Suppose we can randomly sample quadruples (A, A', B, B') with each component = +/-1

Do this n times and compute 4 averages <AB>, <AB'>, <A'B>, <A'B'>

They will certainly satisfy the CHSH inequality <AB> + <AB'> + <A'B> - <A'B'> is less than or equal to 2 (by some simple algebra)

Now suppose that I only use a random sub-sample of size about n/4 to compute <AB>

Suppose I use a disjoint random sub-sample of size about n/4 to compute <AB'> and so on.

The averages taken over four disjoint but completely random sub-samples will be close to the averages based on the whole sample, if n is large (by some elementary probability theory). So when based on four disjoint random subsamples, <AB> + <AB'> + <A'B> - <A'B'> is very unlikely to be much larger than 2.

In real Bell-CHSH experiments, we can't observe quadruples (A, A', B, B'). It is only the hypothesis of local realism that says that they do exist. We only observe one of the four pairs (A,B), (A,B'), etc. We calculate <AB> + <AB'> + <A'B> - <A'B'> using a different set of runs for each of the four averages. Of course, in principle it is possible that the observed value of <AB> + <AB'> + <A'B> - <A'B'> is exactly equal to 4 (the algebraic bound which FrediFizzx likes to remind us of). However, as anyone can easily verify by doing their own computer experiments, the result won't often be outside of the interval [-2, +2] by more than a few times 1 / sqrt n

Real experiments report error bars as well as correlations. There is more to Bell-CHSH than just algebra. There is also a little bit of statistics. It is however very commonly overlooked.

Do this n times and compute 4 averages <AB>, <AB'>, <A'B>, <A'B'>

They will certainly satisfy the CHSH inequality <AB> + <AB'> + <A'B> - <A'B'> is less than or equal to 2 (by some simple algebra)

Now suppose that I only use a random sub-sample of size about n/4 to compute <AB>

Suppose I use a disjoint random sub-sample of size about n/4 to compute <AB'> and so on.

The averages taken over four disjoint but completely random sub-samples will be close to the averages based on the whole sample, if n is large (by some elementary probability theory). So when based on four disjoint random subsamples, <AB> + <AB'> + <A'B> - <A'B'> is very unlikely to be much larger than 2.

In real Bell-CHSH experiments, we can't observe quadruples (A, A', B, B'). It is only the hypothesis of local realism that says that they do exist. We only observe one of the four pairs (A,B), (A,B'), etc. We calculate <AB> + <AB'> + <A'B> - <A'B'> using a different set of runs for each of the four averages. Of course, in principle it is possible that the observed value of <AB> + <AB'> + <A'B> - <A'B'> is exactly equal to 4 (the algebraic bound which FrediFizzx likes to remind us of). However, as anyone can easily verify by doing their own computer experiments, the result won't often be outside of the interval [-2, +2] by more than a few times 1 / sqrt n

Real experiments report error bars as well as correlations. There is more to Bell-CHSH than just algebra. There is also a little bit of statistics. It is however very commonly overlooked.