Kurtosis measures the propensity of a probability distribution to produce extreme outliers in its tails.
It evaluates whether the tails of a distribution contain more or fewer observations than a standard normal distribution.
Mathematically, kurtosis is derived from the fourth standardised moment of the distribution around its mean.
Raising deviations to the fourth power heavily penalises values situated far from the distribution centre.
A standard normal distribution has a kurtosis value of exactly three, which serves as the universal baseline.
Excess kurtosis subtracts three from the raw kurtosis, setting the baseline for a normal curve at zero.
Distributions with positive excess kurtosis are called leptokurtic, characterised by heavy tails that hold more probability than the normal curve.
Leptokurtic data produces extreme outliers far more frequently than would be expected under a normal distribution.
Financial asset returns are notoriously leptokurtic, exhibiting sudden black-swan crashes that normal models fail to anticipate.
Distributions with negative excess kurtosis are called platykurtic, characterised by thin tails and fewer extreme values.
A uniform distribution is platykurtic because its flat profile generates no outliers beyond its bounded limits.
Distributions with zero excess kurtosis are called mesokurtic, matching the tail behaviour of the normal curve.
A widespread misconception is that kurtosis measures peak sharpness, whereas it primarily reflects tail probability weight.
Risk managers rely on kurtosis to parameterise value-at-risk models and prepare institutions for market turbulence.
High kurtosis warns data scientists that normal assumptions will underestimate the probability of extreme operational failures.