Variance equation

Defining the sample s variance with a denominator of n, as in the case of the population s variance leads to a biased estimation of O. The denominators of the variance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample s variance, however, substituting X for p, removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the member can always be deduced from the remaining - 1 members andX For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be... 

Equation 4.7 is referred to as the variance equation and is eommonly used in error analysis (Fraser and Milne, 1990), variational design (Morrison, 1998), reliability... 

In the probabilistic design calculations, the value of Kt would be determined from the empirical models related to the nominal part dimensions, including the dimensional variation estimates from equations 4.19 or 4.20. Norton (1996) models Kt using power laws for many standard cases. Young (1989) uses fourth order polynomials. In either case, it is a relatively straightforward task to include Kt in the probabilistic model by determining the standard deviation through the variance equation. 

Another consideration when using the approach is the assumption that stress and strength are statistically independent however, in practical applications it is to be expected that this is usually the case (Disney et al., 1968). The random variables in the design are assumed to be independent, linear and near-Normal to be used effectively in the variance equation. A high correlation of the random variables in some way, or the use of non-Normal distributions in the stress governing function are often sources of non-linearity and transformations methods should be considered. 

The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to find the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the parameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L). This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S). 

The formulations for the failure governing stress for most stress systems can be found in Young (1989). Using the variance equation and the parameters for the dimensional variation estimates and applied load, a statistical failure theory can be formulated for a probabilistic analysis of stress rupture. 

Determining the stress distribution using the variance equation... 

The variance equation to determine the standard deviation of the change in temperature can be written as ... 

The variance equation with second order terms is ... 

Using the first and second order terms in the variance equation gives exactly the same answer. For different conditions, say where one variable is not dominating the situation as above for the load, then the use of the variance equation with second order terms will be more effective. 

Rather than solve the variance equation for a number of variables directly, this method allows us to simulate the output of the variance, for example the simulated dispersion of a stress variable given that the random variables in the problem can be characterized. 

The EOT flow has a flat profile, almost as a piston therefore, its contribution to dispersion of the migrating zones is small. The EOT positively contributes to the axial diffusion variance (Equation (31)), when it moves in the same direction as the analytes. However, when the capillary wall is not uniformly charged, local turbulence may occur and cause irreproducible dispersion. ... 

The value of N is the only parameter in Equation 8-143. It can be computed from the RTD experimental data from its variance (Equation 8-107). 

Duplicate values of Y for all ten points are used to determine the pooled error variance (Equation 11) and the replication error variance (Equation 12) ... 

Another approach to weighting the data uses the data itself to develop the variance equation. This is the extended least squares (ELS) method. Parameters for the variance equations are included in the fitting... 

Turbulence closure models for the unknown terms in the flux- and variance equations for single phase reactor systems are discussed extensively by Baldyga and Bourne [5] and Fox [49] [50]. However, the application of these model closures should be treated with care as most of these parameterizations are expressed in terms of the turbulent eddy and other scalar dissipation rates (and dissipation energy spectra) which are about the weakest links in turbulence modeling. The predictive capabilities of the Re3molds averaged models... 

In scalar mixing studies and for infinite-rate reacting flows controlled by mixing, the variance of inert scalars is of interest since it is a measure of the local instantaneous departure of concentration from its local instantaneous mean value. For non-reactive flows the variance can be interpreted as a departure from locally perfect mixing. In this case the dissipation of concentration variance can be interpreted as mixing on the molecular scale. The scalar variance equation (1.462) can be derived from the scalar transport equation... 

The square of the standard deviation is called the variance. Equation (16.4-26) can also be written... 

Of course, (2.93) is the variance equation (2.68) at the unstable point x = Xq. It is observed that the variance (2.92) of the distribution increases exponentially, starting with an order of magnitude O (e) and reaching magnitude O (1) within a fluctuation enhancement period. After this the drift-dominated period sets in, for which the full drift term is needed but from which point on the fluctuation term in the Fokker-Planck equation can be neglected by writing... 

Furthermore it is easily checked that the mean value and variance equations derived from (4.32) by proceeding as in Sect. 3.2.4 agree exactly with the results (4.29, 31) derived directly from the master equation (4.18). 

The explicit but lengthy variance equations obtained analogously from (4.31, 51, 52) need not be derived. It should be pointed out, however, that for vanishing preference parameters, i.e. for Jif, = = 0, the equations (4.53) are... 

In order to estimate how long a uni-modal distribution can survive in the migration process the approximate mean value and variance equations (4.53, 31) are solved, as an example, for parameters chosen as for the results of Fig. 4.8 a, b and for a reasonably rhosen initial variance. 

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