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The Pearson median skewness, or second skewness coefficient, [12] [13] is defined as 3 ( mean − median ) / standard deviation . Which is a simple multiple of the nonparametric skew .
Skewness [ + ] () ... This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, ...
The nonparametric skew is one third of the Pearson 2 skewness coefficient and lies between −1 and +1 for any distribution. [5] [6] This range is implied by the fact that the mean lies within one standard deviation of any median. [7] Under an affine transformation of the variable (X), the value of S does not change except for a possible change ...
The first is the square of the skewness: β 1 = γ 1 where γ 1 is the skewness, or third standardized moment. The second is the traditional kurtosis, or fourth standardized moment: β 2 = γ 2 + 3. (Modern treatments define kurtosis γ 2 in terms of cumulants instead of moments, so that for a normal distribution we have γ 2 = 0 and β 2 = 3.
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 .
In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. The test statistic is always nonnegative. If it is far from zero, it signals the data do not have a normal distribution.
Mardia's tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives. [39] The BHEP test [40] computes the norm of the difference between the empirical characteristic function and the theoretical characteristic function of the normal distribution.
Sarle's bimodality coefficient b is [25] = + where γ is the skewness and κ is the kurtosis. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of b lies between 0 and 1. [26]