Excess coefficient

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coefficient of excess, excess

A scalar characteristic of the pointedness of the graph of the probability density of a unimodal distribution. It is used as a certain measure of the deviation of the distribution in question from the normal one. The excess is defined by the formula

where is the second Pearson coefficient (cf. Pearson distribution), and and are the second and fourth central moments of the probability distribution. In terms of the second- and fourth-order semi-invariants (cumulants) and , the excess has the form

If , then one says that the density of the probability distribution has normal excess, because for a normal distribution the excess is . When , one says that the probability distribution has positive excess, which corresponds, as a rule, to the fact that the graph of the density of the relevant distribution in a neighbourhood of the mode has a more pointed and higher vertex then a normal curve. When , one talks of a negative excess of the density, and then the probability density in a neighbourhood of the mode has a lower and flatter vertex than the density of a normal law.

If are independent random variables subject to one and same continuous probability law, then the statistic

is called the sample excess, where

The sample excess is used as a statistical point estimator of when the distribution law of the is not known. In the case of a normal distribution of the random variables , the sample excess is asymptotically normally distributed, as , with parameters


This is the reason why, when the observed value of differs substantially from , one must assume that the distribution of the is not normal. This is used in practice to verify the hypothesis : , which is equivalent to the fact that the distribution of the deviates from the normal distribution.


[1] M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 3. Design and analysis , Griffin (1969)
[2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[3] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)


The (coefficient of) excess is usually called the coefficient of kurtosis, or simply the kurtosis.

A density of normal, positive or negative excess is usually called a density of zero, positive or negative kurtosis, while a density of positive (negative) kurtosis is also said to be leptokurtic (respectively, platykurtic).

How to Cite This Entry:
Excess coefficient. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article