Basic sampling statistics

The overall random error of an analytical process, expressed as the variance v, can be regarded as the sum of two other variances, that due to sampling x), and that due to the remaining measurement components of the process sb. These variances are estimates of the corresponding population variances cr and its components (tT and 75. The overall standard deviation s is calculated as usual from single measurements on each of h sample increments, and the confidence limits of the true mean value of the population 11 are obtained from  

From Equation (I), the variance of v is. v //t. used as an estimate of o /h. To reduce, i.e.. improve this variance, it is necessary to take n measurements of each of the h increments. This allows the. separation of the sampling and measurement variances by one-way analysis of variance. The replication of the measurements is expected to reduce the measurement variance 05, and the overall variance of the mean becomes o- n/i-i-(tT//i. This relationship provides general guidance on the best practical way of reducing the overall variance. The measurement variance can be mini- 

When a bulk material is examined, the size of each increment is also of importance. Clearly, if each increment is too large, it may conceal the extent of variation within the bulk material if it is too small, many increments are necessary to reveal the extent of the sampling variance. Inoameu.s 5] utilized the fact that. v) decreased as the increment size increased to develop the equation  

The broken line shows the result for an idealized sampling plan 

Equations analogous to (2) and (3) can be derived for the more complex situations that arise in stratified random sampling [1], [6]. A computer program has been developed [7] to assist in the solution of sampling problems it is especially directed at geochemical and other areas where the sizes and shapes of particulate solids may affect R.  

---

Basic summary statistics and graphical assessments of the data distributions are often invaluable to this initial examination. For example. Table 3.1 shows summary statistics for the distribution of protein expression levels measured in a proteomics experiment involving the three patients sampled before and after a chemotherapeutic treatment (Cho et al., 2007). 

The first part of this chapter contains a short introduction to statistical mechanics of continuum models of fluids and macromolecules. The next section presents a discussion of basic sampling theory (importance sampling) and the Metropolis Monte Carlo and molecular dynamics methods. The remainder of the chapter is devoted to descriptions of methods for calculating F and S, including those that were mentioned above as well as others. 

Population and sample are discussed in Sect. 20.2. The properties of a population are studied in a representative random sample taken from that population. In pharmacy preparation practice populations are for instance batches of dosage units. Their properties are measured by analytical or biological assays and summarised as means, standard deviations and many other sample statistics. Some basic notions of probability distributions are briefly discussed. 

Table 1 presents basic descriptive statistics of obtained data. Finally, the stochastic sample was consisting of 2838 cases and the total number of obtained cases of all considered variables was equal to 23,967. However, there are many missing data, so in further study the set of 333 cases was considered. This set consists of these sample points for which the chloroform concentration at network point and at water treatment plant were given (laboratory analyzes are made according to the monitoring plan which takes into account high costs of gas chromatography). 

A critical issue in statistical conformational sampling is the reliability and accuracy of the samples. In order to ensure the reliability of the statistical sampling of a particular conformational state, the relevant conformational region must be sampled sufficiently. A measure of the accuracy of statistical results is the variances of the samples. In order for the statistical results to be accurate, several independent samples have to be taken and the variances of the results calculated from these samples must be within given tolerances. Conformational sampling using the standard MC or MD procedures does not always meet the requirements of reliability and accuracy in all problems. Therefore, efforts have to be made to improve upon the basic sampling procedures in order to obtain satisfactory results. 

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

Part A, dealing with the Fundamentals of Quantitative Chemical Analysis, has been extended to incorporate sections of basic theory which were originally spread around the body of the text. This has enabled a more logical development of theoretical concepts to be possible. Part B, concerned with errors, statistics, and sampling, has been extensively rewritten to cover modern approaches to sampling as well as the attendant difficulties in obtaining representative samples from bulk materials. The statistics has been restructured to provide a logical, stepwise approach to a subject which many people find difficult. 

The data in the training set are used to derive the calibration which we use on the spectra of unknown samples (i.e. samples of unknown composition) to predict the concentrations in those samples. In order for the calibration to be valid, the data in the training set which is used to find the calibration must meet certain requirements. Basically, the training set must contain data which, as a group, are representative, in all ways, of the unknown samples on which the analysis will be used. A statistician would express this requirement by saying, "The training set must be a statistically valid sample of the population... 

Microarray experiments generate large and complex data sets that constitute e.g. lists of spot intensities and intensity ratios. Basically, the data obtained from microarray experiments provide information on the relative expression of genes corresponding to the mRNA sample of interest. Computational and statistical tools are required to analyze the large amount of data to address biological questions. To this end, a variety of analytical platforms are available, either free on the Web or via purchase of a commercially available product. 

A sterility test is basically a test which assesses whether a sterilized pharmaceutical or medical product is free from contaminating microorganisms, by incubation of either the whole or a part of that product with a nuhient medium. It thus becomes a destructive test and raises the question as to its suitability for testing large, expensive or delicate products or equipment. Furthermore, by its very nature such a test is a statistical process in which part of a batch is randomly sampled and the chance of the batch being passed for use then depends on the sample passing the sterility test. 

The results of environmental monitoring exercises will be influenced by a variety of variables including the objectives of the study, the sampling regime, the technical methods adopted, the calibre of staff involved, etc. Detailed advice about sampling protocols (e.g. where and when to sample, the volume and number of samples to collect, the use of replicates, controls, statistical interpretation of data, etc.) and of individual analytical techniques are beyond the scope of this book. Some basic considerations include the following, with examples of application for employee exposure and incident investigation. 

Statistical experimental design is characterized by the three basic principles Replication, Randomization and Blocking (block division, planned grouping). Latin square design is especially useful to separate nonrandom variations from random effects which interfere with the former. An example may be the identification of (slightly) different samples, e.g. sorts of wine, by various testers and at several days. To separate the day-to-day and/or tester-to-tester (laboratory-to-laboratory) variations from that of the wine sorts, an m x m Latin square design may be used. In case of m = 3 all three wine samples (a, b, c) are tested be three testers at three days, e.g. in the way represented in Table 5.8 ... 

Analysis of Variance (ANOVA) is a useful tool to compare the difference between sets of analytical results to determine if there is a statistically meaningful difference between a sample analyzed by different methods or performed at different locations by different analysts. The reader is referred to reference [1] and other basic books on statistical methods for discussions of the theory and applications of ANOVA examples of such texts are [2, 3],... 

Statistics establish quality limits for the answers derived from a given method. A given laboratory result, or a sample giving rise to a given result, is considered good if it is within these limits. In order to understand how these limits are established, and therefore how it is known if a given result is unacceptable, some basic knowledge of statistics is needed. We now present a limited treatment of elementary statistics. 

FIGURE 2.9 Basic statistics of multivariate data and covariance matrix. xT, transposed mean vector vT, transposed variance vector vXOtal. total variance (sum of variances vb. .., vm). C is the sample covariance matrix calculated from mean-centered X. 

---