Kurtosis, coefficient

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. 

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. 

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. 

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. 

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. 

Coefficient of skewness Coefficient of kurtosis Outlier test (4 SD)... 

B, negative skewness),The two lower figures show distributions with nonGaussian peakedness (C, positive kurtosis D, negative kurtosis).The Gaussian distribution (dos/ied curve) is shown in all graphs for comparison. The values of the coefficients of skewness (gs) and kurtosis (g( ) are also shown. 

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) ... 

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

The coefficient is approximately zero for the Gaussian distribution. The sign of a nonzero coefficient indicates the type of kurtosis present in the data (Figure 16-5, C and D). 

These experiments show that correlation coefficients between individual descriptors and the ASDs are valuable for indicating similarity and diversity of data sets and compounds. Since the effects of symmetry of distribution may be significant, it is recommended that correlation coefficients be handled together with skewness or kurto-sis. The skewness and kurtosis of the descriptors both show the same trends, although the kurtosis — in particular, the Harness of distribution — is generally more sensitive... 

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. 

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. 

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. 

Moments can be used to calculate any Z),y, and should be positive and finite. For example, D32 = (D30) ( >20). Note that characteristic diameters are not necessarily the same as the common statistical moments mean, standard deviation, coefficient of skewness, and coefficient of kurtosis. The exception is Djo, which is the statistical mean of/o(D), and D43, which is the statistical mean of/3(D). 

The next step in the analysis involved assessment of the model. Given that the multivariate kurtosis in the data was elevated, as indicated by the Mardia coefficient, the robust method was used in this analysis (Bentler, 2006). A specified model is generally indieated as a good fit with the data when the /df ratio is less than 3, the RCFI (Robust Comparative Fit Index) and NNFl (Bentler-Bonnet Non-Normed Fit Index) are above. 90, and when RMSEA (Root Mean-Square Error of Approximation) is below. 07 (Byrne, 2006). Two multi-sample analyses (Byrne, 2006) were carried out contrasting Canadian and Swedish students and, female and male students. 

---