Kurtosis

The curvature, Kyrtosis or kurtosis (Greek κύρτωσις kýrtōsis " curving ", " slumping ") is a measure of the steepness or " peakedness " of a ( unimodal ) probability function, statistical density function or frequency distribution. The bulge is the central moment of order 4. Distributions with low camber spread relatively evenly; for distributions with high curvature, the scattering results from more extreme but rare events.

The excess indicates the difference of the curvature of the function under consideration to the curvature of the density function of a normally distributed random variable.

  • 2.1 Types of excess

Curvature

Empirical curvature

For calculating the curvature of a frequency distribution of the empirical formula is used:

So that the buckle is independent of the unit of the variable, the observed values ​​with the aid of the arithmetic mean and the standard deviation

Standardized. Applies by standardizing

Since the curvature can only assume non-negative values ​​indicated a small value that the standardized observations are strongly concentrated near the mean ( with a variance of 1), ie the distribution is flachgipflig.

Curvature of a random variable

Analogous to the empirical curvature of a frequency distribution is the arch or kurtosis of the density function or probability function of a random variable defined as its on the fourth power of the standard deviation normalized fourth central moment.

Estimate the curvature of a population

In order to estimate the unknown curvature of a population by sampling data ( the sample size ), the expected value and the variance of the sample to be estimated, ie the theoretical be replaced by the empirical moments:

With the sample mean and sample standard deviation.

Excess

In order to assess the extent of convexity better it is compared to the curvature of a normal distribution applies. The excess (also: Überkurtosis ) is therefore defined as

Not infrequently, the curvature is incorrectly referred to as excess.

Types of excess

Distributions according to their excess, divided into:

  • : Normalgipflig or mesokurtisch. The normal distribution has kurtosis and according to the excess.
  • : Steilgipflig, supergaußförmig or leptokurtic. This is compared to the normal distribution sharper distributions, ie Distributions with strong peaks.
  • : Flachgipflig, subgaußförmig or platykurtisch. One speaks of a flattened compared to the normal distribution.
Probability mass function Probability density function Moment (mathematics)
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