Among-samples variance

Evaluation Statistical tests can be used to evaluate relative homogeneity based on observed variations in spot sample composition. For a simple binaiy mixture such as that shown in Fig. 19-8, it can be shown (see Ref. 9) that the expected variance among samples containing n particles each is given by... 

If a sample has been analyzed by k laboratories n times each, the sample variances of the n results from each laboratory can be tested for homogeneity —that is, any variance outliers among the laboratories can be detected. The ISO-recommended test is the Cochran test. The statistic that is tested is... 

Statistical calculations were carried out using Stata 8 (Stata Corp LP 2005) as a program to analyze data. To verify the homogeneity of variance, the Barlett test was applied, then One-way analysis of Variance (ANOVA) was performed, and significant differences (P < 0.05) among samples were determined with Tukey s honestly significant difference range. 

Tables in this chapter contain published pressure-volume-temperature data for amorphous homopolymers. Measurements below the melting temperatures for semi-crystalline materials are not included because of the potentially large variance among samples with differing degrees of crystallinity. Rogers [1] and Zoller [2] have also compiled equation-of-state data for amorphous polymers.

Among the variance reduction techniques for simulation the Importance sampling procedure proved to be most advantageous. The importance Sampling Rrocedure Using Resign Points (ISPUD) - for which an elaborate software environment is already available - Is shown to be very flexible in practical application. Its accuracy and efficiency is quite satisfactory. 

As was mentioned earlier, the estimator of a sample mean is not robust and has a breakdown point of 0%. As the sample mean is used in calculating the sample standard deviation, sample variance and other estimators such as kur-tosis and skewness are also not robust. Sample median is the most popular among several different robust estimators of location. Depending on the number of elements in a variable (odd or even), a sample median is calculated according to two schemes. When a variable contains an odd number of elements, the sample median corresponds to the middle element selected from... 

The data were finally arranged in a 80x5200 matrix and analysed by means of PCA, in order to understand the relationships among samples and investigate the presence of trends or clusters. A 2 PCs model, with 78.26% of explained variance, was chosen. 

Collaborative testing provides a means for estimating the variability (or reproducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (Orand) and that due to systematic differences between the analysts (Osys). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate Grand and Osys (or oJand and Osys). We have not considered, however, what is a reasonable value for a method s reproducibility. 

If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E dY), is zero. The variances are given by the following equation... 

Protocols should require documentation of actual application practices and times. Samples of applied pesticides should be collected to document application rates to study sites. The test substance must be applied with typical equipment used for the crop, and the application must be made in accordance with the labeled use. Another variable that impacts such studies is the fact that most landowners have their own application equipment, which increases the variance in actual application rates among fields and may cause differential intra-field heterogeneity in application rates. 

How can statistical error be estimated for a single observable from independent simulations There seems little choice but to calculate the standard error in the mean values estimated from each simulation using Equation (1), where the variance is computed among the averages from the independent simulations and NJ L> is set to the number of simulations. In essence, each simulation is treated as a single measurement, and presumed to be totally independent of the other trajectories. Importantly, one can perform a "reality check" on such a calculation because the variance of the observable can also be calculated from all data from all simulations — rather than from the simulation means. The squared ratio of this absolute variance to the variance of the means yields a separate (albeit crude) estimate of the number of independent samples. This latter estimate should be of the same order as, or greater than, the number of... 

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