Distribution kurtosis

The kurtosis, the fourth moment, is a measure of peakedness or flatness of a distribution. It has no common notation (k is used here) and is given for a continuous random variable by... 

Find the standard deviation of the Flory distribution as given by Equation (13.26) and relate it to the polydispersity. Extend the calculations in Problem 13.5 to /X3. Find the kurtosis of the distribution in the limit of high conversion. 

Gaussian input sequence r/(i,/) should be transformed first to another sequence (i,/) with appropriate skewness (SK ) and kurtosis K ) using the Johnson translator system of distribution, then let rj i, j) pass through the filter to obtain the output sequence z(j,/) which possesses the specified autocor-... 

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. 

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. 

The results obtained by a number of workers, using 5 different methods, are represented in Fig. 4. The method of Kirkman and Bynum (K10), and method D of Hsia et al. (H12), give such similar results that they are combined. In Fig. 4, normal distribution curves are drawn for normal homozygotes and heterozygotes, using published values for the means and standard deviations (no allowance is made for possible skewness or kurtosis) as far as possible, the same scale is used for all the methods. For galactosemics a smooth curve was drawn from the values published for individual patients. 

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. 

Kurtosis K4 characterizes the proportion of the tails in relation to the center. When compared with the normal distribution, platykurtic distributions have more values in the tails and leptokurtic distribution have less. 

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. 

There is a fourth parameter to describe the characteristics of a distribution, which is the kurtosis. This is the peakedness of the data set. If the data set has a flat peak in the distribution curve it is called platykurtic. If the peak is very sharp it is called leptokurtic. Distributions with peaks in-between are called mesokurtic. 

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

Some literature has defined kurtosis in terms of pdfs that are relatively flat versus relatively peaked at the mode. A tendency in more recent literature is to emphasize the idea of tail weight. Leptokurtic distributions have relatively heavier pdf tails, while platykurtic distributions have relatively lighter tails (Balanda and MacGillivray 1998). 

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. 

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... 

Asa rule, accurate estimation of lower distribution moments will require fewer data than accurate estimation of higher moments (e.g., fewer data are needed for a decent estimate of the mean than for a decent estimate of kurtosis). 

In the hrst situation we hope to dehne a generic distribution based on information from multiple studies, and no study is treated as more representative than another, for the situations where the distribution will be used. Generic assumptions may relate to type of distribution or to distribution parameters (e.g., coefficient of variation, skewness, or kurtosis). An important case is the determination of multiplicative safety factor based on a generic coefficient of variation, and assuming log-normality. 

Note that Equation (9) implies that the square of the standard deviation a2 is the second moment of d relative to the mean d. Higher order moments can be used to represent additional information about the shape of a distribution. For example, the third moment is a measure of the skewness or lopsidedness of a distribution. It equals zero for symmetrical distributions and is positive or negative, depending on whether a distribution contains a higher proportion of particles larger or smaller, respectively, than the mean. The fourth moment (called kurtosis) purportedly measures peakedness, but this quantity is of questionable value. 

The skewness coefficient is. 14192 and the kurtosis is 1.8447. (These are the third and fourth moments divided by the third and fourth power of the sample standard deviation.) Inserting these in the expression above produces L = 10. 141922/6 + (1.8447 - 3)2/24 =. 59. The critical value from the chi-squared distribution with 2 degrees of freedom (95%) is 5.99. Thus, the hypothesis of normality cannot be rejected. 

Exercise. If a distribution is obtained from a set of observations it often consists of a single hump. The first and second cumulant are rough indications of its position and its width. Further information about its shape is contained in its skewness , defined by y3 = k3/k212, and its kurtosis y4 = k4/k2. Prove510... 

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. 

It does, however, indicate the way in which the normal distribution is attained. The absolute skewness mllml and the kurtosis mjml both tend to zero as x-1. The difficulty in the practical application lies in the truncation error and the fact that the higher terms decrease in a somewhat irregular manner. It may often be of value, however, to estimate the skewness from the third moment and even in such a complicated example as that of 4 this is not impracticable. 

Fig. 22a and b. Dependence of the measured elution volume V = VD (in cm3, Fig. a), and of the standard deviation ctd (in cm3), skewness yD, and kurtosis 5D (Fig. b) on the weight-average of the polymerization degree Pw of very narrowly distributed polystyrene samples (BW-middle fractions of the anionic polystyrene standards), injected into the PDC-column 3) at 28 °C where the resolution of the column can be neglected in the Pw-range as indicated (polystyrene/cyclohexane, theta temperature 34 °C)... 

From these basic parameters others are derived which characterize the real shape of a normal distribution skewness and excess (kurtosis) (see Section 2.1.5). 

Positive or negative kurtosis occurs if the convexity of the empirical distribution is more acute or smoother than the curvature of the ideal normal distribution. 

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. 

How plain a distribution is can be evaluated in terms of excess or kurtosis. 

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... 

Tetrahedral A1 and Si, though showing a normal-type distribution, have relatively high kurtosis values. This indicates a relatively narrow range of values for the majority of the samples (73% of the tetrahedal A1 values fall in the 0.25—0.45 range). Some glauconite analyses have insufficient A1 to completely fill all the tetrahedral positions and it is necessary to assign some ferric iron to the tetrahedral sheet. Whether ferric iron occurs in tetrahedral coordination or not when sufficient A1 is available, has not been demonstrated. 

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