A question: what does “mean temperature” actually mean? I argue that it has no physically sensible meaning.
Perhaps ask a related, but substantially easier question first: I have several bank accounts; what is the total amount of money in those accounts? That is clearly a well-posed question: just add up all the account balances and we’re done.
Similar questions can be asked about physical systems: I have several different batteries; what is the total energy stored in those batteries? Just add up all the energy in the batteries, obviously.
We can also ask: what is the mean energy stored in those batteries. Answer: total energy stored divided by the number of batteries.
These questions are meaningful because things like “money” or “energy” are extensive quantities. They can be added up into a physically meaningful quantity: “total money”, or “total energy.”
Not all measurable quantities are extensive. The colour of the batteries is a measurable quantity, and each battery will have some colour. However, it makes no sense to ask about the “total colour” of the batteries, or the average colour.
That is a stupid example really. Let’s make it a bit more sensible. OK: we have several rooms in a house, each with a different temperature. What is the total temperature of the house? This clearly makes no sense at all. Temperature is a so-called intensive quantity; essentially a measurable property of a system, which cannot be added up, or accumulated.
There have been newspaper headlines about weather along the lines of “today is twice as warm as last week” (for example, when today it was 20C and last week it was 10C). This is of course complete rubbish. For example, if we were in America we would have reported temperatures in Fahrenheit, going from 50F last week to 68F today. Same change in temperatures, but apparently not a doubling anymore.
If you think this only happens to hapless newspaper hacks, I can assure you that I have heard similar statements made by trained weather forecasters on national TV. I am sure they were a slip of the tongue or a mind-blank. It is quite stressful to do a live weather forecast slot on national news.
The silliness of the newspaper headlines hides the fundamental property of temperature: it is an intensive quantity and cannot be added up, multiplied by 3, or divided by 5. It is nonsensical.
Now for mean or average temperature of a number of systems: to calculate a mean temperature, we add them first up (nonsense) and divide by number of systems (nonsense.)
So we combine two nonsense operations and expect to get a physically sensible outcome? I think, fundamentally, mean temperature is a nonsense concept.
Technically, mean temperature of a number of systems seems less objectionable than something like total temperature of a number of systems. The mean temperature does not depend on the units in which the temperature is expressed, whereas total temperature does. But eagle-eyed readers will have noted that this is the result of different temperature scales being linear functions of each other.
How about this example: all the rooms in the house have a certain humidity. We can of course add up the total amount of water vapour in the rooms, and then divide by the number of rooms to get the mean amount of water vapour in any room. Water vapour amount is indeed an extensive quantity.
But we cannot do the same trick for relative humidity, RH. RH is an intensive quantity, which is a nonlinear function of vapour amount and temperature. We cannot find the total relative humidity in the house –it just makes no sense– nor can we find the mean relative humidity. How would you calculate it anyway? First determine the mean temperature and then calculate RH, or first determine the RH in each room, and then calculate the mean? The outcomes will be different.
Of course, we can simply define what we mean by “mean temperature” or “mean relative humidity”. It would be a statistical property of the set of systems we are considering. But it has no strict meaning in physics; it is purely a statistical construct.
Of course, we climate scientists always work with ideas like “global mean temperature.” It is worth remembering that this has strictly speaking no physical meaning, and we should be really careful in constructing physical arguments based on it. We should be less concerned about the rising global mean temperature (a physically meaningless concept) and more concerned with the measurable consequences of climate change (e.g., increasing number of heatwaves.)
Reading physics 
One thought on “Mean temperatures don’t mean much”