List of formulas for skewness and kurtosis of empirical distribution:
1. Asymmetry (Skewness)
Skewness measures the degree and direction of a distribution’s deviation from symmetry. The formula for calculating skewness is:
[
\text{Асимметрия} (g_1) = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left( \frac{x_i – \bar{x}}{s} \right)^3
]
Where:
- ( n ) — number of observations,
- ( x_i ) — each value,
- ( \bar{x} ) is the mean value of the sample,
- ( s ) — standard deviation of the sample.
2. Kurtosis
Kurtosis measures the “peakiness” of a distribution. The formula for calculating kurtosis is:
[
\text{Эксцесс} (g_2) = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n} \left( \frac{x_i – \bar{x}}{s} \right)^4 – \frac{3(n-1)^2}{(n-2)(n-3)}
]
where the same designations as above.
3. Notes
- A skewness of 0 indicates a symmetric distribution.
- Positive asymmetry indicates a “long tail” to the right, while negative asymmetry indicates a “long tail” to the left.
- A kurtosis of 0 corresponds to a normal distribution. A positive kurtosis indicates a “higher” peak, while a negative kurtosis indicates a “flatter” peak.