In probability theory and statistics, kurtosis (from the Greek word κυρτός, kyrtos or kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a realvalued random variable, although some sources are insistent that heavy tails, and not peakedness, is what is really being measured by kurtosis^{[1]}. Higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations.
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Definition
The fourth standardized moment is defined as
where μ_{4} 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.
Kurtosis is more commonly defined as the fourth cumulant divided by the square of the second cumulant, which is equal to the fourth moment around the mean divided by the square of the variance of the probability distribution minus 3,
which is also 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 μ_{4} / σ^{4}.
More generally, if X_{1}, ..., X_{n} are independent random variables all having the same variance, then
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 −2 or more. This lower bound is realized by the Bernoulli distribution with p = ½, or "coin toss". There is no upper limit to the excess kurtosis and it may be infinite.
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