  Motivated by a problem arising when analysing data from quarantine searches,
we explore properties of distributions of sums of independent means of
independent lattice-valued random variables. The aim is to determine the extent
to which approximations to those sums require continuity corrections. We show
that, in cases where there are only two different means, the main effects of
distribution smoothness can be understood in terms of the ratio
$\rho_{12}=(e_2n_1)/(e_1n_2)$, where $e_1$ and $e_2$ are the respective maximal
lattice edge widths of the two populations, and $n_1$ and $n_2$ are the
respective sample sizes used to compute the means. If $\rho_{12}$ converges to
an irrational number, or converges sufficiently slowly to a rational number;
and in a number of other cases too, for example those where $\rho_{12}$ does
not converge; the effects of the discontinuity of lattice distributions are of
smaller order than the effects of skewness. However, in other instances, for
example where $\rho_{12}$ converges relatively quickly to a rational number,
the effects of discontinuity and skewness are of the same size. We also treat
higher-order properties, arguing that cases where $\rho_{12}$ converges to an
algebraic irrational number can be less prone to suffer the effects of
discontinuity than cases where the limiting irrational is transcendental. These
results are extended to the case of three or more different means, and also to
problems where distributions are estimated using the bootstrap. The results
have practical interpretation in terms of the accuracy of inference for, among
other quantities, the sum or difference of binomial proportions.
