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

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

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