The non-central t distribution

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Distribution name

(Standard) non-central t distribution (NCT)

Common notation


 = degrees of freedom (, usually  is an integer although in some situations a non-integral  can arise)

 = non-centrality parameter


Probability density function

where  is the NCT cumulative distribution function

Cumulative distribution function




Other comments

 has a standard non-central  distribution with  degrees of freedom and non-centrality parameter  if , ,  and  and  are independent. It has the following non-central moments:


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “non-central t”. 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 “non-central t4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.


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