---
title: "Student-t"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575254/Student-t"
format: markdown
---
Student(Midpoint,Scale,Degrees of freedom) or Student(m,s,**n**)

 

* The Standard Student-t distribution is defined as a Student(0,1,n) distribution, which is also often written as Student(n)

 

[Student-t equations](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26576421)

[Crystal Ball parameter restrictions](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575280)

 

 

The Student distribution plays an important part in statistics, as explained below. Examples of the Standard Student-t (first figure) and the general Student-t distribution (second figure):

 

![image](media://49bde4f0-65f7-45c7-8cdb-8638a2fb7768)

![image](media://9f0f4edf-3404-4f96-b775-9efdea253129)

 

#### Uses

The most common use of the Student distribution is for the estimation of the mean of a (n assumed Normally distributed) population where random samples from that population have been observed, and its standard deviation is unknown. The relationship:

  (1)

![image-20240528-074925.png](media://7c87cd22-fe65-44c9-95b6-8fae50eaf8c2)

   


 > Macro (mathinline)

                      

 

is at the center of the method, discussed in more detail [here](https://modelassist.epixanalytics.com/space/EA/26575350/Classical+statistics+estimation+of+the+Normal+distribution+mean+when+the+standard+deviation+is+not+known) (also an example model of how to use the Student distribution is provided there). This is equivalent to a t-test in classical statistics.

 

Other sample statistics can be approximated to a Student distribution, and thus a Student distribution can be used to describe one's uncertainty about the parameter's true value: in regression analysis, [for example](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574959).

 

#### Generation

The Student-t distribution is not directly available with Crystal Ball 5.5 and before, but can be produced in two ways.

 

The first (Equation 1) using the fact that a Student-t distribution is a standard Normal distribution multiplied by a factor that includes a Chi Squared distribution as explained [here](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575350). When using this method, you will lose the benefits of [Latin Hypercube sampling](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575293) because two distributions are being used to create the random samples.

 

The second method uses the TINV( ) function in Microsoft Excel which returns the inverse of the Student-t distribution. Because the TINV function in Excel only returns positive values, we need to add a [Bernoulli](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575202) distribution, with will result in the following:

(2)

![image-20240528-075439.png](media://5d5a2a97-0550-4a6e-9efb-c744e5b9aa85)


Student(0,1,n) = TINV(Uniform(0,1), n) * IF(Binomial(0.5,1)=0,1,-1)       

 

A slightly more elegant version of this formula is:

(3)

![image-20240528-075601.png](media://d37184d4-d565-438b-91ef-0e798c91a028)


 

Student(0,1,n) = TINV(Uniform(0,1), n) * (Binomial(0.5,1)*2-1)             

 

#### Comments

First discovered by the English statistician *William Sealy Gossett* (1876-1937), whose employer (the brewery company, Guinness) forbade employees from publishing their work, so he wrote a [paper](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26579303) under the pseudonym "Student'. As *n* increases, the Student-t distribution [tends](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575269) to a [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246)(0, 1) distribution.

 

 

 

 

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