramics during Vickers indentation. 

The values of x and s vary from sample set to sample set. However, as N increases, they may be expected to become more and more stable. Their limiting values, for very large N, are numbers characteristic of the frequency distribution, and are referred to as the population mean and the population variance, respectively. 

Although a transfer function relation may not be always invertible analytically, it has value in that the moments of the RTD may be derived from it, and it is thus able to represent an RTD curve. For instance, if Gq and Gq are the limits of the first and second derivatives of the transfer function G(.s) as. s 0, the variance is... 

The variance about the mean, and hence, the confidence limits on the predicted values, is calculated from all previous values. The variance, at any time, is the variance at the most recent time plus the variance at the current time. But these are equal because the best estimate of the current time is the most recent time. Thus, the predicted value of period t+2 will have a confidence interval proportional to twice the variance about the mean and, in general, the confidence interval will increase with the square root of the time into the future. 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

Comparing equations (10) and (5), the lUPAC definition for detection limit, the difference is that RMSE is used instead of For dynamic systems, such as chromatography with autointegration systems, RMSE is easier to measure and more reliable than for reasons discussed earlier. Both are measures of variance and, although dissimilar, provide similar information. This is apparent in the equations used to calculate the values of. Tb and RMSE ... 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

Both the determination of the effective number of scatterers and the associated rescaling of variances are still in progress within BUSTER. The value of n at the moment is fixed by the user at input preparation time for charge density studies, variances are also kept fixed and set equal to the observational c2. An approximate optimal n can be determined empirically by means of several test runs on synthetic data, monitoring the rms deviation of the final density from the reference model density (see below). This is of course only feasible when using synthetic data, for which the perfect answer is known. We plan to overcome this limitation in the future by means of cross-validation methods. 

By means of these variances the limits of detection XkLD of the k analytes tmder investigation can be estimated in analogy to the univariate case (see Sect. 7.5), namely on the basis of the critical values ynetkyC of the net signals their vector is denoted yk,c in the last term of Eq. (6.116b) ... 

In fact, we have already used a modeling strategy when Po(AU) was approximated as a Gaussian. This led to the second-order perturbation theory, which is only of limited accuracy. A simple extension of this approach is to represent Pq(AU) as a linear combination of n Gaussian functions, p, (AU), with different mean values and variances [40]... 

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