Kurtosis

Kurtosis is a standardized fourth central moment used to describe a probability distribution’s tail weight. It is often reported as excess kurtosis, which subtracts three so the normal distribution has value zero.

Central moment: A moment is an expected power of a random variable; a central moment uses deviations from the mean. Kurtosis is the fourth central moment divided by the fourth power of the standard deviation.
Skewness: Skewness is a standardized third central moment that measures distributional asymmetry. Skewness describes asymmetry, while kurtosis concerns the standardized fourth moment and tail weight.
Sample kurtosis: Sample kurtosis is a statistic computed from observed data to estimate a population’s kurtosis. Finite-sample formulas adjust the fourth standardized sample moment in different ways.
Moment problem: The moment problem asks whether a sequence of moments uniquely determines a probability distribution. Different distributions can share kurtosis, so this one moment cannot determine the full shape.
Standard deviation: The standard deviation is the square root of a distribution’s variance, expressing spread in the variable’s units. Its fourth power standardizes the fourth central moment and makes kurtosis dimensionless.
Variance: Variance is the expected squared deviation of a random variable from its mean. Variance measures overall squared spread; kurtosis measures a standardized fourth-power feature.
Jarque–Bera test: The Jarque–Bera test assesses normality using sample skewness and excess kurtosis. Kurtosis supplies one of the two shape components in its test statistic.
Heavy-tailed distribution: A heavy-tailed distribution has tail probabilities that decay more slowly than those of an exponential distribution. Some heavy-tailed distributions lack a finite fourth moment, leaving conventional kurtosis undefined.
Excess kurtosis: Excess kurtosis is kurtosis minus three, making the normal distribution’s value zero. This convention shifts the scale without changing comparisons between distributions.
Tail risk: Tail risk is the possibility and potential severity of outcomes in the extreme ends of a probability distribution. Kurtosis is sometimes used as a summary of tail weight, but it does not specify tail risk completely.

Linked from 2 pages