Student's t-distribution

Student's t-distribution has the probability density function (PDF) given by

${\displaystyle f(t)={\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2},}$

where ${\displaystyle \nu }$ is the number of degrees of freedom and ${\displaystyle \Gamma }$ is the gamma function. This may also be written as

${\displaystyle f(t)={\frac {1}{{\sqrt {\nu }}\,\mathrm {B} ({\frac {1}{2}},{\frac {\nu }{2}})}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2},}$

where B is the Beta function. In particular for integer valued degrees of freedom ${\displaystyle \nu }$ we have:

For ${\displaystyle \nu >1}$ even,

${\displaystyle {\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}={\frac {(\nu -1)(\nu -3)\cdots 5\cdot 3}{2{\sqrt {\nu }}(\nu -2)(\nu -4)\cdots 4\cdot 2\,}}\cdot }$

For ${\displaystyle \nu >1}$ odd,

${\displaystyle {\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\nu \pi }}\,\Gamma ({\frac {\nu }{2}})}}={\frac {(\nu -1)(\nu -3)\cdots 4\cdot 2}{\pi {\sqrt {\nu }}(\nu -2)(\nu -4)\cdots 5\cdot 3\,}}\cdot \!}$

The probability density function is symmetric, and its overall shape resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1. For this reason ${\displaystyle {\nu }}$ is also known as the normality parameter.[14]

The following images show the density of the t-distribution for increasing values of ${\displaystyle \nu }$. The normal distribution is shown as a blue line for comparison. Note that the t-distribution (red line) becomes closer to the normal distribution as ${\displaystyle \nu }$ increases.

Density of the t-distribution (red) for 1, 2, 3, 5, 10, and 30 degrees of freedom compared to the standard normal distribution (blue).
Previous plots shown in green.

1 degree of freedom

2 degrees of freedom

3 degrees of freedom

5 degrees of freedom

10 degrees of freedom

30 degrees of freedom

The cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function. For t > 0,

${\displaystyle F(t)=\int _{-\infty }^{t}f(u)\,du=1-{\tfrac {1}{2}}I_{x(t)}\left({\tfrac {\nu }{2}},{\tfrac {1}{2}}\right),}$

where

${\displaystyle x(t)={\frac {\nu }{t^{2}+\nu }}.}$

Other values would be obtained by symmetry. An alternative formula, valid for ${\displaystyle t^{2}<\nu }$, is

${\displaystyle \int _{-\infty }^{t}f(u)\,du={\tfrac {1}{2}}+t{\frac {\Gamma \left({\tfrac {1}{2}}(\nu +1)\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\tfrac {\nu }{2}}\right)}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}}(\nu +1);{\tfrac {3}{2}};-{\tfrac {t^{2}}{\nu }}\right),}$

where ₂F₁ is a particular case of the hypergeometric function.

For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution.

Certain values of ${\displaystyle \nu }$ give a simple form for Student's t-distribution.

${\displaystyle \nu }$	PDF	CDF	notes
1	${\displaystyle {\frac {1}{\pi (1+t^{2})}}}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\arctan(t)}$	See Cauchy distribution
2	${\displaystyle {\frac {1}{2{\sqrt {2}}\left(1+{\frac {t^{2}}{2}}\right)^{3/2}}}}$	${\displaystyle {\frac {1}{2}}+{\frac {t}{2{\sqrt {2}}{\sqrt {1+{\frac {t^{2}}{2}}}}}}}$
3	${\displaystyle {\frac {2}{\pi {\sqrt {3}}\left(1+{\frac {t^{2}}{3}}\right)^{2}}}}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}{\left[{\frac {1}{\sqrt {3}}}{\frac {t}{1+{\frac {t^{2}}{3}}}}+\arctan \left({\frac {t}{\sqrt {3}}}\right)\right]}}$
4	${\displaystyle {\frac {3}{8\left(1+{\frac {t^{2}}{4}}\right)^{5/2}}}}$	${\displaystyle {\frac {1}{2}}+{\frac {3}{8}}{\frac {t}{\sqrt {1+{\frac {t^{2}}{4}}}}}{\left[1-{\frac {1}{12}}{\frac {t^{2}}{1+{\frac {t^{2}}{4}}}}\right]}}$
5	${\displaystyle {\frac {8}{3\pi {\sqrt {5}}\left(1+{\frac {t^{2}}{5}}\right)^{3}}}}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}{\left[{\frac {t}{{\sqrt {5}}\left(1+{\frac {t^{2}}{5}}\right)}}\left(1+{\frac {2}{3\left(1+{\frac {t^{2}}{5}}\right)}}\right)+\arctan \left({\frac {t}{\sqrt {5}}}\right)\right]}}$
${\displaystyle \infty }$	${\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-t^{2}/2}}$	${\displaystyle {\frac {1}{2}}{\left[1+\operatorname {erf} \left({\frac {t}{\sqrt {2}}}\right)\right]}}$	See Normal distribution, Error function

For ${\displaystyle \nu >1}$, the raw moments of the t-distribution are

${\displaystyle \operatorname {E} (T^{k})={\begin{cases}0&k{\text{ odd}},\quad 0<k<\nu \\{\frac {1}{{\sqrt {\pi }}\Gamma \left({\frac {\nu }{2}}\right)}}\left[\Gamma \left({\frac {k+1}{2}}\right)\Gamma \left({\frac {\nu -k}{2}}\right)\nu ^{\frac {k}{2}}\right]&k{\text{ even}},\quad 0<k<\nu .\\\end{cases}}}$

Moments of order ${\displaystyle \nu }$ or higher do not exist.[15]

The term for ${\displaystyle 0<k<\nu }$, k even, may be simplified using the properties of the gamma function to

${\displaystyle \operatorname {E} (T^{k})=\nu ^{\frac {k}{2}}\,\prod _{i=1}^{k/2}{\frac {2i-1}{\nu -2i}}\qquad k{\text{ even}},\quad 0<k<\nu .}$

For a t-distribution with ${\displaystyle \nu }$ degrees of freedom, the expected value is 0 if ${\displaystyle \nu >1}$, and its variance is ${\displaystyle {\frac {\nu }{\nu -2}}}$ if ${\displaystyle \nu >2}$. The skewness is 0 if ${\displaystyle \nu >3}$ and the excess kurtosis is ${\displaystyle {\frac {6}{\nu -4}}}$ if ${\displaystyle \nu >4}$.

Student's t-distribution generalizes to the three parameter location-scale t-distribution ${\displaystyle lst(\mu ,\tau ^{2},\nu )}$ by introducing a location parameter ${\displaystyle \mu }$ and a scale parameter ${\displaystyle \tau }$. With

${\displaystyle T\sim t_{\nu }}$

and location-scale family transformation

${\displaystyle X=\mu +\tau T}$

we get

${\displaystyle X\sim lst(\mu ,\tau ^{2},\nu )}$

The resulting distribution is also called the non-standardized Student's t-distribution.

The location-scale t distribution has a density defined by:[16]

${\displaystyle p(x\mid \nu ,\mu ,\tau )={\frac {\Gamma ({\frac {\nu +1}{2}})}{\Gamma ({\frac {\nu }{2}}){\sqrt {\pi \nu }}\tau \,}}\left(1+{\frac {1}{\nu }}\left({\frac {x-\mu }{\tau }}\right)^{2}\right)^{-(\nu +1)/2}}$

Equivalently, the density can be written in terms of ${\displaystyle \tau ^{2}}$:

${\displaystyle p(x\mid \nu ,\mu ,\tau ^{2})={\frac {\Gamma ({\frac {\nu +1}{2}})}{\Gamma ({\frac {\nu }{2}}){\sqrt {\pi \nu \tau ^{2}}}}}\left(1+{\frac {1}{\nu }}{\frac {(x-\mu )^{2}}{{\tau }^{2}}}\right)^{-(\nu +1)/2}}$

Other properties of this version of the distribution are:[16]

${\displaystyle {\begin{aligned}\operatorname {E} (X)&=\mu &{\text{ for }}\nu >1\\\operatorname {var} (X)&=\tau ^{2}{\frac {\nu }{\nu -2}}&{\text{ for }}\nu >2\\\operatorname {mode} (X)&=\mu \end{aligned}}}$

• If ${\displaystyle X}$ follows a location-scale t-distribution ${\displaystyle X\sim \mathrm {lst} \left(\mu ,\tau ^{2},\nu \right)}$ then for ${\displaystyle \nu \rightarrow \infty }$ ${\displaystyle X}$ is normally distributed ${\displaystyle X\sim \mathrm {N} \left(\mu ,\tau ^{2}\right)}$ with mean ${\displaystyle \mu }$ and variance ${\displaystyle \tau ^{2}}$.
• The location-scale t-distribution ${\displaystyle \mathrm {lst} \left(\mu ,\tau ^{2},\nu =1\right)}$ with degree of freedom ${\displaystyle \nu =1}$ is equivalent to the Cauchy distribution ${\displaystyle \mathrm {Cau} \left(\mu ,\tau \right)}$.
• The location-scale t-distribution ${\displaystyle \mathrm {lst} \left(\mu =0,\tau ^{2}=1,\nu \right)}$ with ${\displaystyle \mu =0}$ and ${\displaystyle \tau ^{2}=1}$ reduces to the Student's t-distribution ${\displaystyle t_{\nu }}$

The t-distribution arises as the sampling distribution of the t-statistic. Below the one-sample t-statistic is discussed, for the corresponding two-sample t-statistic see Student's t-test.

Let ${\displaystyle x_{1},\ldots ,x_{n}\sim N(\mu ,\sigma ^{2})}$ be independent and identically distributed samples from a normal distribution with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$. The sample mean and unbiased sample variance are given by:

${\displaystyle {\begin{aligned}{\bar {x}}&={\frac {x_{1}+\cdots +x_{n}}{n}},\\[5pt]s^{2}&={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.\end{aligned}}}$

The resulting (one sample) t-statistic is given by

${\displaystyle t={\frac {{\bar {x}}-\mu }{\sqrt {s^{2}/n}}}\sim t_{n-1}.}$

and is distributed according to a Student's t-distribution with ${\displaystyle n-1}$ degrees of freedom.

Thus for inference purposes the t-statistic is a useful "pivotal quantity" in the case when the mean and variance ${\displaystyle (\mu ,\sigma ^{2})}$ are unknown population parameters, in the sense that the t-statistic has then a probability distribution that depends on neither ${\displaystyle \mu }$ nor ${\displaystyle \sigma ^{2}}$.

Instead of the unbiased estimate ${\displaystyle s^{2}}$ we may also use the maximum likelihood estimate

${\displaystyle s_{ML}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}$

yielding the statistic

${\displaystyle t_{ML}={\frac {{\bar {x}}-\mu }{\sqrt {s_{ML}^{2}/n}}}={\sqrt {\frac {n}{n-1}}}t.}$

This is distributed according to the location-scale t-distribution:

${\displaystyle t_{ML}\sim lst(0,\tau ^{2}=n/(n-1),n-1).}$

The location-scale t-distribution results from compounding a Gaussian distribution (normal distribution) with mean ${\displaystyle \mu }$ and unknown variance, with an inverse gamma distribution placed over the variance with parameters ${\displaystyle a=\nu /2}$ and ${\displaystyle b=\nu {\tau }^{2}/2}$. In other words, the random variable X is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is marginalized out (integrated out).

Equivalently, this distribution results from compounding a Gaussian distribution with a scaled-inverse-chi-squared distribution with parameters ${\displaystyle \nu }$ and ${\displaystyle {\tau }^{2}}$. The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. ${\displaystyle \nu =2a,\;{\tau }^{2}={\frac {b}{a}}}$.

The reason for the usefulness of this characterization is that in Bayesian statistics the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian distribution. As a result, the location-scale t-distribution arises naturally in many Bayesian inference problems.[17]

Student's t-distribution is the maximum entropy probability distribution for a random variate X for which ${\displaystyle \operatorname {E} (\ln(\nu +X^{2}))}$ is fixed.[18]

There are various approaches to constructing random samples from the Student's t-distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples; e.g., in the multi-dimensional applications basis of copula-dependency. In the case of stand-alone sampling, an extension of the Box–Muller method and its polar form is easily deployed.[19] It has the merit that it applies equally well to all real positive degrees of freedom, ν, while many other candidate methods fail if ν is close to zero.[19]

The function A(t | ν) is the integral of Student's probability density function, f(t) between −t and t, for t ≥ 0. It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function A(t | ν) can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of t and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in t-tests. For the statistic t, with ν degrees of freedom, A(t | ν) is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t ≥ 0). It can be easily calculated from the cumulative distribution function F_ν(t) of the t-distribution:

${\displaystyle A(t\mid \nu )=F_{\nu }(t)-F_{\nu }(-t)=1-I_{\frac {\nu }{\nu +t^{2}}}\left({\frac {\nu }{2}},{\frac {1}{2}}\right),}$

where I_x is the regularized incomplete beta function (a, b).

For statistical hypothesis testing this function is used to construct the p-value.

Student's t-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the t-distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of the t-distribution.

Confidence intervals and hypothesis tests are two statistical procedures in which the quantiles of the sampling distribution of a particular statistic (e.g. the standard score) are required. In any situation where this statistic is a linear function of the data, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's t-distribution. Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form.

Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student's t-distribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.

A number of statistics can be shown to have t-distributions for samples of moderate size under null hypotheses that are of interest, so that the t-distribution forms the basis for significance tests. For example, the distribution of Spearman's rank correlation coefficient ρ, in the null case (zero correlation) is well approximated by the t distribution for sample sizes above about 20.

Suppose the number A is so chosen that

${\displaystyle \Pr(-A<T<A)=0.9,}$

when T has a t-distribution with n − 1 degrees of freedom. By symmetry, this is the same as saying that A satisfies

${\displaystyle \Pr(T<A)=0.95,}$

so A is the "95th percentile" of this probability distribution, or ${\displaystyle A=t_{(0.05,n-1)}}$. Then

${\displaystyle \Pr \left(-A<{\frac {{\overline {X}}_{n}-\mu }{S_{n}/{\sqrt {n}}}}<A\right)=0.9,}$

and this is equivalent to

${\displaystyle \Pr \left({\overline {X}}_{n}-A{\frac {S_{n}}{\sqrt {n}}}<\mu <{\overline {X}}_{n}+A{\frac {S_{n}}{\sqrt {n}}}\right)=0.9.}$

Therefore, the interval whose endpoints are

${\displaystyle {\overline {X}}_{n}\pm A{\frac {S_{n}}{\sqrt {n}}}}$

is a 90% confidence interval for μ. Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the t-distribution to examine whether the confidence limits on that mean include some theoretically predicted value – such as the value predicted on a null hypothesis.

It is this result that is used in the Student's t-tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the t-distribution can be used to examine whether that difference can reasonably be supposed to be zero.

If the data are normally distributed, the one-sided (1 − α)-upper confidence limit (UCL) of the mean, can be calculated using the following equation:

${\displaystyle \mathrm {UCL} _{1-\alpha }={\overline {X}}_{n}+t_{\alpha ,n-1}{\frac {S_{n}}{\sqrt {n}}}.}$

The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, ${\displaystyle {\overline {X}}_{n}}$ being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL_1−α is equal to the confidence level 1 − α.

The t-distribution can be used to construct a prediction interval for an unobserved sample from a normal distribution with unknown mean and variance.

The Student's t-distribution, especially in its three-parameter (location-scale) version, arises frequently in Bayesian statistics as a result of its connection with the normal distribution. Whenever the variance of a normally distributed random variable is unknown and a conjugate prior placed over it that follows an inverse gamma distribution, the resulting marginal distribution of the variable will follow a Student's t-distribution. Equivalent constructions with the same results involve a conjugate scaled-inverse-chi-squared distribution over the variance, or a conjugate gamma distribution over the precision. If an improper prior proportional to σ⁻² is placed over the variance, the t-distribution also arises. This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to a conjugate normally distributed prior, or is unknown distributed according to an improper constant prior.

Related situations that also produce a t-distribution are:

• The marginal posterior distribution of the unknown mean of a normally distributed variable, with unknown prior mean and variance following the above model.
• The prior predictive distribution and posterior predictive distribution of a new normally distributed data point when a series of independent identically distributed normally distributed data points have been observed, with prior mean and variance as in the above model.

The t-distribution is often used as an alternative to the normal distribution as a model for data, which often has heavier tails than the normal distribution allows for; see e.g. Lange et al.[22] The classical approach was to identify outliers (e.g., using Grubbs's test) and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the t-distribution is a natural choice of model for such data and provides a parametric approach to robust statistics.

A Bayesian account can be found in Gelman et al.[23] The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.