The following article is from motls.blogspot (The Reference Frame) by - Vincent Gray and Alan Siddons Average temperature vs average irradiance
They point out the maths of the solar irradiance problem.
Vincent Gray and Alan Siddons have been emphasizing an important point that the (arithmetic) average temperature is not the relevant quantity that should be substituted into various calculations of the heat and energy budget of Earth.
Why? Because what matters for the energy budget is the average radiated (or absorbed) energy. According to the Stefan-Boltzmann law, the energy radiated by a black body per unit surface area (also called "radiant emittance") is proportional to the fourth power of the absolute temperature.
When we divide the solar constant by four, to obtain 342 Watts per squared meter, we really average the irradiance over latitudes, seasons, and parts of a day. If the Earth were a black body, that would be equivalent to averaging the fourth power of the absolute temperature.
But the average value of the fourth power of temperature is something different than the fourth power of the average temperature!
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Average temperature vs average irradiance