  A. R\'enyi \cite{Renyi} made a definition that gives a generalization of
simple normality in the context of $Q$-Cantor series. In \cite{Mance}, a
definition of $Q$-normality was given that generalizes the notion of normality
in the context of $Q$-Cantor series. In this work, we examine both
$Q$-normality and $Q$-distribution normality, treated in \cite{Laffer} and
\cite{Salat}. Specifically, while the non-equivalence of these two notions is
implicit in \cite{Laffer}, in this paper, we give an explicit construction
witnessing the nontrivial direction. That is, we construct a base $Q$ as well
as a real $x$ that is $Q$-normal yet not $Q$-distribution normal. We next
approach the topic of simultaneous normality by constructing an explicit
example of a base $Q$ as well as a real $x$ that is both $Q$-normal and
$Q$-distribution normal.
