Variance total

Estimations based on statistics can be made for total accuracy, precision, and reproducibility of results related to the sampling procedure being applied. Statistical error is expressed in terms of variance. Total samphng error is the sum of error variance from each step of the process. However, discussions herein will take into consideration only step (I)—mechanical extraction of samples. Mechanical-extracdion accuracy is dependent on design reflecding mechanical and statistical factors in carrying out efficient and practical collection of representative samples S from a bulk quantity B,... 

Project steps Budget Actual Variance Total Planned Actual Variance Total... 

These observations may be summarized conveniently in an analysis-of-variance table-. Table 26-7 illustrates this type of table for the above case. The overall variance (total mean square) Sj(N — 1) contains contributions due to variances within as well as between classes. The variation between classes contains both variation within classes and a variation associated with the classes themselves and is given by the expected mean square aj + not. Whether not is significant can be determined by the F test. Under the null hypothesis, = 0. Whether the ratio... 

On a traite, dans cette serie d analyses des correspondances, les memes parametres physiques et chimiques mais sur les eaux des quatre piezometres et de la resurgence uniquement. Les axes 1, 2, 3, 4, 5 absorbent successive-ment 43, 23, 17, 7 et 5% soit le 95% de la variance totale. Dans le plan des axes 2 et 2 (Fig. 5) on distingue de nouveau deux tendances et leur intersection. 

Once numerical estimates of the weight of a trajectory and its variance (2cr ) are known we are able to use sampled trajectories to compute observables of interest. One such quantity on which this section is focused is the rate of transitions between two states in the system. We examine the transition between a domain A and a domain B, where the A domain is characterized by an inverse temperature - (3. The weight of an individual trajectory which is initiated at the A domain and of a total time length - NAt is therefore... 

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... 

Total variance measured from chromatogram = column variance + variance due to instrument volumes + variance due to electronic response time... 

Standard Costs for Budgetary Control For convenience and simplicity, we shall consider the total cost of a manufac tured product to be the sum of the material, labor, and overhead costs. Standard costs are those that have been predetermined and budgeted for the manufacture of a given amount of product in a given time. The deviation of the actual cost from the standard cost is called the variance. It is far easier to make comparisons between periods by using variances than by using actual production data. The different variances for materi, labor, and overhead costs are listed in Table 9-38. 

The total variances in each category are listed in Table 9-40. 

The a s are dimension constants, with a value of 1. is the multiple correlation coefficient, the fraction of total variance in the data accounted for by the model. 

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

The different dispersion processes (1, 2, 3,...) that occur in a column will now be considered theoretically, their individual contributions to the variance per unit length of the column (Hi, H2, H3...) evaluated and then summed to provide an expression for the total variance per unit length of the column (H), i.e.,... 

Equation (1) can be used in a general way to determine the variance resulting from the different dispersion processes that occur in an LC column. However, although the application of equation (1) to physical chemical processes may be simple, there is often a problem in identifying the average step and, sometimes, the total number of steps associated with the particular process being considered. To illustrate the use of the Random Walk model, equation (1) will be first applied to the problem of radial dispersion that occurs when a sample is placed on a packed LC column in the manner of Horne et al. [3]. 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

Dividing the total variance by the column length (1), the multi-path contribution (Hm) to the overall height of the theoretical plate (H) is obtained. 

Equation (5), however, would apply only to a perfectly packed column so Van Deemter introduced a constant (2X) to account for the inhomogeneity of real packing (for ideal packing (X) would take the value of 0.5). Consequently, his expression for the multi-path contribution to the total variance per unit length for the column (Hm) is... 

Thus, treating the diffusion process in a similar way to that shown in Figure 4 the total variance due to longitudinal diffusion in a column of length (1) is given by equation (7), viz.,... 

Then, the contribution to the total variance per unit length for the column from longitudinal diffusion in the stationary phase will be... 

It is seen that by a simple curve fitting process, the individual contributions to the total variance per unit length can be easily extracted. It is also seen that there is minimum value for the HETP at a particular velocity. Thus, the maximum number of theoretical plates obtainable from a given column (the maximum efficiency) can only be obtained by operating at the optimum mobile phase velocity. 

There are four major sources of extra-column dispersion which can be theoretically examined and/or experimentally measured in terms of their variance contribution to the total extra-column variance. They are as follows ... 

A low volume (0.2 pi) Valeo sample valve was employed with one end of the open tube connected directly to the valve and the other connected directly to the sensor cell of the detector. The UV detector was the LC 85B manufactured by Perkin Elmer, and specially designed to provide low dispersion with a sensor volume of about 1.4 pi. The total variance due to extra-column dispersion was maintained at... 

Another critical instrument specification is the total extra-column dispersion. The subject of extra-column dispersion has already been discussed in chapter 9. It has been shown that the extra-column dispersion determines the minimum column radius and, thus, both the solvent consumption per analysis and the mass sensitivity of the overall chromatographic system. The overall extra-column variance, therefore, must be known and quantitatively specified. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

Often, it is not necessary to know the total CSD since some mean particle characteristic and possibly its variance may suffice. In general, such characteristics of the CSD can be obtained from the moment equation, below, by... 

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