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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.
Deviations from a straight line suggest departures from normality. The plotting can be manually performed by using a special graph paper, called normal probability paper. With modern computers normal plots are commonly made with software. The normal probability plot is a special case of the Q–Q probability plot for a normal distribution.
In statistics, the concept of the shape of a probability distribution arises in questions of finding an appropriate distribution to use to model the statistical properties of a population, given a sample from that population.
The Jarque–Bera test is itself derived from skewness and kurtosis estimates. Mardia's multivariate skewness and kurtosis tests generalize the moment tests to the multivariate case. [7] Other early test statistics include the ratio of the mean absolute deviation to the standard deviation and of the range to the standard deviation. [8]
The kurtosis is here defined to be the standardised fourth moment around the mean. The value of b lies between 0 and 1. [26] The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient. The formula for a finite sample ...
The first two are very similar, while the last, with one degree of freedom, has "heavier tails" meaning that the values farther away from the mean occur relatively more often (i.e. the kurtosis is higher). The Cauchy distribution is also symmetric. Skew distributions to the right. Skewness to left and right
In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve). Main article: Central limit theorem The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution.
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. [when defined as?] In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed ...