  Contextuality and entanglement are valuable resources for quantum computing
and quantum information. Bell inequalities are used to certify entanglement;
thus, it is important to understand why and how they are violated. Quantum
mechanics and behavioral sciences teach us that random variables measuring the
same content (the answer to the same Yes or No question) may vary, if measured
jointly with other random variables. Alice and Bob raw data confirm Einsteinian
non-signaling, but setting dependent experimental protocols are used to create
samples of coupled pairs of distant outcomes and to estimate correlations.
Marginal expectations, estimated using these final samples, depend on distant
settings. Therefore, a system of random variables measured in Bell tests is
inconsistently connected and it should be analyzed using a
Contextuality-by-Default approach, what is done for the first time in this
paper. The violation of Bell inequalities and inconsistent connectedness may be
explained using a contextual locally causal probabilistic model in which
setting dependent variables describing measuring instruments are correctly
incorporated. We prove that this model does not restrict experimenters freedom
of choice which is a prerequisite of science. Contextuality seems to be the
rule and not an exception; thus, it should be carefully tested.
