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In
probability theory and
statistics, '''kurtosis''' is a measure of the peakedness of the
probability distribution of a
real-valued
random variable.
The fourth
standardized moment is defined as μ
4 / σ
4, where μ
4 is the fourth
moment about the mean and σ is the
standard deviation. This is sometimes used as the definition of kurtosis in older works, but is not the definition used here.
Kurtosis is more commonly defined as μ
4 / σ
4 − 3, which is also known as '''kurtosis excess'''. The minus 3 at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables. If ''Y'' is the sum of ''n''
independent random variables, all with the same distribution as ''X'', then Kurt[''Y''] = Kurt[''X''] / ''n'', while the formula would be more complicated if kurtosis were defined as μ
4 / σ
4.
A
normal distribution has a kurtosis of zero (distributions with zero kurtosis are called ''mesokurtic''). A distribution with positive kurtosis is called ''leptokurtic'', and one with negative kurtosis ''platykurtic''.
For a sample of ''n'' values the '''sample kurtosis''' is Σ
''i''(''x''
''i'' − μ)
4 / ''n''σ
4 − 3, where ''x''
''i'' is the ''i''
th value and μ is the
mean.
Given a sub-set of samples from a population, the sample kurtosis above is a
biased estimator of the population kurtosis. An
unbiased estimator of the population kurtosis is
:<math>
\mbox{Kurt} = \frac{n(n+1)}{(n-1)(n-2)(n-3)}
\sum_{i=1}^n \left( \frac{x_i - \bar{x}}{\sigma} \right)^4
- \frac{3(n-1)^2}{(n-2)(n-3)}
</math>
where σ is the sample standard deviation and μ is the sample mean.
==See also==
mean,
variance,
skewness.
== External links ==
*
Free Online Software (Calculator) computes various types of Skewness and Kurtosis statistics for any dataset (includes small and large sample tests).
de:Wlbung
