Kurtosis

Definition

Kurtosis is a measure of the "tailedness" of a probability distribution, describing the heaviness of the tails relative to a normal distribution. A mesokurtic distribution has kurtosis similar to the normal curve. A leptokurtic distribution has heavier tails and a sharper peak, indicating more extreme values than expected under normality. A platykurtic distribution has lighter tails and a flatter peak.

Why It Matters

Kurtosis alerts researchers to the presence of outliers and the risk of extreme events. In finance, leptokurtic return distributions imply that large market movements are more likely than normal-distribution models predict. In quality control, high kurtosis may signal intermittent process failures. Together with skewness, kurtosis is a standard diagnostic for assessing whether data meet the normality assumptions of parametric tests.

Example

A stock analyst models daily returns assuming a normal distribution. However, the empirical kurtosis is 6, far above the normal value of 3 (or 0, depending on the definition used). This leptokurtic pattern reveals that extreme daily gains and losses occur much more frequently than the normal model predicts. The analyst therefore switches to a GARCH model, which explicitly accounts for fat tails and volatility clustering.

Related Terms

• Normal Distribution
• Skewness
• Standard Deviation
• Outlier

Software Notes

• SPSS: Analyze > Descriptive Statistics > Frequencies or Descriptives. Check "Kurtosis" under Statistics. SPSS uses the excess kurtosis definition (subtracting 3), so a normal distribution yields 0.
• R: moments::kurtosis(x) returns the raw kurtosis (3 for normal). e1071::kurtosis(x) returns excess kurtosis by default (0 for normal).
• Stata: summarize varname, detail reports kurtosis in the "Kurtosis" row. Stata uses excess kurtosis (0 for normal).