The Chi-squared distribution
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The chi-squared distribution with 
degrees
of freedom is the distribution of a sum of the squares of independent
standard normal random variables. A consequence is that the sum of independent
chi-squared variables is also chi-squared distributed. It is widely used in
hypothesis testing, goodness of fit analysis or in constructing confidence
intervals. It is a special case of the gamma distribution.
![[SmartChart]](I/ChiSquaredDistribution_files/image002.gif)
![[SmartChart]](I/ChiSquaredDistribution_files/image003.gif)
![[SmartChart]](I/ChiSquaredDistribution_files/image004.gif)
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Distribution name
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Chi-squared
distribution
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Common notation
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Parameters
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=
degrees of freedom (positive integer)
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Domain
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Probability density
function
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Cumulative distribution
function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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Its median is approximately  . Its mode is  . Is also known as the
central chi-squared distribution (when there is a need to contrast it with
the noncentral chi-squared distribution).
In the special case of  the cumulative
distribution function simplifies to  .
As  ,  and 
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Nematrian web functions
Functions relating to the above distribution may be accessed
via the Nematrian
web function library by using a DistributionName of “chi-squared”.
Functions relating to a generalised version of this distribution including
additional location (i.e. shift) and scale parameters may be accessed by using
a DistributionName of “chi-squared3” ”, see also including
additional shift and scale parameters. For details of other supported
probability distributions see here.
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