  We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the
order parameter and its dynamic fluctuations near the onset of the
synchronization transition, paying particular attention to effects induced by
the randomness of the intrinsic frequencies of oscillators. For a population of
size $N$, we study two ways of sampling the intrinsic frequencies according to
the {\it same} given unimodal distribution $g(\omega)$. In the `{\em random}'
case, frequencies are generated independently in accordance with $g(\omega)$,
which gives rise to oscillator number fluctuation within any given frequency
interval. In the `{\em regular}' case, the $N$ frequencies are generated in a
deterministic manner that minimizes the oscillator number fluctuations, leading
to quasi-uniformly spaced frequencies in the population. We find that the two
samplings yield substantially different finite-size properties with clearly
distinct scaling exponents. Moreover, the hyperscaling relation between the
order parameter and its fluctuations is valid in the regular case, but is
violated in the random case. In this last case, a self-consistent mean-field
theory that completely ignores dynamic fluctuations correctly predicts the FSS
exponent of the order parameter but not its critical amplitude.
