What do skewness and kurtosis measure, and what are their practical implications?
Skewness measures the asymmetry of a distribution's tails — positive skew means a longer right tail, negative skew a longer left tail. Kurtosis measures the heaviness of the tails relative to a normal distribution; excess kurtosis above zero indicates more probability mass in the tails and peak than a Gaussian, which matters for risk and outlier frequency.
How to think about it
Give the definitions, then the practical implications in your domain — interviewers in finance or risk especially care about kurtosis because it determines tail event frequency.
Skewness
Skewness is the standardised third central moment:
skewness = E[ ((X - mu) / sigma)^3 ]
- Positive skew (right-skewed): Long right tail. Mean > Median > Mode. Example: income distributions, insurance claim sizes.
- Negative skew (left-skewed): Long left tail. Mean < Median < Mode. Example: exam scores when most students score high with a few failures.
- Zero skew: Symmetric distribution (normal is one example, but symmetric does not imply normal).
The cube preserves sign, so it captures direction of asymmetry, not just magnitude.
Kurtosis
Kurtosis is the standardised fourth central moment:
kurtosis = E[ ((X - mu) / sigma)^4 ]
Excess kurtosis subtracts 3 (the normal distribution’s kurtosis) to allow direct comparison:
excess kurtosis = kurtosis - 3
| Excess kurtosis | Term | Tail behaviour |
|---|---|---|
| = 0 | Mesokurtic | Normal-like tails |
| > 0 | Leptokurtic | Heavy tails, sharp peak |
| < 0 | Platykurtic | Thin tails, flat top |
Financial return series are almost always leptokurtic (excess kurtosis 3-10), meaning extreme market moves are far more frequent than a Gaussian model predicts.
Why it matters in practice
- Model risk: A Gaussian assumption for leptokurtic data underestimates the probability of tail events by orders of magnitude.
- Feature engineering: Highly skewed features benefit from log or Box-Cox transforms before feeding into linear models or distance-based algorithms.
- Quality checks: Sudden increases in skew or kurtosis in production data often flag distribution shift or sensor anomalies.