Kurtosis deviations

Variable Mean SE Standard SE Skewness Kurtosis deviation No. name Rang ... 

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. 

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. 

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. 

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. 

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)... 

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. 

No. name Mean SE Standard deviation SE Skewness Kurtosis... 

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. 

Investigate the numerical stability of the (a) central moments (mean, standard deviation, skewness and kurtosis), and (b) the tails of the output distribution of the simulation. 

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

Applying the first method, four different criteria, namely Dixon s test, Grubbs test, the coefficient of dewness test and the coefficient of kurtosis test are used at a significance level of a = 0.05. If a laboratory mean for each element as single unweighted value was declared to be an outlier by any criterion, it is rejected and the whole procedure repeated until no more outliers could be identified. The remaining laboratory means are then combined in the usual way to provide estimates of the overall mean (consensus value) and its associated standard deviation, standard error and 95% confidence interval. 

The subset coefficient of kurtosis, gk, and its asymptotic standard deviation, s, are computed as follows ... 

Figure 11.12 A histogram of a Raman image. Total number of spectra = 6561 mean score = -0.35 standard deviation of scores = 1.3 skew of scores = 1.3 kurtosis of scores = 1.4.

The skewness as well as the kurtosis of measured or calculated values will be non-zero in many cases, even if the underlying distribution is in fact more or less symmetrical. The meaning of S could be estimated by the standard deviation of S itself Unfortunately, this depends on the shape of the distribution. As an approximation, the standard deviation for Equation 4.22 in the idealized case of a normal distribution... 

The mean and standard deviation of correlation coefficients seems to be a reliable diversity measure. However, as mentioned in the theoretical section, the reliability of the correlation coefficient itself depends on the symmetry of distribution within a descriptor skewness or kurtosis should be regarded if a data set has to be classified as similar or diverse. 

What is more important for a diversity evaluation is that with increasing diversity of a descriptor collection, the mean deviation in skewness should increase. In fact, the skewness of the two data sets investigated is similar and does not clearly indicate a difference between the sets (Table 6.1), whereas the deviation in skewness of the high-diversity data set is about twice the one in the low-diversity data set. The distribution of the kurtosis of the data sets leads to a similar result. 

Whereas the mean correlation coefficient is significantly lower in the arbitrary data set, the mean skewness and mean kurtosis are similar. Though the latter values do not indicate clearly a difference between the data sets — they just indicate a similar symmetry and flatness of distribution — the deviations from the average behavior describes properly the diversity of the data set The average deviations in skewness and kurtosis are about twice as high in the arbitrary data set as those of the benzene derivatives. The ASD and the combination of deviations in correlation coefficients, skewness, and kurtosis provide the most reliable measure for similarity and diversity of data sets. 

---