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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 μ
3 / σ
3, where μ
3 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 Σ
''i''(''x''
''i'' − μ)
3 / ''N''σ
3, where ''x''
''i'' is the ''i''
th 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 - \mu}{\sigma} \right)^3
</math>
where σ is the sample standard deviation and μ is the sample mean.
See also:
mean,
variance,
kurtosis,
cumulant.
de:Schiefe
== External links ==
*
Free Online Software (Calculator) computes various types of Skewness and Kurtosis statistics for any dataset (includes small and large sample tests).
