We introduce an equitable version of proper labellings of graphs, where the notion of equitability is with respect to the resulting vertex sums. That is, we are interested in $k$-labellings where, when computing the sums of labels incident to the vertices, we get a vertex-colouring that is not proper only, but also equitable. For a given graph $G$, we are interested in the parameter $\overline{\chi}_\Sigma(G)$, which is the smallest $k \geq 1$ (if any) such that $G$ admits such $k$-labellings. Through examples of particular graph classes, we observe that this new parameter $\overline{\chi}_\Sigma$ behaves sort of similarly to the parameters $\chi_\Sigma$ and $s$, which parameters lie behind the 1-2-3 Conjecture and the irregularity strength of graphs, in a more or less strong way, depending on the graphs considered. We then prove general bounds on $\overline{\chi}_\Sigma$, showing that, in some contexts (trees and connected graphs with large minimum degree), this parameter is bounded above by roughly $\frac{3n}{4}$ for an $n$-graph. We also prove that determining $\overline{\chi}_\Sigma$ is \textsf{NP}-hard in general, and finish off with directions for further work on the topic.