|
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 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 sample of N values the sample skewness is 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{\sqrt{n(n-1)}}{(n-2)}
\left(\frac{\sum_{i=1}^n \left( x_i - \bar{x} \right)^3}{n\sigma^3}\right)
<math>
where <math>\sigma<math> is the sample standard deviation and <math>\bar{x}<math> is the sample mean.
See also
External links
- Free Online Software (Calculator) (http://www.wessa.net/skewkurt.wasp) computes various types of Skewness and Kurtosis statistics for any dataset (includes small and large sample tests).
| 
