Kurtosis

In probability theory and statistics, kurtosis (from the Greek word kurtos, meaning bulging) 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.

Definition of kurtosis
The fourth standardized moment is defined as
 * $$ \frac{\mu_4}{\sigma^4},\! $$

where &mu;4 is the fourth moment about the mean and &sigma; is the standard deviation. This is sometimes used as the definition of kurtosis in older works, but is not the definition used here.

Kurtosis is more commonly defined as the fourth cumulant divided by the square of the variance of the probability distribution,


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

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 &mu;4 / &sigma;4.

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.

The fourth standardized moment must be at least 1, so the excess kurtosis must be &minus;2 or more; there is no upper limit and it may be infinite.

Terminology and examples
A high kurtosis distribution has a sharper "peak" and flatter "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 $$p = 1/2 \pm \sqrt{1/12}$$.

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 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 Pearson type VII family


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


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

where a is a scale parameter and m is a shape parameter.

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 $$a^2 = 2m-3$$ 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 $$m = 5/2 + 3/\gamma_2$$, where $$\gamma_2$$ 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


 * $$g(x) = 3 \left(2 + x^2\right)^{-5/2}, \!$$

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

In the other direction as $$\gamma_2 \to 0$$ one obtains 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 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 &gamma;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 $$\gamma_2 = \infty$$ (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").

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:


 * D: Laplace distribution, a.k.a. double exponential distribution, red curve (two straight lines in the log-scale plot), kurtosis = 3


 * S: hyperbolic secant distribution, orange curve, kurtosis = 2


 * L: logistic distribution, green curve, kurtosis = 1.2


 * N: normal distribution, black curve (inverted parabola in the log-scale plot), kurtosis = 0
 * C: raised cosine distribution, cyan curve, kurtosis = −0.593762…


 * W: Wigner semicircle distribution, blue curve, kurtosis = −1


 * U: uniform distribution, magenta curve (shown for clarity as a rectangle in both images), kurtosis = −1.2

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, and there exist leptokurtic densities with finite support, for example 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.

Sample kurtosis
For a sample of n values the sample kurtosis is


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

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.

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, Minitab, SPSS, and Excel but not by BMDP) is G2, defined as follows: 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, $$G_2$$ is itself generally biased. For the normal distribution it is unbiased because its expected value is then zero.