The Student’s t distribution

[this page | pdf | back links]

The Student’s t distribution (more simply the t distribution) arises when estimating the mean of a normally distributed population when sample sizes are small and the population standard deviation is unknown.


[SmartChart] [SmartChart] [SmartChart]


Distribution name

(Standard) Student’s t distribution

Common notation


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


Probability density function

Cumulative distribution function





(Excess) kurtosis

Characteristic function

where  is a Bessel function

Other comments

It is a special case of the generalised hyperbolic distribution.


Its non-central moments if  is even and  are:


If  is even and  then , if  is odd and  then  and if  is odd and   then  is undefined.


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “student's 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 “student's t3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.


Contents | Prev | Next

Desktop view | Switch to Mobile