Variance, definitions

The experiment is done by the design shown in Table 2.44, and the obtained results processed by analysis of variance. Definitions with calculation forms of analysis of variance are shown in Table 2.45. 

Variance, analysis, 277, 283-288 definition and equation, 268, 269 Voltage, adjustment, use in avoidance of line interference, 149 minimum desirable for excitation, 102... 

As an example of these techniques, we shall calculate the characteristic function of the gaussian distribution with zero mean and unit variance and then use it to calculate moments. Starting from the definition of the characteristic function, we obtain18 ... 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

The standard deviation, 5, is by definition the square root of the variance,... 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

The degree of branching by transfer with polymer obviously will increase with the conversion since the relative incidence of branching must depend on the ratio of polymer to monomer in the system. To examine the matter from the point of view of reaction rates, let 6 represent the fraction of monomer molecules which have polymerized out of a total of iVo in the system, and let v represent the total number of branches. (At variance with the definition used elsewhere. No is the total number of units polymerized and unpolymerized as well.) The rates of generation of branches and of polymerization can then be written... 

It can be shown that all symmetric matrices of the form X X and XX are positive semi-definite [2]. These cross-product matrices include the widely used dispersion matrices which can take the form of a variance-covariance or correlation matrix, among others (see Section 29.7). 

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 ... 

Note that VTD-variance of Loss Tangent, and that SDTD is the standard deviation of Loss Tangent with similar definitions for GSP (G or real modulus) and GDP (G or loss modulus). 

We remind our readers here that AE, as we have been using it in this derivation is, as you will recall, the difference between AE and AE 0 in equation 41-4 and the expected value in the statistical nomenclature is therefore 21/2 times as large as AE (due to the fact that it is the result of the difference between random variables with equal variance), a difference that should be taken note of when comparing results with the original definition of S/N in equation 41-2. 

This, however, leads to another problem subtracting equation 44-64a from equation 44-51 leaves us with the result that T = 0. Furthermore, the definition of T gives us the result that Es is zero, and that therefore AT is in fact equal to the expression given by equation 44-52b anyway despite our efforts to include the contribution to the variance of the first term in equation 44-51. 

Thus we conclude that we must compute the variance of AT directly from equation 44-68a and the definition of variance ... 

A multivariate ANOVA, however, has some properties different than the univariate ANOVA. In order to be multivariate, obviously there must be more than one variable involved. As we like to do, then, we consider the simplest possible case and the simplest case beyond univariate is obviously to have two variables. The ANOVA for the simplest multivariate case, that is, the partitioning of sums of squares of two variables (X and Y), proceeds as follows. From the definition of variance ... 

We will address aspects of reproducibility, which has previously been defined as, the precision between laboratories . It has also been defined as total between-laboratory precision . This is a measure of the ability of different laboratories to evaluate each other. Reproducibility includes all the measurement errors or variances, including the within-laboratory error. Other terms include precision, defined as the closeness of agreement between independent test results obtained under stipulated conditions [3] and repeatability, or the precision for the same analyst within the same laboratory, or within-laboratory precision . Note that for none of these definitions do we require the true value for an analytical sample . In practice we do not know the true analyte value unless we have created the sample, and then it is only known to a given certainty (i.e., within a determined uncertainty). 

This involves obtaining the mean-residence time, 0, and the variance, (t, of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform,... 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

Note that from its definition, the scalar spatial correlation function is related to the scalar variance by... 

By definition, the scalar variance can be found directly from the scalar energy spectrum by integrating over wavenumber space ... 

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