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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 indicates a platykurtic distribution, which doesn’t necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution.
Normal probability plots are made of raw data, residuals from model fits, and estimated parameters. A normal probability plot. In a normal probability plot (also called a "normal plot"), the sorted data are plotted vs. values selected to make the resulting image look close to a straight line if the data are approximately normally distributed.
The exponentially modified normal distribution is another 3-parameter distribution that is a generalization of the normal distribution to skewed cases. The skew normal still has a normal-like tail in the direction of the skew, with a shorter tail in the other direction; that is, its density is asymptotically proportional to for some positive .
About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. [8] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
For instance, a mixed distribution consisting of very thin Gaussians centred at −99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness of about −9.77, but in a sample of 3 has an expected value of about 0.32, since usually all three samples are in the positive-valued part of the distribution, which is skewed the other way.
The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central and where types of departure from this include: mounded (or unimodal), U-shaped, J-shaped, reverse-J shaped and multi-modal. [1] A bimodal distribution would have two high points rather than one. The shape of a distribution is ...
The Lévy skew alpha-stable distribution or stable distribution is a family of distributions often used to characterize financial data and critical behavior; the Cauchy distribution, Holtsmark distribution, Landau distribution, Lévy distribution and normal distribution are special cases. The Linnik distribution; The logistic distribution
The sample skewness g 1 and kurtosis g 2 are both asymptotically normal. However, the rate of their convergence to the distribution limit is frustratingly slow, especially for g 2 . For example even with n = 5000 observations the sample kurtosis g 2 has both the skewness and the kurtosis of approximately 0.3, which is not negligible.