Friday, January 29, 2010

Skewness

Skewness is a measure of the degree of asymmetry of a distribution. If the left tail (tail at small end of the distribution) is more pronounced than the right tail (tail at the large end of the distribution), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness.

Several types of skewness are defined, the terminology and notation of which are unfortunately rather confusing. "The" skewness of a distribution is defined to be

 gamma_1=(mu_3)/(mu_2^(3/2)),
(1)
Positively Skewed Distribution: - The value of the arithmetic mean is greater than the mode; then the distribution is called Positively Skewed.

Nagatively Skewed Distribution: - If the value of the mode is greater than the arithmetic mean; the distribution is called Negatively Skewed.
http://upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Skewness_Statistics.svg/446px-

Several forms of skewness are also defined. The momental skewness is defined by

 alpha^((m))=1/2gamma_1.
(2)

The Pearson mode skewness is defined by

 ((mean-mode))/sigma.
(3)

Pearson's skewness coefficients are defined by

 (3(mean-mode))/sigma
(4)

and

 (3(mean-median))/sigma.
(5)

The Bowley skewness (also known as quartile skewness coefficient) is defined by

 ((Q_3-Q_2)-(Q_2-Q_1))/(Q_3-Q_1)=(Q_1-2Q_2+Q_3)/(Q_3-Q_1),
(6)

where the Qs denote the interquartile ranges. The momental skewness is

 alpha^((m))=1/2gamma=(mu_3)/(2mu^(3/2)).

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