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 (right-skewed) if the higher tail is longer and negative skew (left-skewed) if the lower tail is longer; getting these the wrong way round is a common error.
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
where σ is the sample standard deviation and 
is the sample mean.
Pearson Skewness Coefficients
Karl Pearson suggested two simpler calculations as a measure of skewness:
though there is no guarantee that these will be the same sign as each other or as the ordinary definition of skewness.
See also
External links
Last updated: 10-12-2005 21:02:46
