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.

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