Skewness is computed as the average cubed deviation from the mean standardized by dividing the standard deviation cubed to make the measure free of scale. The skewness is positive for positively skewed distribution, negative for negatively skewed distribution, and zero for symmetrical distribution.
Sample skewness = Sk = [n/{(n-1)(n-2)}]∑(Xi-X-bar)3/s3 ≈ (1/n)∑(Xi-X-bar)3/s3 (for large n)
Some researchers believe that investors should prefer positive skewness and should avoid negative skewness. For a sample size of 100 or larger, a skewness coefficient of ±0.5 is considered unusually large.
Kurtosis is a statistical measure that measures the peakedness of distribution. The normal distribution has a kurtosis of three. A distribution that is more peaked than a normal distribution is called leptokurtic. A distribution that is less peaked than a normal distribution is called platykurtic.
The excess kurtosis is the kurtosis of a sample minus the kurtosis of a normal distribution or the kurtosis of a sample minus three. A leptokurtic distribution has a kurtosis greater than three and thus has a positive excess kurtosis. A platykurtic distribution has a kurtosis lesser than three and thus has a negative excess kurtosis. A mesokurtic distribution is identical to the normal distribution in peak and has a kurtosis of three and an excess kurtosis of zero.
A leptokurtic distribution has higher kurtosis and fatter tails. Therefore, it is considered riskier. Most of the returns from equity investments and hedge funds are leptokurtic in distribution.
The kurtosis of a sample is calculated by finding the average of deviations from the mean raised to the fourth power and then standardizing that average by dividing by the standard deviation raised to the fourth power.
Sample excess kurtosis = Sample kurtosis - (3(n-1)2)/[(n-2)(n-3)] = Sample kurtosis - 3 (for large n)
Sample kurtosis = [(n(n-1))/{((n-1)(n-2)(n-3)}]∑(Xi-X-bar)4/s4 ≈ (1/n)∑(Xi-X-bar)3/s3 (for large n)
For a sample size of 100 or larger, an excess kurtosis of greater than one is considered unusually high.
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Deviation from mean raised to the power
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Mean absolute deviation
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1
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Variance
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2
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Skewness
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3
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Kurtosis
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4
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