The synthesis mechanism given in Jackson et al. (2022) uses saturated models, along with overdispersed count distributions, to generate synthetic categorical data. The mechanism is controlled by tuning parameters, which can be tuned according to a specific risk or utility metric. Thus expected properties of synthetic data sets can be determined analytically a priori, that is, before they are generated. While Jackson et al. (2022) considered the case of generating m = 1 data set, this paper considers generating m > 1 data sets. In effect, m becomes a tuning parameter and the role of m in relation to the risk-utility trade-off can be shown analytically. The paper introduces a pair of risk metrics, τ₃(k,d) and τ₄(k,d) that are suited to m > 1 data sets; and also considers the more general issue of how best to analyse categorical data sets: average the data sets pre-analysis or average results post-analysis. Finally, the methods are demonstrated empirically with the synthesis of a constructed data set which is used to represent the English School Census.