Measurement of variance

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 ... 

PCA is sensitive with respect to outliers. Outliers are unduly increasing classical measures of variance (that means nonrobust measures), and since the PCs are following directions of maximum variance, they will be attracted by outliers. Figure 3.8 (left) shows this effect for classical PCA. In Figure 3.8 (right), a robust version of PCA was taken (the method is described in Section 3.5). The PCs are defined as directions maximizing A robust measure of variance (see Section 2.3) which is not inflated by the outlier group. As a result, the PCs explain the variability of the nonoutliers which refer to the reliable data information. 

As already noted in Section 3.4, outliers can be influential on PCA. They are able to artificially increase the variance in an otherwise uninformative direction which will be determined as PCA direction. Especially for the goal of dimension reduction this is an undesired feature, and it will mainly appear with classical estimation of the PCs. Robust estimation will determine the PCA directions in such a way that a robust measure of variance is maximized instead of the classical variance. Essential features of robust PCA can be summarized as follows ... 

The last measure of dispersion to be noted here is the range, which is the difference between the largest and the smallest value in any data that are reported. For small quantities of data (less than 10) the range is a useful number for comparison and is used in sampling and quality control. However, since this measure of variance is only affected by two of the data points, it... 

Weighted variance. A weighted measure of variance, defined as... 

Order and factors of variance performance measurement % of variance Performance measurement... 

EXTERNAL VARIANCE A weighted measure of variance that takes into account actual variability of data. (Compare Internal variance)... 

Repeatability Measure of variance between results from testing identical samples with the same methods. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

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). 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

The larger variance is placed in the numerator. For example, the F test allows judgment regarding the existence of a significant difference in the precision between two sets of data or between two analysts. The hypothesis assumed is that both variances are indeed alike and a measure of the same a. 

If improvement in precision is claimed for a set of measurements, the variance for the set against which comparison is being made should be placed in the numerator, regardless of magnitude. An experimental F smaller than unity indicates that the claim for improved precision cannot be supported. The technique just given for examining whether the precision varies with the two different analytical procedures, also serves to compare the precision with different materials, or with different operators, laboratories, or sets of equipment. 

Variance Another common measure of spread is the square of the standard deviation, or the variance. The standard deviation, rather than the variance, is usually reported because the units for standard deviation are the same as that for the mean value. 

Precision is a measure of the spread of data about a central value and may be expressed as the range, the standard deviation, or the variance. Precision is commonly divided into two categories repeatability and reproducibility. Repeatability is the precision obtained when all measurements are made by the same analyst during a single period of laboratory work, using the same solutions and equipment. Reproducibility, on the other hand, is the precision obtained under any other set of conditions, including that between analysts, or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the reproducibility of an analysis can be no better than its repeatability. 

Variance was introduced in Chapter 4 as one measure of a data set s spread around its central tendency. In the context of an analysis of variance, it is useful to see that variance is simply a ratio of the sum of squares for the differences between individual values and their mean, to the degrees of freedom. For example, the variance, s, of a data set consisting of n measurements is given as... 

More recently, a number of tests of chemical stabihty of the latex concentrate have been developed. Chemical stabihty variance in the raw concentrate has considerable effect on the dipping characteristics of latex compounds, and can also affect mechanical stabihty of the compound. A broad rule is that, while latex MST can be increased or decreased without necessarily affecting its chemical stabihty, any change in the latter always is reflected in the MST. A new test, in which chemical stabihty is deterrnined by measurement of the effect of weak 2inc acetate solution added to a second mechanical stabihty sample and the result contrasted with the original MST, is available to numerically quantify chemical stabihty (56). 

If the normalized method is used in addition, the value of Sjj is 3.8314 X 10 /<3 , where <3 is the variance of the measurement of y. The values of a and h are, of course, the same. The variances of a and h are <3 = 0.2532C , cf = 2.610 X 10" <3 . The correlation coefficient is 0.996390, which indicates that there is a positive correlation between x and y. The small value of the variance for h indicates that this parameter is determined very well by the data. The residuals show no particular pattern, and the predictions are plotted along with the data in Fig. 3-58. If the variance of the measurements of y is known through repeated measurements, then the variance of the parameters can be made absolute. 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

One measure of the performance of a control system is the variance of the controlled variable from the target. Both improving the control svstem and reducing the disturbances will lead to a lower variance in the controlled variable. 

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... 

Other measures of efficiency are derived from the experimental RTD, which is characterized at least approximately by the variance This quantity is zero for plug flow and unity for complete mixing, and thus affords natural bounds to an efficiency eqiiated to the variance. It is possible, however, for the variance to fall out of the range (0,1) when stagnancy or bypassing occurs. 

A related measure of efficiency is the equivalent number of stages erkngof CSTR battery with the same variance as the measured RTD. Practically, in some cases 5 or 6 stages may be taken to approximate plug flow. The dispersion coefficient also is a measure of deviation... 

To express the measure of dispersion in the original scale of measurement, it is usual to take the square root of the variance to give the standard deviation ... 

A measure of the variability of the differences is the variance S, which is the second moment of the distribution of these differences ... 

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