  We present an alternative approach to finite-size effects around the
synchronization transition in the standard Kuramoto model. Our main focus lies
on the conditions under which a collective oscillatory mode is well defined.
For this purpose, the minimal value of the amplitude of the complex Kuramoto
order parameter appears as a proper indicator. The dependence of this minimum
on coupling strength varies due to sampling variations and correlates with the
sample kurtosis of the natural frequency distribution. The skewness of the
frequency sample determines the frequency of the resulting collective mode. The
effects of kurtosis and skewness hold in the thermodynamic limit of infinite
ensembles. We prove this by integrating a self-consistency equation for the
complex Kuramoto order parameter for two families of distributions with
controlled kurtosis and skewness, respectively.
