= Solution
The supplied plot gives surface <heat flux>, not a whole-crust mean <temperature>. At young ages its scatter is substantial. Representative readings suitable for an order-of-magnitude conductive calculation are $q_J\simeq0.20\,\mathrm{W\,m^{-2}}$ near $5$ Ma and $q_N\simeq0.070\,\mathrm{W\,m^{-2}}$ near $50$ Ma; plausible graphical ranges are roughly $0.12$–$0.23$ and $0.06$–$0.09\,\mathrm{W\,m^{-2}}$, respectively. These are approximate choices within the plotted population, not uniquely determined statistical means or a digitization of its red curve.
Assume the same normal oceanic <crustal thickness> $h_c=7$ km, seafloor temperature $T_s\simeq0\,{}^{\circ}\mathrm{C}$, negligible internal crustal heating and a linear conductive profile. These are additional assumptions; the paper supplies conductivity but not thickness. With depth $z$ positive downward and $q$ the magnitude of the upward flux, <Fourier's law> gives
$$
T(z)=T_s+\frac{qz}{k},\qquad
\overline T_c=\frac1{h_c}\int_0^{h_c}T(z)\,dz
=T_s+\frac{qh_c}{2k}.
$$
This is the <crustal mean temperature from conductive heat flow>, not the basal temperature, which is twice the excess above $T_s$. Substitution of $k=1.3\,\mathrm{W\,m^{-1}\,K^{-1}}$ gives
$$
\overline T_J\simeq\frac{0.20\times7000}{2\times1.3}=538\,{}^{\circ}\mathrm{C},\qquad
\overline T_N\simeq\frac{0.070\times7000}{2\times1.3}=188\,{}^{\circ}\mathrm{C}.
$$
A sensible rounded answer is therefore \b[about $540\,{}^{\circ}\mathrm{C}$ for the young crust and $190\,{}^{\circ}\mathrm{C}$ for the old crust], with a representative contrast \b[$\Delta\overline T_c\simeq350$ K]. The corresponding basal values are about $1080$ and $380\,{}^{\circ}\mathrm{C}$. The quoted graphical ranges instead yield mean temperatures about $320$–$620$ and $160$–$240\,{}^{\circ}\mathrm{C}$, at the same assumed thickness.
\Image[/past-exam-of-the-plate-tectonics-course-of-the-university-of-oxford/2022-crustal-temperature-and-buoyancy.png]
{title=Original conductive temperature estimates and their crust-only thermal-buoyancy scaling; thickness, linear profiles and representative fluxes are explicit assumptions}
The temperatures scale linearly with $h_c/k$. Hydrothermal circulation can redistribute heat, especially in young permeable crust, and measurements through sediment need not equal the purely conductive cooling-model flux everywhere. The supplied study explicitly addresses the resulting oceanic heat-flow deficit.<Oceanic heat flow: implications for global heat loss by Derrick Hasterok and others>[Abstract, hydrothermal circulation and low conductive measurements in young seafloor; examination Figure 2 supplies the filtered observations]{ref} Thus the calculation gives plausible incoming-crust averages under a stated model, not the entire plate's temperature or its later temperature at depth in the subduction zone.
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