Variance identity for independent copies ($\frac{1}{2}\mathbb{E}|Q-Q^{\prime }{|}^2=\mathbb{E}|Q{|}^2-|\mathbb{E}Q{|}^2$)

For independent identically distributed random vectors $Q,Q^{\prime }$ with finite second absolute moments, expand $|Q-Q^{\prime }{|}^2=|Q{|}^2+|Q^{\prime }{|}^2-2Q\cdot Q^{\prime }$. Their independence makes $\mathbb{E}(Q\cdot Q^{\prime })=|\mathbb{E}Q{|}^2$, proving the identity. The right side is the sum of the coordinate variances. Applied to the displacement in an optimal transport plan, it separates motion of the barycenter of a measure from relative particle motion.