The correlation function G between two spins is usually defined as
$$ G=\langle \sigma_a \sigma_b\rangle - \langle \sigma_a\rangle \langle\sigma_b\rangle $$
The $\sigma$ are the value of the spins (+-1).That's supposed to be equal to
$$ = \left\langle\ (\sigma_a - \langle \sigma_a\rangle) (\sigma_b - \langle \sigma_b\rangle) \ \right\rangle $$
And I'm not quote sure why that is. I tried multiplying the second version out and thought that the negative terms would disappear when averaged. That leaves the almost correct formula, but without the minus sign. So where is the trick?
Sources for these equations can be found in for example here on page 4