Information About

Skewed Distribution
  CATEGORIES ABOUT SKEWNESS
   
probability theory
   
statistical deviation and dispersion
   
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Skewness, the third Standardized Moment , is written as \gamma_1 and defined as

:\gamma_1 = rac{\mu_3}{\sigma^3}, \!

where \mu_3 is the third Moment About The Mean and \sigma is the Standard Deviation . Equivalently, skewness can be defined as the ratio of the third Cumulant \kappa_3 and the third power of the square root of the second cumulant \kappa_2:

:\gamma_1 = rac{\kappa_3}{\kappa_2^{3/2}}. \!

This is analogous to the definition of Kurtosis , which is expressed as the fourth cumulant divided by the fourth power of the square root of the second cumulant.

For a sample of ''N'' values the ''sample skewness'' is

:g_1 = rac{m_3}{m_2^{3/2}} = rac{\sqrt{n\,}\sum_{i=1}^N (x_i-\bar{x})^3}{\left(\sum_{i=1}^N (x_i-\bar{x})^2 ight)^{3/2}}, \!

where x_i is the ''i''th value, \bar{x} is the Sample Mean , m_3 is the sample third Central Moment , and m_2 is the Sample Variance .

Given samples from a population, the equation for the sample skewness g_1 above is a Biased Estimator of the population skewness. The usual estimator of skewness is

:G_1 = rac{k_3}{k_2^{3/2}}
= rac{\sqrt{n\,(n-1)}}{n-2}\; g_1, \!

where k_3 is the unique symmetric unbiased estimator of the third cumulant and k_2 is the symmetric unbiased estimator of the second cumulant. Unfortunately G_1 is, nevertheless, generally biased. Its expected value can even have the opposite sign from the true skewness.

The skewness of a random variable ''X'' is sometimes denoted Skew If ''Y'' is the sum of ''n'' Independent random variables, all with the same distribution as ''X'', then it can be shown that Skew[''Y'' = Skew[''X''] / √''n''.

Skewness affects Mean the most and Mode the least. For a positivevely skewed distribution, Mean > Median > Mode and for a negatively skewed distribution, Mean < Median < Mode

Skewness has benefits in many areas. Many simplistic models assume normal distribution i.e. data is symmetric about the mean. But in reality, data points are not perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.

Section to develop: Why should we care about skew? what difference does it make!


PEARSON SKEWNESS COEFFICIENTS


Karl Pearson suggested two simpler calculations as a measure of skewness:



SEE ALSO




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