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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

 

                                                        

Microsoft Excel Framework

Within the Microsoft Excel computing environment, using the natural logarithm of the gamma function Gammaln circumvents this difficulty, since the number e raised to the n power, if n is an integer, returns the same result as n decremented by one, factorial. However, the Gammaln function also works with arguments that are not integer numbers. For five degrees of freedom, Microsoft Excel's formula for t-distribution can be written as

 

 

Microsoft Excel

(1 / Sqrt ( 5 * Pi() )) * (2 / Exp ( Gammaln (2.5))) * (1 + (A1 ^ 2) / 5) ^ -3.

t-Distribution

 

and plotted as 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.