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{{Short description|fourth standardized moment in statistics}}
{{Use American English|date = January 2019}}

In [[probability theory]] and [[statistics]], '''kurtosis''' (from {{lang-el|κυρτός}}, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the [[probability distribution]] of a [[real number|real]]-valued [[random variable]]. 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. Depending on the particular measure of kurtosis that is used, there are various [[#Interpretation|interpretations]] of kurtosis, and of how particular measures should be interpreted.

The standard measure of kurtosis, originating with [[Karl Pearson]],{{r|Pearson1905}} is based on a scaled version of the fourth [[moment (statistics)|moment]] of the data or population. This number is related to the tails of the distribution, not its peak;{{r|Westfall2014}} hence, the sometimes-seen characterization as "peakedness" is mistaken. For this measure, higher kurtosis is the result of infrequent extreme [[Deviation (statistics)|deviations]] (or outliers), as opposed to frequent modestly sized deviations.

The kurtosis of any univariate [[normal distribution]] is&nbsp;3.  It is common to compare the kurtosis of a distribution to this value.  Distributions with kurtosis less than&nbsp;3 are said to be ''platykurtic'', although this does not imply the distribution is "flat-topped" as sometimes reported. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution. An example of a platykurtic distribution is the [[uniform distribution (continuous)|uniform distribution]], which does not produce outliers.  Distributions with kurtosis greater than 3 are said to be ''leptokurtic''.  An example of a leptokurtic distribution is the [[Laplace distribution]], which has tails that asymptotically approach zero more slowly than a Gaussian, and therefore produces more outliers than the normal distribution. It is also common practice to use an adjusted version of Pearson's kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the [[normal distribution]].  Some authors use "kurtosis" by itself to refer to the excess kurtosis.  For the reason of clarity and generality, however, this article follows the non-excess convention and explicitly indicates where excess kurtosis is meant.

Alternative measures of kurtosis are: the [[L-kurtosis]], which is a scaled version of the fourth [[L-moment]]; measures based on four population or sample [[quantiles]].{{r|Joanes1998}} These are analogous to the alternative measures of [[skewness]] that are not based on ordinary moments.{{r|Joanes1998}}

== Pearson moments ==
The kurtosis is the fourth [[standardized moment]], defined as
:<math>
\operatorname{Kurt}[X] = \operatorname{E}\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{\operatorname{E}[(X-\mu)^4]}{(\operatorname{E}[(X-\mu)^2])^2} = \frac{\mu_4}{\sigma^4},
</math>
where ''μ''<sub>4</sub> is the fourth [[central moment]] and σ is the [[standard deviation]]. Several letters are used in the literature to denote the kurtosis.  A very common choice is ''κ'', which is fine as long as it is clear that it does not refer to a [[cumulant]].  Other choices include ''γ''<sub>2</sub>, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis.

The kurtosis is bounded below by the squared [[skewness]] plus 1:{{r|Pearson1916|p=432}}
:<math> \frac{\mu_4}{\sigma^4} \geq \left(\frac{\mu_3}{\sigma^3}\right)^2 + 1,</math>
where ''μ''<sub>3</sub> is the third [[central moment]]. The lower bound is realized by the [[Bernoulli distribution]]. There is no upper limit to the kurtosis of a general probability distribution, and it may be infinite.

A reason why some authors favor the excess kurtosis is that cumulants are [[intensive and extensive properties|extensive]].  Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis.  For example, let ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> be independent random variables for which the fourth moment exists, and let ''Y'' be the random variable defined by the sum of the ''X''<sub>''i''</sub>.  The excess kurtosis of ''Y'' is

:<math>\operatorname{Kurt}[Y] - 3 =  \frac{1}{( \sum_{j=1}^n \sigma_j^{\,2})^2} \sum_{i=1}^n \sigma_i^{\,4} \cdot \left(\operatorname{Kurt}[X_i] - 3\right),</math>
where <math>\sigma_i</math> is the standard deviation of <math>X_i</math>.  In particular if all of the ''X''<sub>''i''</sub> have the same variance, then this simplifies to

:<math>\operatorname{Kurt}[Y] - 3 = {1 \over n^2} \sum_{i=1}^n (\operatorname{Kurt}[X_i] - 3).</math>

The reason not to subtract off 3 is that the bare [[fourth moment]] better generalizes to [[multivariate distribution]]s, especially when independence is not assumed.  The [[cokurtosis]] between pairs of variables is an order four [[tensor]].  For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to "correct" for an excess becomes confusing.  It is true, however, that the joint cumulants of degree greater than two for any [[multivariate normal distribution]] are zero.

For two random variables, ''X'' and ''Y'', not necessarily independent, the kurtosis of the sum, ''X''&nbsp;+&nbsp;''Y'', is
:: <math>
\begin{align}
\operatorname{Kurt}[X+Y] = {1 \over \sigma_{X+Y}^4} \big( & \sigma_X^4\operatorname{Kurt}[X] + 4\sigma_X^3\sigma_Y\operatorname{Cokurt}[X,X,X,Y] \\
& {} + 6\sigma_X^2\sigma_Y^2\operatorname{Cokurt}[X,X,Y,Y] \\[6pt]
& {} + 4\sigma_X\sigma_Y^3\operatorname{Cokurt}[X,Y,Y,Y] + \sigma_Y^4\operatorname{Kurt}[Y] \big).
\end{align}
</math>
Note that the [[binomial coefficient]]s appear in the above equation.

=== Interpretation ===
The exact interpretation of the Pearson measure of kurtosis (or excess kurtosis) used to be disputed, but is now settled. As Westfall notes in 2014{{r|Westfall2014}}, ''"...its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution)."'' The logic is simple: Kurtosis is the average (or [[expected value]]) of the [[standardized data]] raised to the fourth power. Any standardized values that are less than 1 (i.e., data within one standard deviation of the mean, where the "peak" would be), contribute virtually nothing to kurtosis, since raising a number that is less than 1 to the fourth power makes it closer to zero. The only data values (observed or observable) that contribute to kurtosis in any meaningful way are those outside the region of the peak; i.e., the outliers. Therefore, kurtosis measures outliers only; it measures nothing about the "peak".

Many incorrect interpretations of kurtosis that involve notions of peakedness have been given. One is that kurtosis measures both the "peakedness" of the distribution and the [[heavy-tailed distribution|heaviness of its tail]].{{r|Balanda1988}} Various other incorrect interpretations have been suggested, 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".{{r|Darlington1970}} Balanda and MacGillivray assert that the standard definition of kurtosis "is a poor measure of the kurtosis, peakedness, or tail weight of a distribution"{{r|Balanda1988|p=114}} 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".{{r|Balanda1988}}

===Moors' interpretation===

In 1986 Moors gave an interpretation of kurtosis.{{r|Moors1986}} Let
  
:<math> Z = \frac{ X - \mu } \sigma, </math> 
 
where ''X'' is a random variable, ''μ'' is the mean and ''σ'' is the standard deviation.

Now by definition of the kurtosis <math> \kappa </math>, and by the well-known identity <math> E[V^2] = \operatorname{var}[V] + [E[V]]^2, </math>

:<math> \kappa = E[ Z^4 ]  = \operatorname{var}[ Z^2 ] + [E[Z^2]]^2  = \operatorname{var}[ Z^2 ] + [\operatorname{var}[Z]]^2 = \operatorname{var}[ Z^2 ] + 1 </math>.

The kurtosis can now be seen as a measure of the dispersion of ''Z''<sup>2</sup> around its expectation. Alternatively it can be seen to be a measure of the dispersion of ''Z'' around +1 and&nbsp;−1. ''κ'' attains its minimal value in a symmetric two-point distribution. In terms of the original variable ''X'', the kurtosis is a measure of the dispersion of ''X'' around the two values ''μ''&nbsp;±&nbsp;''σ''.

High values of ''κ'' arise in two circumstances:
 
* where the probability mass is concentrated around the mean and the data-generating process produces occasional values far from the mean, 
* where the probability mass is concentrated in the tails of the distribution.

== Excess kurtosis ==

The ''excess kurtosis'' is defined as kurtosis minus&nbsp;3. There are 3 distinct regimes as described below.

=== Mesokurtic ===
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>.

=== Leptokurtic ===
A distribution with [[Positive number|positive]] excess kurtosis is called '''leptokurtic''', or leptokurtotic. "Lepto-" means "slender".<ref>{{Cite web | url=http://medical-dictionary.thefreedictionary.com/lepto- | title=Lepto-}}</ref> In terms of shape, a leptokurtic distribution has ''[[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''.{{r|Beneviste1980}}

=== Platykurtic ===
[[File:1909 US Penny.jpg|thumb|The [[coin toss]] is the most platykurtic distribution]]
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 ''thinner tails''. Examples of platykurtic distributions include the [[Continuous uniform distribution|continuous]] and [[discrete uniform distribution]]s, and the [[raised cosine distribution]]. The most platykurtic distribution of all is the [[Bernoulli distribution]] with ''p'' = 1/2 (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'', originally proposed by [[Jean-Pierre Kahane]]{{r|Kahane1960}} and further described by Buldygin and Kozachenko.{{r|Buldygin1980}}
{{clear}}

== Graphical examples ==

=== The Pearson type VII family ===
[[Image:Pearson type VII distribution PDF.png|300px|thumb|[[Probability density function|pdf]] for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); and 0 (black)]]
[[Image:Pearson type VII distribution log-PDF.png|300px|thumb|log-pdf for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)]]

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

:<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>

where ''a'' is a [[scale parameter]] and ''m'' is a [[shape parameter]].

All densities in this family are symmetric. The ''k''th moment exists provided ''m''&nbsp;>&nbsp;(''k''&nbsp;+&nbsp;1)/2. For the kurtosis to exist, we require ''m''&nbsp;>&nbsp;5/2. Then the mean and [[skewness]] exist and are both identically zero. Setting ''a''<sup>2</sup>&nbsp;=&nbsp;2''m''&nbsp;−&nbsp;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 excess kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary non-negative excess kurtosis. The reparameterized density is

:<math>g(x; \gamma_2) = f\left(x;\; a = \sqrt{2 + \frac{6}{\gamma_2}},\; m = \frac{5}{2} + \frac{3}{\gamma_2}\right). \!</math>

In the limit as <math>\gamma_2 \to \infty</math> one obtains the density

:<math>g(x) = 3\left(2 + x^2\right)^{-\frac{5}{2}}, \!</math>

which is shown as the red curve in the images on the right.

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.

In the images on the right, the blue curve represents the density <math>x \mapsto g(x; 2)</math> with excess kurtosis of 2.  The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density, although this conclusion is only valid for this select family of distributions. 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 excess kurtosis of 2.  Between the blue curve and the black are other Pearson type VII densities with ''γ''<sub>2</sub>&nbsp;=&nbsp;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").
{{clear}}

=== Other well-known distributions ===
[[Image:Standard symmetric pdfs.png|300px|thumb|[[Probability density function]]s for selected distributions with [[expected value|mean]] 0, [[variance]] 1 and different excess kurtosis]]
[[Image:Standard symmetric pdfs logscale.png|300px|thumb|[[Logarithm]]s of [[probability density function]]s for selected distributions with [[expected value|mean]] 0, [[variance]] 1 and different excess kurtosis]]

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]]:

* D: [[Laplace distribution]], also known as the double exponential distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3
* S: [[hyperbolic secant distribution]], orange curve, excess kurtosis = 2
* L: [[logistic distribution]], green curve, excess kurtosis = 1.2
* N: [[normal distribution]], black curve (inverted parabola in the log-scale plot), excess kurtosis = 0
* C: [[raised cosine distribution]], cyan curve, excess kurtosis = −0.593762...
* W: [[Wigner semicircle distribution]], blue curve, excess kurtosis = −1
* U: [[uniform distribution (continuous)|uniform distribution]], magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.

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]].

There exist platykurtic densities with infinite support,
*e.g., [[exponential power distribution]]s with sufficiently large shape parameter ''b''

and there exist leptokurtic densities with finite support.
*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
{{clear}}

== Sample kurtosis ==

=== Definition ===
For a [[sample (statistics)|sample]] of ''n'' values the '''sample excess kurtosis''' is

:<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>

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]].

This formula has the simpler representation,

:<math> g_2 = \frac{1}{n} \sum_{i=1}^n z_i^4  - 3 </math>

where the <math> z_i </math> values are the standardized data values using the standard deviation defined using ''n'' rather than ''n''&nbsp;−&nbsp;1 in the denominator.

For example, suppose the data values are 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999.

Then the  <math> z_i </math> values are −0.239, −0.225, −0.221, −0.234, −0.230, −0.225, −0.239, −0.230, −0.234, −0.225, −0.230, −0.239, −0.230, −0.230, −0.225, −0.230, −0.216, −0.230, −0.225, 4.359

and the <math> z_i^4 </math> values are  0.003, 0.003, 0.002, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.002, 0.003, 0.003, 360.976.

The average of these values is 18.05 and the excess kurtosis is thus 18.05&nbsp;−&nbsp;3&nbsp;=&nbsp;15.05.  This example makes it clear that data near the "middle" or "peak" of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measure "peakedness". It is simply a measure of the outlier, 999 in this example.

=== Upper bound ===
An upper bound for the sample kurtosis of ''n'' (''n'' > 2) real numbers is{{r|Sharma2015}}
:<math> g_2 \le \frac{1}{2} \frac{n-3}{n-2} g_1^2 + \frac{n}{2} - 3.</math>
where <math>g_1</math> is the sample skewness <math>m_3/m_2^{3/2}</math>.

=== Variance under normality ===
The variance of the sample kurtosis of a sample of size ''n'' from the [[normal distribution]] is{{r|Fisher1930}}
:<math>  \operatorname{var}(g_2) = \frac{24n(n-1)^2}{(n-3)(n-2)(n+3)(n+5)} </math>

Stated differently, under the assumption that the underlying random variable <math>X</math> is normally distributed, it can be shown that <math>\sqrt{n} g_2 \xrightarrow{d} \mathcal{N}(0, 24)</math>.{{r|Kendall1969|p=Page number needed}}

== Estimators of population kurtosis ==
Given a sub-set of samples from a population, the sample excess kurtosis above is a [[biased estimator]] of the population excess kurtosis. An alternative estimator of the population excess kurtosis is defined as follows:

: <math>
\begin{align}
G_2 & = \frac{k_4}{k_2^2} \\[6pt]
& = \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} \\[6pt]
& = \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right] \\[6pt]
& = \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,g_2 + 6 \right] \\[6pt]
& = \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)} \\[6pt]
& = \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)}
\end{align}
</math>

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.{{r|Joanes1998}}

== Applications ==
{{Expand section|date=December 2009}}

The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods.

[[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.

For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see [[Variance#Distribution of the sample variance|variance]].

Pearson's definition of kurtosis is used as an indicator of intermittency in [[turbulence]].{{r|Sandborn1959}}

===Kurtosis convergence===
Applying [[band-pass filter]]s to [[digital image]]s, kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termed ''kurtosis convergence'', can be used to detect image splicing in [[forensic analysis]].{{r|Pan2012}}

== Other measures ==
A different measure of "kurtosis" is provided by using [[L-moment]]s instead of the ordinary moments.{{r|Hosking1992|Hosking2006}}

== See also ==
{{Commons category|Kurtosis}}
* [[Kurtosis risk]]
* [[Maximum entropy probability distribution]]

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== Further reading ==
*{{cite journal | last1 = Kim | first1 = Tae-Hwan | last2 = White | first2 = Halbert | year = 2003 | title = On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index | url = http://escholarship.org/uc/item/7b52v07p | journal = Finance Research Letters | volume = 1 | issue = | pages = 56–70 | doi = 10.1016/S1544-6123(03)00003-5 }} [https://web.archive.org/web/20111118123903/http://weber.ucsd.edu/~hwhite/pub_files/hwcv-092.pdf Alternative source] (Comparison of kurtosis estimators)
*{{cite journal | last1 = Seier | first1 = E. | last2 = Bonett | first2 = D.G. | year = 2003 | title = Two families of kurtosis measures | url = | journal = Metrika | volume = 58 | issue = | pages = 59–70 | doi = 10.1007/s001840200223 }}

== External links ==
{{Wikiversity}}
* {{springer|title=Excess coefficient|id=p/e036800}}
* [http://www.fxsolver.com/solve/share/RMqwaVp85T_5rbacksPD4g==/ Kurtosis calculator]
* [https://archive.is/20121208231710/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)..
* [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]
* [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}}

[[Category:Moment (mathematics)]]
[[Category:Statistical deviation and dispersion]]