The Calculation of Variance

It is often troublesome to calculate the mean S and then find the deviation of each observation x from it before squaring and summing these deviations. 

To lighten the arithmetic in calculating variances, it is frequently worth while to take an arbitrary zero and transform all the numbers on to the new scale. Thus with the above set of numbers we might shift the axis by 9 units so that they now become 

If the numbers had all been 100 larger, then the sum of squares would have been of the order of 100,000, i.e. the arithmetic would have been rather unpleasant, and would have been very much lightened by shifting the zero by 100 units, giving us our first calculation in which the sum of squares was 1470, or by 109 units, giving us otir second calculation in which the sum of squares was only 120. 

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Just as variance describes the spread of data about its mean value for a single variable, so the distribution of multivariate data can be assessed from the covariance. The procedure employed for the calculation of variance can be extended to multivariate analysis by computing the extent of the mutual variability of the variates about some common mean. The measure of this interaction is the covariance. 

Just as in the calculation of variance, there are formulae which simplify the calculation of the individual sums of squares. These formulae are summarized below ... 

Example 2 Calculation of Variance In mixed-hed deionization of a solution of a single salt, there are 8 concentration variables 2 each for cation, anion, hydrogen, and hydroxide. There are 6 connecting relations 2 for ion exchange and 1 for neutralization equilibrium, and 2 ion-exchanger and 1 solution electroneiitrahty relations. The variance is therefore 8 — 6 = 2. 

Example 2 Calculation of Error with Doubled Sample Weight Repeated measurements from a lot of anhydrous alumina for loss on ignition established test standard error of 0.15 percent for sample weight of 500 grams, noting V is the square of s.e. Calculation of variance V and s.e. for a 1000 gram sample is... 

Three commonly used dispersion calculation methods for the prediction of ground level concentrations are based on the above expression. The variance in each method is the calculation of plume rise, Ah, and the horizontal and vertical plume dispersion parameters. These methods are ... 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

The calculation of characteristic functions is sometimes facilitated by first normalizing the random variable involved to have zero mean and unit variance. The transformation that accomplishes this is... 

Table VII contains the weight-average particle diameters as calculated by this technique for all the standards employed using Column Set I. There is some difference between the values of variance used to obtain these averages and those cited in Table V. This is in fact due to the necessity to correct for skewness by...

In the context of data analysis we divide by n rather than by (n - 1) in the calculation of the variance. This procedure is also called autoscaling. It can be verified in Table 31.5 how these transformed data are derived from those of Table 31.4. 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

In summary, after each new measurement a cycle of the algorithm starts with the calculation of the new gain vector (eq. (41.4)). With this gain vector the variance-... 

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) ... 

Variance in root size of 5-day old plants was not taken into account in either the calculation of pmol/root or pmold exuded/root. The weight of filtered and lyophilized exudate of 375 plants was used to extrapolate to the figure of 3245 individuals used for exudate collection in the analysis of exudation/root. 

This approximation has already proven very effective in the calculation of likelihood functions for maximum likelihood refinement of parameters of the heavy-atom model, when phasing macromolecular structure factor amplitudes with the computer program SHARP [53]. A similar approach was also used in computing the variances to be used in evaluation of a %2 criterion in [54]. 

For the basic evaluation of a linear calibration line, several parameters can be used, such as the relative process standard deviation value (Vxc), the Mandel-test, the Xp value [28], the plot of response factor against concentration, the residual plot, or the analysis of variance (ANOVA). The lowest concentration that has been used for the calibration curve should not be less than the value of Xp (see Fig. 4). Vxo (in units of %) and Xp values of the linear regression line Y = a + bX can be calculated using the following equations [28] ... 

One can obtain an exact analytic solution to the first Pontryagin equation only in a few simple cases. That is why in practice one is restricted by the calculation of moments of the first passage time of absorbing boundaries, and, in particular, by the mean and the variance of the first passage time. 

The degrees of freedom of the estimate of variance is given by the divisor (N — 1). This is the number of independent comparisons that can be made between N observations since x is calculated from the observations. If x and (TV — 1) of the values of x are given, the other can be determined. 

Eq. 17.42 is the expression of the resolution for CE in electrophoretic terms. However, the application of this expression for the calculation of Rs in practice is limited because of D,. The diffusion coefficient of different compounds in different media is not always available. Therefore, the resolution is frequently calculated with an expression that employs the width of the peaks obtained in an electropherogram. This way of working results in resolution values that are more realistic as all possible variances are considered (not only longitudinal diffusion in Eq. 17.42). Assuming that the migrating zones have a Gaussian distribution, the resolution can be expressed as follows ... 

At the opposite extreme from the ab initio SCF methods is the Wolfberg-Helmholtz approximation which Hoffmann 6> has applied extensively to organic problems under the term extended Hiickel method . While this has the advantage of requiring very little computation time, the results are so unreliable that the method is essentially useless for the calculation of potential surfaces. Not only are the errors in heats of atomization comparable with those given by ab initio SCF but they are not even the same for isomers. A good example is provided by cyclopropanone (1) which is predicted 7> to be less stable than the isomeric zwitterion 2, a result at variance with the available evidence ) concerning the... 

However, with improper transformation the calculation of confidence bands and amount interval estimates is erroneous because of the non-constant variance ... 

A second reason is that the use of local variance versus global variance can result in markedly different bands. The separate calculations of variance at levels throughout the range of standards produces a wider confidence interval at lower values as seen in Kurtz method. If a common variance is used as the variance estimate then a lower confidence interval is calculated at each point as is probably the case in Wegscheider s method. 

Equations 5.28 and 5.30 provide a general matrix approach to the calculation of the sum of squares of residuals. This sum of squares, SS divided by its associated number of degrees of freedom, DF, is the sample estimate, s, of the population variance of residuals, CJ. ... 

The test of this hypothesis makes use of the calculated Fisher variance ratio, F. 

The optimal number of components from the prediction point of view can be determined by cross-validation (10). This method compares the predictive power of several models and chooses the optimal one. In our case, the models differ in the number of components. The predictive power is calculated by a leave-one-out technique, so that each sample gets predicted once from a model in the calculation of which it did not participate. This technique can also be used to determine the number of underlying factors in the predictor matrix, although if the factors are highly correlated, their number will be underestimated. In contrast to the least squares solution, PLS can estimate the regression coefficients also for underdetermined systems. In this case, it introduces some bias in trade for the (infinite) variance of the least squares solution. 

Although molecular orbital predictions (Section III, A) are often at variance with each other, they are generally agreed that substitution should not take place exclusively at the 2-position. The calculations of Kikuchi do in fact predict the order 2 > 4 > 3,1, which is what is actually observed in the two cases described above. 

There is a statistical test to check the homogeneity of variances. We repeatedly measure the highest and the lowest standard samples (10 times each) and calculate the variances for both data sets. The F-test gives us an answer on the question, whether they are significantly different or not. 

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