## Student's t Distribution

### Overview

The Student’s *t* distribution is a one-parameter family of
curves. This distribution is typically used to test a hypothesis regarding the
population mean when the population standard deviation is unknown.

Statistics and Machine Learning Toolbox™ offers multiple ways to work with the Student’s *t* distribution.

Use distribution-specific functions (

`tcdf`

,`tinv`

,`tpdf`

,`trnd`

,`tstat`

) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple Student’s*t*distributions.Use generic distribution functions (

`cdf`

,`icdf`

,`pdf`

,`random`

) with a specified distribution name (`'T'`

) and parameters.

### Parameters

The Student’s *t* distribution uses the following
parameter.

Parameter | Description | Support |
---|---|---|

nu (ν) | Degrees of freedom | ν` = 1, 2, 3,...` |

### Probability Density Function

The pdf of the Student's *t* distribution is

$$y=f(x|\nu )=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}\frac{1}{\sqrt{\nu \pi}}\frac{1}{{\left(1+\frac{{x}^{2}}{\nu}\right)}^{\frac{\nu +1}{2}}},$$

where *ν* is the degrees of freedom and Γ( · ) is the Gamma
function. The result *y* is the probability of observing a particular value
of *x* from the Student’s *t* distribution with
*ν* degrees of freedom.

For an example, see Compute and Plot Student's t Distribution pdf.

### Cumulative Distribution Function

The cdf of the Student’s *t* distribution is

$$p=F(x|\nu )={\displaystyle {\int}_{-\infty}^{x}\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}\frac{1}{\sqrt{\nu \pi}}\frac{1}{{\left(1+\frac{{t}^{2}}{\nu}\right)}^{\frac{\nu +1}{2}}}dt},$$

where *ν* is the degrees of freedom and Γ( · ) is the Gamma
function. The result *p* is the probability that a single observation from
the *t* distribution with *ν* degrees of freedom falls in
the interval [–∞, *x*].

For an example, see Compute and Plot Student's t Distribution cdf.

### Inverse Cumulative Distribution Function

The *t* inverse function is defined in terms of the Student's
*t* cdf as

$$x={F}^{-1}(p|\nu )=\{x:F(x|\nu )=p\},$$

where

$$p=F(x|\nu )={\displaystyle {\int}_{-\infty}^{x}\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}\frac{1}{\sqrt{\nu \pi}}\frac{1}{{\left(1+\frac{{t}^{2}}{\nu}\right)}^{\frac{\nu +1}{2}}}dt},$$

*ν* is the degrees of freedom, and Γ( · ) is the Gamma
function. The result *x* is the solution of the integral equation where you
supply the probability *p*.

For an example, see Compute Student's t icdf.

### Descriptive Statistics

The mean of the Student’s *t* distribution is *μ* = 0 for degrees of freedom *ν* greater than 1. If
*ν* equals 1, then the mean is undefined.

The variance of the Student’s *t* distribution is $$\frac{\nu}{\nu -2}$$ for degrees of freedom *ν* greater than 2. If
*ν* is less than or equal to 2, then the variance is undefined.

### Examples

#### Compute and Plot Student's `t`

Distribution pdf

`t`

Compute the pdf of a Student's *t* distribution with degrees of freedom equal to `5`

, `10`

, and `50`

.

x = [-5:.1:5]; y1 = tpdf(x,5); y2 = tpdf(x,10); y3 = tpdf(x,50);

Plot the pdf for all three choices `nu`

on the same axis.

figure; plot(x,y1,'Color','black','LineStyle','-') hold on plot(x,y2,'Color','red','LineStyle','-.') plot(x,y3,'Color','blue','LineStyle','--') xlabel('Observation') ylabel('Probability Density') legend({'nu = 5','nu = 10','nu = 50'}) hold off

#### Compute and Plot Student's `t`

Distribution cdf

`t`

Compute the cdf of a Student's *t* distribution with degrees of freedom equal to `5`

, `10`

, and `50`

.

x = [-5:.1:5]; y1 = tcdf(x,5); y2 = tcdf(x,10); y3 = tcdf(x,50);

Plot the cdf for all three choices of `nu`

on the same axis.

figure; plot(x,y1,'Color','black','LineStyle','-') hold on plot(x,y2,'Color','red','LineStyle','-.') plot(x,y3,'Color','blue','LineStyle','--') xlabel('Observation') ylabel('Cumulative Probability') legend({'nu = 5','nu = 10','nu = 50'}) hold off

#### Compute Student's *t* icdf

Find the 95th percentile of the Student's *t* distribution with `50`

degrees of freedom.

p = .95; nu = 50; x = tinv(p,nu)

x = 1.6759

#### Compare Student's `t`

and Normal Distribution pdfs

`t`

The Student’s *t* distribution is a family of curves depending on a single parameter *ν* (the degrees of freedom). As the degrees of freedom *ν* approach infinity, the *t* distribution approaches the standard normal distribution.

Compute the pdfs for the Student's *t* distribution with the parameter `nu = 5`

and the Student's *t* distribution with the parameter `nu = 15`

.

x = [-5:0.1:5]; y1 = tpdf(x,5); y2 = tpdf(x,15);

Compute the pdf for a standard normal distribution.

z = normpdf(x,0,1);

Plot the Student's *t* pdfs and the standard normal pdf on the same figure.

plot(x,y1,'-.',x,y2,'--',x,z,'-') legend('Student''s t Distribution with \nu=5', ... 'Student''s t Distribution with \nu=15', ... 'Standard Normal Distribution','Location','best') xlabel('Observation') ylabel('Probability Density') title('Student''s t and Standard Normal pdfs')

The standard normal pdf has shorter tails than the Student's *t* pdfs.

### Related Distributions

Beta Distribution — The beta distribution is a two-parameter continuous distribution that has parameters

*a*(first shape parameter) and*b*(second shape parameter). If*Y*has a Student's*t*distribution with*ν*degrees of freedom, then $$X=\frac{1}{2}+\frac{1}{2}\frac{Y}{\sqrt{\nu +{Y}^{2}}}$$ has beta distribution with the shape parameters*a*=*ν*/2 and*b*=*ν*/2. This relationship is used to compute values of the*t*cdf and inverse functions, and to generate*t*distributed random numbers.Cauchy Distribution — The Cauchy distribution is a two-parameter continuous distribution with the parameters

*γ*(scale) and*δ*(location). It is a special case of the Stable Distribution with the shape parameters*α*= 1 and*β*= 0. The standard Cauchy distribution (unit scale and location zero) is the Student’s*t*distribution with degrees of freedom*ν*equal to 1. The standard Cauchy distribution has an undefined mean and variance.For an example, see Generate Cauchy Random Numbers Using Student's t.

Chi-Square Distribution — The chi-square distribution is a one-parameter continuous distribution that has the parameter

*ν*(degrees of freedom). If*Z*has a standard normal distribution and*χ*^{2}has a chi-square distribution with degrees of freedom*ν*, then $$\text{t=}\frac{Z}{\sqrt{{\chi}^{2}/\nu}}$$ has a Student's*t*distribution with degrees of freedom*ν*.Noncentral t Distribution — The noncentral

*t*distribution is a two-parameter continuous distribution that generalizes the Student's*t*distribution and has the parameters*ν*(degrees of freedom) and*δ*(noncentrality). Setting*δ*= 0 yields the Student's*t*distribution.Normal Distribution — The normal distribution is a two-parameter continuous distribution with the parameters

*μ*(mean) and*σ*(standard deviation).As the degrees of freedom

*ν*approach infinity, the Student's*t*distribution approaches the standard normal distribution (zero mean and unit standard deviation).For an example, see Compare Student's t and Normal Distribution pdfs

If

*x*is a random sample of size*n*from a normal distribution with mean*μ*, then the statistic $$t=\frac{\overline{x}-\mu}{s/\sqrt{n}}$$, where $$\overline{x}$$ is the sample mean and*s*is the sample standard deviation, has a Student's*t*distribution with*n*—1 degrees of freedom.For an example, see Compute Student's t Distribution cdf.

t Location-Scale Distribution — The

*t*location-scale distribution is a three-parameter continuous distribution with the parameters*μ*(mean),*σ*(scale), and*ν*(shape). If*x*has a*t*location-scale distribution with the parameters*µ*,*σ*, and*ν*, then $$\frac{x-\mu}{\sigma}$$ has a Student's*t*distribution with*ν*degrees of freedom.

## References

[1] Abramowitz, Milton, and
Irene A. Stegun, eds. *Handbook of Mathematical Functions: With Formulas,
Graphs, and Mathematical Tables*. 9. Dover print.; [Nachdr. der Ausg.
von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Devroye, Luc.
*Non-Uniform Random Variate Generation*. New York, NY:
Springer New York, 1986. https://doi.org/10.1007/978-1-4613-8643-8

[3] Evans, Merran, Nicholas
Hastings, and Brian Peacock. *Statistical Distributions*. 2nd
ed. New York: J. Wiley, 1993.

[4] Kreyszig, Erwin.
*Introductory Mathematical Statistics: Principles and
Methods*. New York: Wiley, 1970.

## See Also

`tcdf`

| `tpdf`

| `tinv`

| `tstat`

| `trnd`

| `ttest`

| `ttest2`