Basic Statistical Parameters

The mean is an average over the whole distribution. There are different types of means that can be calculated from the same set of data. See the next subsection (Section 1.3.3) for a more detailed discussion. 

The median is the x value that divides the population into two equal halves. Just like the mean value, populations with different weightings will have different median values. 

The mode is the most common value ofthe distribution, i.e., the highest point of the distribution curve. It is also commonly called the peak value. If there are more than one high frequency region in the distribution, the distribution is called multi-modal. 

A measure of distribution broadness defined as, for arithmetic and geometric, respectively  

The square root of the variance. Note that the geometric standard deviation is not a standard deviation in its true sense. 

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A complete set of descriptors can be calculated by highlighting the node of the corresponding file and selecting data set in the context menu. If not already performed, the data set will be calculated and displayed as a table of all molecules and their basic statistical parameters, like variance, skewness, or kurtosis. Additionally, an ASD is calculated and displayed at the top of the table. The data set table can be saved in different file formats including binary files for fast searches. 

At this point, it is appropriate to introduce the definitions of basic statistical parameters which are extensively covered in textbooks on biochemical statistics (Mandelhall ei a/., 1981 Ratkowsky, 1983 Wiener eta/., 1991 Comish-Bowden, 1995 Daniel, 1995 Rosner, 1996 Zhr, 1996 Punch Allison, 2000) and applied here to enzyme kinetics. 

The main concept addressed in this new multi-part series is the idea of correlation. Correlation may be referred to as the apparent degree of relationship between variables. The term apparent is used because there is no true inference of cause-and-effect when two variables are highly correlated. One may assume that cause-and-effect exists, but this assumption cannot be validated using correlation alone as the test criteria. Correlation has often been referred to as a statistical parameter seeking to define how well a linear or other fitting function describes the relationship between variables however, two variables may be highly correlated under a specific set of test conditions, and not correlated under a different set of experimental conditions. In this case the correlation is conditional and so also is the cause-and-effect phenomenon. If two variables are always perfectly correlated under a variety of conditions, one may have a basis for cause-and-effect, and such a basic relationship permits a well-defined mathematical description. 

This chapter deals with handling the data generated by analytical methods. The first section describes the key statistical parameters used to summarize and describe data sets. These parameters are important, as they are essential for many of the quality assurance activities described in this book. It is impossible to carry out effective method validation, evaluate measurement uncertainty, construct and interpret control charts or evaluate the data from proficiency testing schemes without some knowledge of basic statistics. This chapter also describes the use of control charts in monitoring the performance of measurements over a period of time. Finally, the concept of measurement uncertainty is introduced. The importance of evaluating uncertainty is explained and a systematic approach to evaluating uncertainty is described. 

From the thermodynamic viewpoint, the basic statistical theory is still too complex to provide useful working equations, but it does suggest forms of equations with some purely theoretical terms, and other terms including parameters to be evaluated empirically. In general, the theoretical terms arise from the electrostatic interactions which are simple and well-known while the empirical, terms relate to short-range interionic forces whose characteristics are qualitatively but not quantitatively known from independent sources. But, as we shall see, this division is not complete - there are interactions between the two categories. 

Palm s group has continued to develop statistical procedures for treating solvent effects. In a previous paper, a set of nine basic solvent parameter scales was proposed. Six of them were then purifled via subtraction of contributions dependent on other scales. This set of solvent parameters has now been applied to an extended compilation of experimental data for solvent effects on individual processes. Overall, the new procedure gives a signiflcantly better flt than the well-known equations of Kamlet, Abboud, and Taft, or Koppel and Palm. 

The basic criterion for successful validation was that a method should come within 25% of the "true value" at the 95% confidence level. To meet this criterion, the protocol for experimental testing and method validation was established with a firm statistical basis. A statistical protocol provided methods of data analysis that allowed the accuracy criterion to be evaluated with statistical parameters estimated from the laboratory test data. It also gave a means to evaluate precision and bias, independently and in combination, to determine the accuracy of sampling and analytical methods. The substances studied in the second phase of the study are summarized in Table I. 

Section 5.1 presents the fundamental method as the heart of the chapter— the statistical thermodynamics approach to hydrate phase equilibria. The basic statistical thermodynamic equations are developed, and relationships to measurable, macroscopic hydrate properties are given. The parameters for the method are determined from both macroscopic (e.g., temperature and pressure) and microscopic (spectroscopic, diffraction) measurements. A Gibbs free energy calculation algorithm is given for multicomponent, multiphase systems for comparison with the methods described in Chapter 4. Finally, Section 5.1 concludes with ab initio modifications to the method, along with an assessment of method accuracy. 

This basic assumption is checked by means of the statistical parameters which result from solution of the matrix, which expresses the assumption stated above in the following equation for each compound ... 

Basic statistical analysis on the sensory properties (variance, kurtosis and min-max) showed that 4 sensory parameters where not influenced by the inputs and hence excluded from the analysis (later on, they will be indicated as constant). 

The definitions, methods, and parameters used to validate analytical and bioana-lytical methods are not universal they vary with the type of assay and the regulatory agency. Here, we introduce the more broadly used figures of merit with their generally accepted definitions. Basic statistical procedures are also presented. More technical sources of further information are offered in the Suggested Reading section. 

The HCI software packages arriving with the instruments generally allow basic visual representations with bar/line graphs, together with basic analysis as IC50 calculations and some statistical parameters. However, the user may also export the data obtained to... 

Experimental design is a large topic and we can only mention several of the important issues here. To keep this discussion focused on parameter estimation for reactor models, we must assume the. reader has had exposure to a course in basic statistics [4]. We assume the reader understands the source of experimental error or noise, and knows the difference between correlation and causation. The process of estimating parameters in reactor models is part of the classic, iterative scientific method hypothesize, collect experimental data, compare data and model predictions, modify hypothesis, and repeat. The goal of experimental design is to make this iterative learning process efficient. 

Table 38.1 Basic statistical data on soy isoflavone levels and major laboratory parameters of thyroid function in children...

Many statistical tools have been developed to control critical process parameters. The most commonly used is the control chart, which is an effective way to monitor and control processes and can be defined for both vtuiables and attributes data. The selection of variables data will typically make basic statistical tools more efficient (i.e., lower sample size requirements to achieve necessary confidence levels). 

In basic statistics we learn that probability density functions can be defined by certain constants called distribution parameters. These parameters in turn can be used to characterize random variables through measures of location, shape, and variability of random phenomena. The most important parameters are the mean p and the variance The parameter /r is a measure of the center of the distribution (an analogy is the center of gravity of a mass) while is a measme of its spread or range (an analogy being the moment of inertia of a mass). Hence, when we speak of the mean and the variance of a random variable, we refer to two statistical parameters (constants) that greatly characterize or influence the probabilistic behavior of the random variable. The mean or expected value of a random variable x is defined as... 

This section gives some of the more elementary statistical parameters that may be used to charaeterize and analyze data and discover the underlying relationships among variables that may be hidden by the overlaid variation or noise. Although everything discussed in this section is available in standard texts, a review of the more elementary statistical principles is presented to address the basic problems of measurement quality. This is given... 

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