  The merit factor of a $\{-1, 1\}$ binary sequence measures the collective
smallness of its non-trivial aperiodic autocorrelations. Binary sequences with
large merit factor are important in digital communications because they allow
the efficient separation of signals from noise. It is a longstanding open
question whether the maximum merit factor is asymptotically unbounded and, if
so, what is its limiting value. Attempts to answer this question over almost
sixty years have identified certain classes of binary sequences as particularly
important: skew-symmetric sequences, symmetric sequences, and anti-symmetric
sequences. Using only elementary methods, we find an exact formula for the mean
and variance of the reciprocal merit factor of sequences in each of these
classes, and in the class of all binary sequences. This provides a much deeper
understanding of the distribution of the merit factor in these four classes
than was previously available. A consequence is that, for each of the four
classes, the merit factor of a sequence drawn uniformly at random from the
class converges in probability to a constant as the sequence length increases.
