Should S always be more than 2 to violate CHSH inequality?

Should the measured value always be about $2\sqrt{2}$ or is it just a maximum value and it is possible to measure something like $S = \{1.90, 1.8, 2.0, 2.4, 2.6, 1.8\}$ (with average equal 2) to violate the CHSH inequality?

EDIT: By $S$ I mean $S$ received from statistically significant number of measurements.

Answer

The CHSH inequality is a statement about a linear combination of expectation values of different products of observables. It is not a statement about the outcome of a single measurement. Thus, violation of the CHSH inequality can only be inferred in a statistical sense after performing many repetitions of the experiment.

See: What exactly does $S$ represent in the CHSH inequality $-2\leq S\leq 2$?