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)
Crystal Ball parameter restrictions
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):
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:
Student(0,1,n) = Normal(0,1)^*(n/ChiSq(n)))^{1/2} (1)
is at the center of the method, discussed in more detail here (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.
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. When using this method, you will lose the benefits of Latin Hypercube sampling 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 distribution, with will result in the following:
Student(0,1,n) = TINV(Uniform(0,1), n) * IF(Binomial(0.5,1)=0,1,-1) (2)
A slightly more elegant version of this formula is:
Student(0,1,n) = TINV(Uniform(0,1), n) * (Binomial(0.5,1)*2-1) (3)
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 under the pseudonym "Student'. As n increases, the Student-t distribution tends to a Normal(0, 1) distribution.
