The inclusion of the dispersion in the vertical action J_z estimates in the Ting & Rix (2019) relation. Upper panel: the Ting & Rix (2019) relation between vertical action J_z and stellar age computed at 8 kpc is indicated with the magenta line. The ‘new’ vertical action of Eq. (5) including observed dispersion for σ_Jz/J_z = 0.5 and σ_Jz/J_z = 1 cases is reported with the dark and light-blue points, respectively. The solid dark-blue (light-blue) line indicates the median values for σ_Jz/J_z = 0.5 (σ_Jz/J_z = 1). Lower panel: computed initial vertical velocity v_0, z(τ) (which satisfies the condition of Eq. (4)) as a function of Galactic age τ for the SSPs born at different evolutionary times is reported with the solid magenta line. As in the upper panel, the computed v_0, z including observed dispersion for σ_Jz/J_z = 0.5 and σ_Jz/J_z = 1 cases is indicated with the dark and light-blue points, respectively.