Excess Kurtosis
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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
of a real-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. Similar to skewness, 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’s important to note that different measures of kurtosis can yield varying interpretations. The standard measure of a distribution's kurtosis, originating with
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English biostatistician and mathematician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university ...
, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as " peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or
outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s), and not the configuration of data near the mean. Excess kurtosis, typically compared to a value of 0, characterizes the “tailedness” of a distribution. A univariate
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
has an excess kurtosis of 0. Negative excess kurtosis indicates a platykurtic distribution, which doesn’t necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution. For instance, the uniform distribution (i.e. one that is uniformly finite over some bound and zero elsewhere) is platykurtic. On the other hand, positive excess kurtosis signifies a leptokurtic distribution. The
Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
, for example, has tails that decay more slowly than a Gaussian, 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 use “kurtosis” to refer specifically to excess kurtosis, but this article distinguishes between the two for clarity. 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. These are analogous to the alternative measures of skewness that are not based on ordinary moments.


Pearson moments

The kurtosis is the fourth standardized moment, defined as \operatorname = \operatorname\left 4\right= \frac = \frac, where is the fourth central moment and is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
. Several letters are used in the literature to denote the kurtosis. A very common choice is , which is fine as long as it is clear that it does not refer to a cumulant. Other choices include , 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: \frac \geq \left(\frac\right)^2 + 1, where is the third central moment. The lower bound is realized by the
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with pro ...
. 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 extensive. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let be independent random variables for which the fourth moment exists, and let be the random variable defined by the sum of the . The excess kurtosis of is \operatorname - 3 = \frac \sum_^n \sigma_i^ \cdot \left(\operatorname\left _i\right- 3\right), where \sigma_i is the standard deviation of . In particular if all of the have the same variance, then this simplifies to \operatorname - 3 = \frac \sum_^n \left(\operatorname\left _i\right- 3\right). The reason not to subtract 3 is that the bare moment better generalizes to multivariate distributions, especially when independence is not assumed. The cokurtosis between pairs of variables is an order four
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
. 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 In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
are zero. For two random variables, and , not necessarily independent, the kurtosis of the sum, , is \begin \operatorname +Y= \frac \big( & \sigma_X^4\operatorname \\ & + 4\sigma_X^3 \sigma_Y \operatorname ,X,X,Y\\ pt& + 6\sigma_X^2 \sigma_Y^2 \operatorname ,X,Y,Y\\ pt& + 4\sigma_X \sigma_Y^3 \operatorname ,Y,Y,Y\\ pt& + \sigma_Y^4 \operatorname \big). \end Note that the fourth-power
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s (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, "...''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 (or
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
) 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 the heaviness of its tail . Other incorrect interpretations include notions like “lack of shoulders” (where the “shoulder” refers vaguely to the area between the peak and the tail, or more specifically, the region about one
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
from the mean) or “bimodality.” Balanda and MacGillivray 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 of probability mass from the distribution’s shoulders into its center and tails.


Moors' interpretation

In 1986, Moors gave an interpretation of kurtosis. Let Z = \frac \sigma, where is a random variable, is the mean and is the standard deviation. Now by definition of the kurtosis \kappa , and by the well-known identity \operatorname\left ^2\right= \operatorname + \operatorname 2, \begin \kappa & = \operatorname\left Z^4 \right\\ & = \operatorname\left Z^2 \right+ \operatorname^2 \\ & = \operatorname\left Z^2 \right+ \operatorname 2 = \operatorname\left Z^2 \right+ 1. \end The kurtosis can now be seen as a measure of the dispersion of around its expectation. Alternatively it can be seen to be a measure of the dispersion of around and . attains its minimal value in a symmetric two-point distribution. In terms of the original variable , the kurtosis is a measure of the dispersion of around the two values . High values of 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.


Maximal entropy

The
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
of a distribution is -\!\int p(x) \ln p(x) \, dx. For any \mu \in \R^n, \Sigma \in \R^ with \Sigma positive definite, among all probability distributions on \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, by
Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the conditio ...
method, for any prescribed first n moments, if there exists some probability distribution of form p(x) \propto e^ that has the prescribed moments (if it is feasible), then it is the maximal entropy distribution under the given constraints. By serial expansion, \begin & \int \frac e^ x^ \, dx \\ pt&= \frac \int e^ x^ \, dx \\ pt&= \sum_k \frac \left(-\frac\right)^k (2n+4k-1)!! \\ pt&= (2n-1)!! - \tfrac g (2n+3)!! + O(g^2) \end so if a random variable has probability distribution p(x) = e^/Z, where Z is a normalization constant, then its kurtosis is


Excess kurtosis

The ''excess kurtosis'' is defined as kurtosis minus 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 A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. A few other well-known distributions can be mesokurtic, depending on parameter values: for example, the
binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
is mesokurtic for p = 1/2 \pm \sqrt.


Leptokurtic

A distribution with positive excess kurtosis is called leptokurtic, or leptokurtotic. "Lepto-" means "slender". A leptokurtic distribution has '' fatter tails''. Examples of leptokurtic distributions include the
Student's t-distribution In probability theory and statistics, Student's  distribution (or simply the  distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
, Rayleigh distribution,
Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
,
exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
,
Poisson distribution In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
and the logistic distribution. Such distributions are sometimes termed ''super-Gaussian''.


Platykurtic

A distribution with negative excess kurtosis is called platykurtic, or platykurtotic. "Platy-" means "broad". A platykurtic distribution has ''thinner tails''. Examples of platykurtic distributions include the continuous and discrete uniform distributions, and the raised cosine distribution. The most platykurtic distribution of all is the
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with pro ...
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.


Graphical examples


The Pearson type VII family

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 type VII family, which is a special case of the Pearson type IV family restricted to symmetric densities. The
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
is given by f(x; a, m) = \frac \left +\left(\frac\right)^2 \right, where is a scale parameter and is a
shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. th ...
. All densities in this family are symmetric. The -th moment exists provided . For the kurtosis to exist, we require . Then the mean and skewness exist and are both identically zero. Setting makes the variance equal to unity. Then the only free parameter is , which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize with 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. In the limit as \gamma_2 \to \infty one obtains the density g(x) = 3\left(2 + x^2\right)^, which is shown as the red curve in the images on the right. In the other direction as \gamma_2 \to 0 one obtains the standard 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 a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
. 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  = 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 ("has fat tails").


Other well-known distributions

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 A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved. Unlike a linear Scale (measurement) ...
: * D:
Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
, 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 In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
, 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 bounded support, whereas the densities with positive or zero excess kurtosis are supported on the whole
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
. 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 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 Also, there exist platykurtic densities with infinite peakedness, *e.g., an equal mixture of the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') an ...
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 In probability theory and statistics, Student's  distribution (or simply the  distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
, with mixing probabilities 0.999 and 0.001.


Sample kurtosis


Definitions


A natural but biased estimator

For a sample of ''n'' values, a method of moments estimator of the population excess kurtosis can be defined as g_2 = \frac -3 = \frac - 3 where is the fourth sample moment about the mean, ''m''2 is the second sample moment about the mean (that is, the
sample variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
), is the -th value, and \overline is the sample mean. This formula has the simpler representation, g_2 = \frac \sum_^n z_i^4 - 3 where the z_i values are the standardized data values using the standard deviation defined using rather than 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 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 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 . 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.


Standard unbiased estimator

Given a sub-set of samples from a population, the sample excess kurtosis g_2 above is a biased 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: \begin G_2 & = \frac = \frac \; \frac \\ pt& = \frac \left n+1)\,\frac - 3\,(n-1) \right\\ pt& = \frac \left n+1)\,g_2 + 6 \right\\ pt& = \frac \; \frac - 3\,\frac \\ pt& = \frac \; \frac - 3\,\frac \end where is the unique symmetric
unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...
estimator of the fourth cumulant, is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), is the fourth sample moment about the mean, is the second sample moment about the mean, is the -th value, and \bar is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient G_2 is the version found in Excel and several statistical packages including Minitab, SAS, and SPSS.Doane DP, Seward LE (2011) J Stat Educ 19 (2) Unfortunately, in nonnormal samples G_2 is itself generally biased.


Upper bound

An upper bound for the sample kurtosis of () real numbers is g_2 \le \frac \frac g_1^2 + \frac - 3. where g_1 = m_3/m_2^ is the corresponding sample skewness.


Variance under normality

The variance of the sample kurtosis of a sample of size from the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
is \operatorname(g_2) = \frac Stated differently, under the assumption that the underlying random variable X is normally distributed, it can be shown that \sqrt g_2 \,\xrightarrow\, \mathcal(0, 24).


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 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 In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
. Pearson's definition of kurtosis is used as an indicator of intermittency in
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
. It is also used in magnetic resonance imaging to quantify non-Gaussian diffusion. A concrete example is the following lemma by He, Zhang, and Zhang: Assume a random variable has expectation \operatorname = \mu, variance \operatorname\left X - \mu)^2\right= \sigma^2 and kurtosis \kappa = \tfrac \operatorname\left X - \mu)^4\right Assume we sample n = \tfrac \kappa \log\tfrac many independent copies. Then \Pr\left max_^n X_i \le \mu\right\le \delta \quad\text\quad \Pr\left min_^n X_i \ge \mu\right\le \delta. This shows that with \Theta(\kappa\log\tfrac\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, we might see a lot values either all below or above the mean.


Kurtosis convergence

Applying
band-pass filter A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range. It is the inverse of a '' band-stop filter''. Description In electronics and s ...
s to
digital image A digital image is an image composed of picture elements, also known as pixels, each with '' finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions f ...
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.


Seismic signal analysis

Kurtosis can be used in
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
to distinguish different types of seismic signals. It is particularly effective in differentiating seismic signals generated by human footsteps from other signals. This is useful in security and surveillance systems that rely on seismic detection.


Weather prediction

In
meteorology Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...
, kurtosis is used to analyze weather data distributions. It helps predict extreme weather events by assessing the probability of outlier values in historical data, which is valuable for long-term climate studies and short-term weather forecasting.


Other measures

A different measure of "kurtosis" is provided by using L-moments instead of the ordinary moments.


See also

* Kurtosis risk * Maximum entropy probability distribution


References


Further reading


Alternative source
(Comparison of kurtosis estimators) *


External links

*
Kurtosis calculator

Free Online Software (Calculator)
computes various types of skewness and kurtosis statistics for any dataset (includes small and large sample tests)..

on th


Celebrating 100 years of Kurtosis
a history of the topic, with different measures of kurtosis. {{Statistics, descriptive Moments (mathematics) Statistical deviation and dispersion