Skewness and Kurtosis

An important prerequisite for the use of descriptive statistics is the shape of the distribution of variables, that is, the frequency of values from different ranges of the variable. It is assumed in multiple regression analysis that the residuals — predicted minus observed values — are normally distributed. 

The skewness, S, of a descriptor reflects symmetry of distribution of its components related to the Gaussian distribution 

FIGURE 4.3 Graphical representation of kurtosis, K, and skewness, S, in comparison to the Gaussian (standard) distribution (upper left). The right-hand side shows leptokurtic (peaked) or platykurtic (flatted) distribution as well as positive skewed distribution (fronting) and negative skewed distribution (tailing). 

The kurtosis, K, measures the flatness of a distribution related to the Gaussian distribution 

---

Figure 20 shows a 3-D view of a generated non-Gaussian rough surface with an exponential autocorrelation and desired skewness and kurtosis of -1.75 and 5.0, respectively. The surface shows an outlook of a typical worn surface due to the negative skewness. The real values of SK and K were calculated as -1.7827 and 5.1104, a good agreement between specihed and real values. 

To examine the effects of height distribution on mixed lubrication, rough surfaces with the same exponential autocorrelation function but different combinations of skewness and kurtosis have been generated, following the procedure described in the previous section. Simulations were performed for the point contact problem with geometric parameters of... 

In summary, the height distribution of surface roughness, characterized by the skewness and kurtosis, may present a significant influence on certain performances of mixed lubrication, such as the real contact area, the load carried by asperities, and pressure distribution, while the average film thickness and surface temperature are relatively unaffected. 

The first four terms called, respectively, the average (or expectation value), variance, skewness, and kurtosis, are equal to... 

To facilitate interpretation of the outputs, the authors also created two simulation data sets with identical distributional properties (number of indicators, number of levels, indicator intercorrelations, skew and kurtosis) one taxonic set and one dimensional set. The taxonic data set was created to have a base rate of. 23, which corresponds to the proportion of cases falling at or above a BDI threshold of 10 in the undergraduate data set. Ruscio and Ruscio tried to ensure that indicator validities and nuisance correlations matched the estimated parameters of the real indicators, but they did not indicate how successful this was. 

According to Table 1, semi-invariants of higher order characterize the shape of the profile in terms of variance, skewness, and kurtosis. The outstanding merit of the Weibull distribution is that its shape parameter a provides a summarizing measure for this property. For other distributions, the characterization of the shape is less obvious. 

Characterizing Distribution Shape in Terms of Skewness and Kurtosis... 

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). 

Systems of distributions, such as the Pearson system (Pearson 1894) and the Johnson system (Johnson et al. 1994), can be used to select a distribution based on the skewness and kurtosis, as well as mean and variance. The Student t and logistic distributions are symmetric (like the normal distribution) but have heavier tails than the normal distribution. 

The most familiar estimation procedure is to assume that the population mean and variance are equal to the sample mean and variance. More generally, the method of moments (MOM) approach is to equate sample moments (mean, variance, skewness, and kurtosis) to the corresponding population. Software such as Crystal Ball (Oracle Corporation, Redwood Shores, CA) uses MOM to fit the gamma and beta distributions (see also Johnson et al. 1994). Use of higher moments is exemplified by fitting of the... 

Besides the calculation of average molecular weights, several other means of characterizing the distribution were produced. These include tables of percentile fractions vs. molecular weights, standard deviation, skewness, and kurtosis. The data for the tables were obtained on punched cards as well as printed output. The punched cards were used as input to a CAL COMP plotter to obtain a curve as shown in Figure 3. This plot is normalized with respect to area. No corrections were made for axial dispersion. 

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. 

Fig.l and Fig.2 contain histograms showing the distribution of the oxides and the ions in their various structural positions. Most of the Si02, A1203, MgO, Na20, and K2 O have a normal type distribution. The interlayer cations have a log-normal distribution and the tetrahedral and octahedral cations have a normal-type distribution. Calculated skewness and kurtosis values are listed in Tables III and VI. The data are too limited to draw any significant conclusions. Ahrens (1954) and others have shown... 

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). 

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... 

The standardized coefficients, both skewness and kurtosis, indicate significant deviations from the normal distribution. The data depart significantly from a normal distribution when the standardized coefficients are outside the range — 2.0 to + 2.0. 

The average value and variance of tc2==C) are, of course, much smaller than those for tni C2, but its distribution is far from normal (high standardized skewness and kurtosis). The restriction of the first atom of R1- 5 substituents to carbon significantly reduces the variance and the range of this parameter and makes the distribution closer to normal. 

---