Encyclopedia > Skewness

  Article Content

Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew if the if the positive tail is longer and negative skew if the negative tail is longer.

Skewness, the third standardized moment, is defined as μ3 / σ3, where μ3 is the third moment about the mean and σ is the standard deviation. The skewness of a random variable X is sometimes denoted Skew[X].

For a set of N values the skewness can be calculated as Σi(xi - μ)3 / Nσ3, where xi is the ith value and μ is the mean.

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.

Given samples from a population, the equation for population skewness above is a biased estimator of the population skewness. An unbiased estimator of skewness is

<math> \mbox{Skew} = \frac{n}{(n-1)(n-2)}
\sum_{i=1}^N \left( \frac{x_i - \bar{x}}{\sigma} \right)^3 </math>

where σ is the sample standard deviation and μ is the sample mean.

See also: mean, variance, kurtosis.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Conscience

... be based over these three years. What was registered in the unconsciousness will not be accessible anymore, but it will be always registered in itself. It will fit ...