Kurtosis
What is kurtosis? Provide the definition and intuitive explanation.
Answer
\[
\text{Kurtosis}(X) \;=\; \frac{\mathbb{E}\big[(X - \mu)^4\big]}{\sigma^4}
\]
Kurtosis measures the tailedness of a distribution, usually compared to a normal distribution.
Often one uses excess kurtosis (normal distribution has a kurtosis of 3.):
\[
\text{Excess Kurtosis}(X) = \text{Kurtosis}(X) - 3
\]
In general (not always) we have:
Excess kurtosis \(> 0\): heavy tails / more extreme outliers (leptokurtic).
Excess kurtosis \(= 0\): tails similar to normal (mesokurtic).
Excess kurtosis \(< 0\): lighter tails; fewer extreme outliers (platykurtic).
Notes and comments
Comment 1: High kurtosis does not necessarily mean a very “sharp peak”and it is not related to "peakdness".