Overall variance

Segregation variance of increment collection variance caused by nonrandom distribution of ash content or other constituent in the lot (ASTM D-2234). Total variance overall variance resulting from collecting single increments and including division and analysis of the single increments (ASTM D-2013 ASTM D-2234). 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

Percent of overall variance (So) due to the method as a function of the relative magnitudes of the standard deviation of the method and the standard deviation of sampling (Sm/Ss). The dotted lines show that the variance due to the method accounts for 10% of the overall variance when Ss= 3 xs . 

Thus, a 10% improvement in the method s standard deviation changes the overall variance by approximately 4%. 

To determine which step has the greatest effect on the overall variance, both si, and si must be known. The analysis of replicate samples can be used to estimate the overall variance. The variance due to the method is determined by analyzing a standard sample, for which we may assume a negligible sampling variance. The variance due to sampling is then determined by difference. 

The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 

Unfortunately, the simple situations just described are often the exception. In many cases, both the sampling variance and method variance are significant, and both multiple samples and replicate analyses of each sample are required. The overall error in this circumstance is given by... 

In this experiment the overall variance for the analysis of potassium hydrogen phthalate (KHP) in a mixture of KHP and sucrose is partitioned into that due to sampling and that due to the analytical method (an acid-base titration). By having individuals analyze samples with different % w/w KHP, the relationship between sampling error and concentration of analyte can be explored. 

This experiment introduces random sampling. The experiment s overall variance is divided into that due to the instrument, that due to sample preparation, and that due to sampling. 

The overall standard deviation, S, is the square root of the average variance for the samples used to establish the control plot. 

More attention to selecting and obtaining a representative sample. The design of a statistically based sampling plan and its implementation are discussed earlier, and in more detail than in other textbooks. Topics that are covered include how to obtain a representative sample, how much sample to collect, how many samples to collect, how to minimize the overall variance for an analytical method, tools for collecting samples, and sample preservation. 

Actual Costs versus Standard Costs Let us consider the sales, profits, and manufacduring-cost data in Table 9-40. The gross profit is 33,129 per period better than e)mected. Clearly, there is less incentive to investigate overall costs when the profit variance is favorable than if the profit were less than expected. However, standard costing enables an objective analysis of the data, whether good or bad, to be made. 

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

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. 

The composite curve from the Huber equation is similar to that obtained from that of Van Deemter but the individual contributions to the overall variance are different. The contributions from the resistance to mass transfer in the stationary phase and... 

The sum of the variances will give the overall variance for the extra-column dispersion (og). Thus,... 

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. 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

To confirm the pertinence of a particular dispersion equation, it is necessary to use extremely precise and accurate data. Such data can only be obtained from carefully designed apparatus that provides minimum extra-column dispersion. In addition, it is necessary to employ columns that have intrinsically large peak volumes so that any residual extra-column dispersion that will contribute to the overall variance is not significant. Such conditions were employed by Katz et al. (E. D. Katz, K. L. Ogan and R. P. W. Scott, J. Chromatogr., 270(1983)51) to determine a large quantity of column dispersion data that overall had an accuracy of better than 3%. The data they obtained are as follows and can be used confidently to evaluate other dispersion equations should they appear in the literature. 

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