  Let $\Gamma$ denote a $Q$-polynomial distance-regular graph, with vertex set
$X$ and diameter $D\geq 3$.
  The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$,
where ${\hat x}$ denotes column $x$ of the identity matrix $I \in {\rm
Mat}_X(\mathbb C)$. Let $E$ denote a $Q$-polynomial primitive idempotent of
$\Gamma$. The eigenspace $EV$ is spanned by the vectors
  $\lbrace E {\hat x} \vert x \in X\rbrace$.
  It was previously known that these vectors satisfy a condition called the
balanced set condition. In this paper, we introduce a variation on the balanced
set condition called the Norton-balanced condition. The Norton-balanced
condition involves the Norton algebra product on $EV$. We define $\Gamma$ to be
Norton-balanced whenever $\Gamma$ has a $Q$-polynomial primitive idempotent $E$
such that the set $\lbrace E {\hat x} \vert x \in X\rbrace$ is Norton-balanced.
  We show that $\Gamma$ is Norton-balanced in the following cases:
  (i)
  $\Gamma$ is bipartite; (ii) $\Gamma$ is almost bipartite; (iii) $\Gamma$ is
dual-bipartite; (iv) $\Gamma$ is almost dual-bipartite; (v) $\Gamma$ is tight;
(vi)
  $\Gamma$ is a Hamming graph; (vii) $\Gamma$ is a Johnson graph; (viii)
$\Gamma$ is the Grassmann graph $J_q(2D,D)$;
  (ix) $\Gamma$ is a halved bipartite dual-polar graph; (x) $\Gamma$ is a
halved Hemmeter graph; (xi) $\Gamma$ is a halved hypercube; (xii) $\Gamma$ is a
folded-half hypercube;
  (xiii) $\Gamma$ has $q$-Racah type and affords a spin model.
  Some theoretical results about the Norton-balanced condition are obtained,
and some open problems are given.
