Curtosis

Curtosis, more commonly spelled kurtosis in statistics, is a measure of the shape of a distribution’s tails and peak relative to a normal distribution. It is defined as the fourth standardized moment: kurtosis = E[(X − μ)4] / σ4, where μ is the mean and σ is the standard deviation of the variable X. In practice, the concept is often discussed using excess kurtosis, which is kurtosis minus 3. A normal distribution has an excess kurtosis of 0. Distributions are described as leptokurtic if they have a high peak and heavy tails (excess kurtosis > 0), mesokurtic if they resemble the normal distribution (excess kurtosis ≈ 0), or platykurtic if they have a flatter peak and lighter tails (excess kurtosis < 0).

The term curtosis is a less common variant spelling found in some texts or translations; the standard English term is kurtosis.

Estimation from data involves computing the sample kurtosis, which approximates the population kurtosis. Several formulas exist, and small-sample bias can affect estimates, especially for excess kurtosis. In practice, kurtosis is sensitive to outliers because the fourth power dramatically amplifies extreme values.

Applications of kurtosis include assessing the propensity for extreme events in a dataset, comparing distributions in finance and risk management, quality control, and environmental studies. It complements skewness, which measures asymmetry, by providing information about tail heaviness and the likelihood of outliers relative to a normal baseline.