Kurtosis

In probability theorystatistics, kurtosis ismeasure ofpeakedness ofprobability distribution ofreal-valued random variable.

The fourth standardized momentdefined as μ₄ / σ⁴, where μ₄ isfourth moment aboutmeanσ isstandard deviation. Thissometimes used asdefinitionkurtosisolder works, butnotdefinition used here.

Kurtosismore commonly defined as μ₄ / σ⁴ − 3. The minus 3 atendthis formulaoften explained ascorrectionmakekurtosis ofnormal distribution equalzero. Another reason can be seen by looking atformula forkurtosis ofsumrandom variables. If Y issumn independent random variables, all withsame distribution as X, then Kurt[Y] = Kurt[X] / n, whileformula would be more complicated if kurtosis were defined as μ₄ / σ⁴.

A normal distribution haskurtosiszero (distributionszero kurtosiscalled mesokurtic). A distributionpositive kurtosiscalled leptokurtic,onenegative kurtosis platykurtic.

ForsampleN valuessample kurtosis canΣ_i(x_i  −  μ)⁴ / Nσ⁴ − 3, where x_i isi^th valueμ ismean.

Givensub-setsamples frompopulation,sample kurtosis above isbiased estimator ofpopulation kurtosis. An unbiased estimator ofpopulation kurtosis is

where σ issample standard deviationμ issample mean.

See also: mean, variance, skewness.