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If we have a standard normal variable, and a variable with a chi-squared distribution with k degrees of freedom, then the ratio of the normal variable to the square root of the chi-squared variable is another random variable, which has a distribution proportional to the t-distribution with k degrees of freedom.
This is also the same as the square root of an F-distribution with 1, k degrees of freedom, if k>0.
i.e. given n observations x_i from a normal distribution with mean μ, we define

t = - x   − μ s / √ n ,

where

s2 = n ∑ i=1  ( xi − - x   ) 2 n−1 .

Then t has a t-distribution with n-1 degrees of freedom.