agw - for thinkers not blind followers

Global warming is usually portrayed as changes in the AVERAGE global temperature, and unusually high temperatures are frequently used to give credence to the assertion of AGW.
But are the average temperatures even relevant?

The greenhouse effect, and particularly the role CO2 plays, is only relevant to one of what I see as two separate sides that make up one coin.
The first side is the incoming radiation from the Sun that is the source of the heat. This obviously only occurs during daylight hours. CO2 plays no role in absorbing or limiting incoming solar radiation. Incoming solar radiation is absorbed Rayleigh Scattering, Oxygen and Ozone, and the major greenhouse gas, water vapour.
The second side of the coin is the loss of heat through upgoing thermal radiation as heat is radiated off into space, generally during the hours of darkness. The reason that the nights are generally coldest just before the dawn is because heat is constantly being lost throughout the night, so logically the maximum loss occurs immediately before incoming radiation resumes as the sun appears over the horizon.
It is this side of the coin, the radiation of heat upwards, where the greenhouse effect occurs. The theory is that the presence of greenhouse gases in the atmosphere absorb this upgoing thermal radiation thus preventing the complete loss of the accumulated heat from the surface. We certainly know both from theory and our own observations how water vapour works in this situation, and how a clear night with no clouds and low humidity can become bitterly cold, whilst an overcast night, with heavy cloud cover can be quite warm, or even hot if the day has been hot and humid.
The theory is that increased atmospheric CO2 increases this greenhouse effect and reduces the amount of accumulated heat that is able to be radiated off.
This is where I want the thinkers to put on their thinking caps.
What is the most relevant measurement of temperature for measuring the greenhouse effect, the average highest temperature, the average mean temperature, or the average lowest temperature?
I put it that it should be the average lowest temperature.
What are your thoughts, and reasons why?