  We give a quantitative estimate for the quantum mean absolute deviation on
hyperbolic surfaces of finite area in terms of geometric parameters such as the
genus, number of cusps and injectivity radius. It implies a delocalisation
result of quantum ergodicity type for eigenfunctions of the Laplacian on
hyperbolic surfaces of finite area that Benjamini-Schramm converge to the
hyperbolic plane. We show that this is generic for Mirzakhani's model of random
surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending
on the particular sequence of surfaces considered this gives a result of
delocalisation of most cusp forms or of Eisenstein series.
