Visual Statistics Studio

 

The t- Distribution

The t-distribution belongs to the Euler's family of the gamma distributions. The density function for the t-distribution, associated with certain number of degrees of freedoms signified by the Greek letter n, is shown below.

 

                                        

 

In the above equation . When this argument is an integer, the gamma function is just a factorial offset by one; however, the gamma function returns values of factorial for all positive real numbers.

 

In terms of the Euler's Gamma function

 

                                                        

 

the constant a equals

 

                                                 

 

 

This constant, for degrees of freedom approaching infinity, approximates .3989. The constant b equals zero, the e for a given degrees of freedom equals

 

                                                        

 

This constant, for degrees of freedom approaching infinity, approximates 2.71828182845905. The constant c equals

 

                                                        

 

and for large number of degrees of freedom approximates .5. The constant d equals 2. Thus the ordinate of the t-distribution can be written as

 

                                                

 

 

As the degrees of freedom grow large, the t-distribution gradually changes to normal distribution

 

                                                        

 

plotted in the figure below.

 

 

Theoretically, the normal distribution and the t-distribution are identical only for the infinite number of the degrees of freedom. Practically, you may see for yourself that the differences between the normal and t-distributions are not so large.

 

                                                      

 

Arguably, for sample size where n is greater than 30, and undoubtedly, for sample sizes greater than 60, the difference between the t-distribution and the normal distribution are negligible.