In probability theory and statistics, Student's t distribution (or simply the t distribution) ${\displaystyle t_{\nu }}$ is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

However, ${\displaystyle t_{\nu }}$ has heavier tails, and the amount of probability mass in the tails is controlled by the parameter ${\displaystyle \nu }$. For ${\displaystyle \nu =1}$ the Student's t distribution ${\displaystyle t_{\nu }}$ becomes the standard Cauchy distribution, which has very "fat" tails; whereas for ${\displaystyle \nu \to \infty }$ it becomes the standard normal distribution ${\displaystyle {\mathcal {N}}(0,1),}$ which has very "thin" tails.

The name "Student" is a pseudonym used by William Sealy Gosset in his scientific paper publications during his work at the Guinness Brewery in Dublin, Ireland.

The Student's t distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.

In the form of the location-scale t distribution ${\displaystyle \operatorname {\ell st} (\mu ,\tau ^{2},\nu )}$ it generalizes the normal distribution and also arises in the Bayesian analysis of data from a normal family as a compound distribution when marginalizing over the variance parameter.

Student's t distribution has the probability density function (PDF) given by $${\displaystyle f(t)={\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2},}$$ where ${\displaystyle \nu }$ is the number of degrees of freedom, and ${\displaystyle \Gamma }$ is the gamma function. This may also be written as $${\displaystyle f(t)={\frac {1}{{\sqrt {\nu }}\,\mathrm {B} {\left({\frac {1}{2}},{\frac {\nu }{2}}\right)}}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2},}$$ where ${\displaystyle \mathrm {B} }$ is the beta function. In particular for integer valued degrees of freedom ${\displaystyle \nu }$ we have:

For ${\displaystyle \nu >1}$ and even, $${\displaystyle {\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}={\frac {1}{2{\sqrt {\nu }}}}\cdot {\frac {(\nu -1)\cdot (\nu -3)\cdots 5\cdot 3}{(\nu -2)\cdot (\nu -4)\cdots 4\cdot 2}}.}$$

For ${\displaystyle \nu >1}$ and odd, $${\displaystyle {\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}={\frac {1}{\pi {\sqrt {\nu }}}}\cdot {\frac {(\nu -1)\cdot (\nu -3)\cdots 4\cdot 2}{(\nu -2)\cdot (\nu -4)\cdots 5\cdot 3}}.}$$