Skewness and Kurtosis: Shape of Distribution

Third and Fourth Order: Shape of Distribution

Skewness (Third Moment)

\[\text{Skewness} = \frac{1}{N \sigma_x^3} \sum_{n=0}^{N-1} (x[n] - \mu_x)^3\]

Skewness	Distribution Shape	Example
0	Symmetric (Gaussian)	Thermal noise
> 0	Right-skewed (tail to +∞)	Speech envelopes, seismic impulses
< 0	Left-skewed (tail to -∞)	Clipped signals

Kurtosis (Fourth Moment)

\[\text{Kurtosis} = \frac{1}{N \sigma_x^4} \sum_{n=0}^{N-1} (x[n] - \mu_x)^4\]

Kurtosis	Distribution Shape	Example
3	Gaussian (mesokurtic)	Thermal noise
> 3	Heavy tails, sharp peak (leptokurtic)	Speech, impulsive noise, radar clutter
< 3	Light tails, flat peak (platykurtic)	Uniform noise, oversampled signals

Excess Kurtosis = Kurtosis - 3 (Gaussian = 0)

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