Kurtosis

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real number-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations. ==Definition of kurtosis== The fourth standardized moment is defined as μ₄ / σ⁴, where μ₄ is the fourth moment about the mean and σ is the standard deviation. This is sometimes used as the definition of kurtosis in older works, but is not the definition used here. Kurtosis is more commonly defined as :

\gamma_2 = \frac{\kappa_4}{\kappa_2^2} = \frac{\mu_4}{\sigma^4} - 3, \!

which is also known as kurtosis excess. The minus 3 at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables. If ''Y'' is the sum of ''n'' statistical independence random variables, all with the same distribution as ''X'', then Kurt[''Y''] = Kurt[''X''] / ''n'', while the formula would be more complicated if kurtosis were defined as μ₄ / σ⁴. This is because the kurtosis as we have defined it is the ratio of the fourth cumulant and the square of the second cumulant of the probability distribution. == Terminology and examples == A high kurtosis distribution has a sharper "peak" and fatter "tails", while a low kurtosis distribution has a more rounded peak with wider "shoulders". Distributions with zero kurtosis are called mesokurtic. The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of its parameters. A few other well-known distributions can be mesokurtic, depending on parameter values: for example the binomial distribution is mesokurtic for

p = 1/2 \pm \sqrt{1/12}

. A distribution with positive kurtosis is called leptokurtic. In terms of shape, a leptokurtic distribution has a more acute "peak" around the mean (that is, a higher probability than a normally distributed variable of values near the mean) and "fat tails" (that is, a higher probability than a normally distributed variable of extreme values). Examples of leptokurtic distributions include the Laplace distribution and the logistic distribution. A distribution with negative kurtosis is called platykurtic. In terms of shape, a platykurtic distribution has a smaller "peak" around the mean (that is, a lower probability than a normally distributed variable of values near the mean) and "thin tails" (that is, a lower probability than a normally distributed variable of extreme values). Examples of platykurtic distributions include the Uniform distribution (continuous), and the Maxwell-Boltzmann distribution. == Sample kurtosis == For a sample of ''n'' values the sample kurtosis is :

g_2 = \frac{m_4}{m_{2}^2} -3 = \frac{n\,\sum_{i=1}^n (x_i - \overline{x})^4}{\left(\sum_{i=1}^n (x_i - \overline{x})^2\right)^2} - 3

where ''m''₄ is the fourth sample moment about the mean, ''m''₂ is the second sample moment about the mean (that is, the sample variance), ''x''_''i'' is the ''i''^th value, and

\overline{x}

is the sample mean. == Estimators of population kurtosis == Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis. An unbiased estimator of the population kurtosis is ''G''₂, defined as follows: {|- |

G_2 \!\!\!\!

= \frac{k_4}{k_{2}^2}\!

|- | |

= \frac{n^2\,((n+1)\,m_4 - 3\,(n-1)\,m_{2}^2)}{(n-1)\,(n-2)\,(n-3)} \; \frac{(n-1)^2}{n^2\,m_{2}^2}\!

|- | |

= \frac{n-1}{(n-2)\,(n-3)} \left( (n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right)\!

|- | |

= \frac{n-1}{(n-2) (n-3)} \left( (n+1)\,g_2 + 6 \right)\!

|- | |

= \frac{(n+1)\,n\,(n-1)}{(n-2)\,(n-3)} \; \frac{\sum_{i=1}^n (x_i - \bar{x})^4}{\left(\sum_{i=1}^n (x_i - \bar{x})^2\right)^2} - 3\,\frac{(n-1)^2}{(n-2)\,(n-3)}\!

|- | |

= \frac{(n+1)\,n}{(n-1)\,(n-2)\,(n-3)} \; \frac{\sum_{i=1}^n (x_i - \bar{x})^4}{k_{2}^2} - 3\,\frac{(n-1)^2}{(n-2) (n-3)} \!

|} where ''k''₄ is the unique symmetric unbiased estimator of the fourth cumulant, ''k''₂ is the unbiased estimator of the population variance, ''m''₄ is the fourth sample moment about the mean, ''m''₂ is the sample variance, ''x''_''i'' is the ''i''^th value, and

\bar{x}

is the sample mean. ==See also== * skewness ==References== * Joanes, D. N. & Gill, C. A. (1998) Comparing measures of sample skewness and kurtosis. ''Journal of the Royal Statistical Society (Series D): The Statistician'' 47 (1), 183–189. [http://dx.doi.org/10.1111/1467-9884.00122 doi:10.1111/1467-9884.00122] == External links == * [http://www.wessa.net/skewkurt.wasp Free Online Software (Calculator)] computes various types of skewness and kurtosis statistics for any dataset (includes small and large sample tests). Probability theory Statistics su:Kurtosis

Kurtosis

If this is the "fourth standardized moment", what are the other 3 and what is a standardized moment anyway? do we need an article on it? -- User:Tarquin 10:39 Feb 6, 2003 (UTC) :The first three are the mean, standard deviation, and skewness, if I recall correctly. :::Actually, the word "standarized" refers to the fact that the fourth moment is divided by the 4th power of the standard deviation. — User:Miguel 15:53, 2005 Apr 19 (UTC) :: Thank you :-) It's nice when wikipedia comes up with answers so quickly! -- User:Tarquin 11:04 Feb 6, 2003 (UTC) :::I think the term "central moments" is also used. See also http://planetmath.org/encyclopedia/Moment.htm Kurtosis is a measure of the peakedness ... so what does that mean? If I have a positive kurtosis, is my distribution pointy? Is it flat? -- JohnFouhy, 01:53, 11 Nov 2004 I've tried to put the answer to this in the article: high kurtosis is 'peaked' or 'pointy', low kurtosis is 'rounded'. User:Kappa 05:15, 9 Nov 2004 (UTC) ==Unbiasedness== I have just added a discussion to the skewness page. Similar comments apply here. Unbiasedness of the given kurtosis estimator requires independence of the observations and does not therefore apply to a finite population. It is still biased, but the bias is small. This is because, although we can make the numerator and denominator unbiased separately, the ratio will still be biased. Removing this bias can be done only for specific populations. The best we can do is either: 1 use an unbiased estimate for the fourth moment about the mean,
2 use an unbiased estimate of the fourth cumulant,
in the numerator; and either: 3 use an unbiased estimate for the variance,
4 use an unbiased estimate for the square of the variance, in the denominator. According to the article, the given formula is 2 and 3 but I have not checked this. User:Terry Moore 11 Jun 2005 == So who's Kurt? == I mean, what is the etymology of the term? -User:Finn-Zoltan 19:48, 22 Jun 2005 (UTC) :It's obviously a modern term of Greek language origin (κυρτωσις, fem.). The Oxford English Dictionary gives the non-specialized meaning as "a bulging, convexity". The Henry Liddell-Scott-Jones A Greek-English Lexicon has "bulging, of blood-vessels", "convexity of the sea's surface" and "being humpbacked". According to the OED (corroborated by [http://members.aol.com/jeff570/k.html "Earliest Known Uses of Some of the Words of Mathematics"] and by a search on JSTOR), the first occurrence in print of the modern technical term is in an article by Karl Pearson from June 1905. --User:MarkSweep 21:05, 22 Jun 2005 (UTC)

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Words begining with Kurtosis:

Kurtosis
Kurtosis
Kurtosis_excess