a normalized fourth moment of a distribution or dataset, or other aspects of fat tails
Kurtosis refers to the fat-tailed-ness of a distribution. It is often defined as a normalized fourth central moment $\mu_4$ of a distribution or dataset. It can be any non-negative real number or even (for distributions) infinite.
There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted $\beta_2$ (Abramowitz and Stegun 1972, p. 928) or $\alpha_4$ (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1961, pp. 99-102) and defined by:
$$\beta_2 = \frac{\mu_4}{\mu_2^2}$$
where $\mu_i$ denotes the $i$th central moment (and in particular, $\mu_2$ is the variance).
Note that kurtosis does not measure the "peakedness" of a distribution (Westfall, 2014), as is commonly believed.
Sometimes "kurtosis" refers to the excess kurtosis, defined as $\beta_2 - 3$. This is the amount by which the kurtosis differs from that of any Normal distribution.
An alternative measure of fat-tailed-ness is the L-kurtosis:
$$\frac{EX_{4:4}-3EX_{3:4}+3EX_{2:4}-EX_{1:4}}{2(EX_{2:2}-EX_{1:2})}$$
where, e.g., $EX_{2:4}$ is the expectation for the second-smallest among four draws from the distribution.
Reference: mathworld.wolfram.com
Excerpt reference: statistics.about.com
Westfall, P. H. (2014). Kurtosis as Peakedness, 1905–2014. R.I.P. The American Statistician, 68(3):191-195, DOI: 10.1080/00031305.2014.917055
Reference for L-kurtosis: Wikipedia on L-moments and their ratios.
