Variance of sampling

Harnhy, N., The Estimation of the Variance of Samples Withdrawn from a Random Mixture of Multi-Sized Particles, Chem. Eng. No. 214 CE270-71 (1967). 

Figure 7.6 Principle components analysis (PCA) of PCB congener concentrations in technical Aroclor mixtures, contaminated water, caged brown trout, SPMDs, and hexane filled dialysis bags. The plot shows that 77% of the variance of samples within the 95% confidence ellipse is explained by PCI and PC2 and that caged fish and SPMDs are clustered together (PCA plot courtesy of Kathy Echols, USGS-CERC, Columbia, MO, USA).

When executed improperly, sample preparation is the source of the second-largest component of the overall variance of sampling and analysis. Although it is not specified as a requirement, it is generally recognized that the variance of sample division and analysis is not more than 20% of the total variance of sampling, division, and analysis (ASTM D-2013). The particle size distribution (size consist) of the laboratory sample depends on its intended use in the laboratory and the nature of the test methods to be applied. The minimum allowable... 

Minimization of the variance of sampling and analysis shall be discussed using an example from environmental monitoring according to the scheme of SHARAF et al. [1986]. 

Plan 2 After mixing all twenty samples, one sample, being one twentieth of the whole, is analyzed. The variance of sampling is definitely unchanged after mixing. Then, the variance of the mean value is given by ... 

The description of an object in the sense of environmental investigation may be the determination of the gross composition of an environmental compartment, for example the mean state of a polluted area or particular location. If this is the purpose, the number of individual samples required and the required mass or size of these increments have to be determined. The relationship between the variance of sampling and that of analysis must be known and both have to be optimized. The origin of the variance of the samples can be investigated by the study of variance contribution of the different steps of the analytical process by means of the law of error propagation (Eq. 4-21) according to Section 4.3.4. 

If a sample is to describe a population averaged over space and time, spatial and temporal changes should be taken into account and included in the total variance of sampling. The later steps, that is, sample preservation, transport, storage, preparation for analysis, and the measurement proper, will be dealt with in the subsequent chapters of this book. Taylor9 uses the terms sample... 

Table 3.4 F-Test values for comparison of variance of samples with equal degrees of freedom,...

A formula is presented for calculating the theoretical variance of samples taken from a randomly mixed batch of two materials (P and Q), each of which has its own size distribution ... 

As in the t distribution that describes the distribution of sample means, a specific probability distribution is necessary to describe the distribution of the variance of samples. The chi-square distribution [2] describes the probability distribution of the sample variance for a sample of size n, where the test statistic is... 

If the selection of the sample elements is strictly random there will be no correlation between the observations, and we can use Eq. (2.15) to calculate the variance. Moreover, since we are assiuning that the observations are always made on the same population, all observations are individually distributed with the same population variance, a. Therefore, by substituting 1/n for Oj and for ct in Eq. (2.15) we can express the variance of sample average values as... 

Table 1 Analysis of variances of sampling steps carried out in accordance with Figure 2 for bulk material comprising z lots...

We have already found that the estimation of O is bias-free, i.e., its exact average (O) is identical with O) [see Eq, (4,3)]. However, we have also convinced ourselves that variances of sampled data are not free of bias. Combining Eqs. (4,10) and (4.11), we consequently find that the expected value of the simplest nonlinear function of O, AO) = o ... 

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