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The density for Student's t is given by (Shao, 2003):

where Γ is the gamma function and v are degrees of freedom. For small degrees of freedom v, the t distribution is quite distinct from the standard normal. However, as degrees of freedom increase, the t distribution converges to that of a normal density (Figure 2.11). That is, in the limit, f(t) → f(z), or a bit more formally, .

The fact that t converges to z for large degrees of freedom but is quite distinct from z for small degrees of freedom is one reason why t distributions are often used for small sample problems. When sample size is large, and so consequently are degrees of freedom, whether one treats a random variable as t or z will make little difference in terms of computed p‐values and decisions on respective null hypotheses. This is a direct consequence of the convergence of the two distributions for large degrees of freedom. For a historical overview of how t‐distributions came to be, consult Zabell (2008).