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| In [[probability theory]] and [[statistics]], '''kurtosis''' (from the Greek word κυρτός, ''kyrtos'' or ''kurtos'', meaning curved, arching) is any measure of the "peakedness" of the [[probability distribution]] of a [[real number|real]]-valued [[random variable]].<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref> In a similar way to the concept of [[skewness]], ''kurtosis'' is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. There are various [[#Interpretation|interpretations]] of kurtosis, and of how particular measures should be interpreted; these are primarily peakedness (width of peak), tail weight, and lack of shoulders (distribution primarily peak and tails, not in between).
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| One common measure of kurtosis, originating with [[Karl Pearson]], is based on a scaled version of the fourth [[moment (statistics)|moment]] of the data or population, but it has been argued that this really measures [[Heavy-tailed distribution|heavy tails]], and not peakedness.<ref>[http://support.sas.com/onlinedoc/913/getDoc/en/proc.hlp/a002473332.htm SAS Elementary Statistics Procedures], SAS Institute (section on Kurtosis)</ref> For this measure, higher kurtosis means more of the [[variance]] is the result of infrequent extreme [[Deviation (statistics)|deviations]], as opposed to frequent modestly sized deviations. It is common practice to use an adjusted version of Pearson's kurtosis, the '''excess kurtosis''', to provide a comparison of the shape of a given distribution to that of the [[normal distribution]]. Distributions with negative or positive excess kurtosis are called '''platykurtic distributions''' or '''leptokurtic distributions''' respectively.
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| Alternative measures of kurtosis are: the [[L-kurtosis]], which is a scaled version of the fourth [[L-moment]]; measures based on 4 population or sample [[quantiles]].<ref name=JG>Joanes & Gill (1998)</ref> These correspond to the alternative measures of [[skewness]] that are not based on ordinary moments.<ref name=JG/>
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| [[Image:KurtosisChanges.png|thumb|200px|The "Darkness" data is platykurtic (−0.194), while "Far Red Light" shows leptokurtosis (0.055)]]
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| == Pearson moments==
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| The fourth [[standardized moment]] is defined as
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| :<math>
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| {\beta_2=}\frac{\operatorname{E}[(X-{\mu})^4]}{(\operatorname{E}[(X-{\mu})^2])^2} {=} \frac{\mu_4}{\sigma^4}
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| </math>
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| where μ<sub>4</sub> is the fourth [[moment about the mean]] and σ is the [[standard deviation]].
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| The fourth [[standardized moment]] is lower bounded by the squared [[skewness]] plus 1
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| <ref name="petitjean">Petitjean M. (2013), "The Chiral Index: Applications to Multivariate Distributions and to 3D molecular graphs", Proceedings of 12th International Symposium on Operational Research in Slovenia SOR’13, pp. 11-16, L. Zadnik Stirn, J. Zerovnik, J. Povh, S. Drobne, A. Lisec, Eds., Slovenian Society INFORMATIKA (SDI), Section for Operations Research (SOR), ISBN 978-961-6165-40-2</ref>
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| :<math> \frac{\mu_4}{\sigma^4} \geq \left(\frac{\mu_3}{\sigma^3}\right)^2 + 1 </math>
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| where μ<sub>3</sub> is the third [[moment about the mean]]. | |
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| The fourth [[standardized moment]] is sometimes used as the definition of kurtosis in older works, but is not the definition used here.
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| Kurtosis is more commonly defined as the fourth [[cumulant]] divided by the square of the second cumulant{{Citation needed|date=November 2011}}, which is equal to the fourth moment around the mean divided by the square of the [[variance]] of the probability distribution minus 3,
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| :<math>\gamma_2 = \frac{\kappa_4}{\kappa_2^2} = \frac{\mu_4}{\sigma^4} - 3</math>
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| which is also known as '''{{visible anchor|excess kurtosis}}'''. 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. Suppose that ''Y'' is the sum of ''n'' identically distributed [[statistical independence|independent]] random variables all with the same distribution as ''X''. Then
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| :<math> \operatorname{Kurt}[Y] = \frac{\kappa_4(Y)}{\kappa_2(Y)^2} = \frac{n \kappa_4(X)}{(n \kappa_2(X))^2} = \frac{1}{n} \frac{\kappa_4(X)}{\kappa_2(X)^2} = \frac{1}{n}\operatorname{Kurt}[X] .</math>
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| This formula would be much more complicated if kurtosis were defined just as μ<sub>4</sub> / σ<sup>4</sup> (without the minus 3).
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| More generally, if ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are independent random variables, not necessarily identically distributed, but all having the same variance, then
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| :<math>\operatorname{Kurt}\left(\sum_{i=1}^n X_i \right) = {1 \over n^2} \sum_{i=1}^n \operatorname{Kurt}(X_i),</math>
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| whereas this identity would not hold if the definition did not include the subtraction of 3.
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| <!-- Kurtosis can range from −2 to +infinity. -->
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| The fourth standardized moment must be at least 1, so the excess kurtosis must be −2 or more. This lower bound is realized by the [[Bernoulli distribution]] with ''p'' = ½, or "coin toss". There is no upper limit to the excess kurtosis and it may be infinite.
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| ===Interpretation===
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| The exact interpretation of the Pearson measure of kurtosis (or excess kurtosis) is disputed. The "classical" interpretation, which applies only to symmetric and [[unimodal]] distributions (those whose [[skewness]] is 0), is that kurtosis measures both the "peakedness" of the distribution and the [[heavy-tailed distribution|heaviness of its tail]].<ref name="balanda">Balanda, Kevin P. and H.L. MacGillivray (1988), "Kurtosis: A Critical Review", ''The American Statistician'', 42:2, pp. 111–119.</ref> Various statisticians have proposed other interpretations, such as "lack of shoulders" (where the "shoulder" is defined vaguely as the area between the peak and the tail, or more specifically as the area about one [[standard deviation]] from the mean) or "bimodality".<ref name="darlington">Darlington, Richard B. (1970), "Is Kurtosis Really 'Peakedness'?", ''The American Statistician'', 24:2, pp. 19–22.</ref> Balanda and MacGillivray assert that the standard definition of kurtosis "is a poor measure of the kurtosis, peakedness, or tail weight of a distribution"<ref>Balanda and MacGillivray, p. 114.</ref> and instead propose to "define kurtosis vaguely as the location- and scale-free movement of probability mass from the shoulders of a distribution into its center and tails".<ref name="balanda"/>
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| == Terminology and examples ==
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| A high kurtosis distribution has a sharper ''peak'' and longer, fatter ''tails'', while a low kurtosis distribution has a more rounded peak and shorter, thinner tails.
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| Distributions with zero excess kurtosis are called '''mesokurtic''', or mesokurtotic. The most prominent example of a mesokurtic distribution is the [[normal distribution]] family, regardless of the values of its [[parameter]]s. A few other well-known distributions can be mesokurtic, depending on parameter values: for example the [[binomial distribution]] is mesokurtic for <math>p = 1/2 \pm \sqrt{1/12}</math>.
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| A distribution with [[Positive number|positive]] excess kurtosis is called '''leptokurtic''', or leptokurtotic. "Lepto-" means "slender".<ref>http://medical-dictionary.thefreedictionary.com/lepto-</ref> In terms of shape, a leptokurtic distribution has a more acute ''peak'' around the [[mean]] and ''[[Fat-tailed distribution|fatter tails]]''. Examples of leptokurtic distributions include the [[Student's t-distribution]], [[Rayleigh distribution]], [[Laplace distribution]], [[exponential distribution]], [[Poisson distribution]] and the [[logistic distribution]]. Such distributions are sometimes termed ''super Gaussian''.{{citation needed|date=January 2012}}
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| [[File:1909_US_Penny.jpg|thumb|The [[coin toss]] is the most platykurtic distribution]]
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| A distribution with [[Negative number|negative]] excess kurtosis is called '''platykurtic''', or platykurtotic. "Platy-" means "broad".<ref>http://www.yourdictionary.com/platy-prefix</ref> In terms of shape, a platykurtic distribution has a lower, wider ''peak'' around the mean and ''thinner tails''. Examples of platykurtic distributions include the continuous or discrete [[Uniform distribution (discrete)|uniform distributions]], and the [[raised cosine distribution]]. The most platykurtic distribution of all is the [[Bernoulli distribution]] with ''p'' = ½ (for example the number of times one obtains "heads" when flipping a coin once, a [[coin toss]]), for which the excess kurtosis is −2. Such distributions are sometimes termed ''sub-Gaussian''.<ref>The original paper presenting sub-Gaussians J.P. Kahane, "Local properties of functions interms of random Fourier series," Stud. Math., 19, No. i, 1-25 (1960). See also Buldygin, V. V., & Kozachenko, Y. V. (1980). "Sub-Gaussian random variables". Ukrainian Mathematical Journal, 32(6), 483-489.</ref>
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| {{clear}}
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| == Graphical examples ==
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| === The Pearson type VII family ===
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| [[Image:Pearson type VII distribution PDF.png|300px|thumb|[[Probability_density_function|pdf]] for the Pearson type VII distribution with kurtosis of infinity (red); 2 (blue); and 0 (black)]]
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| [[Image:Pearson type VII distribution log-PDF.png|300px|thumb|log-pdf for the Pearson type VII distribution with kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)]]
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| The effects of kurtosis are illustrated using a [[parametric family]] of distributions whose kurtosis can be adjusted while their lower-order moments and cumulants remain constant. Consider the [[Pearson distribution|Pearson type VII family]], which is a special case of the [[Pearson distribution|Pearson type IV family]] restricted to symmetric densities. The probability density function is given by
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| :<math>f(x; a, m) = \frac{\Gamma(m)}{a\,\sqrt{\pi}\,\Gamma(m-1/2)} \left[1+\left(\frac{x}{a}\right)^2 \right]^{-m}, \!</math>
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| where ''a'' is a [[scale parameter]] and ''m'' is a [[shape parameter]].
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| All densities in this family are symmetric. The ''k''th moment exists provided ''m'' > (''k'' + 1)/2. For the kurtosis to exist, we require ''m'' > 5/2. Then the mean and [[skewness]] exist and are both identically zero. Setting ''a''<sup>2</sup> = 2''m'' − 3 makes the variance equal to unity. Then the only free parameter is ''m'', which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize with <math>m = 5/2 + 3/\gamma_2</math>, where <math>\gamma_2</math> is the kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary positive kurtosis. The reparameterized density is
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| :<math>g(x; \gamma_2) = f(x;\; a=\sqrt{2+6/\gamma_2},\; m=5/2+3/\gamma_2). \!</math>
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| In the limit as <math>\gamma_2 \to \infty</math> one obtains the density
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| :<math>g(x) = 3 \left(2 + x^2\right)^{-5/2}, \!</math>
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| which is shown as the red curve in the images on the right.
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| In the other direction as <math>\gamma_2 \to 0</math> one obtains the [[normal distribution|standard normal]] density as the limiting distribution, shown as the black curve.
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| In the images on the right, the blue curve represents the density <math>x \mapsto g(x; 2)</math> with kurtosis of 2. The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is a [[parabola]]. One can see that the normal density allocates little probability mass to the regions far from the mean ("has thin tails"), compared with the blue curve of the leptokurtic Pearson type VII density with kurtosis of 2. Between the blue curve and the black are other Pearson type VII densities with γ<sub>2</sub> = 1, 1/2, 1/4, 1/8, and 1/16. The red curve again shows the upper limit of the Pearson type VII family, with <math>\gamma_2 = \infty</math> (which, strictly speaking, means that the fourth moment does not exist). The red curve decreases the slowest as one moves outward from the origin ("has fat tails").
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| == Kurtosis of well-known distributions ==
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| [[Image:Standard symmetric pdfs.png|300px|thumb]]
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| [[Image:Standard symmetric pdfs logscale.png|300px|thumb]]
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| Several well-known, unimodal and symmetric distributions from different parametric families are compared here. Each has a mean and skewness of zero. The parameters have been chosen to result in a variance equal to 1 in each case. The images on the right show curves for the following seven densities, on a linear scale and logarithmic scale:
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| * D: [[Laplace distribution]], also known as the double exponential distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3
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| * S: [[hyperbolic secant distribution]], orange curve, excess kurtosis = 2
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| * L: [[logistic distribution]], green curve, excess kurtosis = 1.2
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| * N: [[normal distribution]], black curve (inverted parabola in the log-scale plot), excess kurtosis = 0
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| * C: [[raised cosine distribution]], cyan curve, excess kurtosis = −0.593762...
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| * W: [[Wigner semicircle distribution]], blue curve, excess kurtosis = −1
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| * U: [[uniform distribution (continuous)|uniform distribution]], magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.
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| Note that in these cases the platykurtic densities have bounded [[Support (mathematics)|support]], whereas the densities with positive or zero excess kurtosis are supported on the whole [[real line]].
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| There exist platykurtic densities with infinite support,
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| *e.g., [[exponential power distribution]]s with sufficiently large shape parameter ''b''
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| and there exist leptokurtic densities with finite support.
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| *e.g., a distribution that is uniform between −3 and −0.3, between −0.3 and 0.3, and between 0.3 and 3, with the same density in the (−3, −0.3) and (0.3, 3) intervals, but with 20 times more density in the (−0.3, 0.3) interval
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| == Sample kurtosis ==
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| For a [[sample (statistics)|sample]] of ''n'' values the '''sample excess kurtosis''' is
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| :<math> g_2 = \frac{m_4}{m_{2}^2} -3 = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^4}{\left(\tfrac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2\right)^2} - 3 </math>
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| where ''m''<sub>4</sub> is the fourth sample [[moment about the mean]], ''m''<sub>2</sub> is the second sample moment about the mean (that is, the [[sample variance]]), ''x''<sub>''i''</sub> is the ''i''<sup>th</sup> value, and <math>\overline{x}</math> is the [[sample mean]].
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| The variance of the sample kurtosis of a sample of size ''n'' from the [[normal distribution]] is<ref name=Duncan1997>Duncan Cramer (1997) ''Fundamental Statistics for Social Research''. Routledge. ISBN13: 9780415172042 (p 89)</ref>
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| :<math> \frac{24n(n-1)^2}{(n-3)(n-2)(n+3)(n+5)} </math>
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| An approximate alternative is 24/''n'' but this is inaccurate for small samples.
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| == Estimators of population kurtosis ==
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| Given a sub-set of samples from a population, the sample excess kurtosis above is a [[biased estimator]] of the population excess kurtosis. The usual estimator of the population excess kurtosis (used in [[DAP (software)|DAP]]/[[SAS System|SAS]], [[Minitab]], [[PSPP]]/[[SPSS]], and [[Microsoft Excel|Excel]] but not by [[BMDP]]) is ''G''<sub>2</sub>, defined as follows:
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| : <math> | |
| \begin{align}
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| G_2 & = \frac{k_4}{k_{2}^2} \\
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| & = \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} \\
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| & = \frac{n-1}{(n-2)\,(n-3)} \left( (n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right) \\
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| & = \frac{n-1}{(n-2) (n-3)} \left( (n+1)\,g_2 + 6 \right) \\
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| & = \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)} \\
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| & = \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)}
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| \end{align}
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| </math>
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| where ''k''<sub>4</sub> is the unique symmetric [[bias of an estimator|unbiased]] estimator of the fourth [[cumulant]], ''k''<sub>2</sub> is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), ''m''<sub>4</sub> is the fourth sample moment about the mean, ''m''<sub>2</sub> is the second sample moment about the mean, ''x''<sub>''i''</sub> is the ''i''<sup>th</sup> value, and <math>\bar{x}</math> is the sample mean. Unfortunately, <math>G_2</math> is itself generally biased. For the [[normal distribution]] it is unbiased.{{Citation needed|date=October 2010}}
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| For computationally efficient ways of calculating the sample kurtosis see [[Algorithms for calculating variance#Higher-order statistics|Algorithms for calculating higher-order statistics]].
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| == Applications ==
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| {{Expand section|date=December 2009}}
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| [[D'Agostino's K-squared test]] is a [[goodness-of-fit]] [[normality test]] based on a combination of the sample skewness and sample kurtosis, as is the [[Jarque–Bera test]] for normality.
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| For non-normal samples, the variance of the variance depends on the kurtosis; for details, please see [[Variance#Distribution of the sample variance|variance]].
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| Pearson's definition of kurtosis is used as an indicator of intermittency in [[turbulence]].<ref>Sandborn 1958 http://dx.doi.org/10.1017/S0022112059000581</ref>
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| ==Other measures of kurtosis==
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| A different measure of "kurtosis", that is of the "peakedness" of a distribution, is provided by using [[L-moment]]s instead of the ordinary moments.<ref name=hos:92>{{cite journal | last=Hosking | first= J.R.M. | year=1992 | title=Moments or L moments? An example comparing two measures of distributional shape | journal=The American Statistician | volume=46 | number=3 | pages=186–189 | jstor=2685210}}</ref><ref name=hos:96>{{cite journal | last=Hosking | first=J.R.M. | year=2006 | title=On the characterization of distributions by their L-moments | journal=Journal of Statistical Planning and Inference | volume=136 | pages=193–198}}</ref>
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| == See also ==
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| {{Commons category|Kurtosis}}
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| * [[Kurtosis risk]]
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| {{More footnotes|date=November 2010}}
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| == References ==
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| {{reflist}}
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| ==Further reading==
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| * Joanes,<!--sic--> 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. {{doi|10.1111/1467-9884.00122}}
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| *Kim, Tae-Hwan; & White, Halbert. (2003/4). [http://escholarship.org/uc/item/7b52v07p "On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index".] Finance Research Letters, 1, 56–70 {{doi|10.1016/S1544-6123(03)00003-5}} [http://weber.ucsd.edu/~hwhite/pub_files/hwcv-092.pdf Alternative source] (Comparison of kurtosis estimators)
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| *Seier, E. & Bonett, D.G. (2003). Two families of kurtosis measures. ''Metrika'', 58, 59–70.
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| == External links ==
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| {{wikiversity}}
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| * {{springer|title=Excess coefficient|id=p/e036800}}
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| * [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)..
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| *[http://jeff560.tripod.com/k.html Kurtosis] on the [http://jeff560.tripod.com/mathword.html Earliest known uses of some of the words of mathematics]
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| *[http://faculty.etsu.edu/seier/doc/Kurtosis100years.doc Celebrating 100 years of Kurtosis] a history of the topic, with different measures of kurtosis.
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| {{Statistics|descriptive}}
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| [[Category:Theory of probability distributions]]
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| [[Category:Statistical deviation and dispersion]]
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