  In this paper, we study distributional properties of the sequence of partial
quotients in the continued fraction expansion of fractions $a/N$, where $N$ is
fixed and $a$ runs through the set of mod $N$ residue classes which are coprime
with $N$. Our methods cover statistics such as the sum of partial quotients,
the maximal partial quotient, the empirical distribution of partial quotients,
Dedekind sums, and much more. We prove a sharp concentration inequality for the
sum of partial quotients, and sharp tail estimates for the maximal partial
quotient and for Dedekind sums, all matching the tail behavior in the limit
laws which are known under an extra averaging over the set of possible
denominators $N$. We show that the distribution of partial quotients of reduced
fractions with fixed denominator gives a very good fit to the Gauss-Kuzmin
distribution. As corollaries we establish the existence of reduced fractions
with a small sum of partial quotients resp. a small maximal partial quotient.
