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 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 μ is the sample mean.
See also: mean, variance, kurtosis, cumulant.
Referenced By
Frequency distribution | List of mathematical topics (S-U) | List of probability topics | List of statistical topics | ProbabilityApplications | Probability Applications | Skew | Standard score
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