Several Variances

Suppose, extending our previous example, we had a number of determinations of variability for each method on a number of different samples. Suppose a second sample analysed by Method B gave results 93.1, 91.2 and 92.6. Its variance would then (using the revised zero by subtracting 90 from every observation) be [3.1 + 1.2 + 2.6 — (3.1 + 1.2 -f 2.6) /3]/(3 — 1) = (17.81 — 6.9 /3)/2 == 1.94/2 = 0.97. 

To form the average of the two variances, we weight them in accordance with their degrees of freedom. Thus the average would be 

This average is the same as obtained by pooling the sums of squares and degrees of freedom for the two samples, i.e. (30.82 + 1.94)/(6 + 2). 

This is the correct method for obtaining an average variance. It is incorrect 

If we have the variances expressed as standard deviations, we should square them to convert them to variances, and take the average as indicated. 

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Numerically evaluated average mass transfer coefficients, k, based on the average of 200 different realizations of the log-normal hydraulic conductivity, as a function of C,JC,X for several variances of Y=lnK (a =0.1,0.2,0.3,0.4, and 0.5) for a hydraulic gradient of dhldx=0.01 and mean log-transformed hydraulic conductivity of 7=0.8 are shown in Fig. 7a. The results indicate that for increasing C,JC,X there is a significant increase in k. Low values of the anisotropy... 

Fig. 7a, b Average mass transfer coefficient as a function of a aquifer anisotropy ratio for several variances of the log-transformed hydraulic conductivity distribution b variance of the log-transformed hydraulic conductivity distribution where open circles represent numerically generated data and solid lines represent linear fits. All model parameter values are identical with those used in Fig. 6... 

A more general form of Bartlett s test, for the case where the several variances being compared have different degrees of freedom, is available, but will not be discussed here. 

If you wish to compare the variances of two sets of data that are normally distributed, use the F-test. For comparing more than two samples, it may be sufficient to use the F max-test, on the highest and lowest variances. The Scheff Box (log-ANOVA) test is recommended for testing the significance of differences between several variances. Non-parametric tests exist but are not widely available you may need to transform the data and use a test based on the normal distribution. 

The test of significance for several variances, whereby an analytical method is applied to several samples or where several methods are to be compared after analyzing the same sample, is generally known as Bartlett s test. Specifically we wish to know whether the several variances could reasonably exist in the same population. 

We suppose that a measurement signal is a mix of r (unknown) independent sources Sj with variance cf. A detector give p> r measurement signals /W/t obtained with several frequencies. The relations between sources and measurement signals are supposed to be linear, but the transfer matrix T is unknown. If we get n >p>r samples m i) of measurement signals, mix of n... 

Let s consider the following problem. Two sets of blood samples have been collected from a patient receiving medication to lower her concentration of blood glucose. One set of samples was drawn immediately before the medication was administered the second set was taken several hours later. The samples are analyzed and their respective means and variances reported. ITow do we decide if the medication was successful in lowering the patient s concentration of blood glucose ... 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E y), the true mean, with variance C /n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day s measurement, n. Of course, other fac tors (weather, weekends) may make the observations on different days not distributed identically. 

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. 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

FIG. 23 14 Comp arison of maximiim mixed, segregated, and ping flows, (a) Relative volumes as functions of variance or n, for several reaction orders, (h) Second-order reaction with n = 2 or 3. (c) Second-order, n = 2. (d) Second-order, n = 5. 

Surface roughness to process risk FMEA Severity Rating, strength Ultimate tensile strength Uniaxial yield strength Bilateral tolerance Unilateral tolerance Tolerance to process risk Variance Class width... 

Variances in resin performance and capacities can be expected from normal annual attrition rates of ion-exchange resins. Typical attrition losses that can be expected include (1) Strong cation resin 3 percent per year for three years or 1,000,000 gals/ cu.ft (2) Strong anion resin 25 percent per year for two years or 1,000,000 gals/ cu.ft (3) Weak cation/anion 10 percent per year for two years or 750,000 gals/ cu. ft. A steady falloff of resin-exchange capacity is a matter of concern to the operator and is due to several conditions ... 

Table 2.5-2 Mean Variance and Moment-Generating Functions for Several Distributions ...

However, although it allowed a correct description of the current-voltage characteristics, this model presents several inconsistencies. The main one concerns the mechanism of trap-free transport. As noted by Wu and Conwell [1191, the MTR model assumes a transport in delocalized levels, which is at variance with the low trap-free mobility found in 6T and DH6T (0.04 cm2 V-1 s l). Next, the estimated concentrations of traps are rather high as compared to the total density of molecules in the materials (see Table 14-4). Finally, recent measurements on single ciystals [15, 80, 81] show that the trap-free mobility of 6T could be at least ten times higher than that given in Table 14-4. 

Variance scaling is performed on a variable by variable basis. In other words, we would variance scale a set the concentration values of a data set on a component by component basis. Starting with the first component, we compute the total variance of the concentrations of that component. There are several variations on variance scaling. First, we will consider the most the method which adjusts all the variables to exactly unit variance. To do this we compute the variance of the variable, and then use the variance to scale all the concentrations of all the samples so that the new variance for the component is equal to unity. 

As Equations 10-1 and 10-2 show, the variance (s2) can lay claim to being a more fundamental variable than the standard deviation. In any event, when s2 significantly exceeds sc2, there may be several important sources of variation. It is then advisable to discover what these sources are. Because analysis of variance is a systematic procedure for making this discovery, it is expedient to use it, and to stop thinking exclusively in terms of the standard deviation. 

Several examples will be given to show how analysis of variance can be used in x-ray emission spectrography. The data will be summarized in tabular form under headings already given except for sums of squares, under which are listed data that correspond to the numerator of the fraction in Equation 10-2. 

The mixing time will be that required for the mixture composition to come within a specified deviation from the equilibrium value and this will be dependent upon the way in which the tracer is added and the location of the detector. It may therefore be desirable to record the tracer concentration at several locations, and to define the variance... 

The prediction of the present naive theory is thus seen to contrast with that of Bartell s perturbation treatment and is also at variance with experimental results. This fact lends some support to the ordinary view that the transition state will survive several periods of ordinary vibrational motion. 

Analysis of variance (ANOVA) tests whether one group of subjects (e.g., batch, method, laboratory, etc.) differs from the population of subjects investigated (several batches of one product different methods for the same parameter several laboratories participating in a round-robin test to validate a method, for examples see Refs. 5, 9, 21, 30. Multiple measurements are necessary to establish a benchmark variability ( within-group ) typical for the type of subject. Whenever a difference significantly exceeds this benchmark, at least two populations of subjects are involved. A graphical analogue is the Youden plot (see Fig. 2.1). An additive model is assumed for ANOVA. 

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