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Encyclopedia > Kurtosis
The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0.194 → 0.055)
The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0.194 → 0.055)

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Species T. aestivum T. boeoticum T. compactum T. dicoccoides T. dicoccon T. durum T. monococcum T. spelta T. sphaerococcum T. timopheevii References:   ITIS 42236 2002-09-22 For the indie rock group see: Wheat (band). ... Schematic image of wheat coleoptile (above) and flag leave (below) Coleoptile is the pointed protective sheath covering the emerging shoot in monocotyledons such as oats and grasses. ... It has been suggested that this article or section be merged with Probability axioms. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

Contents

Definition of kurtosis

The fourth standardized moment is defined as μ4 / σ4, where μ4 is the fourth moment about the mean and σ is the standard deviation. This is sometimes used as the definition of kurtosis in older works, but is not the definition used here. In probability theory and statistics, the kth standardized moment of a probability distribution is μk/σk, where μk is the kth moment about the mean and σ is the standard deviation. ... In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...


Kurtosis is more commonly defined as the fourth cumulant divided by the square of the variance of the probability distribution, // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

gamma_2 = frac{kappa_4}{kappa_2^2} = frac{mu_4}{sigma^4} - 3, !

which is known as excess kurtosis. The "minus 3" at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables. Because of the use of the cumulant, if Y is the sum of n independent random variables, all with the same distribution as X, then Kurt[Y] = Kurt[X] / n, while the formula would be more complicated if kurtosis were defined as μ4 / σ4.


More generally, if X1, ..., Xn are independent random variables all having the same variance, then

operatorname{Kurt}left(sum_{i=1}^n X_i right) = {1 over n^2} sum_{i=1}^n operatorname{Kurt}(X_i),

whereas this identity would not hold if the definition did not include the subtraction of 3.


Terminology and examples

A high kurtosis distribution has a sharper "peak" and fatter "tails", while a low kurtosis distribution has a more rounded peak with wider "shoulders".


Distributions with zero 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 is mesokurtic for p = 1/2 pm sqrt{1/12}. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ... The factual accuracy of this article is disputed. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ...


A distribution with positive kurtosis is called leptokurtic, or leptokurtotic. In terms of shape, a leptokurtic distribution has a more acute "peak" around the mean (that is, a higher probability than a normally distributed variable of values near the mean) and "fat tails" (that is, a higher probability than a normally distributed variable of extreme values). Examples of leptokurtic distributions include the Laplace distribution and the logistic distribution. Such distributions are sometimes termed "super Gaussian". A negative number is a number that is less than zero, such as −3. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... It has been suggested that this article or section be merged with Long-range dependency. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ...


A distribution with negative kurtosis is called platykurtic, or platykurtotic. In terms of shape, a platykurtic distribution has a smaller "peak" around the mean (that is, a lower probability than a normally distributed variable of values near the mean) and "thin tails" (that is, a lower probability than a normally distributed variable of extreme values). Examples of platykurtic distributions include the continuous or discrete uniform distributions, and the raised cosine distribution. The most platykurtic distribution of all is the Bernoulli distribution with p = ½ (for example the number of times one obtains "heads" when flipping a coin once), for which the kurtosis is -2. Such distributions are sometimes termed "sub Gaussian". The largest and the smallest element of a set are called extreme values, or extreme records. ... In mathematics, the uniform distributions are simple probability distributions. ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ...


Graphical examples

The Pearson type VII family

pdf for the Pearson type VII distribution with kurtosis of infinity (red); 2 (blue); and 0 (black)
pdf for the Pearson type VII distribution with kurtosis of infinity (red); 2 (blue); and 0 (black)
log-pdf for the Pearson type VII distribution with kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)
log-pdf for the Pearson type VII distribution with kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)

We illustrate the effects of kurtosis 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 Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1600 × 1200 pixel, file size: 234 KB, MIME type: image/png) // Probability density function of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the blue curve shows... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1600 × 1200 pixel, file size: 234 KB, MIME type: image/png) // Probability density function of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the blue curve shows... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1600 × 1200 pixel, file size: 282 KB, MIME type: image/png) Natural logarithm of the probability density of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1600 × 1200 pixel, file size: 282 KB, MIME type: image/png) Natural logarithm of the probability density of the Pearson type VII distribution The red curve shows the limiting density with infinite kurtosis; the... The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution. ... The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution. ...

f(x; a, m) = frac{Gamma(m)}{a,sqrt{pi},Gamma(m-1/2)} left[1+left(frac{x}{a}right)^2 right]^{-m}, !

where a is a scale parameter and m is a shape parameter. In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ...


All densities in this family are symmetric. The kth 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 a2 = 2m − 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 m = 5 / 2 + 3 / γ2, where γ2 is the kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary positive kurtosis. The reparameterized density is

g(x; gamma_2) = f(x;; a=sqrt{2+6/gamma_2},; m=5/2+3/gamma_2). !

In the limit as gamma_2 to infty one obtains the density

g(x) = 3 left(2 + x^2right)^{-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 the standard normal density as the limiting distribution, shown as the black curve. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ...


In the images on the right, the blue curve represents the density x mapsto g(x; 2) with kurtosis of 2. The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is an inverted parabola. One can see that the normal density allocates little probability mass to the regions far from the mean ("has thin tails"), compared with the blue curve of the leptokurtic Pearson type VII density with kurtosis of 2. Between the blue curve and the black are other Pearson type VII densities with γ2 = 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"). A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...


Kurtosis of well-known distributions

In this example we compare several well-known distributions from different parametric families. All densities considered here are unimodal and symmetric. Each has a mean and skewness of zero. Parameters were chosen to result in a variance of unity in each case. The images on the right show curves for the following seven densities, on a linear scale and logarithmic scale: Image File history File links Size of this preview: 800 × 571 pixel Image in higher resolution (2100 × 1500 pixel, file size: 280 KB, MIME type: image/png) Plot of several symmetric unimodal probability densities with unit variance From highest to lowest peak: red, kurtosis 3, Laplace (D)ouble exponential distribution... Image File history File links Size of this preview: 800 × 571 pixel Image in higher resolution (2100 × 1500 pixel, file size: 280 KB, MIME type: image/png) Plot of several symmetric unimodal probability densities with unit variance From highest to lowest peak: red, kurtosis 3, Laplace (D)ouble exponential distribution... Image File history File links Size of this preview: 800 × 571 pixel Image in higher resolution (2100 × 1500 pixel, file size: 350 KB, MIME type: image/png) Logscale plot of several symmetric unimodal probability densities with unit variance From highest to lowest peak: red, kurtosis 3, Laplace (D)ouble exponential... Image File history File links Size of this preview: 800 × 571 pixel Image in higher resolution (2100 × 1500 pixel, file size: 350 KB, MIME type: image/png) Logscale plot of several symmetric unimodal probability densities with unit variance From highest to lowest peak: red, kurtosis 3, Laplace (D)ouble exponential...

Note that in this case the platykurtic densities have bounded support, whereas the densities with nonnegative kurtosis are supported on the whole real line. In general there exist platykurtic densities with infinite support, for example exponential power distributions with sufficiently large shape parameter b. In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... In mathematics, the real line is simply the set of real numbers. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ...


Sample kurtosis

For a sample of n values the sample kurtosis is A sample is that part of a population which is actually observed. ...

g_2 = frac{m_4}{m_{2}^2} -3 = frac{n,sum_{i=1}^n (x_i - overline{x})^4}{left(sum_{i=1}^n (x_i - overline{x})^2right)^2} - 3

where m4 is the fourth sample moment about the mean, m2 is the second sample moment about the mean (that is, the sample variance), xi is the ith value, and overline{x} is the sample mean. In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ... This article is about mathematics. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ...


The formula

D = {1 over n} sum_{i=1}^n{ (x_i - bar{x})^2},
E = {1 over n D^2} sum_{i=1}^n{ (x_i - bar{x})^4} - 3

is also used, where n - the sample size, D - the pre-computed variance, xi - the value of the x'th measurement and bar{x} - the pre-computed arithmetic mean.


Estimators of population kurtosis

Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis. The usual estimator of the population kurtosis (used in SAS, SPSS, and Excel but not by MINITAB or BMDP) is G2, defined as follows: In statistics, a biased estimator is one that for some reason on average over_ or underestimates what is being estimated. ... The SAS System, originally Statistical Analysis System, is an integrated system of software products provided by SAS Institute that enables the programmer to perform: data entry, retrieval, management, and mining report writing and graphics statistical and mathematical analysis business planning, forecasting, and decision support operations research and project management quality... The computer program SPSS (originally, Statistical Package for the Social Sciences) was released in its first version in 1968, and is among the most widely used programs for statistical analysis in social science. ... This article or section does not adequately cite its references or sources. ... Minitab is a computer program designed to perform basic and advanced statistical functions. ... BMDP is a statistical package developed in 1961 at UCLA. Based on the older BIMED program for biomedical applications, it used keyword parameters in the input instead of fixed-format cards, so the letter P was added to the letters BMD, although the name was later defined as being an...

G_2 !!!! = frac{k_4}{k_{2}^2}!
= 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}!
= frac{n-1}{(n-2),(n-3)} left( (n+1),frac{m_4}{m_{2}^2} - 3,(n-1) right)!
= frac{n-1}{(n-2) (n-3)} left( (n+1),g_2 + 6 right)!
= 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})^2right)^2} - 3,frac{(n-1)^2}{(n-2),(n-3)}!
= 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)} !

where k4 is the unique symmetric unbiased estimator of the fourth cumulant, k2 is the unbiased estimator of the population variance, m4 is the fourth sample moment about the mean, m2 is the sample variance, xi is the ith value, and bar{x} is the sample mean. Unfortunately, G2 is itself generally biased. For the normal distribution it is unbiased because its expected value is then zero. // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...


See also

Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... Skewness denotes that observations are not spread symmetrically around an average value. ... There are very few or no other articles that link to this one. ...

References

  • Joanes, D. N. & Gill, C. A. (1998) Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society (Series D): The Statistician 47 (1), 183–189. doi:10.1111/1467-9884.00122

The Royal Statistical Society is a learned society for statistics and a professional body for statisticians in the UK. Founded in 1834 as the Statistical Society of London, it became the Royal Statistical Society in 1887. ...

External links


  Results from FactBites:
 
Kurtosis - Wikipedia, the free encyclopedia (563 words)
In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable.
This is because the kurtosis as we have defined it is the ratio of the fourth cumulant and the square of the second cumulant of the probability distribution.
Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis.
  More results at FactBites »

 
 

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