  The average genus of a 2-bridge knot with crossing number $c$ approaches
$\frac{c}{4} + \frac{1}{12}$ as $c$ approaches infinity, as proven by Suzuki
and Tran and independently Cohen and Lowrance. In this paper, for the genera of
$2$-bridge knots of a fixed crossing number $c$, we show that the median and
mode are both $\lfloor \frac{c+2}{4} \rfloor$ and that the variance approaches
$\frac{c}{16}-\frac{17}{144}$ as $c$ approaches infinity. We prove that the
distribution of genera of 2-bridge knots is asymptotically normal.
