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Kurtosis

From Wikipedia, the free encyclopedia

Fourth standardized moment in statistics

Kurtosis (fromGreek:κυρτός (kyrtos orkurtos), meaning 'curved, arching') refers to the degree oftailedness in theprobability distribution of areal-valued,random variable inprobability theory andstatistics. Similar toskewness, kurtosis provides insight into specific characteristics of a distribution. Various methods exist for quantifying kurtosis in theoretical distributions, and corresponding techniques allow estimation based on sample data from a population. It is important to note that different measures of kurtosis can yield varyinginterpretations.

The standard measure of a distribution's kurtosis, originating withKarl Pearson,^[1] is a scaled version of the fourthmoment of the distribution. This number is related to the tails of the distribution, not its peak;^[2] hence, the sometimes-seen characterization of kurtosis aspeakedness is incorrect. For this measure, higher kurtosis corresponds to greater extremity ofdeviations (oroutliers), and not the configuration of data near themean.

Excess kurtosis, typically compared to a value of 0, characterizes thetailedness of a distribution. A univariatenormal distribution has an excess kurtosis of 0. Negative excess kurtosis indicates aplatykurtic distribution, which does not necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution. For instance, theuniform distribution (i.e., one that is uniformly finite over some bound and zero elsewhere) is platykurtic. On the other hand, positive excess kurtosis signifies aleptokurtic distribution. TheLaplace distribution for example, has tails that decay more slowly than a normal one, resulting in more outliers. To simplify comparison with the normal distribution, excess kurtosis is calculated as Pearson's kurtosis minus 3. Some authors and software packages usekurtosis to refer specifically to excess kurtosis, but this article distinguishes between the two for clarity.

Alternative measures of kurtosis are: theL-kurtosis, which is a scaled version of the fourthL-moment; measures based on four population or samplequantiles.^[3] These are analogous to the alternative measures of skewness that are not based on ordinary moments.^[3]

Pearson moments

The kurtosis is the fourthstandardized moment, defined as $\operatorname {Kurt} [X]=\operatorname {E} \left[{\left({\frac {X-\mu }{\sigma }}\right)}^{4}\right]={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}},$ whereμ₄ is the fourthcentral moment andσ is thestandard deviation. Several letters are used in the literature to denote the kurtosis. A very common choice isκ, which is fine as long as it is clear that it does not refer to acumulant. Other choices includeγ₂, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis. Pearson is systematically usingβ₂.

The kurtosis is bounded below by the squaredskewness plus 1:^[4]^: 432 ${\frac {\mu _{4}}{\sigma ^{4}}}\geq \left({\frac {\mu _{3}}{\sigma ^{3}}}\right)^{2}+1,$ whereμ₃ is the third central moment. The lower bound is realized by theBernoulli 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 areextensive. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, letX₁, ...,X_n be independent random variables for which the fourth moment exists, and letY be the random variable defined by the sum of theX_i. The excess kurtosis ofY is $\operatorname {Kurt} [Y]-3={\frac {1}{\left(\sum _{j=1}^{n}\sigma _{j}^{\,2}\right)^{2}}}\sum _{i=1}^{n}\sigma _{i}^{\,4}\cdot \left(\operatorname {Kurt} \left[X_{i}\right]-3\right),$ where $\sigma _{i}$ is the standard deviation ofX_i. In particular if all of theX_i have the same variance, then this simplifies to $\operatorname {Kurt} [Y]-3={\frac {1}{n^{2}}}\sum _{i=1}^{n}\left(\operatorname {Kurt} \left[X_{i}\right]-3\right).$

The reason not to subtract 3 is that the baremoment better generalizes tomultivariate distributions, especially when independence is not assumed. Thecokurtosis between pairs of variables is an order fourtensor. 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 anymultivariate normal distribution are zero.

For two random variables,X andY, not necessarily independent, the kurtosis of the sum,X +Y, is ${\begin{aligned}\operatorname {Kurt} [X+Y]={\frac {1}{\sigma _{X+Y}^{4}}}{\big (}&\sigma _{X}^{4}\operatorname {Kurt} [X]\\&{}+4\sigma _{X}^{3}\sigma _{Y}\operatorname {Cokurt} [X,X,X,Y]\\[6pt]&{}+6\sigma _{X}^{2}\sigma _{Y}^{2}\operatorname {Cokurt} [X,X,Y,Y]\\[6pt]&{}+4\sigma _{X}\sigma _{Y}^{3}\operatorname {Cokurt} [X,Y,Y,Y]\\[6pt]&{}+\sigma _{Y}^{4}\operatorname {Kurt} [Y]{\big )}.\end{aligned}}$ Note that the fourth-powerbinomial coefficients (1, 4, 6, 4, 1) appear in the above equation.

Interpretation

The interpretation of the Pearson measure of kurtosis (or excess kurtosis) was once debated, but it is now well-established. As noted by Westfall in 2014^[2], "... its unambiguous interpretation relates to tail extremity". Specifically, it reflects either the presence of existing outliers (for sample kurtosis) or the tendency to produce outliers (for the kurtosis of a probability distribution). The underlying logic is straightforward: kurtosis represents the average (orexpected value) of standardized data raised to the fourth power. Standardized values less than 1—corresponding to data within one standard deviation of the mean (where the peak occurs)—contribute minimally to kurtosis. This is because raising a number less than 1 to the fourth power brings it closer to zero. The meaningful contributors to kurtosis are data values outside the peak region, i.e., the outliers. Therefore, kurtosis primarily measures outliers and provides no information about the central peak.

Numerous misconceptions about kurtosis relate to notions of peakedness. One such misconception is that kurtosis measures both the peakedness of a distribution and theheaviness of its tail.^[5] Other incorrect interpretations include notions likelack of shoulders (where theshoulder refers vaguely to the area between the peak and the tail, or more specifically, the region about one standard deviation from the mean) or bimodality.^[6] Balanda andMacGillivray argue that the standard definition of kurtosis "poorly captures the kurtosis, peakedness, or tail weight of a distribution." Instead, they propose a vague definition of kurtosis as the location- and scale-free movement ofprobability mass from the distribution's shoulders into its center and tails.^[5]

Moors' interpretation

In 1986, Moors gave an interpretation of kurtosis.^[7] Let $Z={\frac {X-\mu }{\sigma }},$ whereX is a random variable,μ is the mean andσ is the standard deviation.

Now by definition of the kurtosis $\kappa$ , and by the well-known identity $\operatorname {E} \left[V^{2}\right]=\operatorname {var} [V]+\operatorname {E} [V]^{2},$ ${\begin{aligned}\kappa &=\operatorname {E} \left[Z^{4}\right]\\&=\operatorname {var} \left[Z^{2}\right]+\operatorname {E} {\!\left[Z^{2}\right]}^{2}\\&=\operatorname {var} \left[Z^{2}\right]+\operatorname {var} [Z]^{2}=\operatorname {var} \left[Z^{2}\right]+1.\end{aligned}}$

The kurtosis can now be seen as a measure of the dispersion ofZ² around its expectation. Alternatively it can be seen to be a measure of the dispersion ofZ around+1 and −1.κ attains its minimal value in a symmetric two-point distribution. In terms of the original variableX, the kurtosis is a measure of the dispersion ofX around the two valuesμ ±σ.

High values ofκ arise where the probability mass is concentrated around the mean and the data-generating process produces occasional values far from the mean, or where the probability mass is concentrated in the tails of the distribution.

Maximal entropy

Theentropy of a distribution is ${\textstyle -\!\int p(x)\ln p(x)\,dx.}$

For any $\mu \in \mathbb {R} ^{n},\Sigma \in \mathbb {R} ^{n\times n}$ with $\Sigma$ positive definite, among all probability distributions on $\mathbb {R} ^{n}$ with mean $\mu$ and covariance $\Sigma$ , the normal distribution ${\mathcal {N}}(\mu ,\Sigma )$ has the largest entropy.

Since mean $\mu$ and covariance $\Sigma$ are the first two moments, it is natural to consider extension to higher moments. In fact, byLagrange multiplier method, for any prescribed first n moments, if there exists some probability distribution of form $p(x)\propto e^{\sum _{i}a_{i}x_{i}+\sum _{ij}b_{ij}x_{i}x_{j}+\cdots +\sum _{i_{1}\cdots i_{n}}x_{i_{1}}\cdots x_{i_{n}}}$ that has the prescribed moments (if it is feasible), then it is the maximal entropy distribution under the given constraints.^[8]^[9]

By serial expansion, ${\begin{aligned}&\int {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]&={\frac {1}{\sqrt {2\pi }}}\int e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}x^{2n}\,dx\\[6pt]&=\sum _{k}{\frac {1}{k!}}\left(-{\frac {g}{4}}\right)^{k}(2n+4k-1)!!\\[6pt]&=(2n-1)!!-{\tfrac {1}{4}}g(2n+3)!!+O(g^{2})\end{aligned}}$ so if a random variable has probability distribution $p(x)=e^{-{\frac {1}{2}}x^{2}-{\frac {1}{4}}gx^{4}}/Z$ , where $Z {\displaystyle Z}$ is a normalization constant, then its kurtosis is $3-6g+O(g^{2})$ .^[10]

Excess kurtosis

Theexcess kurtosis is defined as kurtosis minus 3. There are three distinct regimes as described below.

Mesokurtic

Distributions with zero excess kurtosis are calledmesokurtic, ormesokurtotic. The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of itsparameters. A few other well-known distributions can be mesokurtic, depending on parameter values: for example, thebinomial distribution is mesokurtic for ${\textstyle p=1/2\pm {\sqrt {1/12}}}$ .

Leptokurtic

A distribution withpositive excess kurtosis is calledleptokurtic, orleptokurtotic. A leptokurtic distribution hasfatter tails. (lepto- means 'slender', originally referring to the peak.^[11]) Examples of leptokurtic distributions include theStudent's t-distribution,Rayleigh distribution,Laplace distribution,exponential distribution,Poisson distribution and thelogistic distribution. Such distributions are sometimes termedsuper-Gaussian.^[12]

Three symmetric increasingly leptokurtic probability density functions; their intersections are indicated by vertical lines.

Platykurtic

Thecoin toss is the most platykurtic distribution

A distribution withnegative excess kurtosis is calledplatykurtic, orplatykurtotic. A platykurtic distribution hasthinner tails (platy- means 'broad', originally referring to the peak).^[13] Examples of platykurtic distributions include thecontinuous anddiscrete uniform distributions, and theraised cosine distribution. The most platykurtic distribution of all is theBernoulli distribution withp = 1/2 (for example the number of times one obtains heads when flipping a coin once, acoin toss), for which the excess kurtosis is −2.

Graphical examples

The Pearson type VII family

PDF for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); and 0 (black)

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 aparametric family of distributions whose kurtosis can be adjusted while their lower-order moments and cumulants remain constant. Consider thePearson type VII family, which is a special case of thePearson type IV family restricted to symmetric densities. Theprobability density function (PDF) is given by $f(x;a,m)={\frac {\Gamma (m)}{a\,{\sqrt {\pi }}\,\Gamma (m-1/2)}}\left[1+\left({\frac {x}{a}}\right)^{2}\right]^{-m},$ wherea is ascale parameter andm is ashape parameter.

All densities in this family are symmetric. Thek-th moment exists providedm > (k + 1)/2. For the kurtosis to exist, we requirem > 5/2. Then the mean andskewness exist and are both identically zero. Settinga² = 2m − 3 makes the variance equal to unity. Then the only free parameter ism, which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize with ${\textstyle m=5/2+3/\gamma _{2}}$ , where $\gamma _{2}$ 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 $g(x;\gamma _{2})=f{\left(x;\;a={\sqrt {2+6\gamma _{2}^{-1}}},\;m={\tfrac {5}{2}}+3\gamma _{2}^{-1}\right)}.$

In the limit as $\gamma _{2}\to \infty$ , one obtains the density $g(x)=3\left(2+x^{2}\right)^{-5/2},$ which is shown as the red curve in the images on the right.

In the other direction as $\gamma _{2}\to 0$ one obtains thestandard normal density as the limiting distribution, shown as the black curve.

In the images on the right, the blue curve represents the density $x\mapsto g(x;2)$ 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 aparabola. One can see that the normal density allocates little probability mass to the regions far from the mean (hasthin 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 withγ₂ = 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 $\gamma _{2}=\infty$ (which, strictly speaking, means that the fourth moment does not exist). The red curve decreases the slowest as one moves outward from the origin (hasfat tails).

Other well-known distributions

Probability density functions for selected distributions withmean 0,variance 1 and different excess kurtosis

Logarithms ofprobability density functions for selected distributions withmean 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 alinear scale andlogarithmic 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, magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.

Note that in these cases the platykurtic densities have boundedsupport, whereas the densities with positive or zero excess kurtosis are supported on the wholereal line.

One cannot infer that high or low kurtosis distributions have the characteristics indicated by these examples. There exist platykurtic densities with infinite support, e.g.,exponential power distributions with sufficiently large shape parameterb, and there exist leptokurtic densities with finite support. An example of the latter is 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.

A platykurtic distribution that is infinitely peaked

A leptokurtic distribution that is low and appears flat-topped

Also, one cannot infer from the graphs that higher kurtosis distributions are morepeaked and that lower kurtosis distributions are moreflat. There exist platykurtic densities with infinite peakedness; e.g., an equal mixture of thebeta distribution with parameters 0.5 and 1 with its reflection about 0.0, and there exist leptokurtic densities that appear flat-topped; e.g., a mixture of distribution that is uniform between −1 and 1 with a T(4.0000001) Student's t-distribution, with mixing probabilities 0.999 and 0.001.

Graphs of the standardized versions of these distributions are given to the right.

Sample kurtosis

Definitions

A natural but biased estimator

For asample ofn values, amethod of moments estimator of the population excess kurtosis can be defined as $g_{2}={\frac {m_{4}}{m_{2}^{2}}}-3={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{4}}{\left[{\tfrac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{2}\right]^{2}}}-3$ wherem₄ is the fourth samplemoment about the mean,m₂ is the second sample moment about the mean (that is, thesample variance),x_i is thei-th value, and ${\overline {x}}$ is thesample mean.

This formula has the simpler representation, $g_{2}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{4}-3$ where the $z_{i}$ values are the standardized data values using the standard deviation defined usingn rather thann − 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 thez_i 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 thez_i⁴ 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 thus18.05 − 3 = 15.05. This example makes it clear that data near themiddle orpeak of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measurepeakedness. It is simply a measure of the outlier, 999 in this example.

Standard unbiased estimator

Given a sub-set of samples from a population, the sample excess kurtosis $g_{2}$ above is abiased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis, which is unbiased in random samples of a normal distribution, is defined as follows:^[3] ${\begin{aligned}G_{2}&={\frac {k_{4}}{k_{2}^{2}}}={\frac {n^{2}\,\left[(n+1)\,m_{4}-3\,(n-1)\,m_{2}^{2}\right]}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {(n-1)^{2}}{n^{2}\,m_{2}^{2}}}\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,{\frac {m_{4}}{m_{2}^{2}}}-3\,(n-1)\right]\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,g_{2}+6\right]\\[6pt]&={\frac {(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{4}}{\left(\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right)^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)\,(n-3)}}\\[6pt]&={\frac {(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{4}}{k_{2}^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)(n-3)}}\end{aligned}}$ wherek₄ is the unique symmetricunbiased estimator of the fourthcumulant,k₂ is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance),m₄ is the fourth sample moment about the mean,m₂ is the second sample moment about the mean,x_i is thei-th value, and ${\bar {x}}$ is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient $G_{2}$ is the version found inExcel and several statistical packages includingMinitab,SAS, andSPSS.^[14]

Unfortunately, in non-normal samples $G_{2}$ is itself generally biased.

Upper bound

An upper bound for the sample kurtosis ofn (n > 2) real numbers is^[15] $g_{2}\leq {\frac {1}{2}}{\frac {n-3}{n-2}}g_{1}^{2}+{\frac {n}{2}}-3,$ where $g_{1}=m_{3}/m_{2}^{3/2}$ is the corresponding sample skewness.

Variance under normality

The variance of the sample kurtosis of a sample of sizen from thenormal distribution is^[16] $\operatorname {var} (g_{2})={\frac {24n(n-1)^{2}}{(n-3)(n-2)(n+3)(n+5)}}$

Stated differently, under the assumption that the underlying random variable $X {\displaystyle X}$ is normally distributed, it can be shown that ${\sqrt {n}}g_{2}\,\xrightarrow {d} \,{\mathcal {N}}(0,24)$ .^[17]^{: Page number needed}

Applications

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 agoodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is theJarque–Bera test for normality.

For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please seevariance.

Pearson's definition of kurtosis is used as an indicator of intermittency inturbulence.^[18] It is also used in magnetic resonance imaging to quantify non-Gaussian diffusion.^[19]

A concrete example is the following lemma by He, Zhang, and Zhang:^[20]Assume a random variableX has expectation $\operatorname {E} [X]=\mu$ , variance $\operatorname {E} \left[(X-\mu )^{2}\right]=\sigma ^{2}$ and kurtosis ${\textstyle \kappa ={\tfrac {1}{\sigma ^{4}}}\operatorname {E} \left[(X-\mu )^{4}\right].}$ Assume we sample $n={\tfrac {2{\sqrt {3}}+3}{3}}\kappa \log {\tfrac {1}{\delta }}$ many independent copies. Then $\Pr \left[\max _{i=1}^{n}X_{i}\leq \mu \right]\leq \delta \quad {\text{and}}\quad \Pr \left[\min _{i=1}^{n}X_{i}\geq \mu \right]\leq \delta .$

This shows that with $\Theta (\kappa \log {\tfrac {1}{\delta }})$ many samples, we will see one that is above the expectation with probability at least $1-\delta$ . In other words: If the kurtosis is large, there may be a lot of values either all below or above the mean.

Kurtosis convergence

Applyingband-pass filters todigital images, kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termedkurtosis convergence, can be used to detect image splicing inforensic analysis.^[21]

Seismic signal analysis

Kurtosis can be used ingeophysics to distinguish different types ofseismic signals. It is particularly effective in differentiating seismic signals generated by human footsteps from other signals.^[22] This is useful in security and surveillance systems that rely on seismic detection.

Weather prediction

Inmeteorology, kurtosis is used to analyze weather data distributions. It helps predict extreme weather events by assessing the probability of outlier values in historical data,^[23] which is valuable for long-term climate studies and short-term weather forecasting.

Other measures

A different measure of kurtosis is provided by usingL-moments instead of the ordinary moments.^[24]^[25]

See also

Wikimedia Commons has media related toKurtosis.

References

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^He, Simai; Zhang, Jiawei; Zhang, Shuzhong (2010). "Bounding probability of small deviation: A fourth moment approach".Mathematics of Operations Research.35 (1):208–232.doi:10.1287/moor.1090.0438.S2CID 11298475.
^Pan, Xunyu; Zhang, Xing; Lyu, Siwei (2012), "Exposing Image Splicing with Inconsistent Local Noise Variances",2012 IEEE International Conference on Computational Photography (ICCP), 28-29 April 2012; Seattle, WA, USA: IEEE, pp. 1–10,doi:10.1109/ICCPhot.2012.6215223,ISBN 978-1-4673-1662-0,S2CID 14386924{{citation}}: CS1 maint: location (link)
^Liang, Zhiqiang; Wei, Jianming; Zhao, Junyu; Liu, Haitao; Li, Baoqing; Shen, Jie; Zheng, Chunlei (2008-08-27)."The Statistical Meaning of Kurtosis and Its New Application to Identification of Persons Based on Seismic Signals".Sensors.8 (8):5106–5119.Bibcode:2008Senso...8.5106L.doi:10.3390/s8085106.ISSN 1424-8220.PMC 3705491.PMID 27873804.
^Supraja (2024-05-27)."Kurtosis in Practice: Real-World Applications and Interpretations".Analytics Insight. Retrieved2024-11-11.
^Hosking, Jonathan R. M. (1992), "Moments orL moments? An example comparing two measures of distributional shape",The American Statistician,46 (3):186–189,doi:10.1080/00031305.1992.10475880,JSTOR 2685210
^Hosking, Jonathan R. M. (2006), "On the characterization of distributions by theirL-moments",Journal of Statistical Planning and Inference,136 (1):193–198,doi:10.1016/j.jspi.2004.06.004

Further reading

Kim, Tae-Hwan; White, Halbert (2003)."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.S2CID 16913409.Alternative source (Comparison of kurtosis estimators)
Seier, E.; Bonett, D.G. (2003). "Two families of kurtosis measures".Metrika.58:59–70.doi:10.1007/s001840200223.S2CID 115990880.

External links

Wikiversity has learning resources aboutKurtosis

"Excess coefficient",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Kurtosis calculator
Free Online Software (Calculator) computes various types of skewness and kurtosis statistics for any dataset (includes small and large sample tests)..
Kurtosis on theEarliest known uses of some of the words of mathematics
Celebrating 100 years of Kurtosis a history of the topic, with different measures of kurtosis.

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Descriptive statistics

Continuous data

Center	Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode
Dispersion	Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance
Shape	Central limit theorem Moments Kurtosis L-moments Skewness

Index of dispersion

Summary tables

Data collection

Study design	Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power
Survey methodology	Sampling Cluster Stratified Opinion poll Questionnaire Standard error
Controlled experiments	Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control
Adaptive designs	Adaptive clinical trial Stochastic approximation Up-and-down designs
Observational studies	Cohort study Cross-sectional study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank(Wilcoxon) Hodges–Lehmann estimator Rank sum(Mann–Whitney) Nonparametric anova 1-way(Kruskal–Wallis) 2-way(Friedman) Ordered alternative(Jonckheere–Terpstra) Van der Waerden test

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) Template:Least squares and regression analysis
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity
Generalized linear model	Exponential families Logistic(Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / multivariate / time-series / survival analysis

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic(Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model(Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) (Autoregressive model (AR))
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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