Skewness and kurtosis

In modern data analysis, some high-order moments usually become useful, such as skewness and kurtosis.

• Skewness:
  • Measures the asymmetry of the distribution.
  • $skewness = \mathbb{E}\left[\left( \dfrac{X-\mu}{\sigma} \right)^3\right] =: \gamma$.
  • Gausian has skewness 0, that is, the distribution is symmetric.
• Kurtosis:
  • Measure how heavy-tailed the distribution is.
  • $kurtosis = \mathbb{E}\left[\left( \dfrac{X-\mu}{\sigma} \right)^4\right] = \mathcal{k}$
  • Gaussian has kurtosis 3.
• Empirical approximations

  • $\gamma \approx \dfrac{1}{N} \displaystyle{ \sum_{n=1}^{N} \left( \dfrac{X_n-\mu}{\sigma} \right)^3}$

  • $\mathcal{k} \approx \dfrac{1}{N} \displaystyle{ \sum_{n=1}^{N} \left( \dfrac{X_n-\mu}{\sigma} \right)^4}$