Statistics kurtosis

One major task of statistics is to describe the distribution of a set of data. The most important characteristics of a distribution are the location, the dispersion, the skewness and the kurtosis. These are discussed in the following slides. 

Environmental concentrations and other environmental variables tend to have positive skewness. Therefore, environmental statistics texts often focus on positive skew distributions such as the log-normal, gamma, and Weibull. Discussions of distributions with nonnormal kurtosis are somewhat more scarce. 

Skewness and kurtosis can be characterized using familiar formulae, based on 3rd and 4th centered moments. Alternative, outlier-resistant statistics can be based on quantiles (e.g., Helsel and Hirsch 1992 Hoaglin et al. 1983). 

Some statistical tests are specific for evaluation of normality (log-normality, etc., normality of a transformed variable, etc.), while other tests are more broadly applicable. The most popular test of normality appears to be the Shapiro-Wilk test. Specialized tests of normality include outlier tests and tests for nonnormal skewness and nonnormal kurtosis. A chi-square test was formerly the conventional approach, but that approach may now be out of date. 

Statistical tests (see Section 2.2) exist for both skewness and kurtosis. From the result of such tests one can decide if the deviation of a distribution function based on measurements from an ideal (test) function may be tolerated. 

Approximation methods can be useful, but as the degree of complexity of the input distributions or the model increases, in terms of more complex distribution shapes (as reflected by skewness and kurtosis) and non-linear model forms, one typically needs to carry more terms in the Taylor series expansion in order to produce an accurate estimate of percentiles of the distribution of the model output. Thus, such methods are often most widely used simply to quantify the mean and variance of the model output, although even for these statistics, substantial errors can accrue in some situations. Thus, the use of such methods requires careful consideration, as described elsewhere (e.g. Cullen Frey, 1999). 

Exploratory data analysis (EDA). This analysis, also called pretreatment of data , is essential to avoid wrong or obvious conclusions. The EDA objective is to obtain the maximum useful information from each piece of chemico-physical data because the perception and experience of a researcher cannot be sufficient to single out all the significant information. This step comprises descriptive univariate statistical algorithms (e.g. mean, normality assumption, skewness, kurtosis, variance, coefficient of variation), detection of outliers, cleansing of data matrix, measures of the analytical method quality (e.g. precision, sensibility, robustness, uncertainty, traceability) (Eurachem, 1998) and the use of basic algorithms such as box-and-whisker, stem-and-leaf, etc. 

Electrochemical noise can be characterized by some common statistical parameters including the mean, the variance, and the standard deviation. In particular, the standard deviation, o, is used as a measure of the amplitude of the variation in the noise signal. Skew and kurtosis sometimes give indications of the form of corrosion occurring (140). For unfiltered digitized noise data in a time record, the noise resistance, Rn, is... 

Deviations from linearity reveal skewness and/or kurtosis (see p. 269), the significance of which can be tested statistically (see Miller and Miller, 2000). 

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 coefficient-based tests use statistical measures of skewness and kurtosis (Figure 16.5) 2,34,59,63.66 measures are computed from the second, third, and fourth subset moments about the mean (m2, m3, and m, respectively) ... 

Kurtosis is a statistical measure for describing whether a distribution of values is flatter (platykurtic) or more peaked (leptokurtic) than the Gaussian distribution. 

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. 

The terms similarity and diversity can have quite different meanings in chemical investigations. Describing the diversity of a data collection with a general valid measure is almost impossible. Descriptor flexibility allows the characterization of similarity by means of statistics for different tasks. The statistical evaluation of descriptors shows that it is recommended to interpret correlation coefficients together with the symmetry of distribution. In contrast to correlation coefficients, skewness and kurtosis are sensitive indicators to constitutional and conformational changes in a molecule. This feature allows a more precise evaluation of structural similarity or diversity of molecular data sets. 

Statistics on Skewness and Kurtosis by Atom Type for Myoglobin and Lysozyme ... 

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