The inclusion of the dispersion in the vertical action J_z estimates in the Ting & Rix (2019) relation. Upper panels: vertical maximum excursion |z_max| versus age relation for the ES21 model. The grey points indicate stars within the Galactocentric region between 6 and 10 kpc and |z|< 2 kpc as reported in the astroNN catalogue. The contour lines enclose fractions of 0.95, 0.90, 0.75, 0.60, 0.45, 0.30, 0.20, and 0.05 of the total number of observed stars. The colour coding represents the total number of stars predicted by the ES21 model in that region. In the left panel, we draw model results considering a Gaussian error (see Eq. (5)) with standard deviation σ_Jz = 0.5 ⋅ J_z, whereas in the right panel we assume σ_Jz = 1 ⋅ J_z. The shaded grey area highlights the region with ages of more than 8 Gyr, for which the use of Eq. (3) may not be justified. Lower panels: as in the upper ones, but for the |z_max| versus [Mg/Fe] relations.