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For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean. Excess kurtosis, typically compared to a value of 0, characterizes the “tailedness” of a distribution. A univariate normal distribution has an excess kurtosis of 0. Negative excess kurtosis ...
As with variance, skewness, and kurtosis, these are higher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality.
Example distribution with positive skewness. These data are from experiments on wheat grass growth. In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
The shape of a distribution may be considered either descriptively, using terms such as "J-shaped", or numerically, using quantitative measures such as skewness and kurtosis.
The normal probability plot is a graphical technique to identify substantive departures from normality.This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures.
For instance, the Laplace distribution has a kurtosis of 6 and weak exponential tails, but a larger 4th L-moment ratio than e.g. the student-t distribution with d.f.=3, which has an infinite kurtosis and much heavier tails. As an example consider a dataset with a few data points and one outlying data value.
The plot of excess kurtosis as a function of the variance and the mean shows that the minimum value of the excess kurtosis (−2, which is the minimum possible value for excess kurtosis for any distribution) is intimately coupled with the maximum value of variance (1/4) and the symmetry condition: the mean occurring at the midpoint (μ = 1/2).
where n is the number of items in the sample, g is the sample skewness and k is the sample excess kurtosis. The value of b for the uniform distribution is 5/9. This is also its value for the exponential distribution.