In the averaging process on a graph $G = (V, E)$, a random mass distribution $\eta$ on $V$ is repeatedly updated via transformations of the form $\eta_{v}, \eta_{w} \mapsto (\eta_{v} + \eta_{w})/2$, with updates made according to independent Poisson clocks associated to the edge set $E$. We'll discuss this process when the underlying graph $G$ is the integer lattice $\mathbb{Z}^{d}$. We show that it exhibits tight asymptotic concentration around its mean and use this to obtain a central limit theorem. The proof relies on reducing the original problem to one about an associated random walk on $\mathbb{Z}^{d}$, a technique which is likely adaptable to similar processes.