---
abstract: '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 $\rho_{12}=(e_2 n_1)/(e_1 n_2)$, 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 