  To quantitatively characterize height distributions (HDs), one uses
adimensional ratios of their first central moments ($m_n$) or cumulants
($\kappa_n$), especially the skewness $S$ and kurtosis $K$, whose accurate
estimate demands an averaging over all $L^d$ points of the height profile at a
given time, in translation-invariant interfaces, and over $N$ independent
samples. One way of doing this is by calculating $m_n(t)$ [or $\kappa_n(t)$]
for each sample and then carrying out an average of them for the $N$
interfaces, with $S$ and $K$ being calculated only at the end. Another approach
consists in directly calculating the ratios for each interface and, then,
averaging the $N$ values. It turns out, however, that $S$ and $K$ for the
growth regime HDs display strong finite-size and -time effects when estimated
from these "interface statistics", as already observed in some previous works
and clearly shown here, through extensive simulations of several discrete
growth models belonging to the EW and KPZ classes on 1D and 2D substrates of
sizes $L=const.$ and $L \sim t$. Importantly, I demonstrate that with "1-point
statistics'', i.e., by calculating $m_n(t)$ [or $\kappa_n(t)$] once for all $N
L^d$ heights together, these corrections become very weak. However, I find that
this "1-point'' approach fails in uncovering the universality of the HDs in the
steady state regime (SSR) of systems whose average height, $\bar{h}$, is a
fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt
height evolves as $h(t) = \bar{h}(t) + s_{\lambda} A^{1/2} L^{\alpha} \zeta +
\cdots$ -- where $P(\zeta)$ is the underlying SSR HD -- and the fluctuations in
$\bar{h}$ yield $S_{1pt} \sim t^{-1/2}$ and $K_{1pt} \sim t^{-1}$. Nonetheless,
by analyzing $P(h-\bar{h})$, the cumulants of $P(\zeta)$ can be accurately
determined.
