The Definition of Variance

The definition of variance is formally not L(x — S) /(n — 1), as quoted above, but 2(x — x) /n, and it is quoted as the latter in many statistical texts. In any practical application, however, we should use the formula with (n — 1) as the denominator. 

In calculating S(x — x) we should be calculating the sum of squares of the deviations from the tme mean of the population from which the sample is drawn. We have only an estimate of the mean, the mean of our sample, which will not in general be identical with the true mean of the whole population. The sum of squares of the deviations from the mean of the sample will be smaller than the sum of the squares of the deviations from the true mean. To see this, suppose we have two observations 1 and 3, and that the true population mean is actually 1. The sum of the squares of the deviations from the true mean is (I —1) + (3 — 1) . -= 4 working from the true mean we would use n as the divisor to get the estimate of the variance as 4/2 = 2. The sum of the squares of the deviations from the sample mean is (2 — 1) -i- (3 — 2) = 2 if we used n as the divisor the estimate of the variance is 1/1 1, whereas if we use (n— s the divisor we get 2/(2 — 1) = 2 as our estimate of the variance. It is app. mt that the use of (n — 1) instead of n as the divisor helps to compensate for the use of the sample mean instead of the true mean, 

It is a matter of common experience that we cannot manufacture articles identically successive individuals differ slightly in their size, quality, etc. Similarly, if we make repeat determinations of some physico-chemical quantity, e.g. the yield from some process, the values we obtain will not be exactly similar this variability would be due not only to the (possible) variability of the process, but also to the errors of measurement. 

It can be shown theoretically that if we assume that there is a very large number of very small errors ail operating independently, then we get a distribution of the form shown in figure 2, known as the Gaussian distribution. 

This Gaussian distribution has the following valuable properties — 

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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 ... 

From the definition of variance and correlation coefficient and equation (4.2.18), we find that the covariance matrix is... 

According to the definition of variances and correlation coefficients, the following relationships are valid ... 

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 ... 

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. 

As done below for two examples, expressions can also be derived for the scalar variance starting from the model equations. For the homogeneous flow under consideration, micromixing controls the variance decay rate, and thus y can be chosen to agree with a particular model for the scalar dissipation rate. For inhomogeneous flows, the definitions of G and M(n) must be modified to avoid spurious dissipation (Fox 1998). We will discuss the extension of the model to inhomogeneous flows after looking at two simple examples. 

Because of Eq. (4.31), A is the variance-covariance matrix of the set of unknowns X, which we will refer to as the eigenparameter. The eigenparameters X are, by the definition of the variance-covariance matrix, not correlated. 

This definition is connected with the definition of sensitivity of the method as the concentration of the lowest standard with a coefficient of variance (CV) <20%. 

The definition of sample variance with an (n-1) in the denominator leads to an unbiased estimate of the population variance, as shown above. Sometimes the sample variance is defined as the biased variance ... 

From the definition of d the variance-covariance matrix % is evaluated, taking into account the variance-covariance matrices of the input data x, and of the instrument readings yt [1]. 

The definitions of the degree of mixing presented above aim at a local characterization of the mixture homogeneity in the physical space. There also exist more indirect mixing indices. The segregation index J of Danckwerts (12) is one of the most famous ones. It applies to continuous reactors and relies upon the variance of age J = Var oip/Var a>where a is the age of a molecule,... 

The definition of symbols is in the Table of Nomenclature. Basically SD is a number proportional to the reactor length, made dimensionless by a proper combination of thermal and reaction kinetic paramters. t is proportional to the temperature rise, made dimensionless by a combination of inlet temperature and activation energy, y and a2 are the mean and variance, respectively, of the residence-time distribution in the reactor. 

Some of the concepts used in defining confidence limits are extended to the estimation of uncertainty. The uncertainty of an analytical result is a range within which the true value of the analyte concentration is expected to lie, with a given degree of confidence, often 95%. This definition shows that an uncertainty estimate should include the contributions from all the identifiable sources in the measurement process, i.e. including systematic errors as well as the random errors that are described by confidence limits. In principle, uncertainty estimates can be obtained by a painstaking evaluation of each of the steps in an analysis and a summation, in accord with the principle of the additivity of variances (see above) of all the estimated error contributions any systematic errors identified... 

Very often one does not require as much detail as presented in Figure 2 and the model can be simplified considerably. For example, one may only be interested in the first few moments of the latex particle size distribution, F(V,t) so as to get a mean and variance of the distribution. This can be readily calculated from the definition of the jth moment ... 

This last equation has a form similar to the famous equation of the single direction diffusion of a property in an unsteady state, the property here being the local concentration v . The diffusion coefficient is represented by the variance of the elementary speeds which are given by their individual states Vj, vj,..., It is important to notice the consistency of the definition of the diffusion coefficient. 

In order to complete the selection characterization, we can use the variance or dispersion that shows the displacement of the selection values with respect to the mean value. Relations (5.11) and (5.12) give the definition of the dispersion ... 

Recall that the turbulent part of the turbulent variable is given by a = A —A. Substituting this into the biased definition of variances and comparing the result with the definition of averages gives ... 

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