In
probability theory and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of a
real-valued
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. Like
skewness, kurtosis describes a particular aspect of a probability distribution. There are different ways to quantify kurtosis for a theoretical distribution, and there are corresponding ways of estimating it using a sample from a population. Different measures of kurtosis may have different
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 mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
, is a scaled version of the fourth
moment
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* ''Moments'' (Christine Guldbrand ...
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.
It is common to compare the excess kurtosis (defined below) of a distribution to 0, which is the excess kurtosis of any univariate
normal distribution
In 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 e^
The parameter \mu ...
. Distributions with negative excess kurtosis are said to be ''platykurtic'', although this does not imply the distribution is "flat-topped" as is sometimes stated. Rather, it means the distribution produces fewer and/or less extreme outliers than the normal distribution. An example of a platykurtic distribution is the
uniform distribution
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
, which does not produce outliers. Distributions with a positive excess kurtosis are said to be ''leptokurtic''. An example of a leptokurtic distribution is 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 ...
, which has tails that asymptotically approach zero more slowly than a Gaussian, and therefore produces more outliers than the normal distribution. It is common practice to use excess kurtosis, which is defined as Pearson's kurtosis minus 3, to provide a simple comparison to the
normal distribution
In 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 e^
The parameter \mu ...
. Some authors and software packages use "kurtosis" by itself to refer to the excess kurtosis. For clarity and generality, however, this article explicitly indicates where non-excess kurtosis is meant.
Alternative measures of kurtosis are: the
L-kurtosis
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics ( L-statistics) analogous to conventional moments, and can be used to calculate ...
, which is a scaled version of the fourth
L-moment; measures based on four population or sample
quantiles
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile ...
. 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
:
where ''μ''
4 is the fourth
central moment and σ is the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
. 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''
2, 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:
:
where ''μ''
3 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
extensive. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let ''X''
1, ..., ''X''
''n'' be independent random variables for which the fourth moment exists, and let ''Y'' be the random variable defined by the sum of the ''X''
''i''. The excess kurtosis of ''Y'' is
:
where
is the standard deviation of
. In particular if all of the ''X''
''i'' have the same variance, then this simplifies to
:
The reason not to subtract 3 is that the bare
fourth moment better generalizes to
multivariate distributions, 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'' + ''Y'', is
:
Note that the
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 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, ''"...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
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
) of the standardized data raised to the fourth power. 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
heaviness of its tail. 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
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
from the mean) or "bimodality". Balanda and
MacGillivray assert that the standard definition of kurtosis "is a poor measure of the kurtosis, peakedness, or tail weight of a distribution" and instead propose to "define kurtosis vaguely as the location- and scale-free movement of
probability mass
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
from the shoulders of a distribution into its center and tails".
Moors' interpretation
In 1986 Moors gave an interpretation of kurtosis. Let
:
where ''X'' is a random variable, ''μ'' is the mean and ''σ'' is the standard deviation.
Now by definition of the kurtosis
, and by the well-known identity
:
:
.
The kurtosis can now be seen as a measure of the dispersion of ''Z''
2 around its expectation. Alternatively it can be seen to be a measure of the dispersion of ''Z'' around +1 and −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'' ± ''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 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
parameters. 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 ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
is mesokurtic for
.
Leptokurtic
A distribution with
positive excess kurtosis is called leptokurtic, or leptokurtotic. "Lepto-" means "slender". In terms of shape, a leptokurtic distribution has ''
fatter tails''. Examples of leptokurtic distributions include the
Student's t-distribution
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in sit ...
,
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 is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
,
Poisson distribution 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". In terms of shape, a platykurtic distribution has ''thinner tails''. Examples of platykurtic distributions include the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
and
discrete uniform distribution
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Anothe ...
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.
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 is given by
:
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'' > (''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''
2 = 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
, where
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
:
In the limit as
one obtains the density
:
which is shown as the red curve in the images on the right.
In the other direction as
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
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''
2 = 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
(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 way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
:
* 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 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 e^
The parameter \mu ...
, 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
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
, 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
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
, whereas the densities with positive or zero excess kurtosis are supported on the whole
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
.
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 distribution
The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To dis ...
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
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 , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
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 and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in sit ...
, with mixing probabilities 0.999 and 0.001.
Sample kurtosis
Definitions
A natural but biased estimator
For a
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
of ''n'' values, a
method of moments estimator of the population excess kurtosis can be defined as
:
where ''m''
4 is the fourth sample
moment about the mean
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
, ''m''
2 is the second sample moment about the mean (that is, the
sample variance), ''x''
''i'' is the ''i''
th value, and
is the
sample mean.
This formula has the simpler representation,
:
where the
values are the standardized data values using the standard deviation defined using ''n'' rather than ''n'' − 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
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 18.05 − 3 = 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.
Standard unbiased estimator
Given a sub-set of samples from a population, the sample excess kurtosis
above is a
biased estimator
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In st ...
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:
:
where ''k''
4 is the unique symmetric
unbiased estimator of the fourth
cumulant, ''k''
2 is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), ''m''
4 is the fourth sample moment about the mean, ''m''
2 is the second sample moment about the mean, ''x''
''i'' is the ''i''
th value, and
is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient
is the version found in
Excel
ExCeL London (an abbreviation for Exhibition Centre London) is an exhibition centre, international convention centre and former hospital in the Custom House area of Newham, East London. It is situated on a site on the northern quay of the ...
and several statistical packages including
Minitab,
SAS
SAS or Sas may refer to:
Arts, entertainment, and media
* ''SAS'' (novel series), a French book series by Gérard de Villiers
* ''Shimmer and Shine'', an American animated children's television series
* Southern All Stars, a Japanese rock ba ...
, and
SPSS.
[Doane DP, Seward LE (2011) J Stat Educ 19 (2)]
Unfortunately, in nonnormal samples
is itself generally biased.
Upper bound
An upper bound for the sample kurtosis of ''n'' (''n'' > 2) real numbers is
:
where
is the corresponding sample skewness.
Variance under normality
The variance of the sample kurtosis of a sample of size ''n'' from the
normal distribution
In 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 e^
The parameter \mu ...
is
:
Stated differently, under the assumption that the underlying random variable
is normally distributed, it can be shown that
.
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
In statistics, D'Agostino's ''K''2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to gauge the compatibility of given data with the null hypothesis that the data is a realizatio ...
is a
goodness-of-fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
normality test based on a combination of the sample skewness and sample kurtosis, as is the
Jarque–Bera test
In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera.
The test statistic is always nonnegativ ...
for normality.
For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see
variance.
Pearson's definition of kurtosis is used as an indicator of intermittency in
turbulence. 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
, variance
and kurtosis