  Let random vectors $R^c=\{R_p^c:p\in P_c\}$ represent joint measurements of
certain subsets $P_c$ of properties $p\in P$ in different contexts $c\in C$.
Such a system is traditionally called noncontextual if there exists a jointly
distributed set $\{Q_p:p\in P\}$ of random variables such that $R^c$ has the
same distribution as $\{Q_p:p\in P_c\}$ for all $c\in C$. A trivial necessary
condition for noncontextuality and a precondition for most approaches to
measuring contextuality is that the system is consistently connected, i.e., all
$R_p^c,R_p^{c'},\dots$ measuring the same property $p$ have the same
distribution. The Contextuality-by-Default (CbD) approach allows detecting and
measuring "true" contextuality on top of inconsistent connectedness, but at the
price of a higher computational cost.
  In this paper we propose a novel approach to measuring contextuality that
shares the generality and basic definitions of the CbD approach and the
computational benefits of the previously proposed Negative Probability (NP)
approach. The present approach differs from CbD in that instead of considering
all possible joints of the double-indexed random variables $R_p^c$, it
considers all possible approximating single-indexed systems $\{Q_p:p\in P\}$.
The degree of contextuality is defined based on the minimum possible
probabilistic distance of the actual measurements $R^c$ from $\{Q_p:p\in P_c$}.
We show that the defined measure agrees with a certain measure of contextuality
of the CbD approach for all systems where each property enters in exactly two
contexts and that this measure can be calculated far more efficiently than the
CbD measure and even more efficiently than the NP measure for sufficiently
large systems. The present approach can be modified so as to agree with the NP
measure of contextuality on all consistently connected systems while extending
it to inconsistently connected systems.
