  Proofs of Bell-Kochen-Specker contextuality demonstrate that there exists
sets of projectors that cannot each be assigned either 0 or 1 such that each
basis formed from them contains exactly one 1-assigned projector. Instead, at
least some of the projectors must have a valuation that depends on the
\emph{context} in which they are measured. This motivates the question of
\emph{how many} of the projectors must have contextual valuations. In this
paper, we demonstrate a bound on what fraction of rank-1 projective
measurements on a quantum system must be considered to have context-dependent
valuations as a function of the quantum dimension, and show that quantum
mechanics is not as contextual, by this metric, as other possible physical
theories. Attempts to find quantum mechanical scenarios that yield a high value
of this figure-of-merit can be thought of as generalisations or extensions of
the search for small Kochen-Specker sets. We also extend this result to
projector-valued-measures with projectors of arbitrary equal rank.
