In reality we may never know the variance of the population, and we have only a limited sample size due to economical reason as well as other restraints.
It turns out the pdf of t ratio is not exactly the same as normal distribution. Oxford graduate William Sealy Gossett published a paper in 1908 in which he derived a formula for the pdf, and later in 1924 R. A. Fisher presented a rigorous mathematical derivation of Gossett’s pdf. Consequently, the pdf is now known as the Student t distribution.
