  This paper considers the impact of Lyman-alpha coupling and X-ray heating on
the 21-cm brightness-temperature one-point statistics (as predicted by
semi-numerical simulations). The X-ray production efficiency is varied over
four orders of magnitude and the hardness of the X-ray spectrum is varied from
that predicted for high-mass X-ray binaries, to the softer spectrum expected
from the hot inter-stellar medium. We find peaks in the redshift evolution of
both the variance and skewness associated with the efficiency of X-ray
production. The amplitude of the variance is also sensitive to the hardness of
the X-ray SED. We find that the relative timing of the coupling and heating
phases can be inferred from the redshift extent of a plateau that connects a
peak in the variance's evolution associated with Lyman-alpha coupling to the
heating peak. Importantly, we find that late X-ray heating would seriously
hamper our ability to constrain reionization with the variance. Late X-ray
heating also qualitatively alters the evolution of the skewness, providing a
clean way to constrain such models. If foregrounds can be removed, we find that
LOFAR, MWA and PAPER could constrain reionization and late X-ray heating models
with the variance. We find that HERA and SKA (phase 1) will be able to
constrain both reionization and heating by measuring the variance using
foreground-avoidance techniques. If foregrounds can be removed they will also
be able to constrain the nature of Lyman-alpha coupling.
